mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicTopology.ModelCategory.Transport

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 : ModelCategory D
inst✝² : CategoryWithCofibrations C
inst✝¹ : CategoryWithFibrations C
inst✝ : CategoryWithWeakEquivalences C
e : C ≌ D
h₁ : cofibrations C = (cofibrations D).inverseImage e.functor
h₂ : fibrations C = (fibrations D).inverseImage e.functor
h₃ : weakEquivalences C = (weakEquivalences D).inverseImage e.functor
h₁' : trivialCofibrations C = (trivialCofibrations D).inverseImage e.functor
h₂' : trivialFibrations C = (trivialFibrations D).inverseImage e.functor
this✝¹ : ∀ {X Y : C} (f : X ⟶ Y) [hf : Cofibration f], Cofibration (e.functor.map f)
this✝ : ∀ {X Y : C} (f : X ⟶ Y) [hf : Fibration f], Fibration (e.functor.map f)
this : ∀ {X Y : C} (f : X ⟶ Y) [hf : WeakEquivalence f], WeakEquivalence (e.functor.map f)
⊢ (trivialCofibrations C).HasFactorization (fibrations C)

Tactic:
rw [h₁', h₂]; infer_instance

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

Goal state before:
pp: X : SSet
f₀ f₁ f₃ : Δ[2] ⟶ X
h₀₂ : stdSimplex.δ 2 ≫ f₁ = stdSimplex.δ 1 ≫ f₃
h₁₂ : stdSimplex.δ 2 ≫ f₀ = stdSimplex.δ 0 ≫ f₃
h₂₃ : stdSimplex.δ 0 ≫ f₀ = stdSimplex.δ 0 ≫ f₁
⊢ ι₃ ≫ desc f₀ f₁ f₃ h₀₂ h₁₂ h₂₃ = f₃

Tactic:
by
  rw [← cancel_epi (stdSimplex.faceSingletonComplIso.{u} 3).inv, ← Category.assoc,
    horn.faceSingletonComplIso_inv_ι]
  exact (horn.isColimit 2).fac _ (.right ⟨3, by simp⟩)

Goal state after:
Module: Mathlib.CategoryTheory.Category.ReflQuiv

Goal state before:
pp: V W : Type u
inst✝¹ : ReflQuiver V
inst✝ : ReflQuiver W
e : Quiv.of V ≅ Quiv.of W
h_id : ∀ (X : V), e.hom.map (𝟙rq X) = 𝟙rq (e.hom.obj X)
⊢ { toPrefunctor := e.inv, map_id := ⋯ } ≫ { toPrefunctor := e.hom, map_id := h_id } = 𝟙 (of W)

Tactic:
by
    apply forgetToQuiv.map_injective
    exact e.inv_hom_id

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

Goal state before:
pp: V : Truncated 2
W : MorphismProperty V.HomotopyCategory
inst✝ : W.IsMultiplicative
hW : ∀ {x y : V.obj (op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })} (e : Edge x y), W (homMk e)
⊢ ⊤ ≤ W

Tactic:
by
    rw [← multiplicativeClosure_morphismPropertyHomMk,
      MorphismProperty.multiplicativeClosure_le_iff]
    rintro _ _ _ ⟨_, _, e⟩
    exact hW e

Goal state after:
Module: Mathlib.AlgebraicTopology.Reedy.Basic

Goal state before:
pp: 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
⊢ W₁.MapFactorizationData W₂ f

Tactic:
by
  letI := (r.nonempty_unique f).some
  exact default

Goal state after:
Module: Mathlib.AlgebraicTopology.Reedy.Basic

Goal state before:
pp: 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 Z Y : C
f : X ⟶ Z
g : Z ⟶ Y
hfg : r.deg Z ≤ r.degHom f ∧ r.deg Z ≤ r.degHom g
φ : W₁.MapFactorizationData W₂ (f ≫ g) := { Z := Z, i := f, p := g, fac := ⋯, hi := ⋯, hp := ⋯ }
⊢ r.deg Z ≤ r.degHom (f ≫ g)

Tactic:
rw [r.degHom_eq φ]

