mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicTopology.SimplicialSet.TopAdj

Goal state before:
pp: ⊢ stdSimplexHomeomorphUnitInterval (⦋1⦌.toTopHomeo ((TopCat.Hom.hom (toTop.map (stdSimplex.δ 0))) default)) =
    stdSimplexHomeomorphUnitInterval ⟨Pi.single 1 1, ⋯⟩

Tactic:
]

Goal state after:
⊢ stdSimplexHomeomorphUnitInterval (⦋1⦌.toTopHomeo ((TopCat.Hom.hom (toTop.map (stdSimplex.δ 0))) default)) =
    stdSimplexHomeomorphUnitInterval ⟨Pi.single 1 1, ⋯⟩
Module: Mathlib.AlgebraicTopology.SimplicialSet.TopAdj

Goal state before:
pp: case e_6
⊢ stdSimplex.vertex ((ConcreteCategory.hom (SimplexCategory.δ 0)) 0) = ⟨Pi.single 1 1, ⋯⟩

Tactic:
]

Goal state after:
case e_6
⊢ stdSimplex.vertex ((ConcreteCategory.hom (SimplexCategory.δ 0)) 0) = ⟨Pi.single 1 1, ⋯⟩
Module: Mathlib.AlgebraicTopology.SingularHomology.Basic

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasCoproducts C
inst✝ : Preadditive C
n : ℕ
F : TopCat ⥤ C
X : TopCat
i : TopCat.toSSet.obj X _⦋n⦌
⊢ ((ConcreteCategory.hom ((sSetTopAdj.unit.app (TopCat.toSSet.obj X)).app (Opposite.op ⦋n⦌))) i).down ≫
      sSetTopAdj.counit.app X =
    i.down

Tactic:
]

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasCoproducts C
inst✝ : Preadditive C
n : ℕ
F : TopCat ⥤ C
X : TopCat
i : TopCat.toSSet.obj X _⦋n⦌
⊢ ((ConcreteCategory.hom ((sSetTopAdj.unit.app (TopCat.toSSet.obj X)).app (Opposite.op ⦋n⦌))) i).down ≫
      sSetTopAdj.counit.app X =
    i.down
Module: Mathlib.AlgebraicTopology.SimplicialSet.TopAdj

Goal state before:
pp: X : TopCat
⊢ ι₀ ≫ TopCat.toSSet.obj X ◁ toSSetObjI = TopCat.toSSet.map TopCat.ι₀ ≫ Functor.OplaxMonoidal.δ TopCat.toSSet X TopCat.I

Tactic:
]

Goal state after:
X : TopCat
⊢ ι₀ ≫ TopCat.toSSet.obj X ◁ toSSetObjI = TopCat.toSSet.map TopCat.ι₀ ≫ Functor.OplaxMonoidal.δ TopCat.toSSet X TopCat.I
Module: Mathlib.AlgebraicTopology.SimplicialSet.TopAdj

Goal state before:
pp: X : TopCat
⊢ ι₁ ≫ TopCat.toSSet.obj X ◁ toSSetObjI = TopCat.toSSet.map TopCat.ι₁ ≫ Functor.OplaxMonoidal.δ TopCat.toSSet X TopCat.I

Tactic:
]

Goal state after:
X : TopCat
⊢ ι₁ ≫ TopCat.toSSet.obj X ◁ toSSetObjI = TopCat.toSSet.map TopCat.ι₁ ≫ Functor.OplaxMonoidal.δ TopCat.toSSet X TopCat.I
Module: Mathlib.SetTheory.Cardinal.CountableCover

Goal state before:
pp: α ι : Type u
a : Cardinal.{u}
inst✝¹ : Countable ι
f : ι → Set α
l : Filter ι
inst✝ : l.NeBot
t : Set α
ht : ∀ x ∈ t, ∀ᶠ (i : ι) in l, x ∈ f i
h'f : ∀ (i : ι), #↑(f i) ≤ a
ha : ℵ₀ ≤ a
this : t ⊆ ⋃ i, f i
⊢ ℵ₀ * a ≤ ℵ₀ * a

Tactic:
]

