l3lab/ntp-mathlib huggingface dataset:
theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
    (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
    ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
  suffices
    ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from
    ⟨this, fun h =>
      Or.casesOn h
        (fun h =>
          Set.Subset.trans h <|
            Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _))
        fun h =>
        Or.casesOn h
          (fun h =>
            Set.Subset.trans h <|
              Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _))
          fun ⟨i, his, hi⟩ => by
          refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _;
            exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩
  generalize hn : s.card = n; intro h
  induction' n with n ih generalizing a b s
  · clear hp
    rw [Finset.card_eq_zero] at hn
    subst hn
    rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h
    simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff]
  classical
    

Proof state:
case succ
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
  ∀ {s : Finset ι} {a b : ι},
    (∀ i ∈ s, IsPrime (f i)) →
      Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
s : Finset ι
a b : ι
hp : ∀ i ∈ s, IsPrime (f i)
hn : Finset.card s = Nat.succ n
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i)
⊢ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i

Next tactic:
replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n :=
      Finset.card_eq_succ.1 hn
theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
    (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
    ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
  suffices
    ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from
    ⟨this, fun h =>
      Or.casesOn h
        (fun h =>
          Set.Subset.trans h <|
            Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _))
        fun h =>
        Or.casesOn h
          (fun h =>
            Set.Subset.trans h <|
              Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _))
          fun ⟨i, his, hi⟩ => by
          refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _;
            exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩
  generalize hn : s.card = n; intro h
  induction' n with n ih generalizing a b s
  · clear hp
    rw [Finset.card_eq_zero] at hn
    subst hn
    rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h
    simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff]
  classical
    replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n :=
      Finset.card_eq_succ.1 hn
    

Proof state:
case succ
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
  ∀ {s : Finset ι} {a b : ι},
    (∀ i ∈ s, IsPrime (f i)) →
      Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
s : Finset ι
a b : ι
hp : ∀ i ∈ s, IsPrime (f i)
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i)
hn : ∃ i t, i ∉ t ∧ insert i t = s ∧ Finset.card t = n
⊢ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i

Next tactic:
rcases hn with ⟨i, t, hit, rfl, hn⟩
theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
    (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
    ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
  suffices
    ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from
    ⟨this, fun h =>
      Or.casesOn h
        (fun h =>
          Set.Subset.trans h <|
            Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _))
        fun h =>
        Or.casesOn h
          (fun h =>
            Set.Subset.trans h <|
              Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _))
          fun ⟨i, his, hi⟩ => by
          refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _;
            exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩
  generalize hn : s.card = n; intro h
  induction' n with n ih generalizing a b s
  · clear hp
    rw [Finset.card_eq_zero] at hn
    subst hn
    rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h
    simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff]
  classical
    replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n :=
      Finset.card_eq_succ.1 hn
    rcases hn with ⟨i, t, hit, rfl, hn⟩
    

Proof state:
case succ.intro.intro.intro.intro
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
  ∀ {s : Finset ι} {a b : ι},
    (∀ i ∈ s, IsPrime (f i)) →
      Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b i : ι
t : Finset ι
hit : i ∉ t
hn : Finset.card t = n
hp : ∀ i_1 ∈ insert i t, IsPrime (f i_1)
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1)
⊢ I ≤ f a ∨ I ≤ f b ∨ ∃ i_1 ∈ insert i t, I ≤ f i_1

Next tactic:
replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp
theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
    (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
    ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
  suffices
    ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from
    ⟨this, fun h =>
      Or.casesOn h
        (fun h =>
          Set.Subset.trans h <|
            Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _))
        fun h =>
        Or.casesOn h
          (fun h =>
            Set.Subset.trans h <|
              Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _))
          fun ⟨i, his, hi⟩ => by
          refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _;
            exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩
  generalize hn : s.card = n; intro h
  induction' n with n ih generalizing a b s
  · clear hp
    rw [Finset.card_eq_zero] at hn
    subst hn
    rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h
    simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff]
  classical
    replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n :=
      Finset.card_eq_succ.1 hn
    rcases hn with ⟨i, t, hit, rfl, hn⟩
    replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp
    

Proof state:
case succ.intro.intro.intro.intro
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
  ∀ {s : Finset ι} {a b : ι},
    (∀ i ∈ s, IsPrime (f i)) →
      Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b i : ι
t : Finset ι
hit : i ∉ t
hn : Finset.card t = n
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1)
hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x)
⊢ I ≤ f a ∨ I ≤ f b ∨ ∃ i_1 ∈ insert i t, I ≤ f i_1

Next tactic:
by_cases Ht : ∃ j ∈ t, f j ≤ f i
theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
    (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
    ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
  suffices
    ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from
    ⟨this, fun h =>
      Or.casesOn h
        (fun h =>
          Set.Subset.trans h <|
            Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _))
        fun h =>
        Or.casesOn h
          (fun h =>
            Set.Subset.trans h <|
              Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _))
          fun ⟨i, his, hi⟩ => by
          refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _;
            exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩
  generalize hn : s.card = n; intro h
  induction' n with n ih generalizing a b s
  · clear hp
    rw [Finset.card_eq_zero] at hn
    subst hn
    rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h
    simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff]
  classical
    replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n :=
      Finset.card_eq_succ.1 hn
    rcases hn with ⟨i, t, hit, rfl, hn⟩
    replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp
    by_cases Ht : ∃ j ∈ t, f j ≤ f i
    · 

Proof state:
case pos
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
  ∀ {s : Finset ι} {a b : ι},
    (∀ i ∈ s, IsPrime (f i)) →
      Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b i : ι
t : Finset ι
hit : i ∉ t
hn : Finset.card t = n
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1)
hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x)
Ht : ∃ j ∈ t, f j ≤ f i
⊢ I ≤ f a ∨ I ≤ f b ∨ ∃ i_1 ∈ insert i t, I ≤ f i_1

Next tactic:
obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht
theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
    (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
    ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
  suffices
    ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from
    ⟨this, fun h =>
      Or.casesOn h
        (fun h =>
          Set.Subset.trans h <|
            Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _))
        fun h =>
        Or.casesOn h
          (fun h =>
            Set.Subset.trans h <|
              Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _))
          fun ⟨i, his, hi⟩ => by
          refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _;
            exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩
  generalize hn : s.card = n; intro h
  induction' n with n ih generalizing a b s
  · clear hp
    rw [Finset.card_eq_zero] at hn
    subst hn
    rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h
    simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff]
  classical
    replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n :=
      Finset.card_eq_succ.1 hn
    rcases hn with ⟨i, t, hit, rfl, hn⟩
    replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp
    by_cases Ht : ∃ j ∈ t, f j ≤ f i
    · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht
      

Proof state:
case pos.intro.intro
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
  ∀ {s : Finset ι} {a b : ι},
    (∀ i ∈ s, IsPrime (f i)) →
      Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b i : ι
t : Finset ι
hit : i ∉ t
hn : Finset.card t = n
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1)
hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x)
j : ι
hjt : j ∈ t
hfji : f j ≤ f i
⊢ I ≤ f a ∨ I ≤ f b ∨ ∃ i_1 ∈ insert i t, I ≤ f i_1

Next tactic:
obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t :=
        ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩
theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
    (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
    ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
  suffices
    ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from
    ⟨this, fun h =>
      Or.casesOn h
        (fun h =>
          Set.Subset.trans h <|
            Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _))
        fun h =>
        Or.casesOn h
          (fun h =>
            Set.Subset.trans h <|
              Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _))
          fun ⟨i, his, hi⟩ => by
          refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _;
            exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩
  generalize hn : s.card = n; intro h
  induction' n with n ih generalizing a b s
  · clear hp
    rw [Finset.card_eq_zero] at hn
    subst hn
    rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h
    simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff]
  classical
    replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n :=
      Finset.card_eq_succ.1 hn
    rcases hn with ⟨i, t, hit, rfl, hn⟩
    replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp
    by_cases Ht : ∃ j ∈ t, f j ≤ f i
    · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht
      obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t :=
        ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩
      

Proof state:
case pos.intro.intro.intro.intro
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
  ∀ {s : Finset ι} {a b : ι},
    (∀ i ∈ s, IsPrime (f i)) →
      Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b i j : ι
hfji : f j ≤ f i
u : Finset ι
hju : j ∉ u
hit : i ∉ insert j u
hn : Finset.card (insert j u) = n
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i (insert j u)), ↑(f i_1)
hp : IsPrime (f i) ∧ ∀ x ∈ insert j u, IsPrime (f x)
hjt : j ∈ insert j u
⊢ I ≤ f a ∨ I ≤ f b ∨ ∃ i_1 ∈ insert i (insert j u), I ≤ f i_1

Next tactic:
have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by
        rw [Finset.forall_mem_insert] at hp ⊢
        exact ⟨hp.1, hp.2.2⟩
theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
    (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
    ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
  suffices
    ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from
    ⟨this, fun h =>
      Or.casesOn h
        (fun h =>
          Set.Subset.trans h <|
            Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _))
        fun h =>
        Or.casesOn h
          (fun h =>
            Set.Subset.trans h <|
              Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _))
          fun ⟨i, his, hi⟩ => by
          refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _;
            exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩
  generalize hn : s.card = n; intro h
  induction' n with n ih generalizing a b s
  · clear hp
    rw [Finset.card_eq_zero] at hn
    subst hn
    rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h
    simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff]
  classical
    replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n :=
      Finset.card_eq_succ.1 hn
    rcases hn with ⟨i, t, hit, rfl, hn⟩
    replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp
    by_cases Ht : ∃ j ∈ t, f j ≤ f i
    · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht
      obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t :=
        ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩
      have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by
        

Proof state:
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
  ∀ {s : Finset ι} {a b : ι},
    (∀ i ∈ s, IsPrime (f i)) →
      Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b i j : ι
hfji : f j ≤ f i
u : Finset ι
hju : j ∉ u
hit : i ∉ insert j u
hn : Finset.card (insert j u) = n
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i (insert j u)), ↑(f i_1)
hp : IsPrime (f i) ∧ ∀ x ∈ insert j u, IsPrime (f x)
hjt : j ∈ insert j u
⊢ ∀ k ∈ insert i u, IsPrime (f k)

Next tactic:
rw [Finset.forall_mem_insert] at hp ⊢
theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
    (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
    ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
  suffices
    ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from
    ⟨this, fun h =>
      Or.casesOn h
        (fun h =>
          Set.Subset.trans h <|
            Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _))
        fun h =>
        Or.casesOn h
          (fun h =>
            Set.Subset.trans h <|
              Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _))
          fun ⟨i, his, hi⟩ => by
          refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _;
            exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩
  generalize hn : s.card = n; intro h
  induction' n with n ih generalizing a b s
  · clear hp
    rw [Finset.card_eq_zero] at hn
    subst hn
    rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h
    simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff]
  classical
    replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n :=
      Finset.card_eq_succ.1 hn
    rcases hn with ⟨i, t, hit, rfl, hn⟩
    replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp
    by_cases Ht : ∃ j ∈ t, f j ≤ f i
    · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht
      obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t :=
        ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩
      have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by
        rw [Finset.forall_mem_insert] at hp ⊢
        

Proof state:
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
  ∀ {s : Finset ι} {a b : ι},
    (∀ i ∈ s, IsPrime (f i)) →
      Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b i j : ι
hfji : f j ≤ f i
u : Finset ι
hju : j ∉ u
hit : i ∉ insert j u
hn : Finset.card (insert j u) = n
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i (insert j u)), ↑(f i_1)
hp : IsPrime (f i) ∧ IsPrime (f j) ∧ ∀ x ∈ u, IsPrime (f x)
hjt : j ∈ insert j u
⊢ IsPrime (f i) ∧ ∀ x ∈ u, IsPrime (f x)

Next tactic:
exact ⟨hp.1, hp.2.2⟩
theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
    (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
    ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
  suffices
    ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from
    ⟨this, fun h =>
      Or.casesOn h
        (fun h =>
          Set.Subset.trans h <|
            Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _))
        fun h =>
        Or.casesOn h
          (fun h =>
            Set.Subset.trans h <|
              Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _))
          fun ⟨i, his, hi⟩ => by
          refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _;
            exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩
  generalize hn : s.card = n; intro h
  induction' n with n ih generalizing a b s
  · clear hp
    rw [Finset.card_eq_zero] at hn
    subst hn
    rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h
    simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff]
  classical
    replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n :=
      Finset.card_eq_succ.1 hn
    rcases hn with ⟨i, t, hit, rfl, hn⟩
    replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp
    by_cases Ht : ∃ j ∈ t, f j ≤ f i
    · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht
      obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t :=
        ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩
      have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by
        rw [Finset.forall_mem_insert] at hp ⊢
        exact ⟨hp.1, hp.2.2⟩
      

Proof state:
case pos.intro.intro.intro.intro
R✝ : Type u
ι : Type u_1
inst✝¹ : CommSemiring R✝
I✝ J K L : Ideal R✝
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
  ∀ {s : Finset ι} {a b : ι},
    (∀ i ∈ s, IsPrime (f i)) →
      Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b i j : ι
hfji : f j ≤ f i
u : Finset ι
hju : j ∉ u
hit : i ∉ insert j u
hn : Finset.card (insert j u) = n
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i (insert j u)), ↑(f i_1)
hp : IsPrime (f i) ∧ ∀ x ∈ insert j u, IsPrime (f x)
hjt : j ∈ insert j u
hp' : ∀ k ∈ insert i u, IsPrime (f k)
⊢ I ≤ f a ∨ I ≤ f b ∨ ∃ i_1 ∈ insert i (insert j u), I ≤ f i_1

Next tactic:
have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit