awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
G : Type u_2
inst : AddCommMonoid G
inst_1 : PartialOrder G
inst_2 : CanonicallyOrderedAdd G
m : AddContent G C
hC : IsSetSemiring C
J : Finset (Set α)
h_ss : (↑J : Set (Set α)) ⊆ C
h_mem : ⋃₀ (↑J : Set (Set α)) ∈ C
⊢ LE.le.{u_2} (α := G) ((m : (a : Set α) → G) (⋃₀ (↑J : Set (Set α))) : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
have have : (↑(J.disjiUnion (hC.disjointOfUnion (C := C) h_ss) sorry) : Set (Set α)) ⊆ C
Tactic: classical
  have h1 : (disjiUnion J (hC.disjointOfUnion h_ss)
      (hC.pairwiseDisjoint_disjointOfUnion h_ss) : Set (Set α)) ⊆ C := by
    simp only [disjiUnion_eq_biUnion, coe_biUnion, mem_coe, iUnion_subset_iff]
    exact fun _ x ↦ hC.disjointOfUnion_subset h_ss x
  have h2 : PairwiseDisjoint (disjiUnion J (hC.disjointOfUnion h_ss)
      ((hC.pairwiseDisjoint_disjointOfUnion h_ss)) : Set (Set α)) id := by
    simp only [disjiUnion_eq_biUnion, coe_biUnion, mem_coe]
    exact hC.pairwiseDisjoint_biUnion_disjointOfUnion h_ss
  have h3 : ⋃₀ J = ⋃₀ ((disjiUnion J (hC.disjointOfUnion h_ss)
      (hC.pairwiseDisjoint_disjointOfUnion h_ss)) : Set (Set α)) := by
    simp only [disjiUnion_eq_biUnion, coe_biUnion, mem_coe]
    exact (Exists.choose_spec (hC.disjointOfUnion_props h_ss)).2.2.2.2.2
  rw [h3, addContent_sUnion h1 h2, sum_disjiUnion]
  · gcongr with x hx
    refine sum_addContent_le_of_subset hC (hC.disjointOfUnion_subset h_ss hx)
      (hC.pairwiseDisjoint_disjointOfUnion_of_mem h_ss hx) (h_ss hx)
      (fun _ s ↦ hC.subset_of_mem_disjointOfUnion h_ss hx s)
  · simp only [disjiUnion_eq_biUnion, coe_biUnion, mem_coe] at *
    exact h3.symm ▸ h_mem
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
G : Type u_2
inst : AddCommMonoid G
inst_1 : PartialOrder G
inst_2 : CanonicallyOrderedAdd G
m : AddContent G C
hC : IsSetSemiring C
J : Finset (Set α)
h_ss : (↑J : Set (Set α)) ⊆ C
h_mem : ⋃₀ (↑J : Set (Set α)) ∈ C
have : (↑(J.disjiUnion (hC.disjointOfUnion (C := C) h_ss) ⋯) : Set (Set α)) ⊆ C
⊢ LE.le.{u_2} (α := G) ((m : (a : Set α) → G) (⋃₀ (↑J : Set (Set α))) : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
have have₁ : ∀ (h1 : (↑(J.disjiUnion (hC.disjointOfUnion (C := C) h_ss) sorry) : Set (Set α)) ⊆ C),
  LE.le.{u_2} (α := G) ((m : (a : Set α) → G) (⋃₀ (↑J : Set (Set α))) : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
Tactic: classical
  have h1 : (disjiUnion J (hC.disjointOfUnion h_ss)
      (hC.pairwiseDisjoint_disjointOfUnion h_ss) : Set (Set α)) ⊆ C := by
    simp only [disjiUnion_eq_biUnion, coe_biUnion, mem_coe, iUnion_subset_iff]
    exact fun _ x ↦ hC.disjointOfUnion_subset h_ss x
  have h2 : PairwiseDisjoint (disjiUnion J (hC.disjointOfUnion h_ss)
      ((hC.pairwiseDisjoint_disjointOfUnion h_ss)) : Set (Set α)) id := by
    simp only [disjiUnion_eq_biUnion, coe_biUnion, mem_coe]
    exact hC.pairwiseDisjoint_biUnion_disjointOfUnion h_ss
  have h3 : ⋃₀ J = ⋃₀ ((disjiUnion J (hC.disjointOfUnion h_ss)
      (hC.pairwiseDisjoint_disjointOfUnion h_ss)) : Set (Set α)) := by
    simp only [disjiUnion_eq_biUnion, coe_biUnion, mem_coe]
    exact (Exists.choose_spec (hC.disjointOfUnion_props h_ss)).2.2.2.2.2
  rw [h3, addContent_sUnion h1 h2, sum_disjiUnion]
  · gcongr with x hx
    refine sum_addContent_le_of_subset hC (hC.disjointOfUnion_subset h_ss hx)
      (hC.pairwiseDisjoint_disjointOfUnion_of_mem h_ss hx) (h_ss hx)
      (fun _ s ↦ hC.subset_of_mem_disjointOfUnion h_ss hx s)
  · simp only [disjiUnion_eq_biUnion, coe_biUnion, mem_coe] at *
    exact h3.symm ▸ h_mem
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
G : Type u_2
inst : AddCommMonoid G
inst_1 : PartialOrder G
inst_2 : CanonicallyOrderedAdd G
m : AddContent G C
hC : IsSetSemiring C
J : Finset (Set α)
h_ss : (↑J : Set (Set α)) ⊆ C
h_mem : ⋃₀ (↑J : Set (Set α)) ∈ C
have : (↑(J.disjiUnion (hC.disjointOfUnion (C := C) h_ss) ⋯) : Set (Set α)) ⊆ C
have₁ :
  ∀ (h1 : (↑(J.disjiUnion (hC.disjointOfUnion (C := C) h_ss) ⋯) : Set (Set α)) ⊆ C),
    LE.le.{u_2} (α := G) ((m : (a : Set α) → G) (⋃₀ (↑J : Set (Set α))) : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
have₂ : (↑(J.disjiUnion (hC.disjointOfUnion (C := C) h_ss) ⋯) : Set (Set α)) ⊆ C
have₃ have₄ :
  ∀ (h1 : (↑(J.disjiUnion (hC.disjointOfUnion (C := C) h_ss) ⋯) : Set (Set α)) ⊆ C),
    LE.le.{u_2} (α := G) ((m : (a : Set α) → G) (⋃₀ (↑J : Set (Set α))) : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
⊢ LE.le.{u_2} (α := G) ((m : (a : Set α) → G) (⋃₀ (↑J : Set (Set α))) : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
have have₅ : ∀ (h1 : (↑(J.disjiUnion (hC.disjointOfUnion (C := C) h_ss) sorry) : Set (Set α)) ⊆ C)
  (h2 : (↑(J.disjiUnion (hC.disjointOfUnion (C := C) h_ss) sorry) : Set (Set α)).PairwiseDisjoint id),
  LE.le.{u_2} (α := G) ((m : (a : Set α) → G) (⋃₀ (↑J : Set (Set α))) : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
Tactic: classical
  have h1 : (disjiUnion J (hC.disjointOfUnion h_ss)
      (hC.pairwiseDisjoint_disjointOfUnion h_ss) : Set (Set α)) ⊆ C := by
    simp only [disjiUnion_eq_biUnion, coe_biUnion, mem_coe, iUnion_subset_iff]
    exact fun _ x ↦ hC.disjointOfUnion_subset h_ss x
  have h2 : PairwiseDisjoint (disjiUnion J (hC.disjointOfUnion h_ss)
      ((hC.pairwiseDisjoint_disjointOfUnion h_ss)) : Set (Set α)) id := by
    simp only [disjiUnion_eq_biUnion, coe_biUnion, mem_coe]
    exact hC.pairwiseDisjoint_biUnion_disjointOfUnion h_ss
  have h3 : ⋃₀ J = ⋃₀ ((disjiUnion J (hC.disjointOfUnion h_ss)
      (hC.pairwiseDisjoint_disjointOfUnion h_ss)) : Set (Set α)) := by
    simp only [disjiUnion_eq_biUnion, coe_biUnion, mem_coe]
    exact (Exists.choose_spec (hC.disjointOfUnion_props h_ss)).2.2.2.2.2
  rw [h3, addContent_sUnion h1 h2, sum_disjiUnion]
  · gcongr with x hx
    refine sum_addContent_le_of_subset hC (hC.disjointOfUnion_subset h_ss hx)
      (hC.pairwiseDisjoint_disjointOfUnion_of_mem h_ss hx) (h_ss hx)
      (fun _ s ↦ hC.subset_of_mem_disjointOfUnion h_ss hx s)
  · simp only [disjiUnion_eq_biUnion, coe_biUnion, mem_coe] at *
    exact h3.symm ▸ h_mem
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
G : Type u_2
inst : AddCommMonoid G
inst_1 : PartialOrder G
inst_2 : CanonicallyOrderedAdd G
m : AddContent G C
hC : IsSetSemiring C
J : Finset (Set α)
h_ss : (↑J : Set (Set α)) ⊆ C
h_mem : ⋃₀ (↑J : Set (Set α)) ∈ C
have : (↑(J.disjiUnion (hC.disjointOfUnion (C := C) h_ss) ⋯) : Set (Set α)) ⊆ C
have₁ :
  ∀ (h1 : (↑(J.disjiUnion (hC.disjointOfUnion (C := C) h_ss) ⋯) : Set (Set α)) ⊆ C),
    LE.le.{u_2} (α := G) ((m : (a : Set α) → G) (⋃₀ (↑J : Set (Set α))) : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
have₂ : (↑(J.disjiUnion (hC.disjointOfUnion (C := C) h_ss) ⋯) : Set (Set α)) ⊆ C
have₃ have₄ :
  ∀ (h1 : (↑(J.disjiUnion (hC.disjointOfUnion (C := C) h_ss) ⋯) : Set (Set α)) ⊆ C),
    LE.le.{u_2} (α := G) ((m : (a : Set α) → G) (⋃₀ (↑J : Set (Set α))) : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
have₅ :
  ∀ (h1 : (↑(J.disjiUnion (hC.disjointOfUnion (C := C) h_ss) ⋯) : Set (Set α)) ⊆ C)
    (h2 : (↑(J.disjiUnion (hC.disjointOfUnion (C := C) h_ss) ⋯) : Set (Set α)).PairwiseDisjoint id),
    LE.le.{u_2} (α := G) ((m : (a : Set α) → G) (⋃₀ (↑J : Set (Set α))) : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
⊢ LE.le.{u_2} (α := G) ((m : (a : Set α) → G) (⋃₀ (↑J : Set (Set α))) : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
have have₆ : ∀ (h1 : (↑(J.disjiUnion (hC.disjointOfUnion (C := C) h_ss) sorry) : Set (Set α)) ⊆ C)
  (h2 : (↑(J.disjiUnion (hC.disjointOfUnion (C := C) h_ss) sorry) : Set (Set α)).PairwiseDisjoint id)
  (h3 : ⋃₀ (↑J : Set (Set α)) = ⋃₀ (↑(J.disjiUnion (hC.disjointOfUnion (C := C) h_ss) sorry) : Set (Set α))),
  LE.le.{u_2} (α := G) ((m : (a : Set α) → G) (⋃₀ (↑J : Set (Set α))) : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
Tactic: classical
  have h1 : (disjiUnion J (hC.disjointOfUnion h_ss)
      (hC.pairwiseDisjoint_disjointOfUnion h_ss) : Set (Set α)) ⊆ C := by
    simp only [disjiUnion_eq_biUnion, coe_biUnion, mem_coe, iUnion_subset_iff]
    exact fun _ x ↦ hC.disjointOfUnion_subset h_ss x
  have h2 : PairwiseDisjoint (disjiUnion J (hC.disjointOfUnion h_ss)
      ((hC.pairwiseDisjoint_disjointOfUnion h_ss)) : Set (Set α)) id := by
    simp only [disjiUnion_eq_biUnion, coe_biUnion, mem_coe]
    exact hC.pairwiseDisjoint_biUnion_disjointOfUnion h_ss
  have h3 : ⋃₀ J = ⋃₀ ((disjiUnion J (hC.disjointOfUnion h_ss)
      (hC.pairwiseDisjoint_disjointOfUnion h_ss)) : Set (Set α)) := by
    simp only [disjiUnion_eq_biUnion, coe_biUnion, mem_coe]
    exact (Exists.choose_spec (hC.disjointOfUnion_props h_ss)).2.2.2.2.2
  rw [h3, addContent_sUnion h1 h2, sum_disjiUnion]
  · gcongr with x hx
    refine sum_addContent_le_of_subset hC (hC.disjointOfUnion_subset h_ss hx)
      (hC.pairwiseDisjoint_disjointOfUnion_of_mem h_ss hx) (h_ss hx)
      (fun _ s ↦ hC.subset_of_mem_disjointOfUnion h_ss hx s)
  · simp only [disjiUnion_eq_biUnion, coe_biUnion, mem_coe] at *
    exact h3.symm ▸ h_mem
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
t : Set α
G : Type u_2
inst : AddCommMonoid G
inst_1 : PartialOrder G
inst_2 : CanonicallyOrderedAdd G
m : AddContent G C
hC : IsSetSemiring C
J : Finset (Set α)
h_ss : (↑J : Set (Set α)) ⊆ C
ht : t ∈ C
htJ : t ⊆ ⋃₀ (↑J : Set (Set α))
Jt : Finset (Set α) := Finset.image (fun (u : Set α) ↦ t ∩ u) J
⊢ t = ⋃₀ (↑Jt : Set (Set α))
have rw : t ⊆ ⋃ i ∈ (↑J : Set (Set α)), i
Tactic: rw [coe_image, sUnion_image, ← inter_iUnion₂, inter_eq_self_of_subset_left]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (hypothesis)
Goal:
α : Type u_1
C : Set (Set α)
t : Set α
G : Type u_2
inst : AddCommMonoid G
inst_1 : PartialOrder G
inst_2 : CanonicallyOrderedAdd G
m : AddContent G C
hC : IsSetSemiring C
J : Finset (Set α)
h_ss : (↑J : Set (Set α)) ⊆ C
ht : t ∈ C
htJ : t ⊆ ⋃₀ (↑J : Set (Set α))
Jt : Finset (Set α) := Finset.image (fun (u : Set α) ↦ t ∩ u) J
⊢ LE.le.{u_2} (α := G) ((m : (a : Set α) → G) t : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
have ht_eq : t = ⋃₀ (↑Jt : Set (Set α))
Tactic: have ht_eq : t = ⋃₀ Jt := by
    rw [coe_image, sUnion_image, ← inter_iUnion₂, inter_eq_self_of_subset_left]
    rwa [← sUnion_eq_biUnion]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
t : Set α
G : Type u_2
inst : AddCommMonoid G
inst_1 : PartialOrder G
inst_2 : CanonicallyOrderedAdd G
m : AddContent G C
hC : IsSetSemiring C
J : Finset (Set α)
h_ss : (↑J : Set (Set α)) ⊆ C
ht : t ∈ C
htJ : t ⊆ ⋃₀ (↑J : Set (Set α))
Jt : Finset (Set α) := Finset.image (fun (u : Set α) ↦ t ∩ u) J
ht_eq : t = ⋃₀ (↑Jt : Set (Set α))
⊢ LE.le.{u_2} (α := G) ((m : (a : Set α) → G) t : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
have rw : LE.le.{u_2} (α := G) ((m : (a : Set α) → G) (⋃₀ (↑Jt : Set (Set α))) : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
Tactic: rw [ht_eq]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
t : Set α
G : Type u_2
inst : AddCommMonoid G
inst_1 : PartialOrder G
inst_2 : CanonicallyOrderedAdd G
m : AddContent G C
hC : IsSetSemiring C
J : Finset (Set α)
h_ss : (↑J : Set (Set α)) ⊆ C
ht : t ∈ C
htJ : t ⊆ ⋃₀ (↑J : Set (Set α))
Jt : Finset (Set α) := Finset.image (fun (u : Set α) ↦ t ∩ u) J
ht_eq : t = ⋃₀ (↑Jt : Set (Set α))
⊢ LE.le.{u_2} (α := G) ((m : (a : Set α) → G) (⋃₀ (↑Jt : Set (Set α))) : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
have refine_1 : (↑Jt : Set (Set α)) ⊆ C
Tactic: refine (addContent_sUnion_le_sum hC Jt ?_ ?_).trans ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
t : Set α
G : Type u_2
inst : AddCommMonoid G
inst_1 : PartialOrder G
inst_2 : CanonicallyOrderedAdd G
m : AddContent G C
hC : IsSetSemiring C
J : Finset (Set α)
h_ss : (↑J : Set (Set α)) ⊆ C
ht : t ∈ C
htJ : t ⊆ ⋃₀ (↑J : Set (Set α))
Jt : Finset (Set α) := Finset.image (fun (u : Set α) ↦ t ∩ u) J