Goal state after:
Module: Mathlib.SetTheory.Cardinal.CountableCover

Goal state before:
pp: α : Type u
ι : Type v
a : Cardinal.{u}
inst✝¹ : Countable ι
f : ι → Set α
l : Filter ι
inst✝ : l.NeBot
ht : ∀ (x : α), ∀ᶠ (i : ι) in l, x ∈ f i
h'f : ∀ (i : ι), #↑(f i) ≤ a
⊢ #↑Set.univ ≤ a

Tactic:
]

Goal state after:
α : Type u
ι : Type v
a : Cardinal.{u}
inst✝¹ : Countable ι
f : ι → Set α
l : Filter ι
inst✝ : l.NeBot
ht : ∀ (x : α), ∀ᶠ (i : ι) in l, x ∈ f i
h'f : ∀ (i : ι), #↑(f i) ≤ a
⊢ #↑Set.univ ≤ a
Module: Mathlib.SetTheory.Cardinal.CountableCover

Goal state before:
pp: case a
α : Type u
ι : Type v
a : Cardinal.{u}
inst✝¹ : Countable ι
f : ι → Set α
l : Filter ι
inst✝ : l.NeBot
ht : ∀ (x : α), ∀ᶠ (i : ι) in l, x ∈ f i
h'f : ∀ (i : ι), #↑(f i) = a
i : ι
⊢ #↑(f i) ≤ #α

Tactic:
]

Goal state after:
case a
α : Type u
ι : Type v
a : Cardinal.{u}
inst✝¹ : Countable ι
f : ι → Set α
l : Filter ι
inst✝ : l.NeBot
ht : ∀ (x : α), ∀ᶠ (i : ι) in l, x ∈ f i
h'f : ∀ (i : ι), #↑(f i) = a
i : ι
⊢ #↑(f i) ≤ #α
Module: Mathlib.AlgebraicTopology.SimplicialSet.NerveAdjunction

Goal state before:
pp: case «0»
X Y : Truncated 2
f₀ : X.obj (op { obj := ⦋0⦌, property := _proof_11 }) → Y.obj (op { obj := ⦋0⦌, property := _proof_11 })
f₁ : X.obj (op { obj := ⦋1⦌, property := _proof_12 }) → Y.obj (op { obj := ⦋1⦌, property := _proof_12 })
hδ₁ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 1 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 1 _proof_11 _proof_13).op)) (f₁ x)
hδ₀ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 0 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 0 _proof_11 _proof_13).op)) (f₁ x)
hσ :
  ∀ (x : X.obj (op { obj := ⦋0⦌, property := _proof_11 })),
    f₁ ((ConcreteCategory.hom (X.map (σ₂ 0 _proof_13 _proof_11).op)) x) =
      (ConcreteCategory.hom (Y.map (σ₂ 0 _proof_13 _proof_11).op)) (f₀ x)
hY : Y.StrictSegal
x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })
⊢ f₁
      ((ConcreteCategory.hom (X.map (δ₂ 2 _proof_12 _proof_15).op))
        ((ConcreteCategory.hom (X.map (σ₂ 0 _proof_15 _proof_12).op)) x)) =
    ((hY.spineEquiv 2 f₂._proof_4) ((ConcreteCategory.hom (Y.map (σ₂ 0 _proof_15 _proof_12).op)) (f₁ x))).arrow 0

Tactic:
]

Goal state after:
case «0»
X Y : Truncated 2
f₀ : X.obj (op { obj := ⦋0⦌, property := _proof_11 }) → Y.obj (op { obj := ⦋0⦌, property := _proof_11 })
f₁ : X.obj (op { obj := ⦋1⦌, property := _proof_12 }) → Y.obj (op { obj := ⦋1⦌, property := _proof_12 })
hδ₁ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 1 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 1 _proof_11 _proof_13).op)) (f₁ x)
hδ₀ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 0 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 0 _proof_11 _proof_13).op)) (f₁ x)
hσ :
  ∀ (x : X.obj (op { obj := ⦋0⦌, property := _proof_11 })),
    f₁ ((ConcreteCategory.hom (X.map (σ₂ 0 _proof_13 _proof_11).op)) x) =
      (ConcreteCategory.hom (Y.map (σ₂ 0 _proof_13 _proof_11).op)) (f₀ x)
hY : Y.StrictSegal
x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })
⊢ f₁
      ((ConcreteCategory.hom (X.map (δ₂ 2 _proof_12 _proof_15).op))
        ((ConcreteCategory.hom (X.map (σ₂ 0 _proof_15 _proof_12).op)) x)) =
    ((hY.spineEquiv 2 f₂._proof_4) ((ConcreteCategory.hom (Y.map (σ₂ 0 _proof_15 _proof_12).op)) (f₁ x))).arrow 0
Module: Mathlib.Topology.Algebra.Module.Cardinality

