l3lab/ntp-mathlib huggingface dataset:
theorem ext' (h : c.snd = d.snd) : c = d := by
  obtain ⟨⟨s₁, t₁⟩, _, h₁⟩ := c
  obtain ⟨⟨s₂, t₂⟩, _, h₂⟩ := d
  dsimp at h₁ h₂ h
  substs h h₁ h₂
  

Proof state:
case mk.mk.mk.mk
ι : Sort u_1
α : Type u_2
β : Type u_3
γ : Type u_4
κ : ι → Sort u_5
r : α → β → Prop
s s₁ s₂ : Set α
t t₁✝ t₂ t₁ : Set β
closure_fst✝¹ closure_fst✝ : intentClosure r (extentClosure r t₁, t₁).1 = (extentClosure r t₁, t₁).2
⊢ { toProd := (extentClosure r t₁, t₁), closure_fst := closure_fst✝¹,
      closure_snd := (_ : extentClosure r t₁ = extentClosure r t₁) } =
    { toProd := (extentClosure r t₁, t₁), closure_fst := closure_fst✝,
      closure_snd := (_ : extentClosure r t₁ = extentClosure r t₁) }

Next tactic:
rfl
instance instSupConcept : Sup (Concept α β r) :=
  ⟨fun c d =>
    { fst := extentClosure r (c.snd ∩ d.snd)
      snd := c.snd ∩ d.snd
      closure_fst := by
        

Proof state:
ι : Sort u_1
α : Type u_2
β : Type u_3
γ : Type u_4
κ : ι → Sort u_5
r : α → β → Prop
s s₁ s₂ : Set α
t t₁ t₂ : Set β
c✝ d✝ c d : Concept α β r
⊢ intentClosure r (extentClosure r (c.toProd.2 ∩ d.toProd.2), c.toProd.2 ∩ d.toProd.2).1 =
    (extentClosure r (c.toProd.2 ∩ d.toProd.2), c.toProd.2 ∩ d.toProd.2).2

Next tactic:
rw [← c.closure_fst, ← d.closure_fst, ← intentClosure_union,
          intentClosure_extentClosure_intentClosure]
instance instInfConcept : Inf (Concept α β r) :=
  ⟨fun c d =>
    { fst := c.fst ∩ d.fst
      snd := intentClosure r (c.fst ∩ d.fst)
      closure_fst := rfl
      closure_snd := by
        

Proof state:
ι : Sort u_1
α : Type u_2
β : Type u_3
γ : Type u_4
κ : ι → Sort u_5
r : α → β → Prop
s s₁ s₂ : Set α
t t₁ t₂ : Set β
c✝ d✝ c d : Concept α β r
⊢ extentClosure r (c.toProd.1 ∩ d.toProd.1, intentClosure r (c.toProd.1 ∩ d.toProd.1)).2 =
    (c.toProd.1 ∩ d.toProd.1, intentClosure r (c.toProd.1 ∩ d.toProd.1)).1

Next tactic:
rw [← c.closure_snd, ← d.closure_snd, ← extentClosure_union,
          extentClosure_intentClosure_extentClosure]
@[simp]
theorem snd_subset_snd_iff : c.snd ⊆ d.snd ↔ d ≤ c := by
  

Proof state:
ι : Sort u_1
α : Type u_2
β : Type u_3
γ : Type u_4
κ : ι → Sort u_5
r : α → β → Prop
s s₁ s₂ : Set α
t t₁ t₂ : Set β
c d : Concept α β r
⊢ c.toProd.2 ⊆ d.toProd.2 ↔ d ≤ c

Next tactic:
refine' ⟨fun h => _, fun h => _⟩
@[simp]
theorem snd_subset_snd_iff : c.snd ⊆ d.snd ↔ d ≤ c := by
  refine' ⟨fun h => _, fun h => _⟩
  · 

Proof state:
case refine'_1
ι : Sort u_1
α : Type u_2
β : Type u_3
γ : Type u_4
κ : ι → Sort u_5
r : α → β → Prop
s s₁ s₂ : Set α
t t₁ t₂ : Set β
c d : Concept α β r
h : c.toProd.2 ⊆ d.toProd.2
⊢ d ≤ c

Next tactic:
rw [← fst_subset_fst_iff, ← c.closure_snd, ← d.closure_snd]
@[simp]
theorem snd_subset_snd_iff : c.snd ⊆ d.snd ↔ d ≤ c := by
  refine' ⟨fun h => _, fun h => _⟩
  · rw [← fst_subset_fst_iff, ← c.closure_snd, ← d.closure_snd]
    