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplexCategory.ToMkOne

Goal state before:
pp: case pos
n : ℕ
i : Fin (n + 2)
j : Fin (n + 1)
h : i ≤ j.castSucc
k : Fin (⦋n + 1⦌.len + 1)
hk : k < i
⊢ (ConcreteCategory.hom (toMk₁ i)) (j.predAbove k) = (ConcreteCategory.hom (toMk₁ i.castSucc)) k

pp: case neg
n : ℕ
i : Fin (n + 2)
j : Fin (n + 1)
h : i ≤ j.castSucc
k : Fin (⦋n + 1⦌.len + 1)
hk : i ≤ k
⊢ (ConcreteCategory.hom (toMk₁ i)) (j.predAbove k) = (ConcreteCategory.hom (toMk₁ i.castSucc)) k

Tactic:
· #adaptation_note /-- Before https://github.com/leanprover/lean4/pull/13166
    (replacing grind's canonicalizer with a type-directed normalizer), `grind` closed this goal.
    It is not yet clear whether this is due to defeq abuse in Mathlib or a problem in the new
    canonicalizer; a minimization would help. The original proof was:
    `grind [Fin.castPred, Fin.predAbove_of_le_castSucc, toMk₁_of_castSucc_lt]` -/
    simp; grind [Fin.castPred, Fin.predAbove_of_le_castSucc, toMk₁_of_castSucc_lt]

Goal state after:
case neg
n : ℕ
i : Fin (n + 2)
j : Fin (n + 1)
h : i ≤ j.castSucc
k : Fin (⦋n + 1⦌.len + 1)
hk : i ≤ k
⊢ (ConcreteCategory.hom (toMk₁ i)) (j.predAbove k) = (ConcreteCategory.hom (toMk₁ i.castSucc)) k
Module: Mathlib.Order.UpperLower.Relative

Goal state before:
pp: α : Type u_1
P : α → Prop
inst✝ : LE α
s : Set { x // P x }
⊢ IsUpperSet s ↔ IsRelUpperSet (Subtype.val '' s) P

Tactic:
refine ⟨fun h a x ↦ ?_, fun h a b x y ↦ ?_⟩

Goal state after:
case refine_1
α : Type u_1
P : α → Prop
inst✝ : LE α
s : Set { x // P x }
h : IsUpperSet s
a : α
x : a ∈ Subtype.val '' s
⊢ P a ∧ ∀ ⦃b : α⦄, a ≤ b → P b → b ∈ Subtype.val '' s

case refine_2
α : Type u_1
P : α → Prop
inst✝ : LE α
s : Set { x // P x }
h : IsRelUpperSet (Subtype.val '' s) P
a b : { x // P x }
x : a ≤ b
y : a ∈ s
⊢ b ∈ s
Module: Mathlib.Order.UpperLower.Relative

Goal state before:
pp: α : Type u_1
P : α → Prop
inst✝ : LE α
s : Set { x // P x }
⊢ IsLowerSet s ↔ IsRelLowerSet (Subtype.val '' s) P

Tactic:
refine ⟨fun h a x ↦ ?_, fun h a b x y ↦ ?_⟩

Goal state after:
case refine_1
α : Type u_1
P : α → Prop
inst✝ : LE α
s : Set { x // P x }
h : IsLowerSet s
a : α
x : a ∈ Subtype.val '' s
⊢ P a ∧ ∀ ⦃b : α⦄, b ≤ a → P b → b ∈ Subtype.val '' s

case refine_2
α : Type u_1
P : α → Prop
inst✝ : LE α
s : Set { x // P x }
h : IsRelLowerSet (Subtype.val '' s) P
a b : { x // P x }
x : b ≤ a
y : a ∈ s
⊢ b ∈ s
Module: Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Monoidal

Goal state before:
pp: x✝ y✝ z✝ : AugmentedSimplexCategory
f✝ g✝ : x✝ ⊗ y✝ ⟶ z✝
x y z : SimplexCategory
f g : tensorObjOf x y ⟶ z
h₁ : inl' x y ≫ f = inl' x y ≫ g
h₂ : inr' x y ≫ f = inr' x y ≫ g
i : Fin ((tensorObjOf x y).len + 1)
j : Fin (x.len + 1 + (y.len + 1)) := Fin.cast ⋯ i
⊢ ↑((ConcreteCategory.hom f) i) = ↑((ConcreteCategory.hom g) i)