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
x : 𝕜
x✝ : x ∈ Set.univ
U : Set 𝕜
hU : U ∈ 𝓝 x
c : 𝕜
c_pos : 0 < ‖c‖
hc : ‖c‖ < 1
A : Tendsto (fun n ↦ x + c ^ n) atTop (𝓝 x)
⊢ ∃ y ∈ U ∩ Set.univ, y ≠ x

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
x : 𝕜
x✝ : x ∈ Set.univ
U : Set 𝕜
hU : U ∈ 𝓝 x
c : 𝕜
c_pos : 0 < ‖c‖
hc : ‖c‖ < 1
A : Tendsto (fun n ↦ x + c ^ n) atTop (𝓝 x)
⊢ ∃ y ∈ U ∩ Set.univ, y ≠ x
Module: Mathlib.Topology.Algebra.Module.Cardinality

Goal state before:
pp: E : Type u_1
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : Zero E
inst✝² : MulActionWithZero 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul 𝕜 E
s : Set E
hs : s ∈ 𝓝 0
c : 𝕜
hc : 1 < ‖c‖
cn_ne : ∀ (n : ℕ), c ^ n ≠ 0
x : E
⊢ ‖c‖⁻¹ < 1

Tactic:
]

Goal state after:
E : Type u_1
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : Zero E
inst✝² : MulActionWithZero 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul 𝕜 E
s : Set E
hs : s ∈ 𝓝 0
c : 𝕜
hc : 1 < ‖c‖
cn_ne : ∀ (n : ℕ), c ^ n ≠ 0
x : E
⊢ ‖c‖⁻¹ < 1
Module: Mathlib.Topology.Algebra.Module.Cardinality

Goal state before:
pp: E : Type u_1
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : Zero E
inst✝² : MulActionWithZero 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul 𝕜 E
s : Set E
hs : s ∈ 𝓝 0
c : 𝕜
hc : 1 < ‖c‖
cn_ne : ∀ (n : ℕ), c ^ n ≠ 0
x : E
this : Tendsto (fun n ↦ (c ^ n)⁻¹ • x) atTop (𝓝 0)
⊢ ∀ᶠ (n : ℕ) in atTop, x ∈ c ^ n • s

Tactic:
]

Goal state after:
E : Type u_1
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : Zero E
inst✝² : MulActionWithZero 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul 𝕜 E
s : Set E
hs : s ∈ 𝓝 0
c : 𝕜
hc : 1 < ‖c‖
cn_ne : ∀ (n : ℕ), c ^ n ≠ 0
x : E
this : Tendsto (fun n ↦ (c ^ n)⁻¹ • x) atTop (𝓝 0)
⊢ ∀ᶠ (n : ℕ) in atTop, x ∈ c ^ n • s
Module: Mathlib.Analysis.AbsoluteValue.Equivalence

Goal state before:
pp: case inr
R : Type u_1
S : Type u_2
inst✝³ : Field R
inst✝² : Semifield S
inst✝¹ : LinearOrder S
v w : AbsoluteValue R S
inst✝ : IsStrictOrderedRing S
h : ∀ (x : R), v x < 1 ↔ w x < 1
x y : R
hy₀ : v x ≠ 0
⊢ v (y * x⁻¹) < 1 ↔ w (y * x⁻¹) < 1

Tactic:
]

