l3lab/ntp-mathlib huggingface dataset:
/-- 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
    

Proof state:
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, a ≠ 0 ∧ Φ(p, k) x = a • x

Next tactic:
rw [finrank_eq_one_iff_of_nonzero' x hx] at h_dim
/-- 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
    

Proof state:
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
this✝ : Nontrivial V
x : V
h_dim : ∀ (w : V), ∃ c, c • x = w
hx : x ≠ 0
this : Φ(p, k) x ≠ 0
⊢ ∃ a, a ≠ 0 ∧ Φ(p, k) x = a • x

Next tactic:
obtain ⟨a, ha⟩ := h_dim (Φ(p, k) x)
/-- 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)
    

Proof state:
case intro
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
this✝ : Nontrivial V
x : V
h_dim : ∀ (w : V), ∃ c, c • x = w
hx : x ≠ 0
this : Φ(p, k) x ≠ 0
a : K(p, k)
ha : a • x = Φ(p, k) x
⊢ ∃ a, a ≠ 0 ∧ Φ(p, k) x = a • x

Next tactic:
refine' ⟨a, _, ha.symm⟩
/-- 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⟩
    

Proof state:
case intro
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
this✝ : Nontrivial V
x : V
h_dim : ∀ (w : V), ∃ c, c • x = w
hx : x ≠ 0
this : Φ(p, k) x ≠ 0
a : K(p, k)
ha : a • x = Φ(p, k) x
⊢ a ≠ 0

Next tactic:
intro ha'
/-- 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'
    

Proof state:
case intro
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
this✝ : Nontrivial V
x : V
h_dim : ∀ (w : V), ∃ c, c • x = w
hx : x ≠ 0
this : Φ(p, k) x ≠ 0
a : K(p, k)
ha : a • x = Φ(p, k) x
ha' : a = 0
⊢ False

Next tactic:
apply this
/-- 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
    

Proof state:
case intro
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
this✝ : Nontrivial V
x : V
h_dim : ∀ (w : V), ∃ c, c • x = w
hx : x ≠ 0
this : Φ(p, k) x ≠ 0
a : K(p, k)
ha : a • x = Φ(p, k) x
ha' : a = 0
⊢ Φ(p, k) x = 0

Next tactic:
simp only [← ha, ha', zero_smul]
/-- 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]
  

Proof state:
case intro.intro.intro
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
⊢ ∃ m, Nonempty (StandardOneDimIsocrystal p k m ≃ᶠⁱ[p, k] V)

Next tactic:
obtain ⟨b, hb, m, hmb⟩ := WittVector.exists_frobenius_solution_fractionRing p ha
/-- 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
  

Proof state:
case intro.intro.intro.intro.intro.intro
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 : (IsFractionRing.fieldEquivOfRingEquiv (frobeniusEquiv p k)) b * a = ↑p ^ m * b
⊢ ∃ m, Nonempty (StandardOneDimIsocrystal p k m ≃ᶠⁱ[p, k] V)

Next tactic:
replace hmb : φ(p, k) b * a = (p : K(p, k)) ^ m * b := by convert hmb
/-- 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 

Proof state:
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 : (IsFractionRing.fieldEquivOfRingEquiv (frobeniusEquiv p k)) b * a = ↑p ^ m * b
⊢ φ(p, k) b * a = ↑p ^ m * b

Next tactic:
convert hmb
/-- 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
  

Proof state:
case intro.intro.intro.intro.intro.intro
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
⊢ ∃ m, Nonempty (StandardOneDimIsocrystal p k m ≃ᶠⁱ[p, k] V)

Next tactic:
use m
/-- 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
  

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
⊢ Nonempty (StandardOneDimIsocrystal p k m ≃ᶠⁱ[p, k] V)

Next tactic:
let F₀ : StandardOneDimIsocrystal p k m →ₗ[K(p, k)] V := LinearMap.toSpanSingleton K(p, k) V x
/-- 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
  

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
⊢ Nonempty (StandardOneDimIsocrystal p k m ≃ᶠⁱ[p, k] V)

Next tactic:
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]
/-- 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
    

Proof state:
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
⊢ StandardOneDimIsocrystal p k m ≃ₗ[K(p, k)] V

Next tactic:
refine' LinearEquiv.ofBijective F₀ ⟨_, _⟩
/-- 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 ⟨_, _⟩
    · 

Proof state:
case refine'_1
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
⊢ Function.Injective ⇑F₀

Next tactic:
rw [← LinearMap.ker_eq_bot]
/-- 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]
      

Proof state:
case refine'_1
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
⊢ LinearMap.ker F₀ = ⊥

Next tactic:
exact LinearMap.ker_toSpanSingleton K(p, k) V hx
/-- 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
    · 

Proof state:
case refine'_2
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
⊢ Function.Surjective ⇑F₀

Next tactic:
rw [← LinearMap.range_eq_top]