Tactic:
have : i = j.cast (Nat.succ_add x.len (y.len + 1)) := rfl

Goal state after:
x✝ y✝ z✝ : AugmentedSimplexCategory
f✝ g✝ : x✝ ⊗ y✝ ⟶ z✝
x y z : SimplexCategory
f g : tensorObjOf x y ⟶ z
h₁ : inl' x y ≫ f = inl' x y ≫ g
h₂ : inr' x y ≫ f = inr' x y ≫ g
i : Fin ((tensorObjOf x y).len + 1)
j : Fin (x.len + 1 + (y.len + 1)) := Fin.cast ⋯ i
this : i = Fin.cast ⋯ j
⊢ ↑((ConcreteCategory.hom f) i) = ↑((ConcreteCategory.hom g) i)
Module: Mathlib.AlgebraicTopology.SimplicialObject.II

Goal state before:
pp: n m p : ℕ
f : Fin (n + 1) →o Fin (m + 1)
g : Fin (m + 1) →o Fin (p + 1)
x : Fin (p + 1)
y : Fin (m + 1)
hy : x.castSucc ≤ (g y).castSucc ∧ ∀ i < y, (g i).castSucc < x.castSucc
z : Fin (n + 1)
hz : y.castSucc ≤ (f z).castSucc ∧ ∀ i < z, (f i).castSucc < y.castSucc
i : Fin (n + 1)
hi : i < z
⊢ f i < y

Tactic:
simpa using hz.2 i hi

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

Goal state before:
pp: case inl
n : ℕ
i x : Fin (n + 2)
⊢ map' i.succAboveOrderEmb.toOrderHom x.castSucc = i.predAbove x.castSucc

Tactic:
by_cases! hx : x ≤ i
    · rw [Fin.predAbove_of_le_castSucc _ _ (by simpa), Fin.castPred_castSucc]
      obtain ⟨x, rfl⟩ | rfl := x.eq_castSucc_or_eq_last
      · simp only [map'_eq_castSucc_iff, OrderEmbedding.toOrderHom_coe,
          Fin.succAboveOrderEmb_apply, Fin.castSucc_le_castSucc_iff,
          Fin.castSucc_lt_castSucc_iff]
        constructor
        · obtain hx | rfl := hx.lt_or_eq
          · rwa [Fin.succAbove_of_castSucc_lt]
          · simpa only [Fin.succAbove_castSucc_self] using Fin.castSucc_le_succ x
        · intro j hj
          rwa [Fin.succAbove_of_castSucc_lt _ _ (lt_of_lt_of_le (by simpa) hx),
            Fin.castSucc_lt_castSucc_iff]
      · obtain rfl : i = Fin.last _ := Fin.last_le_iff.1 hx
        simp [map'_eq_last_iff]
    · obtain ⟨x, rfl⟩ := Fin.eq_succ_of_ne_zero (Fin.ne_zero_of_lt hx)
      rw [Fin.predAbove_of_castSucc_lt _ _ (by simpa [Fin.le_castSucc_iff]),
        Fin.pred_castSucc_succ, map'_eq_castSucc_iff]
      simp only [Fin.succAbove_of_lt_succ _ _ hx,
        OrderEmbedding.toOrderHom_coe, Fin.succAboveOrderEmb_apply,
        le_refl, Fin.castSucc_lt_castSucc_iff, true_and]
      intro j hj
      by_cases! h : j.castSucc < i
      · simpa [Fin.succAbove_of_castSucc_lt _ _ h] using hj.le
      · rwa [Fin.succAbove_of_le_castSucc _ _ h, Fin.succ_lt_succ_iff]

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

Goal state before:
pp: p q n : ℕ
z : (Δ[p] ⊗ Δ[q]) _⦋n⦌
⊢ z ∈ (Δ[p] ⊗ Δ[q]).nonDegenerate n ↔ Function.Injective ⇑(objEquiv z)

Tactic:
← nonDegenerate_iff_of_mono (isoNerve p q).hom,

Goal state after:
p q n : ℕ
z : (Δ[p] ⊗ Δ[q]) _⦋n⦌
⊢ (ConcreteCategory.hom ((isoNerve p q).hom.app (Opposite.op ⦋n⦌))) z ∈
      (nerve (ULift.{u, 0} (Fin (p + 1) × Fin (q + 1)))).nonDegenerate n ↔
    Function.Injective ⇑(objEquiv z)
Module: Mathlib.AlgebraicTopology.SimplicialSet.ProdStdSimplex

Goal state before:
pp: p q n : ℕ
z : (Δ[p] ⊗ Δ[q]) _⦋n⦌
⊢ z ∈ (Δ[p] ⊗ Δ[q]).nonDegenerate n ↔ StrictMono ⇑(objEquiv z)

