l3lab/ntp-mathlib huggingface dataset:
/-- As expected, the Fibonacci sequence is a solution of `fibRec`. -/
theorem fib_isSol_fibRec : fibRec.IsSolution (fun x => x.fib : ℕ → α) := by
  rw [fibRec]
  

Proof state:
α : Type u_1
inst✝ : CommSemiring α
⊢ LinearRecurrence.IsSolution { order := 2, coeffs := ![1, 1] } fun x => ↑(Nat.fib x)

Next tactic:
intro n
/-- As expected, the Fibonacci sequence is a solution of `fibRec`. -/
theorem fib_isSol_fibRec : fibRec.IsSolution (fun x => x.fib : ℕ → α) := by
  rw [fibRec]
  intro n
  

Proof state:
α : Type u_1
inst✝ : CommSemiring α
n : ℕ
⊢ (fun x => ↑(Nat.fib x)) (n + { order := 2, coeffs := ![1, 1] }.order) =
    Finset.sum Finset.univ fun i =>
      LinearRecurrence.coeffs { order := 2, coeffs := ![1, 1] } i * (fun x => ↑(Nat.fib x)) (n + ↑i)

Next tactic:
simp only
/-- As expected, the Fibonacci sequence is a solution of `fibRec`. -/
theorem fib_isSol_fibRec : fibRec.IsSolution (fun x => x.fib : ℕ → α) := by
  rw [fibRec]
  intro n
  simp only
  

Proof state:
α : Type u_1
inst✝ : CommSemiring α
n : ℕ
⊢ ↑(Nat.fib (n + 2)) = Finset.sum Finset.univ fun x => Matrix.vecCons 1 ![1] x * ↑(Nat.fib (n + ↑x))

Next tactic:
rw [Nat.fib_add_two, add_comm]
/-- As expected, the Fibonacci sequence is a solution of `fibRec`. -/
theorem fib_isSol_fibRec : fibRec.IsSolution (fun x => x.fib : ℕ → α) := by
  rw [fibRec]
  intro n
  simp only
  rw [Nat.fib_add_two, add_comm]
  

Proof state:
α : Type u_1
inst✝ : CommSemiring α
n : ℕ
⊢ ↑(Nat.fib (n + 1) + Nat.fib n) = Finset.sum Finset.univ fun x => Matrix.vecCons 1 ![1] x * ↑(Nat.fib (n + ↑x))

Next tactic:
simp [Finset.sum_fin_eq_sum_range, Finset.sum_range_succ']
/-- The geometric sequence `fun n ↦ φ^n` is a solution of `fibRec`. -/
theorem geom_gold_isSol_fibRec : fibRec.IsSolution (φ ^ ·) := by
  

Proof state:
α : Type u_1
inst✝ : CommSemiring α
⊢ LinearRecurrence.IsSolution fibRec fun x => φ ^ x

Next tactic:
rw [fibRec.geom_sol_iff_root_charPoly, fibRec_charPoly_eq]
/-- The geometric sequence `fun n ↦ φ^n` is a solution of `fibRec`. -/
theorem geom_gold_isSol_fibRec : fibRec.IsSolution (φ ^ ·) := by
  rw [fibRec.geom_sol_iff_root_charPoly, fibRec_charPoly_eq]
  

Proof state:
α : Type u_1
inst✝ : CommSemiring α
⊢ IsRoot (X ^ 2 - (X + 1)) φ

Next tactic:
simp [sub_eq_zero, - div_pow]
/-- The geometric sequence `fun n ↦ ψ^n` is a solution of `fibRec`. -/
theorem geom_goldConj_isSol_fibRec : fibRec.IsSolution (ψ ^ ·) := by
  

Proof state:
α : Type u_1
inst✝ : CommSemiring α
⊢ LinearRecurrence.IsSolution fibRec fun x => ψ ^ x

Next tactic:
rw [fibRec.geom_sol_iff_root_charPoly, fibRec_charPoly_eq]
/-- The geometric sequence `fun n ↦ ψ^n` is a solution of `fibRec`. -/
theorem geom_goldConj_isSol_fibRec : fibRec.IsSolution (ψ ^ ·) := by
  rw [fibRec.geom_sol_iff_root_charPoly, fibRec_charPoly_eq]
  

Proof state:
α : Type u_1
inst✝ : CommSemiring α
⊢ IsRoot (X ^ 2 - (X + 1)) ψ

Next tactic:
simp [sub_eq_zero, - div_pow]
/-- Binet's formula as a function equality. -/
theorem Real.coe_fib_eq' :
    (fun n => Nat.fib n : ℕ → ℝ) = fun n => (φ ^ n - ψ ^ n) / Real.sqrt 5 := by
  

Proof state:
⊢ (fun n => ↑(Nat.fib n)) = fun n => (φ ^ n - ψ ^ n) / sqrt 5

Next tactic:
rw [fibRec.sol_eq_of_eq_init]
/-- Binet's formula as a function equality. -/
theorem Real.coe_fib_eq' :
    (fun n => Nat.fib n : ℕ → ℝ) = fun n => (φ ^ n - ψ ^ n) / Real.sqrt 5 := by
  rw [fibRec.sol_eq_of_eq_init]
  · 

Proof state:
⊢ Set.EqOn (fun n => ↑(Nat.fib n)) (fun n => (φ ^ n - ψ ^ n) / sqrt 5) ↑(Finset.range fibRec.order)

Next tactic:
intro i hi
/-- Binet's formula as a function equality. -/
theorem Real.coe_fib_eq' :
    (fun n => Nat.fib n : ℕ → ℝ) = fun n => (φ ^ n - ψ ^ n) / Real.sqrt 5 := by
  rw [fibRec.sol_eq_of_eq_init]
  · intro i hi
    

Proof state:
i : ℕ
hi : i ∈ ↑(Finset.range fibRec.order)
⊢ (fun n => ↑(Nat.fib n)) i = (fun n => (φ ^ n - ψ ^ n) / sqrt 5) i

Next tactic:
norm_cast at hi
/-- Binet's formula as a function equality. -/
theorem Real.coe_fib_eq' :
    (fun n => Nat.fib n : ℕ → ℝ) = fun n => (φ ^ n - ψ ^ n) / Real.sqrt 5 := by
  rw [fibRec.sol_eq_of_eq_init]
  · intro i hi
    norm_cast at hi
    

Proof state:
i : ℕ
hi : i ∈ Finset.range fibRec.order
⊢ (fun n => ↑(Nat.fib n)) i = (fun n => (φ ^ n - ψ ^ n) / sqrt 5) i

Next tactic:
fin_cases hi
/-- Binet's formula as a function equality. -/
theorem Real.coe_fib_eq' :
    (fun n => Nat.fib n : ℕ → ℝ) = fun n => (φ ^ n - ψ ^ n) / Real.sqrt 5 := by
  rw [fibRec.sol_eq_of_eq_init]
  · intro i hi
    norm_cast at hi
    fin_cases hi
    · 

Proof state:
case head
⊢ (fun n => ↑(Nat.fib n)) 0 = (fun n => (φ ^ n - ψ ^ n) / sqrt 5) 0

Next tactic:
simp
/-- Binet's formula as a function equality. -/
theorem Real.coe_fib_eq' :
    (fun n => Nat.fib n : ℕ → ℝ) = fun n => (φ ^ n - ψ ^ n) / Real.sqrt 5 := by
  rw [fibRec.sol_eq_of_eq_init]
  · intro i hi
    norm_cast at hi
    fin_cases hi
    · simp
    · 

Proof state:
case tail.head
⊢ (fun n => ↑(Nat.fib n)) 1 = (fun n => (φ ^ n - ψ ^ n) / sqrt 5) 1

Next tactic:
simp only [goldenRatio, goldenConj]
/-- Binet's formula as a function equality. -/
theorem Real.coe_fib_eq' :
    (fun n => Nat.fib n : ℕ → ℝ) = fun n => (φ ^ n - ψ ^ n) / Real.sqrt 5 := by
  rw [fibRec.sol_eq_of_eq_init]
  · intro i hi
    norm_cast at hi
    fin_cases hi
    · simp
    · simp only [goldenRatio, goldenConj]
      

Proof state:
case tail.head
⊢ ↑(Nat.fib 1) = (((1 + sqrt 5) / 2) ^ 1 - ((1 - sqrt 5) / 2) ^ 1) / sqrt 5

Next tactic:
ring_nf
/-- Binet's formula as a function equality. -/
theorem Real.coe_fib_eq' :
    (fun n => Nat.fib n : ℕ → ℝ) = fun n => (φ ^ n - ψ ^ n) / Real.sqrt 5 := by
  rw [fibRec.sol_eq_of_eq_init]
  · intro i hi
    norm_cast at hi
    fin_cases hi
    · simp
    · simp only [goldenRatio, goldenConj]
      ring_nf
      

Proof state:
case tail.head
⊢ 1 = sqrt 5 * (sqrt 5)⁻¹

Next tactic:
rw [mul_inv_cancel]
/-- Binet's formula as a function equality. -/
theorem Real.coe_fib_eq' :
    (fun n => Nat.fib n : ℕ → ℝ) = fun n => (φ ^ n - ψ ^ n) / Real.sqrt 5 := by
  rw [fibRec.sol_eq_of_eq_init]
  · intro i hi
    norm_cast at hi
    fin_cases hi
    · simp
    · simp only [goldenRatio, goldenConj]
      ring_nf
      rw [mul_inv_cancel]; 

Proof state:
case tail.head
⊢ sqrt 5 ≠ 0

Next tactic:
norm_num
/-- Binet's formula as a function equality. -/
theorem Real.coe_fib_eq' :
    (fun n => Nat.fib n : ℕ → ℝ) = fun n => (φ ^ n - ψ ^ n) / Real.sqrt 5 := by
  rw [fibRec.sol_eq_of_eq_init]
  · intro i hi
    norm_cast at hi
    fin_cases hi
    · simp
    · simp only [goldenRatio, goldenConj]
      ring_nf
      rw [mul_inv_cancel]; norm_num
  · 

Proof state:
case hu
⊢ LinearRecurrence.IsSolution fibRec fun n => ↑(Nat.fib n)

Next tactic:
exact fib_isSol_fibRec
/-- Binet's formula as a function equality. -/
theorem Real.coe_fib_eq' :
    (fun n => Nat.fib n : ℕ → ℝ) = fun n => (φ ^ n - ψ ^ n) / Real.sqrt 5 := by
  rw [fibRec.sol_eq_of_eq_init]
  · intro i hi
    norm_cast at hi
    fin_cases hi
    · simp
    · simp only [goldenRatio, goldenConj]
      ring_nf
      rw [mul_inv_cancel]; norm_num
  · exact fib_isSol_fibRec
  · -- Porting note: Rewrote this proof
    

Proof state:
case hv
⊢ LinearRecurrence.IsSolution fibRec fun n => (φ ^ n - ψ ^ n) / sqrt 5

Next tactic:
suffices LinearRecurrence.IsSolution fibRec
        ((fun n ↦ (sqrt 5)⁻¹ * φ ^ n) - (fun n ↦ (sqrt 5)⁻¹ * ψ ^ n)) by
      convert this
      rw [Pi.sub_apply]
      ring
/-- Binet's formula as a function equality. -/
theorem Real.coe_fib_eq' :
    (fun n => Nat.fib n : ℕ → ℝ) = fun n => (φ ^ n - ψ ^ n) / Real.sqrt 5 := by
  rw [fibRec.sol_eq_of_eq_init]
  · intro i hi
    norm_cast at hi
    fin_cases hi
    · simp
    · simp only [goldenRatio, goldenConj]
      ring_nf
      rw [mul_inv_cancel]; norm_num
  · exact fib_isSol_fibRec
  · -- Porting note: Rewrote this proof
    suffices LinearRecurrence.IsSolution fibRec
        ((fun n ↦ (sqrt 5)⁻¹ * φ ^ n) - (fun n ↦ (sqrt 5)⁻¹ * ψ ^ n)) by
      

Proof state:
this : LinearRecurrence.IsSolution fibRec ((fun n => (sqrt 5)⁻¹ * φ ^ n) - fun n => (sqrt 5)⁻¹ * ψ ^ n)
⊢ LinearRecurrence.IsSolution fibRec fun n => (φ ^ n - ψ ^ n) / sqrt 5

Next tactic:
convert this
/-- Binet's formula as a function equality. -/
theorem Real.coe_fib_eq' :
    (fun n => Nat.fib n : ℕ → ℝ) = fun n => (φ ^ n - ψ ^ n) / Real.sqrt 5 := by
  rw [fibRec.sol_eq_of_eq_init]
  · intro i hi
    norm_cast at hi
    fin_cases hi
    · simp
    · simp only [goldenRatio, goldenConj]
      ring_nf
      rw [mul_inv_cancel]; norm_num
  · exact fib_isSol_fibRec
  · -- Porting note: Rewrote this proof
    suffices LinearRecurrence.IsSolution fibRec
        ((fun n ↦ (sqrt 5)⁻¹ * φ ^ n) - (fun n ↦ (sqrt 5)⁻¹ * ψ ^ n)) by
      convert this
      

Proof state:
case h.e'_4.h
this : LinearRecurrence.IsSolution fibRec ((fun n => (sqrt 5)⁻¹ * φ ^ n) - fun n => (sqrt 5)⁻¹ * ψ ^ n)
x✝ : ℕ
⊢ (φ ^ x✝ - ψ ^ x✝) / sqrt 5 = ((fun n => (sqrt 5)⁻¹ * φ ^ n) - fun n => (sqrt 5)⁻¹ * ψ ^ n) x✝

Next tactic:
rw [Pi.sub_apply]
/-- Binet's formula as a function equality. -/
theorem Real.coe_fib_eq' :
    (fun n => Nat.fib n : ℕ → ℝ) = fun n => (φ ^ n - ψ ^ n) / Real.sqrt 5 := by
  rw [fibRec.sol_eq_of_eq_init]
  · intro i hi
    norm_cast at hi
    fin_cases hi
    · simp
    · simp only [goldenRatio, goldenConj]
      ring_nf
      rw [mul_inv_cancel]; norm_num
  · exact fib_isSol_fibRec
  · -- Porting note: Rewrote this proof
    suffices LinearRecurrence.IsSolution fibRec
        ((fun n ↦ (sqrt 5)⁻¹ * φ ^ n) - (fun n ↦ (sqrt 5)⁻¹ * ψ ^ n)) by
      convert this
      rw [Pi.sub_apply]
      

Proof state:
case h.e'_4.h
this : LinearRecurrence.IsSolution fibRec ((fun n => (sqrt 5)⁻¹ * φ ^ n) - fun n => (sqrt 5)⁻¹ * ψ ^ n)
x✝ : ℕ
⊢ (φ ^ x✝ - ψ ^ x✝) / sqrt 5 = (sqrt 5)⁻¹ * φ ^ x✝ - (sqrt 5)⁻¹ * ψ ^ x✝

Next tactic:
ring
/-- Binet's formula as a function equality. -/
theorem Real.coe_fib_eq' :
    (fun n => Nat.fib n : ℕ → ℝ) = fun n => (φ ^ n - ψ ^ n) / Real.sqrt 5 := by
  rw [fibRec.sol_eq_of_eq_init]
  · intro i hi
    norm_cast at hi
    fin_cases hi
    · simp
    · simp only [goldenRatio, goldenConj]
      ring_nf
      rw [mul_inv_cancel]; norm_num
  · exact fib_isSol_fibRec
  · -- Porting note: Rewrote this proof
    suffices LinearRecurrence.IsSolution fibRec
        ((fun n ↦ (sqrt 5)⁻¹ * φ ^ n) - (fun n ↦ (sqrt 5)⁻¹ * ψ ^ n)) by
      convert this
      rw [Pi.sub_apply]
      ring
    

Proof state:
case hv
⊢ LinearRecurrence.IsSolution fibRec ((fun n => (sqrt 5)⁻¹ * φ ^ n) - fun n => (sqrt 5)⁻¹ * ψ ^ n)

Next tactic:
apply (@fibRec ℝ _).solSpace.sub_mem
/-- Binet's formula as a function equality. -/
theorem Real.coe_fib_eq' :
    (fun n => Nat.fib n : ℕ → ℝ) = fun n => (φ ^ n - ψ ^ n) / Real.sqrt 5 := by
  rw [fibRec.sol_eq_of_eq_init]
  · intro i hi
    norm_cast at hi
    fin_cases hi
    · simp
    · simp only [goldenRatio, goldenConj]
      ring_nf
      rw [mul_inv_cancel]; norm_num
  · exact fib_isSol_fibRec
  · -- Porting note: Rewrote this proof
    suffices LinearRecurrence.IsSolution fibRec
        ((fun n ↦ (sqrt 5)⁻¹ * φ ^ n) - (fun n ↦ (sqrt 5)⁻¹ * ψ ^ n)) by
      convert this
      rw [Pi.sub_apply]
      ring
    apply (@fibRec ℝ _).solSpace.sub_mem
    · 

Proof state:
case hv.a
⊢ (fun n => (sqrt 5)⁻¹ * φ ^ n) ∈ LinearRecurrence.solSpace fibRec

Next tactic:
exact Submodule.smul_mem fibRec.solSpace (Real.sqrt 5)⁻¹ geom_gold_isSol_fibRec
/-- Binet's formula as a function equality. -/
theorem Real.coe_fib_eq' :
    (fun n => Nat.fib n : ℕ → ℝ) = fun n => (φ ^ n - ψ ^ n) / Real.sqrt 5 := by
  rw [fibRec.sol_eq_of_eq_init]
  · intro i hi
    norm_cast at hi
    fin_cases hi
    · simp
    · simp only [goldenRatio, goldenConj]
      ring_nf
      rw [mul_inv_cancel]; norm_num
  · exact fib_isSol_fibRec
  · -- Porting note: Rewrote this proof
    suffices LinearRecurrence.IsSolution fibRec
        ((fun n ↦ (sqrt 5)⁻¹ * φ ^ n) - (fun n ↦ (sqrt 5)⁻¹ * ψ ^ n)) by
      convert this
      rw [Pi.sub_apply]
      ring
    apply (@fibRec ℝ _).solSpace.sub_mem
    · exact Submodule.smul_mem fibRec.solSpace (Real.sqrt 5)⁻¹ geom_gold_isSol_fibRec
    · 

Proof state:
case hv.a
⊢ (fun n => (sqrt 5)⁻¹ * ψ ^ n) ∈ LinearRecurrence.solSpace fibRec

Next tactic:
exact Submodule.smul_mem fibRec.solSpace (Real.sqrt 5)⁻¹ geom_goldConj_isSol_fibRec
/-- Binet's formula as a dependent equality. -/
theorem Real.coe_fib_eq : ∀ n, (Nat.fib n : ℝ) = (φ ^ n - ψ ^ n) / Real.sqrt 5 := by
  

Proof state:
⊢ ∀ (n : ℕ), ↑(Nat.fib n) = (φ ^ n - ψ ^ n) / sqrt 5

Next tactic:
rw [← Function.funext_iff, Real.coe_fib_eq']
theorem lpMeas.ae_eq_zero_of_forall_set_integral_eq_zero (hm : m ≤ m0) (f : lpMeas E' 𝕜 m p μ)
    (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    -- Porting note: needed to add explicit casts in the next two hypotheses
    (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn (f : Lp E' p μ) s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, (f : Lp E' p μ) x ∂μ = 0) :
    f =ᵐ[μ] (0 : α → E') := by
  

Proof state:
α : Type u_1
E' : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
m m0 : MeasurableSpace α
μ : Measure α
inst✝⁸ : IsROrC 𝕜
inst✝⁷ : NormedAddCommGroup E'
inst✝⁶ : InnerProductSpace 𝕜 E'
inst✝⁵ : CompleteSpace E'
inst✝⁴ : NormedSpace ℝ E'
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
hm : m ≤ m0
f : ↥(lpMeas E' 𝕜 m p μ)
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ⊤
hf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn (↑↑↑f) s
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∫ (x : α) in s, ↑↑↑f x ∂μ = 0
⊢ ↑↑↑f =ᵐ[μ] 0

Next tactic:
obtain ⟨g, hg_sm, hfg⟩ := lpMeas.ae_fin_strongly_measurable' hm f hp_ne_zero hp_ne_top
theorem lpMeas.ae_eq_zero_of_forall_set_integral_eq_zero (hm : m ≤ m0) (f : lpMeas E' 𝕜 m p μ)
    (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    -- Porting note: needed to add explicit casts in the next two hypotheses
    (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn (f : Lp E' p μ) s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, (f : Lp E' p μ) x ∂μ = 0) :
    f =ᵐ[μ] (0 : α → E') := by
  obtain ⟨g, hg_sm, hfg⟩ := lpMeas.ae_fin_strongly_measurable' hm f hp_ne_zero hp_ne_top
  

Proof state:
case intro.intro
α : Type u_1
E' : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
m m0 : MeasurableSpace α
μ : Measure α
inst✝⁸ : IsROrC 𝕜
inst✝⁷ : NormedAddCommGroup E'
inst✝⁶ : InnerProductSpace 𝕜 E'
inst✝⁵ : CompleteSpace E'
inst✝⁴ : NormedSpace ℝ E'
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
hm : m ≤ m0
f : ↥(lpMeas E' 𝕜 m p μ)
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ⊤
hf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn (↑↑↑f) s
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∫ (x : α) in s, ↑↑↑f x ∂μ = 0
g : α → E'
hg_sm : FinStronglyMeasurable g (Measure.trim μ hm)
hfg : ↑↑↑f =ᵐ[μ] g
⊢ ↑↑↑f =ᵐ[μ] 0

Next tactic:
refine' hfg.trans _
theorem lpMeas.ae_eq_zero_of_forall_set_integral_eq_zero (hm : m ≤ m0) (f : lpMeas E' 𝕜 m p μ)
    (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    -- Porting note: needed to add explicit casts in the next two hypotheses
    (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn (f : Lp E' p μ) s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, (f : Lp E' p μ) x ∂μ = 0) :
    f =ᵐ[μ] (0 : α → E') := by
  obtain ⟨g, hg_sm, hfg⟩ := lpMeas.ae_fin_strongly_measurable' hm f hp_ne_zero hp_ne_top
  refine' hfg.trans _
  -- Porting note: added
  

Proof state:
case intro.intro
α : Type u_1
E' : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
m m0 : MeasurableSpace α
μ : Measure α
inst✝⁸ : IsROrC 𝕜
inst✝⁷ : NormedAddCommGroup E'
inst✝⁶ : InnerProductSpace 𝕜 E'
inst✝⁵ : CompleteSpace E'
inst✝⁴ : NormedSpace ℝ E'
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
hm : m ≤ m0
f : ↥(lpMeas E' 𝕜 m p μ)
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ⊤
hf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn (↑↑↑f) s
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∫ (x : α) in s, ↑↑↑f x ∂μ = 0
g : α → E'
hg_sm : FinStronglyMeasurable g (Measure.trim μ hm)
hfg : ↑↑↑f =ᵐ[μ] g
⊢ g =ᵐ[μ] 0

Next tactic:
unfold Filter.EventuallyEq at hfg
theorem lpMeas.ae_eq_zero_of_forall_set_integral_eq_zero (hm : m ≤ m0) (f : lpMeas E' 𝕜 m p μ)
    (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    -- Porting note: needed to add explicit casts in the next two hypotheses
    (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn (f : Lp E' p μ) s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, (f : Lp E' p μ) x ∂μ = 0) :
    f =ᵐ[μ] (0 : α → E') := by
  obtain ⟨g, hg_sm, hfg⟩ := lpMeas.ae_fin_strongly_measurable' hm f hp_ne_zero hp_ne_top
  refine' hfg.trans _
  -- Porting note: added
  unfold Filter.EventuallyEq at hfg
  

Proof state:
case intro.intro
α : Type u_1
E' : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
m m0 : MeasurableSpace α
μ : Measure α
inst✝⁸ : IsROrC 𝕜
inst✝⁷ : NormedAddCommGroup E'
inst✝⁶ : InnerProductSpace 𝕜 E'
inst✝⁵ : CompleteSpace E'
inst✝⁴ : NormedSpace ℝ E'
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
hm : m ≤ m0
f : ↥(lpMeas E' 𝕜 m p μ)
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ⊤
hf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn (↑↑↑f) s
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∫ (x : α) in s, ↑↑↑f x ∂μ = 0
g : α → E'
hg_sm : FinStronglyMeasurable g (Measure.trim μ hm)
hfg : ∀ᵐ (x : α) ∂μ, ↑↑↑f x = g x
⊢ g =ᵐ[μ] 0

Next tactic:
refine' ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim hm _ _ hg_sm
theorem lpMeas.ae_eq_zero_of_forall_set_integral_eq_zero (hm : m ≤ m0) (f : lpMeas E' 𝕜 m p μ)
    (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    -- Porting note: needed to add explicit casts in the next two hypotheses
    (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn (f : Lp E' p μ) s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, (f : Lp E' p μ) x ∂μ = 0) :
    f =ᵐ[μ] (0 : α → E') := by
  obtain ⟨g, hg_sm, hfg⟩ := lpMeas.ae_fin_strongly_measurable' hm f hp_ne_zero hp_ne_top
  refine' hfg.trans _
  -- Porting note: added
  unfold Filter.EventuallyEq at hfg
  refine' ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim hm _ _ hg_sm
  · 

Proof state:
case intro.intro.refine'_1
α : Type u_1
E' : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
m m0 : MeasurableSpace α
μ : Measure α
inst✝⁸ : IsROrC 𝕜
inst✝⁷ : NormedAddCommGroup E'
inst✝⁶ : InnerProductSpace 𝕜 E'
inst✝⁵ : CompleteSpace E'
inst✝⁴ : NormedSpace ℝ E'
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
hm : m ≤ m0
f : ↥(lpMeas E' 𝕜 m p μ)
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ⊤
hf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn (↑↑↑f) s
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∫ (x : α) in s, ↑↑↑f x ∂μ = 0
g : α → E'
hg_sm : FinStronglyMeasurable g (Measure.trim μ hm)
hfg : ∀ᵐ (x : α) ∂μ, ↑↑↑f x = g x
⊢ ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn g s

Next tactic:
intro s hs hμs
theorem lpMeas.ae_eq_zero_of_forall_set_integral_eq_zero (hm : m ≤ m0) (f : lpMeas E' 𝕜 m p μ)
    (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    -- Porting note: needed to add explicit casts in the next two hypotheses
    (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn (f : Lp E' p μ) s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, (f : Lp E' p μ) x ∂μ = 0) :
    f =ᵐ[μ] (0 : α → E') := by
  obtain ⟨g, hg_sm, hfg⟩ := lpMeas.ae_fin_strongly_measurable' hm f hp_ne_zero hp_ne_top
  refine' hfg.trans _
  -- Porting note: added
  unfold Filter.EventuallyEq at hfg
  refine' ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim hm _ _ hg_sm
  · intro s hs hμs
    

Proof state:
case intro.intro.refine'_1
α : Type u_1
E' : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
m m0 : MeasurableSpace α
μ : Measure α
inst✝⁸ : IsROrC 𝕜
inst✝⁷ : NormedAddCommGroup E'
inst✝⁶ : InnerProductSpace 𝕜 E'
inst✝⁵ : CompleteSpace E'
inst✝⁴ : NormedSpace ℝ E'
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
hm : m ≤ m0
f : ↥(lpMeas E' 𝕜 m p μ)
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ⊤
hf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn (↑↑↑f) s
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∫ (x : α) in s, ↑↑↑f x ∂μ = 0
g : α → E'
hg_sm : FinStronglyMeasurable g (Measure.trim μ hm)
hfg : ∀ᵐ (x : α) ∂μ, ↑↑↑f x = g x
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s < ⊤
⊢ IntegrableOn g s

Next tactic:
have hfg_restrict : f =ᵐ[μ.restrict s] g := ae_restrict_of_ae hfg
theorem lpMeas.ae_eq_zero_of_forall_set_integral_eq_zero (hm : m ≤ m0) (f : lpMeas E' 𝕜 m p μ)
    (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    -- Porting note: needed to add explicit casts in the next two hypotheses
    (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn (f : Lp E' p μ) s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, (f : Lp E' p μ) x ∂μ = 0) :
    f =ᵐ[μ] (0 : α → E') := by
  obtain ⟨g, hg_sm, hfg⟩ := lpMeas.ae_fin_strongly_measurable' hm f hp_ne_zero hp_ne_top
  refine' hfg.trans _
  -- Porting note: added
  unfold Filter.EventuallyEq at hfg
  refine' ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim hm _ _ hg_sm
  · intro s hs hμs
    have hfg_restrict : f =ᵐ[μ.restrict s] g := ae_restrict_of_ae hfg
    

Proof state:
case intro.intro.refine'_1
α : Type u_1
E' : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
m m0 : MeasurableSpace α
μ : Measure α
inst✝⁸ : IsROrC 𝕜
inst✝⁷ : NormedAddCommGroup E'
inst✝⁶ : InnerProductSpace 𝕜 E'
inst✝⁵ : CompleteSpace E'
inst✝⁴ : NormedSpace ℝ E'
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
hm : m ≤ m0
f : ↥(lpMeas E' 𝕜 m p μ)
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ⊤
hf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn (↑↑↑f) s
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∫ (x : α) in s, ↑↑↑f x ∂μ = 0
g : α → E'
hg_sm : FinStronglyMeasurable g (Measure.trim μ hm)
hfg : ∀ᵐ (x : α) ∂μ, ↑↑↑f x = g x
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s < ⊤
hfg_restrict : ↑↑↑f =ᵐ[Measure.restrict μ s] g
⊢ IntegrableOn g s

Next tactic:
rw [IntegrableOn, integrable_congr hfg_restrict.symm]
theorem lpMeas.ae_eq_zero_of_forall_set_integral_eq_zero (hm : m ≤ m0) (f : lpMeas E' 𝕜 m p μ)
    (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    -- Porting note: needed to add explicit casts in the next two hypotheses
    (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn (f : Lp E' p μ) s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, (f : Lp E' p μ) x ∂μ = 0) :
    f =ᵐ[μ] (0 : α → E') := by
  obtain ⟨g, hg_sm, hfg⟩ := lpMeas.ae_fin_strongly_measurable' hm f hp_ne_zero hp_ne_top
  refine' hfg.trans _
  -- Porting note: added
  unfold Filter.EventuallyEq at hfg
  refine' ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim hm _ _ hg_sm
  · intro s hs hμs
    have hfg_restrict : f =ᵐ[μ.restrict s] g := ae_restrict_of_ae hfg
    rw [IntegrableOn, integrable_congr hfg_restrict.symm]
    

Proof state:
case intro.intro.refine'_1
α : Type u_1
E' : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
m m0 : MeasurableSpace α
μ : Measure α
inst✝⁸ : IsROrC 𝕜
inst✝⁷ : NormedAddCommGroup E'
inst✝⁶ : InnerProductSpace 𝕜 E'
inst✝⁵ : CompleteSpace E'
inst✝⁴ : NormedSpace ℝ E'
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
hm : m ≤ m0
f : ↥(lpMeas E' 𝕜 m p μ)
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ⊤
hf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn (↑↑↑f) s
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∫ (x : α) in s, ↑↑↑f x ∂μ = 0
g : α → E'
hg_sm : FinStronglyMeasurable g (Measure.trim μ hm)
hfg : ∀ᵐ (x : α) ∂μ, ↑↑↑f x = g x
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s < ⊤
hfg_restrict : ↑↑↑f =ᵐ[Measure.restrict μ s] g
⊢ Integrable ↑↑↑f

Next tactic:
exact hf_int_finite s hs hμs