Goal state after:
case inr
R : Type u_1
S : Type u_2
inst✝³ : Field R
inst✝² : Semifield S
inst✝¹ : LinearOrder S
v w : AbsoluteValue R S
inst✝ : IsStrictOrderedRing S
h : ∀ (x : R), v x < 1 ↔ w x < 1
x y : R
hy₀ : v x ≠ 0
⊢ v (y * x⁻¹) < 1 ↔ w (y * x⁻¹) < 1
Module: Mathlib.Topology.Algebra.Module.Cardinality

Goal state before:
pp: E : Type u_1
𝕜 : Type u_2
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : AddGroup E
inst✝³ : MulActionWithZero 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul 𝕜 E
s : Set E
x : E
hs : s ∈ 𝓝 x
g : E ≃ₜ E := Homeomorph.addLeft x
t : Set E := ⇑g ⁻¹' s
this : t ∈ 𝓝 0
A : #↑s = #E
B : #↑t = #↑s
⊢ #↑s = #E

Tactic:
]

Goal state after:
E : Type u_1
𝕜 : Type u_2
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : AddGroup E
inst✝³ : MulActionWithZero 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul 𝕜 E
s : Set E
x : E
hs : s ∈ 𝓝 x
g : E ≃ₜ E := Homeomorph.addLeft x
t : Set E := ⇑g ⁻¹' s
this : t ∈ 𝓝 0
A : #↑s = #E
B : #↑t = #↑s
⊢ #↑s = #E
Module: Mathlib.Analysis.AbsoluteValue.Equivalence

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁵ : Field R
inst✝⁴ : Semifield S
inst✝³ : LinearOrder S
v w : AbsoluteValue R S
inst✝² : IsStrictOrderedRing S
inst✝¹ : Archimedean S
inst✝ : ExistsAddOfLE S
hv✝ : v.IsNontrivial
h : ∀ (x : R), v x < 1 → w x < 1
a : R
ha₀ : a ≠ 0
hw : w a < 1
x₀ : R
hx₀ : x₀ ≠ 0 ∧ v x₀ < 1
n : ℕ
hv : 1 ≤ v a
⊢ w (x₀ * (a ^ n)⁻¹) < 1

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁵ : Field R
inst✝⁴ : Semifield S
inst✝³ : LinearOrder S
v w : AbsoluteValue R S
inst✝² : IsStrictOrderedRing S
inst✝¹ : Archimedean S
inst✝ : ExistsAddOfLE S
hv✝ : v.IsNontrivial
h : ∀ (x : R), v x < 1 → w x < 1
a : R
ha₀ : a ≠ 0
hw : w a < 1
x₀ : R
hx₀ : x₀ ≠ 0 ∧ v x₀ < 1
n : ℕ
hv : 1 ≤ v a
⊢ w (x₀ * (a ^ n)⁻¹) < 1
Module: Mathlib.Topology.Algebra.Module.Cardinality

Goal state before:
pp: E : Type u
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : CompleteSpace 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : Nontrivial E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul 𝕜 E
s : Set E
hs : s.Countable
⊢ interior s = ∅

Tactic:
]

Goal state after:
E : Type u
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : CompleteSpace 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : Nontrivial E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul 𝕜 E
s : Set E
hs : s.Countable
⊢ interior s = ∅
Module: Mathlib.Analysis.AbsoluteValue.Equivalence

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁵ : Field R
inst✝⁴ : Semifield S
inst✝³ : LinearOrder S
v w : AbsoluteValue R S
inst✝² : IsStrictOrderedRing S
inst✝¹ : Archimedean S
inst✝ : ExistsAddOfLE S
hv✝ : v.IsNontrivial
h : ∀ (x : R), v x < 1 → w x < 1
a : R
ha₀ : a ≠ 0
hw : w a < 1
x₀ : R
hx₀ : x₀ ≠ 0 ∧ v x₀ < 1
n : ℕ
hv : 1 ≤ v a
⊢ v x₀ < v a ^ n

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁵ : Field R
inst✝⁴ : Semifield S
inst✝³ : LinearOrder S
v w : AbsoluteValue R S
inst✝² : IsStrictOrderedRing S
inst✝¹ : Archimedean S
inst✝ : ExistsAddOfLE S
hv✝ : v.IsNontrivial
h : ∀ (x : R), v x < 1 → w x < 1
a : R
ha₀ : a ≠ 0
hw : w a < 1
x₀ : R
hx₀ : x₀ ≠ 0 ∧ v x₀ < 1
n : ℕ
hv : 1 ≤ v a
⊢ v x₀ < v a ^ n
Module: Mathlib.AlgebraicTopology.SimplicialSet.NerveAdjunction

Goal state before:
pp: case «0»
X Y : Truncated 2
f₀ : X.obj (op { obj := ⦋0⦌, property := _proof_11 }) → Y.obj (op { obj := ⦋0⦌, property := _proof_11 })
f₁ : X.obj (op { obj := ⦋1⦌, property := _proof_12 }) → Y.obj (op { obj := ⦋1⦌, property := _proof_12 })
hδ₁ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 1 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 1 _proof_11 _proof_13).op)) (f₁ x)
hδ₀ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 0 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 0 _proof_11 _proof_13).op)) (f₁ x)
hσ :
  ∀ (x : X.obj (op { obj := ⦋0⦌, property := _proof_11 })),
    f₁ ((ConcreteCategory.hom (X.map (σ₂ 0 _proof_13 _proof_11).op)) x) =
      (ConcreteCategory.hom (Y.map (σ₂ 0 _proof_13 _proof_11).op)) (f₀ x)
hY : Y.StrictSegal
x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })
⊢ (ConcreteCategory.hom (Y.map (σ₂ 0 _proof_13 _proof_11).op))
      ((ConcreteCategory.hom (Y.map (δ₂ 1 _proof_11 _proof_13).op)) (f₁ x)) =
    (ConcreteCategory.hom (Y.map (σ₂ 0 δ₂_zero_comp_σ₂_one._proof_3 δ₂_zero_comp_σ₂_one._proof_4).op))
      ((ConcreteCategory.hom (Y.map (δ₂ 1 δ₂_zero_comp_σ₂_one._proof_4 δ₂_zero_comp_σ₂_one._proof_3).op)) (f₁ x))

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Calculus.IteratedDeriv.ConvergenceOnBall

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : RCLike 𝕜
f : 𝕜 → 𝕜
x : 𝕜
r : ENNReal
hr_pos : 0 < r
h : AnalyticOnNhd 𝕜 f (Metric.eball x r)
⊢ r ≤ (FormalMultilinearSeries.ofScalars 𝕜 fun n ↦ iteratedDeriv n f x / ↑n.factorial).radius →
    HasFPowerSeriesOnBall f (FormalMultilinearSeries.ofScalars 𝕜 fun n ↦ iteratedDeriv n f x / ↑n.factorial) x r

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝ : RCLike 𝕜
f : 𝕜 → 𝕜
x : 𝕜
r : ENNReal
hr_pos : 0 < r
h : AnalyticOnNhd 𝕜 f (Metric.eball x r)
⊢ r ≤ (FormalMultilinearSeries.ofScalars 𝕜 fun n ↦ iteratedDeriv n f x / ↑n.factorial).radius →
    HasFPowerSeriesOnBall f (FormalMultilinearSeries.ofScalars 𝕜 fun n ↦ iteratedDeriv n f x / ↑n.factorial) x r
Module: Mathlib.AlgebraicTopology.SimplicialSet.NerveAdjunction

Goal state before:
pp: case «1»
X Y : Truncated 2
f₀ : X.obj (op { obj := ⦋0⦌, property := _proof_11 }) → Y.obj (op { obj := ⦋0⦌, property := _proof_11 })
f₁ : X.obj (op { obj := ⦋1⦌, property := _proof_12 }) → Y.obj (op { obj := ⦋1⦌, property := _proof_12 })
hδ₁ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 1 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 1 _proof_11 _proof_13).op)) (f₁ x)
hδ₀ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 0 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 0 _proof_11 _proof_13).op)) (f₁ x)
hσ :
  ∀ (x : X.obj (op { obj := ⦋0⦌, property := _proof_11 })),
    f₁ ((ConcreteCategory.hom (X.map (σ₂ 0 _proof_13 _proof_11).op)) x) =
      (ConcreteCategory.hom (Y.map (σ₂ 0 _proof_13 _proof_11).op)) (f₀ x)
hY : Y.StrictSegal
x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })
⊢ f₁
      ((ConcreteCategory.hom (X.map (δ₂ 0 _proof_12 _proof_15).op))
        ((ConcreteCategory.hom (X.map (σ₂ 0 _proof_15 _proof_12).op)) x)) =
    ((hY.spineEquiv 2 f₂._proof_4) ((ConcreteCategory.hom (Y.map (σ₂ 0 _proof_15 _proof_12).op)) (f₁ x))).arrow 1

Tactic:
]

Goal state after:
case «1»
X Y : Truncated 2
f₀ : X.obj (op { obj := ⦋0⦌, property := _proof_11 }) → Y.obj (op { obj := ⦋0⦌, property := _proof_11 })
f₁ : X.obj (op { obj := ⦋1⦌, property := _proof_12 }) → Y.obj (op { obj := ⦋1⦌, property := _proof_12 })
hδ₁ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 1 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 1 _proof_11 _proof_13).op)) (f₁ x)
hδ₀ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 0 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 0 _proof_11 _proof_13).op)) (f₁ x)
hσ :
  ∀ (x : X.obj (op { obj := ⦋0⦌, property := _proof_11 })),
    f₁ ((ConcreteCategory.hom (X.map (σ₂ 0 _proof_13 _proof_11).op)) x) =
      (ConcreteCategory.hom (Y.map (σ₂ 0 _proof_13 _proof_11).op)) (f₀ x)
hY : Y.StrictSegal
x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })
⊢ f₁
      ((ConcreteCategory.hom (X.map (δ₂ 0 _proof_12 _proof_15).op))
        ((ConcreteCategory.hom (X.map (σ₂ 0 _proof_15 _proof_12).op)) x)) =
    ((hY.spineEquiv 2 f₂._proof_4) ((ConcreteCategory.hom (Y.map (σ₂ 0 _proof_15 _proof_12).op)) (f₁ x))).arrow 1
Module: Mathlib.Analysis.Calculus.IteratedDeriv.ConvergenceOnBall

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : RCLike 𝕜
f : 𝕜 → 𝕜
x : 𝕜
r : ENNReal
hr_pos : 0 < r
h : AnalyticOnNhd 𝕜 f (Metric.eball x r)
p : FormalMultilinearSeries 𝕜 𝕜 𝕜 := FormalMultilinearSeries.ofScalars 𝕜 fun n ↦ iteratedDeriv n f x / ↑n.factorial
hr : r ≤ p.radius
g : 𝕜 → 𝕜 := fun t ↦ p.sum (t - x)
hg : HasFPowerSeriesOnBall g p x p.radius
hg' :
  IsPreconnected (Metric.eball x r) →
    ∀ {z₀ : 𝕜}, z₀ ∈ Metric.eball x r → f =ᶠ[nhds z₀] g → Set.EqOn f g (Metric.eball x r)
hf : HasFPowerSeriesAt f (FormalMultilinearSeries.ofScalars 𝕜 fun n ↦ iteratedDeriv n f x / ↑n.factorial) x
⊢ ∃ ε > 0, Metric.eball x ε ⊆ {x | (fun x ↦ f x = g x) x}

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝ : RCLike 𝕜
f : 𝕜 → 𝕜
x : 𝕜
r : ENNReal
hr_pos : 0 < r
h : AnalyticOnNhd 𝕜 f (Metric.eball x r)
p : FormalMultilinearSeries 𝕜 𝕜 𝕜 := FormalMultilinearSeries.ofScalars 𝕜 fun n ↦ iteratedDeriv n f x / ↑n.factorial
hr : r ≤ p.radius
g : 𝕜 → 𝕜 := fun t ↦ p.sum (t - x)
hg : HasFPowerSeriesOnBall g p x p.radius
hg' :
  IsPreconnected (Metric.eball x r) →
    ∀ {z₀ : 𝕜}, z₀ ∈ Metric.eball x r → f =ᶠ[nhds z₀] g → Set.EqOn f g (Metric.eball x r)
hf : HasFPowerSeriesAt f (FormalMultilinearSeries.ofScalars 𝕜 fun n ↦ iteratedDeriv n f x / ↑n.factorial) x
⊢ ∃ ε > 0, Metric.eball x ε ⊆ {x | (fun x ↦ f x = g x) x}
Module: Mathlib.AlgebraicTopology.SimplicialSet.NerveAdjunction

Goal state before:
pp: case «0»
X Y : Truncated 2
f₀ : X.obj (op { obj := ⦋0⦌, property := _proof_11 }) → Y.obj (op { obj := ⦋0⦌, property := _proof_11 })
f₁ : X.obj (op { obj := ⦋1⦌, property := _proof_12 }) → Y.obj (op { obj := ⦋1⦌, property := _proof_12 })
hδ₁ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 1 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 1 _proof_11 _proof_13).op)) (f₁ x)
hδ₀ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 0 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 0 _proof_11 _proof_13).op)) (f₁ x)
hσ :
  ∀ (x : X.obj (op { obj := ⦋0⦌, property := _proof_11 })),
    f₁ ((ConcreteCategory.hom (X.map (σ₂ 0 _proof_13 _proof_11).op)) x) =
      (ConcreteCategory.hom (Y.map (σ₂ 0 _proof_13 _proof_11).op)) (f₀ x)
hY : Y.StrictSegal
x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })
⊢ f₁
      ((ConcreteCategory.hom (X.map (δ₂ 2 _proof_12 _proof_15).op))
        ((ConcreteCategory.hom (X.map (σ₂ 1 _proof_15 _proof_12).op)) x)) =
    ((hY.spineEquiv 2 f₂._proof_4) ((ConcreteCategory.hom (Y.map (σ₂ 1 _proof_15 _proof_12).op)) (f₁ x))).arrow 0

Tactic:
]

Goal state after:
case «0»
X Y : Truncated 2
f₀ : X.obj (op { obj := ⦋0⦌, property := _proof_11 }) → Y.obj (op { obj := ⦋0⦌, property := _proof_11 })
f₁ : X.obj (op { obj := ⦋1⦌, property := _proof_12 }) → Y.obj (op { obj := ⦋1⦌, property := _proof_12 })
hδ₁ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 1 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 1 _proof_11 _proof_13).op)) (f₁ x)
hδ₀ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 0 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 0 _proof_11 _proof_13).op)) (f₁ x)
hσ :
  ∀ (x : X.obj (op { obj := ⦋0⦌, property := _proof_11 })),
    f₁ ((ConcreteCategory.hom (X.map (σ₂ 0 _proof_13 _proof_11).op)) x) =
      (ConcreteCategory.hom (Y.map (σ₂ 0 _proof_13 _proof_11).op)) (f₀ x)
hY : Y.StrictSegal
x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })
⊢ f₁
      ((ConcreteCategory.hom (X.map (δ₂ 2 _proof_12 _proof_15).op))
        ((ConcreteCategory.hom (X.map (σ₂ 1 _proof_15 _proof_12).op)) x)) =
    ((hY.spineEquiv 2 f₂._proof_4) ((ConcreteCategory.hom (Y.map (σ₂ 1 _proof_15 _proof_12).op)) (f₁ x))).arrow 0
Module: Mathlib.LinearAlgebra.Complex.Determinant

Goal state before:
pp: ⊢ 1 * -1 - 0 * 0 = -1

Tactic:
]

Goal state after:
⊢ 1 * -1 - 0 * 0 = -1
Module: Mathlib.Analysis.Normed.Module.Connected

Goal state before:
pp: E : Type u_1
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul ℝ E
h : 1 < Module.rank ℝ E
s : Set E
hs : s.Countable
this : Nontrivial E
a : E
ha : a ∈ sᶜ
b : E
hb : b ∈ sᶜ
hab : a ≠ b
c : E := 2⁻¹ • (a + b)
x : E := 2⁻¹ • (b - a)
Ia : c - x = a
Ib : c + x = b
x_ne_zero : x ≠ 0
y : E
hy : LinearIndependent ℝ ![x, y]
t t' : ℝ
htt' : t ≠ t'
N : {c + x} ∩ s = ∅
⊢ [c + x -[ℝ] c + t • y] ∩ [c + x -[ℝ] c + t' • y] ∩ s ⊆ {c + x} ∩ s

Tactic:
]

Goal state after:
E : Type u_1
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul ℝ E
h : 1 < Module.rank ℝ E
s : Set E
hs : s.Countable
this : Nontrivial E
a : E
ha : a ∈ sᶜ
b : E
hb : b ∈ sᶜ
hab : a ≠ b
c : E := 2⁻¹ • (a + b)
x : E := 2⁻¹ • (b - a)
Ia : c - x = a
Ib : c + x = b
x_ne_zero : x ≠ 0
y : E
hy : LinearIndependent ℝ ![x, y]
t t' : ℝ
htt' : t ≠ t'
N : {c + x} ∩ s = ∅
⊢ [c + x -[ℝ] c + t • y] ∩ [c + x -[ℝ] c + t' • y] ∩ s ⊆ {c + x} ∩ s
Module: Mathlib.AlgebraicTopology.SimplicialSet.NerveAdjunction

Goal state before:
pp: case «1»
X Y : Truncated 2
f₀ : X.obj (op { obj := ⦋0⦌, property := _proof_11 }) → Y.obj (op { obj := ⦋0⦌, property := _proof_11 })
f₁ : X.obj (op { obj := ⦋1⦌, property := _proof_12 }) → Y.obj (op { obj := ⦋1⦌, property := _proof_12 })
hδ₁ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 1 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 1 _proof_11 _proof_13).op)) (f₁ x)
hδ₀ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 0 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 0 _proof_11 _proof_13).op)) (f₁ x)
hσ :
  ∀ (x : X.obj (op { obj := ⦋0⦌, property := _proof_11 })),
    f₁ ((ConcreteCategory.hom (X.map (σ₂ 0 _proof_13 _proof_11).op)) x) =
      (ConcreteCategory.hom (Y.map (σ₂ 0 _proof_13 _proof_11).op)) (f₀ x)
hY : Y.StrictSegal
x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })
⊢ f₁
      ((ConcreteCategory.hom (X.map (δ₂ 0 _proof_12 _proof_15).op))
        ((ConcreteCategory.hom (X.map (σ₂ 1 _proof_15 _proof_12).op)) x)) =
    ((hY.spineEquiv 2 f₂._proof_4) ((ConcreteCategory.hom (Y.map (σ₂ 1 _proof_15 _proof_12).op)) (f₁ x))).arrow 1

Tactic:
]

Goal state after:
case «1»
X Y : Truncated 2
f₀ : X.obj (op { obj := ⦋0⦌, property := _proof_11 }) → Y.obj (op { obj := ⦋0⦌, property := _proof_11 })
f₁ : X.obj (op { obj := ⦋1⦌, property := _proof_12 }) → Y.obj (op { obj := ⦋1⦌, property := _proof_12 })
hδ₁ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 1 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 1 _proof_11 _proof_13).op)) (f₁ x)
hδ₀ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 0 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 0 _proof_11 _proof_13).op)) (f₁ x)
hσ :
  ∀ (x : X.obj (op { obj := ⦋0⦌, property := _proof_11 })),
    f₁ ((ConcreteCategory.hom (X.map (σ₂ 0 _proof_13 _proof_11).op)) x) =
      (ConcreteCategory.hom (Y.map (σ₂ 0 _proof_13 _proof_11).op)) (f₀ x)
hY : Y.StrictSegal
x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })
⊢ f₁
      ((ConcreteCategory.hom (X.map (δ₂ 0 _proof_12 _proof_15).op))
        ((ConcreteCategory.hom (X.map (σ₂ 1 _proof_15 _proof_12).op)) x)) =
    ((hY.spineEquiv 2 f₂._proof_4) ((ConcreteCategory.hom (Y.map (σ₂ 1 _proof_15 _proof_12).op)) (f₁ x))).arrow 1