Proof state:
case refine'_1
ι : Sort u_1
α : Type u_2
β : Type u_3
γ : Type u_4
κ : ι → Sort u_5
r : α → β → Prop
s s₁ s₂ : Set α
t t₁ t₂ : Set β
c d : Concept α β r
h : c.toProd.2 ⊆ d.toProd.2
⊢ extentClosure r d.toProd.2 ⊆ extentClosure r c.toProd.2

Next tactic:
exact extentClosure_anti _ h
@[simp]
theorem snd_subset_snd_iff : c.snd ⊆ d.snd ↔ d ≤ c := by
  refine' ⟨fun h => _, fun h => _⟩
  · rw [← fst_subset_fst_iff, ← c.closure_snd, ← d.closure_snd]
    exact extentClosure_anti _ h
  · 

Proof state:
case refine'_2
ι : Sort u_1
α : Type u_2
β : Type u_3
γ : Type u_4
κ : ι → Sort u_5
r : α → β → Prop
s s₁ s₂ : Set α
t t₁ t₂ : Set β
c d : Concept α β r
h : d ≤ c
⊢ c.toProd.2 ⊆ d.toProd.2

Next tactic:
rw [← c.closure_fst, ← d.closure_fst]
@[simp]
theorem snd_subset_snd_iff : c.snd ⊆ d.snd ↔ d ≤ c := by
  refine' ⟨fun h => _, fun h => _⟩
  · rw [← fst_subset_fst_iff, ← c.closure_snd, ← d.closure_snd]
    exact extentClosure_anti _ h
  · rw [← c.closure_fst, ← d.closure_fst]
    

Proof state:
case refine'_2
ι : Sort u_1
α : Type u_2
β : Type u_3
γ : Type u_4
κ : ι → Sort u_5
r : α → β → Prop
s s₁ s₂ : Set α
t t₁ t₂ : Set β
c d : Concept α β r
h : d ≤ c
⊢ intentClosure r c.toProd.1 ⊆ intentClosure r d.toProd.1

Next tactic:
exact intentClosure_anti _ h
@[simp]
theorem snd_ssubset_snd_iff : c.snd ⊂ d.snd ↔ d < c := by
  

Proof state:
ι : Sort u_1
α : Type u_2
β : Type u_3
γ : Type u_4
κ : ι → Sort u_5
r : α → β → Prop
s s₁ s₂ : Set α
t t₁ t₂ : Set β
c d : Concept α β r
⊢ c.toProd.2 ⊂ d.toProd.2 ↔ d < c

Next tactic:
rw [ssubset_iff_subset_not_subset, lt_iff_le_not_le, snd_subset_snd_iff, snd_subset_snd_iff]
instance instLatticeConcept : Lattice (Concept α β r) :=
  { Concept.instSemilatticeInfConcept with
    sup := (· ⊔ ·)
    le_sup_left := fun c d => snd_subset_snd_iff.1 <| inter_subset_left _ _
    le_sup_right := fun c d => snd_subset_snd_iff.1 <| inter_subset_right _ _
    sup_le := fun c d e => by
      

Proof state:
ι : Sort u_1
α : Type u_2
β : Type u_3
γ : Type u_4
κ : ι → Sort u_5
r : α → β → Prop
s s₁ s₂ : Set α
t t₁ t₂ : Set β
c✝ d✝ : Concept α β r
src✝ : SemilatticeInf (Concept α β r) := instSemilatticeInfConcept
c d e : Concept α β r
⊢ c ≤ e → d ≤ e → c ⊔ d ≤ e

Next tactic:
simp_rw [← snd_subset_snd_iff]
instance instLatticeConcept : Lattice (Concept α β r) :=
  { Concept.instSemilatticeInfConcept with
    sup := (· ⊔ ·)
    le_sup_left := fun c d => snd_subset_snd_iff.1 <| inter_subset_left _ _
    le_sup_right := fun c d => snd_subset_snd_iff.1 <| inter_subset_right _ _
    sup_le := fun c d e => by
      simp_rw [← snd_subset_snd_iff]
      

Proof state:
ι : Sort u_1
α : Type u_2
β : Type u_3
γ : Type u_4
κ : ι → Sort u_5
r : α → β → Prop
s s₁ s₂ : Set α
t t₁ t₂ : Set β
c✝ d✝ : Concept α β r
src✝ : SemilatticeInf (Concept α β r) := instSemilatticeInfConcept
c d e : Concept α β r
⊢ e.toProd.2 ⊆ c.toProd.2 → e.toProd.2 ⊆ d.toProd.2 → e.toProd.2 ⊆ (c ⊔ d).toProd.2

Next tactic:
exact subset_inter
instance : SupSet (Concept α β r) :=
  ⟨fun S =>
    { fst := extentClosure r (⋂ c ∈ S, (c : Concept _ _ _).snd)
      snd := ⋂ c ∈ S, (c : Concept _ _ _).snd
      closure_fst := by
        