Tactic:
← nonDegenerate_iff_of_mono (isoNerve p q).hom,

Goal state after:
p q n : ℕ
z : (Δ[p] ⊗ Δ[q]) _⦋n⦌
⊢ (ConcreteCategory.hom ((isoNerve p q).hom.app (Opposite.op ⦋n⦌))) z ∈
      (nerve (ULift.{u, 0} (Fin (p + 1) × Fin (q + 1)))).nonDegenerate n ↔
    StrictMono ⇑(objEquiv z)
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Rank

Goal state before:
pp: X✝ : SSet
A✝ : X✝.Subcomplex
X : SSet
A : X.Subcomplex
h : A.PairingCore
α : Type v
inst✝ : PartialOrder α
g : h.pairing.RankFunction α
x✝ y✝ : h.ι
hxy : h.pairing.AncestralRel (h.equivII x✝) (h.equivII y✝)
⊢ g.rank (h.equivII x✝) < g.rank (h.equivII y✝)

Tactic:
exact g.lt hxy

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Rank

Goal state before:
pp: X✝ : SSet
A✝ : X✝.Subcomplex
X : SSet
A : X.Subcomplex
h : A.PairingCore
α : Type v
inst✝ : PartialOrder α
g : h.pairing.WeakRankFunction α
x✝ y✝ : h.ι
hxy : h.pairing.AncestralRel (h.equivII x✝) (h.equivII y✝)
⊢ h.dim x✝ = h.dim y✝ → g.rank (h.equivII x✝) < g.rank (h.equivII y✝)

Tactic:
exact g.lt hxy

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.RelativeCellComplex

Goal state before:
pp: X : SSet
A : X.Subcomplex
P : A.Pairing
ι : Type v
inst✝ : LinearOrder ι
f : P.RankFunction ι
i₁ i₂ : ι
h : i₁ ≤ i₂
⊢ ∀ i < i₁, ∀ (i_2 : f.Cell i), (↑(P.p i_2.s)).subcomplex ≤ f.filtration i₂

Tactic:
intro j hj c

Goal state after:
X : SSet
A : X.Subcomplex
P : A.Pairing
ι : Type v
inst✝ : LinearOrder ι
f : P.RankFunction ι
i₁ i₂ : ι
h : i₁ ≤ i₂
j : ι
hj : j < i₁
c : f.Cell j
⊢ (↑(P.p c.s)).subcomplex ≤ f.filtration i₂
Module: Mathlib.CategoryTheory.Limits.FunctorCategory.Shapes.Pullbacks

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F G H : D ⥤ C
f : F ⟶ H
g : G ⟶ H
c : (X : D) → PullbackCone (f.app X) (g.app X)
hc : (X : D) → IsLimit (c X)
k : D
⊢ IsLimit (((evaluation D C).obj k).mapCone (combine f g c hc))

Tactic:
refine IsLimit.equivOfNatIsoOfIso ?_ _ _ ?_ (hc k)

Goal state after:
case refine_1
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F G H : D ⥤ C
f : F ⟶ H
g : G ⟶ H
c : (X : D) → PullbackCone (f.app X) (g.app X)
hc : (X : D) → IsLimit (c X)
k : D
⊢ cospan (f.app k) (g.app k) ≅ cospan f g ⋙ (evaluation D C).obj k

case refine_2
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F G H : D ⥤ C
f : F ⟶ H
g : G ⟶ H
c : (X : D) → PullbackCone (f.app X) (g.app X)
hc : (X : D) → IsLimit (c X)
k : D
⊢ (Cone.postcompose (Iso.hom ?refine_1)).obj (c k) ≅ ((evaluation D C).obj k).mapCone (combine f g c hc)
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.RelativeCellComplex

Goal state before:
pp: case a.notMem
X : SSet
A : X.Subcomplex
P : A.Pairing
ι : Type v
inst✝¹ : LinearOrder ι
f : P.RankFunction ι
inst✝ : P.IsProper
j : ι
c : f.Cell j
s : A.N
hs : s.subcomplex ≤ c.horn.image c.map
t : ↑P.II
ht : s = ↑t ∨ s = ↑(P.p t)
⊢ s.subcomplex ≤ f.filtration (f.rank c.s)

Tactic:
refine le_trans ?_ (f.subcomplex_le_filtration ⟨t, rfl⟩ (f.lt ?_))

Goal state after:
case a.notMem.refine_1
X : SSet
A : X.Subcomplex
P : A.Pairing
ι : Type v
inst✝¹ : LinearOrder ι
f : P.RankFunction ι
inst✝ : P.IsProper
j : ι
c : f.Cell j
s : A.N
hs : s.subcomplex ≤ c.horn.image c.map
t : ↑P.II
ht : s = ↑t ∨ s = ↑(P.p t)
⊢ s.subcomplex ≤ (↑(P.p { s := t, rank_s := ⋯ }.s)).subcomplex

case a.notMem.refine_2
X : SSet
A : X.Subcomplex
P : A.Pairing
ι : Type v
inst✝¹ : LinearOrder ι
f : P.RankFunction ι
inst✝ : P.IsProper
j : ι
c : f.Cell j
s : A.N
hs : s.subcomplex ≤ c.horn.image c.map
t : ↑P.II
ht : s = ↑t ∨ s = ↑(P.p t)
⊢ P.AncestralRel t c.s
Module: Mathlib.CategoryTheory.LiftingProperties.ParametrizedAdjunction

Goal state before:
pp: case h₀
C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
inst✝² : Category.{v₁, u₁} C₁
inst✝¹ : Category.{v₂, u₂} C₂
inst✝ : Category.{v₃, u₃} C₃
F : C₁ ⥤ C₂ ⥤ C₃
G : C₁ᵒᵖ ⥤ C₃ ⥤ C₂
adj₂ : F ⊣₂ G
X₁ Y₁ : C₁
f₁ : X₁ ⟶ Y₁
X₂ Y₂ : C₂
f₂ : X₂ ⟶ Y₂
X₃ Y₃ : C₃
f₃ : X₃ ⟶ Y₃
sq₁₂ : F.PushoutObjObj f₁ f₂
sq₁₃ : G.PullbackObjObj f₁ f₃
α : Arrow.mk sq₁₂.ι ⟶ Arrow.mk f₃
l : Arrow.LiftStruct ((adj₂.arrowHomEquiv sq₁₂ sq₁₃) α)
⊢ sq₁₂.inl ≫ (Arrow.mk sq₁₂.ι).hom ≫ adj₂.homEquiv.symm l.l = sq₁₂.inl ≫ Arrow.Hom.left α

pp: case h₁
C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
inst✝² : Category.{v₁, u₁} C₁
inst✝¹ : Category.{v₂, u₂} C₂
inst✝ : Category.{v₃, u₃} C₃
F : C₁ ⥤ C₂ ⥤ C₃
G : C₁ᵒᵖ ⥤ C₃ ⥤ C₂
adj₂ : F ⊣₂ G
X₁ Y₁ : C₁
f₁ : X₁ ⟶ Y₁
X₂ Y₂ : C₂
f₂ : X₂ ⟶ Y₂
X₃ Y₃ : C₃
f₃ : X₃ ⟶ Y₃
sq₁₂ : F.PushoutObjObj f₁ f₂
sq₁₃ : G.PullbackObjObj f₁ f₃
α : Arrow.mk sq₁₂.ι ⟶ Arrow.mk f₃
l : Arrow.LiftStruct ((adj₂.arrowHomEquiv sq₁₂ sq₁₃) α)
⊢ sq₁₂.inr ≫ (Arrow.mk sq₁₂.ι).hom ≫ adj₂.homEquiv.symm l.l = sq₁₂.inr ≫ Arrow.Hom.left α

Tactic:
· have := l.fac_left
          dsimp at this ⊢
          simp only [sq₁₂.inl_ι_assoc, ← adj₂.homEquiv_symm_naturality_two,
            this, Equiv.symm_apply_apply]

Goal state after:
case h₁
C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
inst✝² : Category.{v₁, u₁} C₁
inst✝¹ : Category.{v₂, u₂} C₂
inst✝ : Category.{v₃, u₃} C₃
F : C₁ ⥤ C₂ ⥤ C₃
G : C₁ᵒᵖ ⥤ C₃ ⥤ C₂
adj₂ : F ⊣₂ G
X₁ Y₁ : C₁
f₁ : X₁ ⟶ Y₁
X₂ Y₂ : C₂
f₂ : X₂ ⟶ Y₂
X₃ Y₃ : C₃
f₃ : X₃ ⟶ Y₃
sq₁₂ : F.PushoutObjObj f₁ f₂
sq₁₃ : G.PullbackObjObj f₁ f₃
α : Arrow.mk sq₁₂.ι ⟶ Arrow.mk f₃
l : Arrow.LiftStruct ((adj₂.arrowHomEquiv sq₁₂ sq₁₃) α)
⊢ sq₁₂.inr ≫ (Arrow.mk sq₁₂.ι).hom ≫ adj₂.homEquiv.symm l.l = sq₁₂.inr ≫ Arrow.Hom.left α
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd

Goal state before:
pp: m : ℕ
k : Fin (m + 1)
n : ℕ
s t : Type₁ k n
h : s.x = t.x
hs : min s.x ⋯ = s.index.succ
ht : min t.x ⋯ = t.index.succ
⊢ s = t

Tactic:
obtain ⟨x, d, hd, l, isIndex⟩ := s

Goal state after:
m : ℕ
k : Fin (m + 1)
n : ℕ
t : Type₁ k n
ht : min t.x ⋯ = t.index.succ
x : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N
d : ℕ
hd : x.dim = d + 1
l : Fin (d + 1)
isIndex : IsIndex x hd l.succ
h : { x := x, d := d, hd := hd, index := l, isIndex := isIndex }.x = t.x
hs :
  min { x := x, d := d, hd := hd, index := l, isIndex := isIndex }.x ⋯ =
    { x := x, d := d, hd := hd, index := l, isIndex := isIndex }.index.succ
⊢ { x := x, d := d, hd := hd, index := l, isIndex := isIndex } = t
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd

Goal state before:
pp: case neg
m : ℕ
k : Fin (m + 1)
n : ℕ
s : Type₁ k n
ht : IsType₂ s.δ
i : Fin (((ht.type₁ ⋯).x.cast ⋯).dim + 1)
hi : i ≠ s.index.castSucc
⊢ φ s.δ ⋯ i = (objEquiv (s.x.cast ⋯).simplex) i

Tactic:
· rw [← s.isIndex.min_δ] at hi
    rw [φ_of_ne _ rfl _ hi]
    change objEquiv (s.x.cast s.hd).simplex
      (s.index.castSucc.succAbove ((min s.δ rfl).predAbove i)) = _
    congr 1
    rw [← s.isIndex.min_δ]
    exact Fin.succAbove_predAbove hi

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

Goal state before:
pp: X : SSet
x : X.π₀
⊢ mapπ₀ (𝟙 X) x = x

Tactic:
by
  induction x
  simp

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

Goal state before:
pp: X Y Z : SSet
f : X ⟶ Y
g : Y ⟶ Z
x : X.π₀
⊢ mapπ₀ (f ≫ g) x = mapπ₀ g (mapπ₀ f x)

Tactic:
by
  induction x
  simp

Goal state after:
Module: Mathlib.Analysis.AbsoluteValue.Equivalence

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁷ : Field R
inst✝⁶ : Field S
inst✝⁵ : LinearOrder S
inst✝⁴ : TopologicalSpace S
inst✝³ : IsStrictOrderedRing S
inst✝² : Archimedean S
inst✝¹ : OrderTopology S
ι : Type u_3
inst✝ : Finite ι
v : ι → AbsoluteValue R S
w : AbsoluteValue R S
a b : R
i : ι
ha : 1 < (v i) a
haj : ∀ (j : ι), j ≠ i → (v j) a < 1
haw : 1 < w a
hb : 1 < (v i) b
hbw : w b < 1
c : ℕ → R := fun n ↦ 1 / (1 + a⁻¹ ^ n) * b
hcᵢ : Tendsto (fun n ↦ (v i) (c n)) atTop (𝓝 ((v i) b))
hcⱼ : ∀ (j : ι), j ≠ i → Tendsto (fun n ↦ (v j) (c n)) atTop (𝓝 0)
⊢ ∃ k, 1 < (v i) k ∧ (∀ (j : ι), j ≠ i → (v j) k < 1) ∧ w k < 1

Tactic:
have hcₙ : atTop.Tendsto (fun n ↦ w (c n)) (𝓝 (w b)) := by
    have : w a⁻¹ < 1 := map_inv₀ w _ ▸ inv_lt_one_of_one_lt₀ haw
    simpa [c] using (tendsto_div_one_add_pow_nhds_one this).mul_const (w b)

