{R : Type u} →
  {Γ₀ : Type v} →
    [inst : CommRing R] →
      [inst_1 : LinearOrderedCommGroupWithZero Γ₀] →
        {v : Valuation R Γ₀} →
          {O : Type w} →
            [inst_2 : CommRing O] →
              [inst_3 : Algebra O R] →
                {motive : v.Integers O → Sort u_1} →
                  (t : v.Integers O) →
                    ((hom_inj : Function.Injective ⇑(algebraMap O R)) →
                        (map_le_one : ∀ (x : O), v ((algebraMap O R) x) ≤ 1) →
                          (exists_of_le_one : ∀ ⦃r : R⦄, v r ≤ 1 → ∃ x, (algebraMap O R) x = r) → motive ⋯) →
                      motive t