Proof state:
ι : Sort u_1
α : Type u_2
β : Type u_3
γ : Type u_4
κ : ι → Sort u_5
r : α → β → Prop
s s₁ s₂ : Set α
t t₁ t₂ : Set β
c d : Concept α β r
S : Set (Concept α β r)
⊢ intentClosure r (extentClosure r (⋂ c ∈ S, c.toProd.2), ⋂ c ∈ S, c.toProd.2).1 =
    (extentClosure r (⋂ c ∈ S, c.toProd.2), ⋂ c ∈ S, c.toProd.2).2

Next tactic:
simp_rw [← closure_fst, ← intentClosure_iUnion₂,
          intentClosure_extentClosure_intentClosure]
instance : InfSet (Concept α β r) :=
  ⟨fun S =>
    { fst := ⋂ c ∈ S, (c : Concept _ _ _).fst
      snd := intentClosure r (⋂ c ∈ S, (c : Concept _ _ _).fst)
      closure_fst := rfl
      closure_snd := by
        

Proof state:
ι : Sort u_1
α : Type u_2
β : Type u_3
γ : Type u_4
κ : ι → Sort u_5
r : α → β → Prop
s s₁ s₂ : Set α
t t₁ t₂ : Set β
c d : Concept α β r
S : Set (Concept α β r)
⊢ extentClosure r (⋂ c ∈ S, c.toProd.1, intentClosure r (⋂ c ∈ S, c.toProd.1)).2 =
    (⋂ c ∈ S, c.toProd.1, intentClosure r (⋂ c ∈ S, c.toProd.1)).1

Next tactic:
simp_rw [← closure_snd, ← extentClosure_iUnion₂,
          extentClosure_intentClosure_extentClosure]
/-- The function `expNegInvGlue` vanishes on `(-∞, 0]`. -/
theorem zero_of_nonpos {x : ℝ} (hx : x ≤ 0) : expNegInvGlue x = 0 := by 

Proof state:
x : ℝ
hx : x ≤ 0
⊢ expNegInvGlue x = 0

Next tactic:
simp [expNegInvGlue, hx]
/-- The function `expNegInvGlue` is positive on `(0, +∞)`. -/
theorem pos_of_pos {x : ℝ} (hx : 0 < x) : 0 < expNegInvGlue x := by
  

Proof state:
x : ℝ
hx : 0 < x
⊢ 0 < expNegInvGlue x

Next tactic:
simp [expNegInvGlue, not_le.2 hx, exp_pos]
/-- The function `expNegInvGlue` is nonnegative. -/
theorem nonneg (x : ℝ) : 0 ≤ expNegInvGlue x := by
  

Proof state:
x : ℝ
⊢ 0 ≤ expNegInvGlue x

Next tactic:
cases le_or_gt x 0 with
  | inl h => exact ge_of_eq (zero_of_nonpos h)
  | inr h => exact le_of_lt (pos_of_pos h)
/-- The function `expNegInvGlue` is nonnegative. -/
theorem nonneg (x : ℝ) : 0 ≤ expNegInvGlue x := by
  

Proof state:
x : ℝ
x✝ : x ≤ 0 ∨ x > 0
⊢ 0 ≤ expNegInvGlue x

Next tactic:
cases le_or_gt x 0 with
  | inl h => exact ge_of_eq (zero_of_nonpos h)
  | inr h => exact le_of_lt (pos_of_pos h)
/-- The function `expNegInvGlue` is nonnegative. -/
theorem nonneg (x : ℝ) : 0 ≤ expNegInvGlue x := by
  cases le_or_gt x 0 with
  

Proof state:
case inl
x : ℝ
h : x ≤ 0
⊢ 0 ≤ expNegInvGlue x

Next tactic:
| inl h => exact ge_of_eq (zero_of_nonpos h)
/-- The function `expNegInvGlue` is nonnegative. -/
theorem nonneg (x : ℝ) : 0 ≤ expNegInvGlue x := by
  cases le_or_gt x 0 with
  | inl h => 

Proof state:
case inl
x : ℝ
h : x ≤ 0
⊢ 0 ≤ expNegInvGlue x

Next tactic:
exact ge_of_eq (zero_of_nonpos h)
/-- The function `expNegInvGlue` is nonnegative. -/
theorem nonneg (x : ℝ) : 0 ≤ expNegInvGlue x := by
  cases le_or_gt x 0 with
  | inl h => exact ge_of_eq (zero_of_nonpos h)
  