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁷ : Field R
inst✝⁶ : Field S
inst✝⁵ : LinearOrder S
inst✝⁴ : TopologicalSpace S
inst✝³ : IsStrictOrderedRing S
inst✝² : Archimedean S
inst✝¹ : OrderTopology S
ι : Type u_3
inst✝ : Finite ι
v : ι → AbsoluteValue R S
w : AbsoluteValue R S
a b : R
i : ι
ha : 1 < (v i) a
haj : ∀ (j : ι), j ≠ i → (v j) a < 1
haw : 1 < w a
hb : 1 < (v i) b
hbw : w b < 1
c : ℕ → R := fun n ↦ 1 / (1 + a⁻¹ ^ n) * b
hcᵢ : Tendsto (fun n ↦ (v i) (c n)) atTop (𝓝 ((v i) b))
hcⱼ : ∀ (j : ι), j ≠ i → Tendsto (fun n ↦ (v j) (c n)) atTop (𝓝 0)
hcₙ : Tendsto (fun n ↦ w (c n)) atTop (𝓝 (w b))
⊢ ∃ k, 1 < (v i) k ∧ (∀ (j : ι), j ≠ i → (v j) k < 1) ∧ w k < 1
Module: Mathlib.Analysis.Analytic.Order

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
n : ℕ
⊢ ∀ (a : 𝕜 → E),
    (AnalyticAt 𝕜 a z₀ ∧ ∀ᶠ (z : 𝕜) in 𝓝 z₀, (-f) z = (z - z₀) ^ n • a z) ↔
      AnalyticAt 𝕜 ((Equiv.neg (𝕜 → E)) a) z₀ ∧ ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ n • (Equiv.neg (𝕜 → E)) a z

Tactic:
by simp [neg_eq_iff_eq_neg]

Goal state after:
Module: Mathlib.Analysis.Analytic.Order

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CharZero 𝕜
inst✝ : CompleteSpace E
f : 𝕜 → E
hf : AnalyticAt 𝕜 f 0
n : ℕ
this : AnalyticAt 𝕜 (fun z ↦ ∑ i ∈ Finset.range n, (z ^ i / ↑i.factorial) • iteratedDeriv i f 0) 0
⊢ ∃ F,
    AnalyticAt 𝕜 F 0 ∧
      ∀ᶠ (z : 𝕜) in 𝓝 0, f z - ∑ i ∈ Finset.range n, (z ^ i / ↑i.factorial) • iteratedDeriv i f 0 = z ^ n • F z

Tactic:
convert! (natCast_le_analyticOrderAt (hf.fun_sub this)).mp ?_

Goal state after:
case e'_5
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CharZero 𝕜
inst✝ : CompleteSpace E
f : 𝕜 → E
hf : AnalyticAt 𝕜 f 0
n : ℕ
this : AnalyticAt 𝕜 (fun z ↦ ∑ i ∈ Finset.range n, (z ^ i / ↑i.factorial) • iteratedDeriv i f 0) 0
x✝¹ : 𝕜 → E
x✝ : 𝕜
⊢ x✝ = x✝ - 0

case convert_2
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CharZero 𝕜
inst✝ : CompleteSpace E
f : 𝕜 → E
hf : AnalyticAt 𝕜 f 0
n : ℕ
this : AnalyticAt 𝕜 (fun z ↦ ∑ i ∈ Finset.range n, (z ^ i / ↑i.factorial) • iteratedDeriv i f 0) 0
⊢ ↑n ≤ analyticOrderAt (fun i ↦ f i - ∑ i_1 ∈ Finset.range n, (i ^ i_1 / ↑i_1.factorial) • iteratedDeriv i_1 f 0) 0
Module: Mathlib.Analysis.SpecialFunctions.OrdinaryHypergeometric

Goal state before:
pp: 𝕂 : Type u_1
𝔸 : Type u_2
inst✝² : RCLike 𝕂
inst✝¹ : NormedDivisionRing 𝔸
inst✝ : NormedAlgebra 𝕂 𝔸
a b c : 𝕂
habc : ∀ (kn : ℕ), ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
this : Tendsto (fun k ↦ (a + ↑k)⁻¹ * (b + ↑k)⁻¹ * (c + ↑k) * (1 + ↑k)) atTop (𝓝 1)
⊢ Tendsto (fun n ↦ ‖a + ↑n‖⁻¹ * ‖b + ↑n‖⁻¹ * ‖c + ↑n‖ * ‖1 + ↑n‖) atTop (𝓝 ↑1)

Tactic:
simp only [← norm_inv, ← norm_mul, NNReal.coe_one]

