-- Module: Mathlib.Analysis.AbsoluteValue.Equivalence
Goal:
R : Type u_1
S : Type u_2
inst : Field R
inst_1 : Field S
inst_2 : LinearOrder S
inst_3 : TopologicalSpace S
inst_4 : IsStrictOrderedRing S
inst_5 : Archimedean S
inst_6 : OrderTopology S
ι : Type u_3
inst_7 : Finite ι
v : ι → AbsoluteValue R S
h : ∀ (i : ι), (v i).IsNontrivial
hv : Pairwise fun (i j : ι) ↦ ¬(v i).IsEquiv (v j)
have have₁ have₂ have₃ :
  ∀ (this : Fintype ι) (i : ι),
    ∃ (a : R),
      LT.lt.{u_2} (α := S) (1 : S) ((v i : (a : R) → S) a : S) ∧
        ∀ (j : ι), j ≠ i → LT.lt.{u_2} (α := S) ((v j : (a : R) → S) a : S) (1 : S)
have₄ have₅ :
  ∀ (this : Fintype ι) (h : ∀ (i : ι), (v i).IsNontrivial) (hv : Pairwise fun (i j : ι) ↦ ¬(v i).IsEquiv (v j)) (i : ι),
    ∃ (a : R),
      LT.lt.{u_2} (α := S) (1 : S) ((v i : (a : R) → S) a : S) ∧
        ∀ (j : ι), j ≠ i → LT.lt.{u_2} (α := S) ((v j : (a : R) → S) a : S) (1 : S)
refine_1 :
  ∀ (this : Fintype ι) (ι : Type u_3) (x : Fintype ι),
    Subsingleton ι →
      ∀ (v : ι → AbsoluteValue R S) (h : ∀ (i : ι), (v i).IsNontrivial)
        (hv : Pairwise fun (i j : ι) ↦ ¬(v i).IsEquiv (v j)) (i : ι),
        ∃ (a : R),
          LT.lt.{u_2} (α := S) (1 : S) ((v i : (a : R) → S) a : S) ∧
            ∀ (j : ι), j ≠ i → LT.lt.{u_2} (α := S) ((v j : (a : R) → S) a : S) (1 : S)
refine_2 refine_2₁ :
  ∀ (this : Fintype ι),
    let P : (ι : Type u_3) → [Fintype ι] → Prop := fun (ι : Type u_3) [Fintype ι] ↦
      ∀ (v : ι → AbsoluteValue R S),
        (∀ (i : ι), (v i).IsNontrivial) →
          (Pairwise fun (i j : ι) ↦ ¬(v i).IsEquiv (v j)) →
            ∀ (i : ι),
              ∃ (a : R),
                LT.lt.{u_2} (α := S) (1 : S) ((v i : (a : R) → S) a : S) ∧
                  ∀ (j : ι), j ≠ i → LT.lt.{u_2} (α := S) ((v j : (a : R) → S) a : S) (1 : S);
    ∀ (ι : Type u_3) (x : Fintype ι),
      Nontrivial ι →
        ∀ (ih : ∀ (β : Type u_3) [inst : Fintype β], card β < card ι → P β) (v : ι → AbsoluteValue R S)
          (h : ∀ (i : ι), (v i).IsNontrivial) (hv : Pairwise fun (i j : ι) ↦ ¬(v i).IsEquiv (v j)) (i : ι),
          ∃ (a : R),
            LT.lt.{u_2} (α := S) (1 : S) ((v i : (a : R) → S) a : S) ∧
              ∀ (j : ι), j ≠ i → LT.lt.{u_2} (α := S) ((v j : (a : R) → S) a : S) (1 : S)
⊢ ∀ (i : ι),
    ∃ (a : R),
      LT.lt.{u_2} (α := S) (1 : S) ((v i : (a : R) → S) a : S) ∧
        ∀ (j : ι), j ≠ i → LT.lt.{u_2} (α := S) ((v j : (a : R) → S) a : S) (1 : S)

Tactic:
classical
  have := Fintype.ofFinite ι
  let P (ι : Type _) [Fintype ι] : Prop :=
    ∀ v : ι → AbsoluteValue R S, (∀ i, (v i).IsNontrivial) →
      (Pairwise fun i j ↦ ¬(v i).IsEquiv (v j)) → ∀ i, ∃ (a : R), 1 < v i a ∧ ∀ j ≠ i, v j a < 1
  -- Use strong induction on the index.
  revert hv h; refine induction_subsingleton_or_nontrivial (P := P) ι (fun ι _ _ v h hv i ↦ ?_)
    (fun ι _ _ ih v h hv i ↦ ?_) v
  · -- If `ι` is trivial this follows immediately from `(v i).IsNontrivial`.
    let ⟨a, ha⟩ := (h i).exists_abv_gt_one
    exact ⟨a, ha, fun j hij ↦ absurd (Subsingleton.elim i j) hij.symm⟩
  · rcases eq_or_ne (card ι) 2 with (hc | hc)
    · -- If `ι` has two elements this is `exists_one_lt_lt_one_of_not_isEquiv`.
      let ⟨j, hj⟩ := (Nat.card_eq_two_iff' i).1 <| card_eq_nat_card ▸ hc
      let ⟨a, ha⟩ := (v i).exists_one_lt_lt_one_of_not_isEquiv (h i) (h j) (hv hj.1.symm)
      exact ⟨a, ha.1, fun _ h ↦ hj.2 _ h ▸ ha.2⟩
    have hlt : 2 < card ι := Nat.lt_of_le_of_ne (one_lt_card_iff_nontrivial.2 ‹_›) hc.symm
    -- Otherwise, choose another distinguished index `j ≠ i`.
    let ⟨j, hj⟩ := exists_ne i
    -- Apply induction first on the subcollection `v i` for `i ≠ j` to get `a : K`
    let ⟨a, ha⟩ := ih {k : ι // k ≠ j} (card_subtype_lt fun a ↦ a rfl) (restrict _ v)
      (fun i ↦ h _) (hv.comp_of_injective val_injective) ⟨i, hj.symm⟩
    -- Then apply induction next to the subcollection `{v i, v j}` to get `b : K`.
    let ⟨b, hb⟩ := ih {k : ι // k = i ∨ k = j} (by linarith [card_subtype_eq_or_eq_of_ne hj.symm])
      (restrict _ v) (fun _ ↦ h _) (hv.comp_of_injective val_injective) ⟨i, .inl rfl⟩
    rcases eq_or_ne (v j a) 1 with (ha₁ | ha₁)
    · -- If `v j a = 1` then take a large enough value from the sequence `a ^ n * b`.
      let ⟨c, hc⟩ := exists_one_lt_lt_one_pi_of_eq_one ha.1 ha.2 ha₁ hb.1 (hb.2 ⟨j, .inr rfl⟩
        (by grind))
      refine ⟨c, hc.1, fun k hk ↦ ?_⟩
      rcases eq_or_ne k j with (rfl | h); try exact hc.2.2; exact hc.2.1 ⟨k, h⟩ (by grind)
    rcases ha₁.lt_or_gt with (ha_lt | ha_gt)
    · -- If `v j a < 1` then `a` works as the divergent point.
      refine ⟨a, ha.1, fun k hk ↦ ?_⟩
      rcases eq_or_ne k j with (rfl | h); try exact ha_lt; exact ha.2 ⟨k, h⟩ (by grind)
    · -- If `1 < v j a` then take a large enough value from the sequence `b / (1 + a ^ (-n))`.
      let ⟨c, hc⟩ := exists_one_lt_lt_one_pi_of_one_lt ha.1 ha.2 ha_gt hb.1 (hb.2 ⟨j, .inr rfl⟩
        (by grind))
      refine ⟨c, hc.1, fun k hk ↦ ?_⟩
      rcases eq_or_ne k j with (rfl | h); try exact hc.2.2; exact hc.2.1 ⟨k, h⟩ (by grind)

Premises used: rfl