Proof state:
case inr
x : ℝ
h : x > 0
⊢ 0 ≤ expNegInvGlue x

Next tactic:
| inr h => exact le_of_lt (pos_of_pos h)
/-- The function `expNegInvGlue` is nonnegative. -/
theorem nonneg (x : ℝ) : 0 ≤ expNegInvGlue x := by
  cases le_or_gt x 0 with
  | inl h => exact ge_of_eq (zero_of_nonpos h)
  | inr h => 

Proof state:
case inr
x : ℝ
h : x > 0
⊢ 0 ≤ expNegInvGlue x

Next tactic:
exact le_of_lt (pos_of_pos h)
/-- Our function tends to zero at zero faster than any $P(x^{-1})$, $P∈ℝ[X]$, tends to infinity. -/
theorem tendsto_polynomial_inv_mul_zero (p : ℝ[X]) :
    Tendsto (fun x ↦ p.eval x⁻¹ * expNegInvGlue x) (𝓝 0) (𝓝 0) := by
  

Proof state:
p : ℝ[X]
⊢ Tendsto (fun x => Polynomial.eval x⁻¹ p * expNegInvGlue x) (𝓝 0) (𝓝 0)

Next tactic:
simp only [expNegInvGlue, mul_ite, mul_zero]
/-- Our function tends to zero at zero faster than any $P(x^{-1})$, $P∈ℝ[X]$, tends to infinity. -/
theorem tendsto_polynomial_inv_mul_zero (p : ℝ[X]) :
    Tendsto (fun x ↦ p.eval x⁻¹ * expNegInvGlue x) (𝓝 0) (𝓝 0) := by
  simp only [expNegInvGlue, mul_ite, mul_zero]
  

Proof state:
p : ℝ[X]
⊢ Tendsto (fun x => if x ≤ 0 then 0 else Polynomial.eval x⁻¹ p * rexp (-x⁻¹)) (𝓝 0) (𝓝 0)

Next tactic:
refine tendsto_const_nhds.if ?_
/-- Our function tends to zero at zero faster than any $P(x^{-1})$, $P∈ℝ[X]$, tends to infinity. -/
theorem tendsto_polynomial_inv_mul_zero (p : ℝ[X]) :
    Tendsto (fun x ↦ p.eval x⁻¹ * expNegInvGlue x) (𝓝 0) (𝓝 0) := by
  simp only [expNegInvGlue, mul_ite, mul_zero]
  refine tendsto_const_nhds.if ?_
  

Proof state:
p : ℝ[X]
⊢ Tendsto (fun x => Polynomial.eval x⁻¹ p * rexp (-x⁻¹)) (𝓝 0 ⊓ 𝓟 {x | ¬x ≤ 0}) (𝓝 0)

Next tactic:
simp only [not_le]
/-- Our function tends to zero at zero faster than any $P(x^{-1})$, $P∈ℝ[X]$, tends to infinity. -/
theorem tendsto_polynomial_inv_mul_zero (p : ℝ[X]) :
    Tendsto (fun x ↦ p.eval x⁻¹ * expNegInvGlue x) (𝓝 0) (𝓝 0) := by
  simp only [expNegInvGlue, mul_ite, mul_zero]
  refine tendsto_const_nhds.if ?_
  simp only [not_le]
  

Proof state:
p : ℝ[X]
⊢ Tendsto (fun x => Polynomial.eval x⁻¹ p * rexp (-x⁻¹)) (𝓝 0 ⊓ 𝓟 {x | 0 < x}) (𝓝 0)

Next tactic:
have : Tendsto (fun x ↦ p.eval x⁻¹ / exp x⁻¹) (𝓝[>] 0) (𝓝 0) :=
    p.tendsto_div_exp_atTop.comp tendsto_inv_zero_atTop
/-- Our function tends to zero at zero faster than any $P(x^{-1})$, $P∈ℝ[X]$, tends to infinity. -/
theorem tendsto_polynomial_inv_mul_zero (p : ℝ[X]) :
    Tendsto (fun x ↦ p.eval x⁻¹ * expNegInvGlue x) (𝓝 0) (𝓝 0) := by
  simp only [expNegInvGlue, mul_ite, mul_zero]
  refine tendsto_const_nhds.if ?_
  simp only [not_le]
  have : Tendsto (fun x ↦ p.eval x⁻¹ / exp x⁻¹) (𝓝[>] 0) (𝓝 0) :=
    p.tendsto_div_exp_atTop.comp tendsto_inv_zero_atTop
  

Proof state:
p : ℝ[X]
this : Tendsto (fun x => Polynomial.eval x⁻¹ p / rexp x⁻¹) (𝓝[>] 0) (𝓝 0)
⊢ Tendsto (fun x => Polynomial.eval x⁻¹ p * rexp (-x⁻¹)) (𝓝 0 ⊓ 𝓟 {x | 0 < x}) (𝓝 0)

Next tactic:
refine this.congr' <| mem_of_superset self_mem_nhdsWithin fun x hx ↦ ?_
/-- Our function tends to zero at zero faster than any $P(x^{-1})$, $P∈ℝ[X]$, tends to infinity. -/
theorem tendsto_polynomial_inv_mul_zero (p : ℝ[X]) :
    Tendsto (fun x ↦ p.eval x⁻¹ * expNegInvGlue x) (𝓝 0) (𝓝 0) := by
  simp only [expNegInvGlue, mul_ite, mul_zero]
  refine tendsto_const_nhds.if ?_
  simp only [not_le]
  have : Tendsto (fun x ↦ p.eval x⁻¹ / exp x⁻¹) (𝓝[>] 0) (𝓝 0) :=
    p.tendsto_div_exp_atTop.comp tendsto_inv_zero_atTop
  refine this.congr' <| mem_of_superset self_mem_nhdsWithin fun x hx ↦ ?_
  

Proof state:
p : ℝ[X]
this : Tendsto (fun x => Polynomial.eval x⁻¹ p / rexp x⁻¹) (𝓝[>] 0) (𝓝 0)
x : ℝ
hx : x ∈ {x | 0 < x}
⊢ x ∈ {x | (fun x => Polynomial.eval x⁻¹ p / rexp x⁻¹ = (fun x => Polynomial.eval x⁻¹ p * rexp (-x⁻¹)) x) x}

Next tactic:
simp [expNegInvGlue, hx.out.not_le, exp_neg, div_eq_mul_inv]
theorem hasDerivAt_polynomial_eval_inv_mul (p : ℝ[X]) (x : ℝ) :
    HasDerivAt (fun x ↦ p.eval x⁻¹ * expNegInvGlue x)
      ((X ^ 2 * (p - derivative (R := ℝ) p)).eval x⁻¹ * expNegInvGlue x) x := by
  

Proof state:
p : ℝ[X]
x : ℝ
⊢ HasDerivAt (fun x => Polynomial.eval x⁻¹ p * expNegInvGlue x)
    (Polynomial.eval x⁻¹ (X ^ 2 * (p - derivative p)) * expNegInvGlue x) x

Next tactic:
rcases lt_trichotomy x 0 with hx | rfl | hx
theorem hasDerivAt_polynomial_eval_inv_mul (p : ℝ[X]) (x : ℝ) :
    HasDerivAt (fun x ↦ p.eval x⁻¹ * expNegInvGlue x)
      ((X ^ 2 * (p - derivative (R := ℝ) p)).eval x⁻¹ * expNegInvGlue x) x := by
  rcases lt_trichotomy x 0 with hx | rfl | hx
  · 

Proof state:
case inl
p : ℝ[X]
x : ℝ
hx : x < 0
⊢ HasDerivAt (fun x => Polynomial.eval x⁻¹ p * expNegInvGlue x)
    (Polynomial.eval x⁻¹ (X ^ 2 * (p - derivative p)) * expNegInvGlue x) x

Next tactic:
rw [zero_of_nonpos hx.le, mul_zero]
theorem hasDerivAt_polynomial_eval_inv_mul (p : ℝ[X]) (x : ℝ) :
    HasDerivAt (fun x ↦ p.eval x⁻¹ * expNegInvGlue x)
      ((X ^ 2 * (p - derivative (R := ℝ) p)).eval x⁻¹ * expNegInvGlue x) x := by
  rcases lt_trichotomy x 0 with hx | rfl | hx
  · rw [zero_of_nonpos hx.le, mul_zero]
    

Proof state:
case inl
p : ℝ[X]
x : ℝ
hx : x < 0
⊢ HasDerivAt (fun x => Polynomial.eval x⁻¹ p * expNegInvGlue x) 0 x

Next tactic:
refine (hasDerivAt_const _ 0).congr_of_eventuallyEq ?_
theorem hasDerivAt_polynomial_eval_inv_mul (p : ℝ[X]) (x : ℝ) :
    HasDerivAt (fun x ↦ p.eval x⁻¹ * expNegInvGlue x)
      ((X ^ 2 * (p - derivative (R := ℝ) p)).eval x⁻¹ * expNegInvGlue x) x := by
  rcases lt_trichotomy x 0 with hx | rfl | hx
  · rw [zero_of_nonpos hx.le, mul_zero]
    refine (hasDerivAt_const _ 0).congr_of_eventuallyEq ?_
    

Proof state:
case inl
p : ℝ[X]
x : ℝ
hx : x < 0
⊢ (fun x => Polynomial.eval x⁻¹ p * expNegInvGlue x) =ᶠ[𝓝 x] fun x => 0

Next tactic:
filter_upwards [gt_mem_nhds hx] with y hy
theorem hasDerivAt_polynomial_eval_inv_mul (p : ℝ[X]) (x : ℝ) :
    HasDerivAt (fun x ↦ p.eval x⁻¹ * expNegInvGlue x)
      ((X ^ 2 * (p - derivative (R := ℝ) p)).eval x⁻¹ * expNegInvGlue x) x := by
  rcases lt_trichotomy x 0 with hx | rfl | hx
  · rw [zero_of_nonpos hx.le, mul_zero]
    refine (hasDerivAt_const _ 0).congr_of_eventuallyEq ?_
    filter_upwards [gt_mem_nhds hx] with y hy
    

Proof state:
case h
p : ℝ[X]
x : ℝ
hx : x < 0
y : ℝ
hy : y < 0
⊢ Polynomial.eval y⁻¹ p * expNegInvGlue y = 0

Next tactic:
rw [zero_of_nonpos hy.le, mul_zero]
theorem hasDerivAt_polynomial_eval_inv_mul (p : ℝ[X]) (x : ℝ) :
    HasDerivAt (fun x ↦ p.eval x⁻¹ * expNegInvGlue x)
      ((X ^ 2 * (p - derivative (R := ℝ) p)).eval x⁻¹ * expNegInvGlue x) x := by
  rcases lt_trichotomy x 0 with hx | rfl | hx
  · rw [zero_of_nonpos hx.le, mul_zero]
    refine (hasDerivAt_const _ 0).congr_of_eventuallyEq ?_
    filter_upwards [gt_mem_nhds hx] with y hy
    rw [zero_of_nonpos hy.le, mul_zero]
  · 

Proof state:
case inr.inl
p : ℝ[X]
⊢ HasDerivAt (fun x => Polynomial.eval x⁻¹ p * expNegInvGlue x)
    (Polynomial.eval 0⁻¹ (X ^ 2 * (p - derivative p)) * expNegInvGlue 0) 0

Next tactic:
rw [expNegInvGlue.zero, mul_zero, hasDerivAt_iff_tendsto_slope]
theorem hasDerivAt_polynomial_eval_inv_mul (p : ℝ[X]) (x : ℝ) :
    HasDerivAt (fun x ↦ p.eval x⁻¹ * expNegInvGlue x)
      ((X ^ 2 * (p - derivative (R := ℝ) p)).eval x⁻¹ * expNegInvGlue x) x := by
  rcases lt_trichotomy x 0 with hx | rfl | hx
  · rw [zero_of_nonpos hx.le, mul_zero]
    refine (hasDerivAt_const _ 0).congr_of_eventuallyEq ?_
    filter_upwards [gt_mem_nhds hx] with y hy
    rw [zero_of_nonpos hy.le, mul_zero]
  · rw [expNegInvGlue.zero, mul_zero, hasDerivAt_iff_tendsto_slope]
    

Proof state:
case inr.inl
p : ℝ[X]
⊢ Tendsto (slope (fun x => Polynomial.eval x⁻¹ p * expNegInvGlue x) 0) (𝓝[≠] 0) (𝓝 0)

Next tactic:
refine ((tendsto_polynomial_inv_mul_zero (p * X)).mono_left inf_le_left).congr fun x ↦ ?_
theorem hasDerivAt_polynomial_eval_inv_mul (p : ℝ[X]) (x : ℝ) :
    HasDerivAt (fun x ↦ p.eval x⁻¹ * expNegInvGlue x)
      ((X ^ 2 * (p - derivative (R := ℝ) p)).eval x⁻¹ * expNegInvGlue x) x := by
  rcases lt_trichotomy x 0 with hx | rfl | hx
  · rw [zero_of_nonpos hx.le, mul_zero]
    refine (hasDerivAt_const _ 0).congr_of_eventuallyEq ?_
    filter_upwards [gt_mem_nhds hx] with y hy
    rw [zero_of_nonpos hy.le, mul_zero]
  · rw [expNegInvGlue.zero, mul_zero, hasDerivAt_iff_tendsto_slope]
    refine ((tendsto_polynomial_inv_mul_zero (p * X)).mono_left inf_le_left).congr fun x ↦ ?_
    

Proof state:
case inr.inl
p : ℝ[X]
x : ℝ
⊢ Polynomial.eval x⁻¹ (p * X) * expNegInvGlue x = slope (fun x => Polynomial.eval x⁻¹ p * expNegInvGlue x) 0 x

Next tactic:
simp [slope_def_field, div_eq_mul_inv, mul_right_comm]
theorem hasDerivAt_polynomial_eval_inv_mul (p : ℝ[X]) (x : ℝ) :
    HasDerivAt (fun x ↦ p.eval x⁻¹ * expNegInvGlue x)
      ((X ^ 2 * (p - derivative (R := ℝ) p)).eval x⁻¹ * expNegInvGlue x) x := by
  rcases lt_trichotomy x 0 with hx | rfl | hx
  · rw [zero_of_nonpos hx.le, mul_zero]
    refine (hasDerivAt_const _ 0).congr_of_eventuallyEq ?_
    filter_upwards [gt_mem_nhds hx] with y hy
    rw [zero_of_nonpos hy.le, mul_zero]
  · rw [expNegInvGlue.zero, mul_zero, hasDerivAt_iff_tendsto_slope]
    refine ((tendsto_polynomial_inv_mul_zero (p * X)).mono_left inf_le_left).congr fun x ↦ ?_
    simp [slope_def_field, div_eq_mul_inv, mul_right_comm]
  · 

Proof state:
case inr.inr
p : ℝ[X]
x : ℝ
hx : 0 < x
⊢ HasDerivAt (fun x => Polynomial.eval x⁻¹ p * expNegInvGlue x)
    (Polynomial.eval x⁻¹ (X ^ 2 * (p - derivative p)) * expNegInvGlue x) x

Next tactic:
have := ((p.hasDerivAt x⁻¹).mul (hasDerivAt_neg _).exp).comp x (hasDerivAt_inv hx.ne')
theorem hasDerivAt_polynomial_eval_inv_mul (p : ℝ[X]) (x : ℝ) :
    HasDerivAt (fun x ↦ p.eval x⁻¹ * expNegInvGlue x)
      ((X ^ 2 * (p - derivative (R := ℝ) p)).eval x⁻¹ * expNegInvGlue x) x := by
  rcases lt_trichotomy x 0 with hx | rfl | hx
  · rw [zero_of_nonpos hx.le, mul_zero]
    refine (hasDerivAt_const _ 0).congr_of_eventuallyEq ?_
    filter_upwards [gt_mem_nhds hx] with y hy
    rw [zero_of_nonpos hy.le, mul_zero]
  · rw [expNegInvGlue.zero, mul_zero, hasDerivAt_iff_tendsto_slope]
    refine ((tendsto_polynomial_inv_mul_zero (p * X)).mono_left inf_le_left).congr fun x ↦ ?_
    simp [slope_def_field, div_eq_mul_inv, mul_right_comm]
  · have := ((p.hasDerivAt x⁻¹).mul (hasDerivAt_neg _).exp).comp x (hasDerivAt_inv hx.ne')
    

Proof state:
case inr.inr
p : ℝ[X]
x : ℝ
hx : 0 < x
this :
  HasDerivAt ((fun y => Polynomial.eval y p * rexp (-y)) ∘ Inv.inv)
    ((Polynomial.eval x⁻¹ (derivative p) * rexp (-x⁻¹) + Polynomial.eval x⁻¹ p * (rexp (-x⁻¹) * -1)) * -(x ^ 2)⁻¹) x
⊢ HasDerivAt (fun x => Polynomial.eval x⁻¹ p * expNegInvGlue x)
    (Polynomial.eval x⁻¹ (X ^ 2 * (p - derivative p)) * expNegInvGlue x) x

Next tactic:
convert this.congr_of_eventuallyEq _ using 1
theorem hasDerivAt_polynomial_eval_inv_mul (p : ℝ[X]) (x : ℝ) :
    HasDerivAt (fun x ↦ p.eval x⁻¹ * expNegInvGlue x)
      ((X ^ 2 * (p - derivative (R := ℝ) p)).eval x⁻¹ * expNegInvGlue x) x := by
  rcases lt_trichotomy x 0 with hx | rfl | hx
  · rw [zero_of_nonpos hx.le, mul_zero]
    refine (hasDerivAt_const _ 0).congr_of_eventuallyEq ?_
    filter_upwards [gt_mem_nhds hx] with y hy
    rw [zero_of_nonpos hy.le, mul_zero]
  · rw [expNegInvGlue.zero, mul_zero, hasDerivAt_iff_tendsto_slope]
    refine ((tendsto_polynomial_inv_mul_zero (p * X)).mono_left inf_le_left).congr fun x ↦ ?_
    simp [slope_def_field, div_eq_mul_inv, mul_right_comm]
  · have := ((p.hasDerivAt x⁻¹).mul (hasDerivAt_neg _).exp).comp x (hasDerivAt_inv hx.ne')
    convert this.congr_of_eventuallyEq _ using 1
    · 

Proof state:
case h.e'_7
p : ℝ[X]
x : ℝ
hx : 0 < x
this :
  HasDerivAt ((fun y => Polynomial.eval y p * rexp (-y)) ∘ Inv.inv)
    ((Polynomial.eval x⁻¹ (derivative p) * rexp (-x⁻¹) + Polynomial.eval x⁻¹ p * (rexp (-x⁻¹) * -1)) * -(x ^ 2)⁻¹) x
⊢ Polynomial.eval x⁻¹ (X ^ 2 * (p - derivative p)) * expNegInvGlue x =
    (Polynomial.eval x⁻¹ (derivative p) * rexp (-x⁻¹) + Polynomial.eval x⁻¹ p * (rexp (-x⁻¹) * -1)) * -(x ^ 2)⁻¹

Next tactic:
simp [expNegInvGlue, hx.not_le]
theorem hasDerivAt_polynomial_eval_inv_mul (p : ℝ[X]) (x : ℝ) :
    HasDerivAt (fun x ↦ p.eval x⁻¹ * expNegInvGlue x)
      ((X ^ 2 * (p - derivative (R := ℝ) p)).eval x⁻¹ * expNegInvGlue x) x := by
  rcases lt_trichotomy x 0 with hx | rfl | hx
  · rw [zero_of_nonpos hx.le, mul_zero]
    refine (hasDerivAt_const _ 0).congr_of_eventuallyEq ?_
    filter_upwards [gt_mem_nhds hx] with y hy
    rw [zero_of_nonpos hy.le, mul_zero]
  · rw [expNegInvGlue.zero, mul_zero, hasDerivAt_iff_tendsto_slope]
    refine ((tendsto_polynomial_inv_mul_zero (p * X)).mono_left inf_le_left).congr fun x ↦ ?_
    simp [slope_def_field, div_eq_mul_inv, mul_right_comm]
  · have := ((p.hasDerivAt x⁻¹).mul (hasDerivAt_neg _).exp).comp x (hasDerivAt_inv hx.ne')
    convert this.congr_of_eventuallyEq _ using 1
    · simp [expNegInvGlue, hx.not_le]
      

Proof state:
case h.e'_7
p : ℝ[X]
x : ℝ
hx : 0 < x
this :
  HasDerivAt ((fun y => Polynomial.eval y p * rexp (-y)) ∘ Inv.inv)
    ((Polynomial.eval x⁻¹ (derivative p) * rexp (-x⁻¹) + Polynomial.eval x⁻¹ p * (rexp (-x⁻¹) * -1)) * -(x ^ 2)⁻¹) x
⊢ (x ^ 2)⁻¹ * (Polynomial.eval x⁻¹ p - Polynomial.eval x⁻¹ (derivative p)) * rexp (-x⁻¹) =
    -((Polynomial.eval x⁻¹ (derivative p) * rexp (-x⁻¹) + -(Polynomial.eval x⁻¹ p * rexp (-x⁻¹))) * (x ^ 2)⁻¹)

Next tactic:
ring
theorem hasDerivAt_polynomial_eval_inv_mul (p : ℝ[X]) (x : ℝ) :
    HasDerivAt (fun x ↦ p.eval x⁻¹ * expNegInvGlue x)
      ((X ^ 2 * (p - derivative (R := ℝ) p)).eval x⁻¹ * expNegInvGlue x) x := by
  rcases lt_trichotomy x 0 with hx | rfl | hx
  · rw [zero_of_nonpos hx.le, mul_zero]
    refine (hasDerivAt_const _ 0).congr_of_eventuallyEq ?_
    filter_upwards [gt_mem_nhds hx] with y hy
    rw [zero_of_nonpos hy.le, mul_zero]
  · rw [expNegInvGlue.zero, mul_zero, hasDerivAt_iff_tendsto_slope]
    refine ((tendsto_polynomial_inv_mul_zero (p * X)).mono_left inf_le_left).congr fun x ↦ ?_
    simp [slope_def_field, div_eq_mul_inv, mul_right_comm]
  · have := ((p.hasDerivAt x⁻¹).mul (hasDerivAt_neg _).exp).comp x (hasDerivAt_inv hx.ne')
    convert this.congr_of_eventuallyEq _ using 1
    · simp [expNegInvGlue, hx.not_le]
      ring
    · 

Proof state:
case inr.inr.convert_2
p : ℝ[X]
x : ℝ
hx : 0 < x
this :
  HasDerivAt ((fun y => Polynomial.eval y p * rexp (-y)) ∘ Inv.inv)
    ((Polynomial.eval x⁻¹ (derivative p) * rexp (-x⁻¹) + Polynomial.eval x⁻¹ p * (rexp (-x⁻¹) * -1)) * -(x ^ 2)⁻¹) x
⊢ (fun x => Polynomial.eval x⁻¹ p * expNegInvGlue x) =ᶠ[𝓝 x] (fun y => Polynomial.eval y p * rexp (-y)) ∘ Inv.inv

Next tactic:
filter_upwards [lt_mem_nhds hx] with y hy