Goal state after:
𝕂 : Type u_1
𝔸 : Type u_2
inst✝² : RCLike 𝕂
inst✝¹ : NormedDivisionRing 𝔸
inst✝ : NormedAlgebra 𝕂 𝔸
a b c : 𝕂
habc : ∀ (kn : ℕ), ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
this : Tendsto (fun k ↦ (a + ↑k)⁻¹ * (b + ↑k)⁻¹ * (c + ↑k) * (1 + ↑k)) atTop (𝓝 1)
⊢ Tendsto (fun n ↦ ‖(a + ↑n)⁻¹ * (b + ↑n)⁻¹ * (c + ↑n) * (1 + ↑n)‖) atTop (𝓝 1)
Module: Mathlib.Analysis.Meromorphic.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
ι : Type u_5
s : Finset ι
G : ι → 𝕜 → E
x : 𝕜
h : ∀ σ ∈ s, MeromorphicAt (G σ) x
⊢ MeromorphicAt (fun z ↦ ∑ n ∈ s, G n z) x

Tactic:
by
  convert! sum h (s := s)
  simp

Goal state after:
Module: Mathlib.Analysis.Meromorphic.Basic

Goal state before:
pp: case pos
𝕜 : Type u_1
𝕜' : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
ι : Type u_5
F : ι → 𝕜 → 𝕜'
x : 𝕜
hF : ∀ (i : ι), MeromorphicAt (F i) x
h₂f : Function.HasFiniteSupport F
⊢ MeromorphicAt (∑ᶠ (i : ι), F i) x

Tactic:
simpa [finsum_eq_sum F h₂f] using sum (by aesop)

Goal state after:
Module: Mathlib.Analysis.Meromorphic.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
x : 𝕜
inst✝ : CompleteSpace E
f : 𝕜 → E
n : ℕ
h : AnalyticAt 𝕜 (fun z ↦ (z - x) ^ n • f z) x
⊢ ∀ᶠ (y : 𝕜) in 𝓝[≠] x, AnalyticAt 𝕜 f y

Tactic:
apply AnalyticAt.eventually_analyticAt at h

Goal state after:
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
x : 𝕜
inst✝ : CompleteSpace E
f : 𝕜 → E
n : ℕ
h : ∀ᶠ (y : 𝕜) in 𝓝 x, AnalyticAt 𝕜 (fun z ↦ (z - x) ^ n • f z) y
⊢ ∀ᶠ (y : 𝕜) in 𝓝[≠] x, AnalyticAt 𝕜 f y
Module: Mathlib.Analysis.Calculus.FDeriv.Extend

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
s : Set E
x : E
f' : E →L[ℝ] F
f_diff : DifferentiableOn ℝ f s
s_conv : Convex ℝ s
s_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
f_cont : ∀ y ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s, ContinuousWithinAt f s y
h : Tendsto (fun y ↦ fderiv ℝ f y) (𝓝[s] x) (𝓝 f')
hx : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
ε : ℝ
ε_pos : 0 < ε
δ : ℝ
δ_pos : δ > 0
hδ : ∀ y ∈ s, dist y x < δ → ‖fderiv ℝ f y - f'‖ < ε
B : Set E := ball x δ
key : ∀ p ∈ (B ∩ s) ×ˢ (B ∩ s), ‖f p.2 - f p.1 - (f' p.2 - f' p.1)‖ ≤ ε * ‖p.2 - p.1‖
u v : E
uv_in : (u, v) ∈ closure[instTopologicalSpaceProd] ((B ∩ s) ×ˢ (B ∩ s))
f_cont' : ∀ y ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s, ContinuousWithinAt (f - ⇑f') s y
this✝ : (B ∩ s) ×ˢ (B ∩ s) ⊆ s ×ˢ s
u_in : u ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
v_in : v ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
this : ∀ (u v : E), f v - f u - (f' v - f' u) = f v - f' v - (f u - f' u)
⊢ Tendsto (fun y ↦ ‖f y.2 - f' y.2 - (f y.1 - f' y.1)‖) (𝓝[s] u ×ˢ 𝓝[s] v) (𝓝 ‖f v - f' v - (f u - f' u)‖)

Tactic:
exact
      Tendsto.comp continuous_norm.continuousAt
        ((Tendsto.comp (f_cont' v v_in) tendsto_snd).sub <|
          Tendsto.comp (f_cont' u u_in) tendsto_fst)

Goal state after: