Theorem: Ring.DirectLimit.of.zero_exact_aux
File: Mathlib/Algebra/DirectLimit.lean

Proof steps:
state_before: R : Type u
inst✝⁵ : Ring R
ι : Type v
inst✝⁴ : Preorder ι
G : ι → Type w
inst✝³ : (i : ι) → CommRing (G i)
f : (i j : ι) → i ≤ j → G i → G j
f' : (i j : ι) → i ≤ j → G i →+* G j
inst✝² : DirectedSystem G fun i j h => ⇑(f' i j h)
inst✝¹ : Nonempty ι
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
x : FreeCommRing ((i : ι) × G i)
H :
  (Ideal.Quotient.mk
        (Ideal.span
          {a |
            (∃ i j,
                ∃ (H : i ≤ j),
                  ∃ x, FreeCommRing.of ⟨j, (fun i j h => ⇑(f' i j h)) i j H x⟩ - FreeCommRing.of ⟨i, x⟩ = a) ∨
              (∃ i, FreeCommRing.of ⟨i, 1⟩ - 1 = a) ∨
                (∃ i x y, FreeCommRing.of ⟨i, x + y⟩ - (FreeCommRing.of ⟨i, x⟩ + FreeCommRing.of ⟨i, y⟩) = a) ∨
                  ∃ i x y, FreeCommRing.of ⟨i, x * y⟩ - FreeCommRing.of ⟨i, x⟩ * FreeCommRing.of ⟨i, y⟩ = a}))
      x =
    0
⊢ ∃ j s,
    ∃ (H : ∀ k ∈ s, k.fst ≤ j),
      x.IsSupported s ∧
        ∀ [inst : DecidablePred fun x => x ∈ s], (lift fun ix => (f' (↑ix).fst j ⋯) (↑ix).snd) ((restriction s) x) = 0
state_after: R : Type u
inst✝⁵ : Ring R
ι : Type v
inst✝⁴ : Preorder ι
G : ι → Type w
inst✝³ : (i : ι) → CommRing (G i)
f : (i j : ι) → i ≤ j → G i → G j
f' : (i j : ι) → i ≤ j → G i →+* G j
inst✝² : DirectedSystem G fun i j h => ⇑(f' i j h)
inst✝¹ : Nonempty ι
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
x : FreeCommRing ((i : ι) × G i)
H :
  (Ideal.Quotient.mk
        (Ideal.span
          {a |
            (∃ i j,
                ∃ (H : i ≤ j),
                  ∃ x, FreeCommRing.of ⟨j, (fun i j h => ⇑(f' i j h)) i j H x⟩ - FreeCommRing.of ⟨i, x⟩ = a) ∨
              (∃ i, FreeCommRing.of ⟨i, 1⟩ - 1 = a) ∨
                (∃ i x y, FreeCommRing.of ⟨i, x + y⟩ - (FreeCommRing.of ⟨i, x⟩ + FreeCommRing.of ⟨i, y⟩) = a) ∨
                  ∃ i x y, FreeCommRing.of ⟨i, x * y⟩ - FreeCommRing.of ⟨i, x⟩ * FreeCommRing.of ⟨i, y⟩ = a}))
      x =
    0
this : (α : Sort ?u.611507) → DecidableEq α
⊢ ∃ j s,
    ∃ (H : ∀ k ∈ s, k.fst ≤ j),
      x.IsSupported s ∧
        ∀ [inst : DecidablePred fun x => x ∈ s], (lift fun ix => (f' (↑ix).fst j ⋯) (↑ix).snd) ((restriction s) x) = 0
tactic: have := Classical.decEq
state_before: R : Type u
inst✝⁵ : Ring R
ι : Type v
inst✝⁴ : Preorder ι
G : ι → Type w
inst✝³ : (i : ι) → CommRing (G i)
f : (i j : ι) → i ≤ j → G i → G j
f' : (i j : ι) → i ≤ j → G i →+* G j
inst✝² : DirectedSystem G fun i j h => ⇑(f' i j h)
inst✝¹ : Nonempty ι
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
x : FreeCommRing ((i : ι) × G i)
H :
  (Ideal.Quotient.mk
        (Ideal.span
          {a |
            (∃ i j,
                ∃ (H : i ≤ j),
                  ∃ x, FreeCommRing.of ⟨j, (fun i j h => ⇑(f' i j h)) i j H x⟩ - FreeCommRing.of ⟨i, x⟩ = a) ∨
              (∃ i, FreeCommRing.of ⟨i, 1⟩ - 1 = a) ∨
                (∃ i x y, FreeCommRing.of ⟨i, x + y⟩ - (FreeCommRing.of ⟨i, x⟩ + FreeCommRing.of ⟨i, y⟩) = a) ∨