ht_eq : t = ⋃₀ (↑Jt : Set (Set α))
refine_1 : (↑Jt : Set (Set α)) ⊆ C
⊢ LE.le.{u_2} (α := G) ((m : (a : Set α) → G) (⋃₀ (↑Jt : Set (Set α))) : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
have refine_2 : ⋃₀ (↑Jt : Set (Set α)) ∈ C
Tactic: refine (addContent_sUnion_le_sum hC Jt ?_ ?_).trans ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
t : Set α
G : Type u_2
inst : AddCommMonoid G
inst_1 : PartialOrder G
inst_2 : CanonicallyOrderedAdd G
m : AddContent G C
hC : IsSetSemiring C
J : Finset (Set α)
h_ss : (↑J : Set (Set α)) ⊆ C
ht : t ∈ C
htJ : t ⊆ ⋃₀ (↑J : Set (Set α))
Jt : Finset (Set α) := Finset.image (fun (u : Set α) ↦ t ∩ u) J
ht_eq : t = ⋃₀ (↑Jt : Set (Set α))
refine_1 : (↑Jt : Set (Set α)) ⊆ C
refine_2 : ⋃₀ (↑Jt : Set (Set α)) ∈ C
⊢ LE.le.{u_2} (α := G) ((m : (a : Set α) → G) (⋃₀ (↑Jt : Set (Set α))) : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
have refine_3 : LE.le.{u_2} (α := G) (∑ u ∈ Jt, (m : (a : Set α) → G) u) (∑ u ∈ J, (m : (a : Set α) → G) u)
Tactic: refine (addContent_sUnion_le_sum hC Jt ?_ ?_).trans ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
t : Set α
G : Type u_2
inst : AddCommMonoid G
inst_1 : PartialOrder G
inst_2 : CanonicallyOrderedAdd G
m : AddContent G C
hC : IsSetSemiring C
J : Finset (Set α)
h_ss : (↑J : Set (Set α)) ⊆ C
ht : t ∈ C
htJ : t ⊆ ⋃₀ (↑J : Set (Set α))
Jt : Finset (Set α) := Finset.image (fun (u : Set α) ↦ t ∩ u) J
ht_eq : t = ⋃₀ (↑Jt : Set (Set α))
s : Set α
⊢ s ∈ (↑Jt : Set (Set α)) → s ∈ C
have refine_1 : ∀ x ∈ J, t ∩ x = s → s ∈ C
Tactic: simp only [Jt, coe_image, Set.mem_image, mem_coe, forall_exists_index, and_imp]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
t : Set α
G : Type u_2
inst : AddCommMonoid G
inst_1 : PartialOrder G
inst_2 : CanonicallyOrderedAdd G
m : AddContent G C
hC : IsSetSemiring C
J : Finset (Set α)
h_ss : (↑J : Set (Set α)) ⊆ C
ht : t ∈ C
htJ : t ⊆ ⋃₀ (↑J : Set (Set α))
Jt : Finset (Set α) := Finset.image (fun (u : Set α) ↦ t ∩ u) J
ht_eq : t = ⋃₀ (↑Jt : Set (Set α))
⊢ LE.le.{u_2} (α := G) (∑ u ∈ Jt, (m : (a : Set α) → G) u) (∑ u ∈ J, (m : (a : Set α) → G) u)
have refine_3 : ∀ (u : Set α) (hu : u ∈ J), LE.le.{u_2} (α := G) ((m : (a : Set α) → G) (t ∩ u) : G) ((m : (a : Set α) → G) u : G)
Tactic: refine (Finset.sum_image_le_of_nonneg fun _ _ ↦ zero_le).trans (sum_le_sum fun u hu ↦ ?_)
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
t : Set α
G : Type u_2
inst : AddCommMonoid G
inst_1 : PartialOrder G
inst_2 : CanonicallyOrderedAdd G
m : AddContent G C
hC : IsSetSemiring C
J : Finset (Set α)
h_ss : (↑J : Set (Set α)) ⊆ C
ht : t ∈ C
htJ : t ⊆ ⋃₀ (↑J : Set (Set α))
let let₁ let₂ : LE.le.{u_2} (α := G) ((m : (a : Set α) → G) t : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
⊢ LE.le.{u_2} (α := G) ((m : (a : Set α) → G) t : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
have let₃ : let Jt : Finset (Set α) := Finset.image (fun (u : Set α) ↦ t ∩ u) J;
∀ (ht_eq : t = ⋃₀ (↑Jt : Set (Set α))),
  LE.le.{u_2} (α := G) ((m : (a : Set α) → G) t : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
Tactic: classical
  let Jt := J.image (fun u ↦ t ∩ u)
  have ht_eq : t = ⋃₀ Jt := by
    rw [coe_image, sUnion_image, ← inter_iUnion₂, inter_eq_self_of_subset_left]
    rwa [← sUnion_eq_biUnion]
  rw [ht_eq]
  refine (addContent_sUnion_le_sum hC Jt ?_ ?_).trans ?_
  · intro s
    simp only [Jt, coe_image, Set.mem_image, mem_coe, forall_exists_index, and_imp]
    rintro u hu rfl
    exact hC.inter_mem _ ht _ (h_ss hu)
  · rwa [← ht_eq]
  · refine (Finset.sum_image_le_of_nonneg fun _ _ ↦ zero_le).trans (sum_le_sum fun u hu ↦ ?_)
    exact addContent_mono hC (hC.inter_mem _ ht _ (h_ss hu)) (h_ss hu) inter_subset_right
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
t : Set α
G : Type u_2
inst : AddCommMonoid G
inst_1 : PartialOrder G
inst_2 : CanonicallyOrderedAdd G
m : AddContent G C
hC : IsSetSemiring C
J : Finset (Set α)
h_ss : (↑J : Set (Set α)) ⊆ C
ht : t ∈ C
htJ : t ⊆ ⋃₀ (↑J : Set (Set α))
let let₁ let₂ : LE.le.{u_2} (α := G) ((m : (a : Set α) → G) t : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
let₃ :
  let Jt : Finset (Set α) := Finset.image (fun (u : Set α) ↦ t ∩ u) J;
  ∀ (ht_eq : t = ⋃₀ (↑Jt : Set (Set α))),
    LE.le.{u_2} (α := G) ((m : (a : Set α) → G) t : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
⊢ LE.le.{u_2} (α := G) ((m : (a : Set α) → G) t : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
have let₄ : let Jt : Finset (Set α) := Finset.image (fun (u : Set α) ↦ t ∩ u) J;
∀ (ht_eq : t = ⋃₀ (↑Jt : Set (Set α))),
  LE.le.{u_2} (α := G) ((m : (a : Set α) → G) (⋃₀ (↑Jt : Set (Set α))) : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
Tactic: classical
  let Jt := J.image (fun u ↦ t ∩ u)
  have ht_eq : t = ⋃₀ Jt := by
    rw [coe_image, sUnion_image, ← inter_iUnion₂, inter_eq_self_of_subset_left]
    rwa [← sUnion_eq_biUnion]
  rw [ht_eq]
  refine (addContent_sUnion_le_sum hC Jt ?_ ?_).trans ?_
  · intro s
    simp only [Jt, coe_image, Set.mem_image, mem_coe, forall_exists_index, and_imp]
    rintro u hu rfl
    exact hC.inter_mem _ ht _ (h_ss hu)
  · rwa [← ht_eq]
  · refine (Finset.sum_image_le_of_nonneg fun _ _ ↦ zero_le).trans (sum_le_sum fun u hu ↦ ?_)
    exact addContent_mono hC (hC.inter_mem _ ht _ (h_ss hu)) (h_ss hu) inter_subset_right
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
t : Set α
G : Type u_2
inst : AddCommMonoid G
inst_1 : PartialOrder G
inst_2 : CanonicallyOrderedAdd G
m : AddContent G C
hC : IsSetSemiring C
J : Finset (Set α)
h_ss : (↑J : Set (Set α)) ⊆ C
ht : t ∈ C
htJ : t ⊆ ⋃₀ (↑J : Set (Set α))
let let₁ let₂ : LE.le.{u_2} (α := G) ((m : (a : Set α) → G) t : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
let₃ :
  let Jt : Finset (Set α) := Finset.image (fun (u : Set α) ↦ t ∩ u) J;
  ∀ (ht_eq : t = ⋃₀ (↑Jt : Set (Set α))),
    LE.le.{u_2} (α := G) ((m : (a : Set α) → G) t : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
let₄ :
  let Jt : Finset (Set α) := Finset.image (fun (u : Set α) ↦ t ∩ u) J;
  ∀ (ht_eq : t = ⋃₀ (↑Jt : Set (Set α))),
    LE.le.{u_2} (α := G) ((m : (a : Set α) → G) (⋃₀ (↑Jt : Set (Set α))) : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
⊢ LE.le.{u_2} (α := G) ((m : (a : Set α) → G) t : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
have refine_1 : let Jt : Finset (Set α) := Finset.image (fun (u : Set α) ↦ t ∩ u) J;
∀ (ht_eq : t = ⋃₀ (↑Jt : Set (Set α))), (↑Jt : Set (Set α)) ⊆ C
Tactic: classical
  let Jt := J.image (fun u ↦ t ∩ u)
  have ht_eq : t = ⋃₀ Jt := by
    rw [coe_image, sUnion_image, ← inter_iUnion₂, inter_eq_self_of_subset_left]
    rwa [← sUnion_eq_biUnion]
  rw [ht_eq]
  refine (addContent_sUnion_le_sum hC Jt ?_ ?_).trans ?_
  · intro s
    simp only [Jt, coe_image, Set.mem_image, mem_coe, forall_exists_index, and_imp]
    rintro u hu rfl
    exact hC.inter_mem _ ht _ (h_ss hu)
  · rwa [← ht_eq]
  · refine (Finset.sum_image_le_of_nonneg fun _ _ ↦ zero_le).trans (sum_le_sum fun u hu ↦ ?_)
    exact addContent_mono hC (hC.inter_mem _ ht _ (h_ss hu)) (h_ss hu) inter_subset_right
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
t : Set α
G : Type u_2
inst : AddCommMonoid G
inst_1 : PartialOrder G
inst_2 : CanonicallyOrderedAdd G
m : AddContent G C
hC : IsSetSemiring C
J : Finset (Set α)
h_ss : (↑J : Set (Set α)) ⊆ C
ht : t ∈ C
htJ : t ⊆ ⋃₀ (↑J : Set (Set α))
let let₁ let₂ : LE.le.{u_2} (α := G) ((m : (a : Set α) → G) t : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
let₃ :
  let Jt : Finset (Set α) := Finset.image (fun (u : Set α) ↦ t ∩ u) J;
  ∀ (ht_eq : t = ⋃₀ (↑Jt : Set (Set α))),
    LE.le.{u_2} (α := G) ((m : (a : Set α) → G) t : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
let₄ :
  let Jt : Finset (Set α) := Finset.image (fun (u : Set α) ↦ t ∩ u) J;
  ∀ (ht_eq : t = ⋃₀ (↑Jt : Set (Set α))),
    LE.le.{u_2} (α := G) ((m : (a : Set α) → G) (⋃₀ (↑Jt : Set (Set α))) : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
refine_1 :
  let Jt : Finset (Set α) := Finset.image (fun (u : Set α) ↦ t ∩ u) J;
  ∀ (ht_eq : t = ⋃₀ (↑Jt : Set (Set α))), (↑Jt : Set (Set α)) ⊆ C
⊢ LE.le.{u_2} (α := G) ((m : (a : Set α) → G) t : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
have refine_2 : let Jt : Finset (Set α) := Finset.image (fun (u : Set α) ↦ t ∩ u) J;
∀ (ht_eq : t = ⋃₀ (↑Jt : Set (Set α))), ⋃₀ (↑Jt : Set (Set α)) ∈ C
Tactic: classical
  let Jt := J.image (fun u ↦ t ∩ u)
  have ht_eq : t = ⋃₀ Jt := by
    rw [coe_image, sUnion_image, ← inter_iUnion₂, inter_eq_self_of_subset_left]
    rwa [← sUnion_eq_biUnion]
  rw [ht_eq]
  refine (addContent_sUnion_le_sum hC Jt ?_ ?_).trans ?_
  · intro s
    simp only [Jt, coe_image, Set.mem_image, mem_coe, forall_exists_index, and_imp]
    rintro u hu rfl
    exact hC.inter_mem _ ht _ (h_ss hu)
  · rwa [← ht_eq]
  · refine (Finset.sum_image_le_of_nonneg fun _ _ ↦ zero_le).trans (sum_le_sum fun u hu ↦ ?_)
    exact addContent_mono hC (hC.inter_mem _ ht _ (h_ss hu)) (h_ss hu) inter_subset_right
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
t : Set α
G : Type u_2
inst : AddCommMonoid G
inst_1 : PartialOrder G
inst_2 : CanonicallyOrderedAdd G
m : AddContent G C
hC : IsSetSemiring C
J : Finset (Set α)
h_ss : (↑J : Set (Set α)) ⊆ C
ht : t ∈ C
htJ : t ⊆ ⋃₀ (↑J : Set (Set α))
let let₁ let₂ : LE.le.{u_2} (α := G) ((m : (a : Set α) → G) t : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
let₃ :
  let Jt : Finset (Set α) := Finset.image (fun (u : Set α) ↦ t ∩ u) J;
  ∀ (ht_eq : t = ⋃₀ (↑Jt : Set (Set α))),
    LE.le.{u_2} (α := G) ((m : (a : Set α) → G) t : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
let₄ :
  let Jt : Finset (Set α) := Finset.image (fun (u : Set α) ↦ t ∩ u) J;
  ∀ (ht_eq : t = ⋃₀ (↑Jt : Set (Set α))),
    LE.le.{u_2} (α := G) ((m : (a : Set α) → G) (⋃₀ (↑Jt : Set (Set α))) : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
refine_1 :
  let Jt : Finset (Set α) := Finset.image (fun (u : Set α) ↦ t ∩ u) J;
  ∀ (ht_eq : t = ⋃₀ (↑Jt : Set (Set α))), (↑Jt : Set (Set α)) ⊆ C
refine_2 :
  let Jt : Finset (Set α) := Finset.image (fun (u : Set α) ↦ t ∩ u) J;
  ∀ (ht_eq : t = ⋃₀ (↑Jt : Set (Set α))), ⋃₀ (↑Jt : Set (Set α)) ∈ C
⊢ LE.le.{u_2} (α := G) ((m : (a : Set α) → G) t : G) (∑ u ∈ J, (m : (a : Set α) → G) u : G)
have refine_3 : let Jt : Finset (Set α) := Finset.image (fun (u : Set α) ↦ t ∩ u) J;
∀ (ht_eq : t = ⋃₀ (↑Jt : Set (Set α))),
  LE.le.{u_2} (α := G) (∑ u ∈ Jt, (m : (a : Set α) → G) u) (∑ u ∈ J, (m : (a : Set α) → G) u)
Tactic: classical
  let Jt := J.image (fun u ↦ t ∩ u)
  have ht_eq : t = ⋃₀ Jt := by
    rw [coe_image, sUnion_image, ← inter_iUnion₂, inter_eq_self_of_subset_left]
    rwa [← sUnion_eq_biUnion]
  rw [ht_eq]
  refine (addContent_sUnion_le_sum hC Jt ?_ ?_).trans ?_
  · intro s
    simp only [Jt, coe_image, Set.mem_image, mem_coe, forall_exists_index, and_imp]
    rintro u hu rfl
    exact hC.inter_mem _ ht _ (h_ss hu)
  · rwa [← ht_eq]
  · refine (Finset.sum_image_le_of_nonneg fun _ _ ↦ zero_le).trans (sum_le_sum fun u hu ↦ ?_)
    exact addContent_mono hC (hC.inter_mem _ ht _ (h_ss hu)) (h_ss hu) inter_subset_right
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.AddContent (goal)
Goal:
α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_subadd :
  ∀ (f : ℕ → Set α),
    (∀ (i : ℕ), f i ∈ C) →
      ⋃ (i : ℕ), f i ∈ C →
        ∀ (_hf_disj : Pairwise (Disjoint on f)),
          LE.le (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
            (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ (i : ℕ), f i ∈ C
hf_disj : Pairwise (Disjoint on f)
⊢ Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞) (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞)
have refine : LE.le (α := ℝ≥0∞) (∑' (i : ℕ), (m : (a : Set α) → ℝ≥0∞) (f i) : ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
Tactic: refine le_antisymm (m_subadd f hf hf_Union hf_disj) ?_