/-- A one-dimensional isocrystal over an algebraically closed field
admits an isomorphism to one of the standard (indexed by `m : ℤ`) one-dimensional isocrystals. -/
theorem isocrystal_classification (k : Type*) [Field k] [IsAlgClosed k] [CharP k p] (V : Type*)
    [AddCommGroup V] [Isocrystal p k V] (h_dim : finrank K(p, k) V = 1) :
    ∃ m : ℤ, Nonempty (StandardOneDimIsocrystal p k m ≃ᶠⁱ[p, k] V) := by
  haveI : Nontrivial V := FiniteDimensional.nontrivial_of_finrank_eq_succ h_dim
  obtain ⟨x, hx⟩ : ∃ x : V, x ≠ 0 := exists_ne 0
  have : Φ(p, k) x ≠ 0 := by simpa only [map_zero] using Φ(p, k).injective.ne hx
  obtain ⟨a, ha, hax⟩ : ∃ a : K(p, k), a ≠ 0 ∧ Φ(p, k) x = a • x := by
    rw [finrank_eq_one_iff_of_nonzero' x hx] at h_dim
    obtain ⟨a, ha⟩ := h_dim (Φ(p, k) x)
    refine' ⟨a, _, ha.symm⟩
    intro ha'
    apply this
    simp only [← ha, ha', zero_smul]
  obtain ⟨b, hb, m, hmb⟩ := WittVector.exists_frobenius_solution_fractionRing p ha
  replace hmb : φ(p, k) b * a = (p : K(p, k)) ^ m * b := by convert hmb
  use m
  let F₀ : StandardOneDimIsocrystal p k m →ₗ[K(p, k)] V := LinearMap.toSpanSingleton K(p, k) V x
  let F : StandardOneDimIsocrystal p k m ≃ₗ[K(p, k)] V := by
    refine' LinearEquiv.ofBijective F₀ ⟨_, _⟩
    · rw [← LinearMap.ker_eq_bot]
      exact LinearMap.ker_toSpanSingleton K(p, k) V hx
    · rw [← LinearMap.range_eq_top]
      rw [← (finrank_eq_one_iff_of_nonzero x hx).mp h_dim]
      rw [LinearMap.span_singleton_eq_range]
  -- Porting note: `refine'` below gets confused when this is inlined.
  let E := (LinearEquiv.smulOfNeZero K(p, k) _ _ hb).trans F
  refine' ⟨⟨E, _⟩⟩
  simp only
  intro c
  rw [LinearEquiv.trans_apply, LinearEquiv.trans_apply, LinearEquiv.smulOfNeZero_apply,
    LinearEquiv.smulOfNeZero_apply, LinearEquiv.map_smul, LinearEquiv.map_smul]
  -- Porting note: was
  -- simp only [hax, LinearEquiv.ofBijective_apply, LinearMap.toSpanSingleton_apply,
  --   LinearEquiv.map_smulₛₗ, StandardOneDimIsocrystal.frobenius_apply, Algebra.id.smul_eq_mul]
  rw [LinearEquiv.ofBijective_apply, LinearEquiv.ofBijective_apply]
  erw [LinearMap.toSpanSingleton_apply K(p, k) V x c, LinearMap.toSpanSingleton_apply K(p, k) V x]
  simp only [hax, LinearEquiv.ofBijective_apply, LinearMap.toSpanSingleton_apply,
    LinearEquiv.map_smulₛₗ, StandardOneDimIsocrystal.frobenius_apply, Algebra.id.smul_eq_mul]
  simp only [← mul_smul]
  

Proof state:
case h
p : ℕ
inst✝⁶ : Fact (Nat.Prime p)
k✝ : Type u_1
inst✝⁵ : CommRing k✝
k : Type u_2
inst✝⁴ : Field k
inst✝³ : IsAlgClosed k
inst✝² : CharP k p
V : Type u_3
inst✝¹ : AddCommGroup V
inst✝ : Isocrystal p k V
h_dim : finrank K(p, k) V = 1
this✝ : Nontrivial V
x : V
hx : x ≠ 0
this : Φ(p, k) x ≠ 0
a : K(p, k)
ha : a ≠ 0
hax : Φ(p, k) x = a • x
b : K(p, k)
hb : b ≠ 0
m : ℤ
hmb : φ(p, k) b * a = ↑p ^ m * b
F₀ : StandardOneDimIsocrystal p k m →ₗ[K(p, k)] V := LinearMap.toSpanSingleton K(p, k) V x
F : StandardOneDimIsocrystal p k m ≃ₗ[K(p, k)] V :=
  LinearEquiv.ofBijective F₀ (_ : Function.Injective ⇑F₀ ∧ Function.Surjective ⇑F₀)
E : StandardOneDimIsocrystal p k m ≃ₗ[K(p, k)] V :=
  LinearEquiv.smulOfNeZero K(p, k) (StandardOneDimIsocrystal p k m) b hb ≪≫ₗ F
c : StandardOneDimIsocrystal p k m
⊢ (φ(p, k) b * (φ(p, k) c * a)) • x = (b * (↑p ^ m * φ(p, k) c)) • x

Next tactic:
congr 1