                  ∃ i x y, FreeCommRing.of ⟨i, x * y⟩ - FreeCommRing.of ⟨i, x⟩ * FreeCommRing.of ⟨i, y⟩ = a}))
      x =
    0
this : (α : Sort ?u.611507) → DecidableEq α
⊢ ∃ j s,
    ∃ (H : ∀ k ∈ s, k.fst ≤ j),
      x.IsSupported s ∧
        ∀ [inst : DecidablePred fun x => x ∈ s], (lift fun ix => (f' (↑ix).fst j ⋯) (↑ix).snd) ((restriction s) x) = 0
state_after: case refine_1
R : Type u
inst✝⁵ : Ring R
ι : Type v
inst✝⁴ : Preorder ι
G : ι → Type w
inst✝³ : (i : ι) → CommRing (G i)
f : (i j : ι) → i ≤ j → G i → G j
f' : (i j : ι) → i ≤ j → G i →+* G j
inst✝² : DirectedSystem G fun i j h => ⇑(f' i j h)
inst✝¹ : Nonempty ι
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
x : FreeCommRing ((i : ι) × G i)
H :
  (Ideal.Quotient.mk
        (Ideal.span
          {a |
            (∃ i j,
                ∃ (H : i ≤ j),
                  ∃ x, FreeCommRing.of ⟨j, (fun i j h => ⇑(f' i j h)) i j H x⟩ - FreeCommRing.of ⟨i, x⟩ = a) ∨
              (∃ i, FreeCommRing.of ⟨i, 1⟩ - 1 = a) ∨
                (∃ i x y, FreeCommRing.of ⟨i, x + y⟩ - (FreeCommRing.of ⟨i, x⟩ + FreeCommRing.of ⟨i, y⟩) = a) ∨
                  ∃ i x y, FreeCommRing.of ⟨i, x * y⟩ - FreeCommRing.of ⟨i, x⟩ * FreeCommRing.of ⟨i, y⟩ = a}))
      x =
    0
this : (α : Sort ?u.611507) → DecidableEq α
⊢ ∀
    x ∈
      {a |
        (∃ i j,
            ∃ (H : i ≤ j), ∃ x, FreeCommRing.of ⟨j, (fun i j h => ⇑(f' i j h)) i j H x⟩ - FreeCommRing.of ⟨i, x⟩ = a) ∨
          (∃ i, FreeCommRing.of ⟨i, 1⟩ - 1 = a) ∨
            (∃ i x y, FreeCommRing.of ⟨i, x + y⟩ - (FreeCommRing.of ⟨i, x⟩ + FreeCommRing.of ⟨i, y⟩) = a) ∨
              ∃ i x y, FreeCommRing.of ⟨i, x * y⟩ - FreeCommRing.of ⟨i, x⟩ * FreeCommRing.of ⟨i, y⟩ = a},
    ∃ j s,
      ∃ (H : ∀ k ∈ s, k.fst ≤ j),
        x.IsSupported s ∧
          ∀ [inst : DecidablePred fun x => x ∈ s], (lift fun ix => (f' (↑ix).fst j ⋯) (↑ix).snd) ((restriction s) x) = 0

case refine_2
R : Type u
inst✝⁵ : Ring R
ι : Type v
inst✝⁴ : Preorder ι
G : ι → Type w
inst✝³ : (i : ι) → CommRing (G i)
f : (i j : ι) → i ≤ j → G i → G j
f' : (i j : ι) → i ≤ j → G i →+* G j
inst✝² : DirectedSystem G fun i j h => ⇑(f' i j h)
inst✝¹ : Nonempty ι
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
x : FreeCommRing ((i : ι) × G i)
H :
  (Ideal.Quotient.mk
        (Ideal.span
          {a |
            (∃ i j,
                ∃ (H : i ≤ j),
                  ∃ x, FreeCommRing.of ⟨j, (fun i j h => ⇑(f' i j h)) i j H x⟩ - FreeCommRing.of ⟨i, x⟩ = a) ∨
              (∃ i, FreeCommRing.of ⟨i, 1⟩ - 1 = a) ∨
                (∃ i x y, FreeCommRing.of ⟨i, x + y⟩ - (FreeCommRing.of ⟨i, x⟩ + FreeCommRing.of ⟨i, y⟩) = a) ∨
                  ∃ i x y, FreeCommRing.of ⟨i, x * y⟩ - FreeCommRing.of ⟨i, x⟩ * FreeCommRing.of ⟨i, y⟩ = a}))
      x =
    0
this : (α : Sort ?u.611507) → DecidableEq α
⊢ ∃ j s,
    ∃ (H : ∀ k ∈ s, k.fst ≤ j),
      IsSupported 0 s ∧
        ∀ [inst : DecidablePred fun x => x ∈ s], (lift fun ix => (f' (↑ix).fst j ⋯) (↑ix).snd) ((restriction s) 0) = 0

case refine_3
R : Type u
inst✝⁵ : Ring R
ι : Type v
inst✝⁴ : Preorder ι
G : ι → Type w
inst✝³ : (i : ι) → CommRing (G i)
f : (i j : ι) → i ≤ j → G i → G j
f' : (i j : ι) → i ≤ j → G i →+* G j
inst✝² : DirectedSystem G fun i j h => ⇑(f' i j h)
inst✝¹ : Nonempty ι
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
x : FreeCommRing ((i : ι) × G i)
H :
  (Ideal.Quotient.mk
        (Ideal.span
          {a |
            (∃ i j,
                ∃ (H : i ≤ j),
                  ∃ x, FreeCommRing.of ⟨j, (fun i j h => ⇑(f' i j h)) i j H x⟩ - FreeCommRing.of ⟨i, x⟩ = a) ∨
              (∃ i, FreeCommRing.of ⟨i, 1⟩ - 1 = a) ∨
                (∃ i x y, FreeCommRing.of ⟨i, x + y⟩ - (FreeCommRing.of ⟨i, x⟩ + FreeCommRing.of ⟨i, y⟩) = a) ∨
                  ∃ i x y, FreeCommRing.of ⟨i, x * y⟩ - FreeCommRing.of ⟨i, x⟩ * FreeCommRing.of ⟨i, y⟩ = a}))
      x =
    0
this : (α : Sort ?u.611507) → DecidableEq α
⊢ ∀ (x y : FreeCommRing ((i : ι) × G i)),
    (∃ j s,
        ∃ (H : ∀ k ∈ s, k.fst ≤ j),
          x.IsSupported s ∧
            ∀ [inst : DecidablePred fun x => x ∈ s],
              (lift fun ix => (f' (↑ix).fst j ⋯) (↑ix).snd) ((restriction s) x) = 0) →
      (∃ j s,
          ∃ (H : ∀ k ∈ s, k.fst ≤ j),
            y.IsSupported s ∧
              ∀ [inst : DecidablePred fun x => x ∈ s],
                (lift fun ix => (f' (↑ix).fst j ⋯) (↑ix).snd) ((restriction s) y) = 0) →
        ∃ j s,
          ∃ (H : ∀ k ∈ s, k.fst ≤ j),
            (x + y).IsSupported s ∧
              ∀ [inst : DecidablePred fun x => x ∈ s],
                (lift fun ix => (f' (↑ix).fst j ⋯) (↑ix).snd) ((restriction s) (x + y)) = 0

case refine_4
R : Type u
inst✝⁵ : Ring R
ι : Type v
inst✝⁴ : Preorder ι
G : ι → Type w
inst✝³ : (i : ι) → CommRing (G i)
f : (i j : ι) → i ≤ j → G i → G j
f' : (i j : ι) → i ≤ j → G i →+* G j
inst✝² : DirectedSystem G fun i j h => ⇑(f' i j h)
inst✝¹ : Nonempty ι
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
x : FreeCommRing ((i : ι) × G i)
H :
  (Ideal.Quotient.mk
        (Ideal.span
          {a |
            (∃ i j,
                ∃ (H : i ≤ j),
                  ∃ x, FreeCommRing.of ⟨j, (fun i j h => ⇑(f' i j h)) i j H x⟩ - FreeCommRing.of ⟨i, x⟩ = a) ∨
              (∃ i, FreeCommRing.of ⟨i, 1⟩ - 1 = a) ∨
                (∃ i x y, FreeCommRing.of ⟨i, x + y⟩ - (FreeCommRing.of ⟨i, x⟩ + FreeCommRing.of ⟨i, y⟩) = a) ∨
                  ∃ i x y, FreeCommRing.of ⟨i, x * y⟩ - FreeCommRing.of ⟨i, x⟩ * FreeCommRing.of ⟨i, y⟩ = a}))
      x =
    0
this : (α : Sort ?u.611507) → DecidableEq α
⊢ ∀ (a x : FreeCommRing ((i : ι) × G i)),
    (∃ j s,
        ∃ (H : ∀ k ∈ s, k.fst ≤ j),
          x.IsSupported s ∧
            ∀ [inst : DecidablePred fun x => x ∈ s],
              (lift fun ix => (f' (↑ix).fst j ⋯) (↑ix).snd) ((restriction s) x) = 0) →
      ∃ j s,
        ∃ (H : ∀ k ∈ s, k.fst ≤ j),
          (a • x).IsSupported s ∧
            ∀ [inst : DecidablePred fun x => x ∈ s],
              (lift fun ix => (f' (↑ix).fst j ⋯) (↑ix).snd) ((restriction s) (a • x)) = 0
tactic: refine span_induction (Ideal.Quotient.eq_zero_iff_mem.1 H) ?_ ?_ ?_ ?_