awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.WittVector.IsPoly (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R
h :
  ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R) (n : ℕ),
    Eq (α := R) ((ghostComponent n : (a : 𝕎 R) → R) (f x y) : R) ((ghostComponent n : (a : 𝕎 R) → R) (g x y) : R)
φ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ
hf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun (n : ℕ) ↦ peval (φ n) ![x.coeff, y.coeff]
ψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ
hg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun (n : ℕ) ↦ peval (ψ n) ![x.coeff, y.coeff]
R : Type u
_Rcr : CommRing R
x y : 𝕎 R
⊢ f x y = g x y
have h₁ : ∀ (n : ℕ), (f x y).coeff n = (g x y).coeff n
Tactic: ext n
-- Lean 4 proof step from Mathlib.RingTheory.WittVector.IsPoly (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R
h :
  ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R) (n : ℕ),
    Eq (α := R) ((ghostComponent n : (a : 𝕎 R) → R) (f x y) : R) ((ghostComponent n : (a : 𝕎 R) → R) (g x y) : R)
φ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ
hf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun (n : ℕ) ↦ peval (φ n) ![x.coeff, y.coeff]
ψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ
hg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun (n : ℕ) ↦ peval (ψ n) ![x.coeff, y.coeff]
R : Type u
_Rcr : CommRing R
x y : 𝕎 R
n : ℕ
⊢ (f x y).coeff n = (g x y).coeff n
have h₁ : ∀ (n : ℕ),
  Eq (α := MvPolynomial (Fin 2 × ℕ) ℤ)
    ((bind₁ φ : (a : MvPolynomial ℕ ℤ) → MvPolynomial (Fin 2 × ℕ) ℤ) (wittPolynomial p ℤ n) :
      MvPolynomial (Fin 2 × ℕ) ℤ)
    ((bind₁ ψ : (a : MvPolynomial ℕ ℤ) → MvPolynomial (Fin 2 × ℕ) ℤ) (wittPolynomial p ℤ n) :
      MvPolynomial (Fin 2 × ℕ) ℤ)
Tactic: rw [hf, hg, poly_eq_of_wittPolynomial_bind_eq' p φ ψ]
-- Lean 4 proof step from Mathlib.RingTheory.WittVector.IsPoly (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R
h :
  ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R) (n : ℕ),
    Eq (α := R) ((ghostComponent n : (a : 𝕎 R) → R) (f x y) : R) ((ghostComponent n : (a : 𝕎 R) → R) (g x y) : R)
φ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ
hf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun (n : ℕ) ↦ peval (φ n) ![x.coeff, y.coeff]
ψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ
hg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun (n : ℕ) ↦ peval (ψ n) ![x.coeff, y.coeff]
R : Type u
_Rcr : CommRing R
x y : 𝕎 R
n k : ℕ
⊢ Eq (α := MvPolynomial (Fin 2 × ℕ) ℤ)
    ((bind₁ φ : (a : MvPolynomial ℕ ℤ) → MvPolynomial (Fin 2 × ℕ) ℤ) (wittPolynomial p ℤ k) :
      MvPolynomial (Fin 2 × ℕ) ℤ)
    ((bind₁ ψ : (a : MvPolynomial ℕ ℤ) → MvPolynomial (Fin 2 × ℕ) ℤ) (wittPolynomial p ℤ k) :
      MvPolynomial (Fin 2 × ℕ) ℤ)
have h₁ : ∀ (x : Fin 2 × ℕ → ℤ),
  Eq (α := ℤ)
    ((MvPolynomial.eval x : (a : MvPolynomial (Fin 2 × ℕ) ℤ) → ℤ)
        ((bind₁ φ : (a : MvPolynomial ℕ ℤ) → MvPolynomial (Fin 2 × ℕ) ℤ) (wittPolynomial p ℤ k) :
          MvPolynomial (Fin 2 × ℕ) ℤ) :
      ℤ)
    ((MvPolynomial.eval x : (a : MvPolynomial (Fin 2 × ℕ) ℤ) → ℤ)
        ((bind₁ ψ : (a : MvPolynomial ℕ ℤ) → MvPolynomial (Fin 2 × ℕ) ℤ) (wittPolynomial p ℤ k) :
          MvPolynomial (Fin 2 × ℕ) ℤ) :
      ℤ)
Tactic: apply MvPolynomial.funext
-- Lean 4 proof step from Mathlib.RingTheory.WittVector.IsPoly (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R
h :
  ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R) (n : ℕ),
    Eq (α := R) ((ghostComponent n : (a : 𝕎 R) → R) (f x y) : R) ((ghostComponent n : (a : 𝕎 R) → R) (g x y) : R)
φ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ
hf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun (n : ℕ) ↦ peval (φ n) ![x.coeff, y.coeff]
ψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ
hg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun (n : ℕ) ↦ peval (ψ n) ![x.coeff, y.coeff]
R : Type u
_Rcr : CommRing R
x_1 y : 𝕎 R
n k : ℕ
x : Fin 2 × ℕ → ℤ
⊢ Eq (α := ℤ)
    ((MvPolynomial.eval x : (a : MvPolynomial (Fin 2 × ℕ) ℤ) → ℤ)
        ((bind₁ φ : (a : MvPolynomial ℕ ℤ) → MvPolynomial (Fin 2 × ℕ) ℤ) (wittPolynomial p ℤ k) :
          MvPolynomial (Fin 2 × ℕ) ℤ) :
      ℤ)
    ((MvPolynomial.eval x : (a : MvPolynomial (Fin 2 × ℕ) ℤ) → ℤ)
        ((bind₁ ψ : (a : MvPolynomial ℕ ℤ) → MvPolynomial (Fin 2 × ℕ) ℤ) (wittPolynomial p ℤ k) :
          MvPolynomial (Fin 2 × ℕ) ℤ) :
      ℤ)
have h₁ : Eq (α := ℤ)
  ((eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun (i : ℕ) ↦
          (MvPolynomial.eval x : (a : MvPolynomial (Fin 2 × ℕ) ℤ) → ℤ) (φ i) :
        (a : MvPolynomial ℕ ℤ) → ℤ)
      (wittPolynomial p ℤ k) :
    ℤ)
  ((eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun (i : ℕ) ↦
          (MvPolynomial.eval x : (a : MvPolynomial (Fin 2 × ℕ) ℤ) → ℤ) (ψ i) :
        (a : MvPolynomial ℕ ℤ) → ℤ)
      (wittPolynomial p ℤ k) :
    ℤ)
Tactic: simp only [hom_bind₁]
-- Lean 4 proof step from Mathlib.RingTheory.WittVector.IsPoly (hypothesis)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R
h :
  ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R) (n : ℕ),
    Eq (α := R) ((ghostComponent n : (a : 𝕎 R) → R) (f x y) : R) ((ghostComponent n : (a : 𝕎 R) → R) (g x y) : R)
φ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ
hf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun (n : ℕ) ↦ peval (φ n) ![x.coeff, y.coeff]
ψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ
hg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun (n : ℕ) ↦ peval (ψ n) ![x.coeff, y.coeff]
R : Type u
_Rcr : CommRing R
x_1 y : 𝕎 R
n k : ℕ
x : Fin 2 × ℕ → ℤ
⊢ Eq (α := ℤ)
    ((eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun (i : ℕ) ↦
            (MvPolynomial.eval x : (a : MvPolynomial (Fin 2 × ℕ) ℤ) → ℤ) (φ i) :
          (a : MvPolynomial ℕ ℤ) → ℤ)
        (wittPolynomial p ℤ k) :
      ℤ)
    ((eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun (i : ℕ) ↦
            (MvPolynomial.eval x : (a : MvPolynomial (Fin 2 × ℕ) ℤ) → ℤ) (ψ i) :
          (a : MvPolynomial ℕ ℤ) → ℤ)
        (wittPolynomial p ℤ k) :
      ℤ)
have h₁ : Eq.{u + 1} (α := ULift.{u, 0} ℤ)
  ((ghostComponent k : (a : 𝕎 (ULift.{u, 0} ℤ)) → ULift.{u, 0} ℤ)
      (f (mk p fun (i : ℕ) ↦ { down := x (0, i) }) (mk p fun (i : ℕ) ↦ { down := x (1, i) })) :
    ULift.{u, 0} ℤ)
  ((ghostComponent k : (a : 𝕎 (ULift.{u, 0} ℤ)) → ULift.{u, 0} ℤ)
      (g (mk p fun (i : ℕ) ↦ { down := x (0, i) }) (mk p fun (i : ℕ) ↦ { down := x (1, i) })) :
    ULift.{u, 0} ℤ)
Tactic: specialize h (ULift ℤ) (mk p fun i => ⟨x (0, i)⟩) (mk p fun i => ⟨x (1, i)⟩) k
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.RingTheory.WittVector.IsPoly (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R
φ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ
hf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun (n : ℕ) ↦ peval (φ n) ![x.coeff, y.coeff]
ψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ
hg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun (n : ℕ) ↦ peval (ψ n) ![x.coeff, y.coeff]
R : Type u
_Rcr : CommRing R
x_1 y : 𝕎 R
n k : ℕ
x : Fin 2 × ℕ → ℤ
h :
  Eq.{u + 1} (α := ULift.{u, 0} ℤ)
    ((eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ))
            (coeff
              (f (mk p fun (i : ℕ) ↦ { down := x (0, i) : ULift.{u, 0} ℤ })
                (mk p fun (i : ℕ) ↦ { down := x (1, i) : ULift.{u, 0} ℤ }))) :
          (a : MvPolynomial ℕ ℤ) → ULift.{u, 0} ℤ)
        (wittPolynomial p ℤ k) :
      ULift.{u, 0} ℤ)
    ((eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ))
            (coeff
              (g (mk p fun (i : ℕ) ↦ { down := x (0, i) : ULift.{u, 0} ℤ })
                (mk p fun (i : ℕ) ↦ { down := x (1, i) : ULift.{u, 0} ℤ }))) :
          (a : MvPolynomial ℕ ℤ) → ULift.{u, 0} ℤ)
        (wittPolynomial p ℤ k) :
      ULift.{u, 0} ℤ)
⊢ Eq (α := ℤ)
    ((eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun (i : ℕ) ↦
            (MvPolynomial.eval x : (a : MvPolynomial (Fin 2 × ℕ) ℤ) → ℤ) (φ i) :
          (a : MvPolynomial ℕ ℤ) → ℤ)
        (wittPolynomial p ℤ k) :
      ℤ)
    ((eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun (i : ℕ) ↦
            (MvPolynomial.eval x : (a : MvPolynomial (Fin 2 × ℕ) ℤ) → ℤ) (ψ i) :
          (a : MvPolynomial ℕ ℤ) → ℤ)
        (wittPolynomial p ℤ k) :
      ℤ)
have h₁ : Eq.{u + 1} (α := ULift.{u, 0} ℤ)
  ((RingEquiv.symm ULift.ringEquiv : (a : ℤ) → ULift.{u, 0} ℤ)
      ((eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun (i : ℕ) ↦
              (MvPolynomial.eval x : (a : MvPolynomial (Fin 2 × ℕ) ℤ) → ℤ) (φ i) :
            (a : MvPolynomial ℕ ℤ) → ℤ)
          (wittPolynomial p ℤ k) :
        ℤ) :
    ULift.{u, 0} ℤ)
  ((RingEquiv.symm ULift.ringEquiv : (a : ℤ) → ULift.{u, 0} ℤ)
      ((eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun (i : ℕ) ↦
              (MvPolynomial.eval x : (a : MvPolynomial (Fin 2 × ℕ) ℤ) → ℤ) (ψ i) :
            (a : MvPolynomial ℕ ℤ) → ℤ)
          (wittPolynomial p ℤ k) :
        ℤ) :
    ULift.{u, 0} ℤ)
Tactic: apply (ULift.ringEquiv.symm : ℤ ≃+* _).injective
-- Lean 4 proof step from Mathlib.RingTheory.WittVector.IsPoly (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R
φ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ
hf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun (n : ℕ) ↦ peval (φ n) ![x.coeff, y.coeff]
ψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ
hg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun (n : ℕ) ↦ peval (ψ n) ![x.coeff, y.coeff]
R : Type u
_Rcr : CommRing R
x_1 y : 𝕎 R
n k : ℕ
x : Fin 2 × ℕ → ℤ
h :
  Eq.{u + 1} (α := ULift.{u, 0} ℤ)
    ((eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ))
            (coeff
              (f (mk p fun (i : ℕ) ↦ { down := x (0, i) : ULift.{u, 0} ℤ })
                (mk p fun (i : ℕ) ↦ { down := x (1, i) : ULift.{u, 0} ℤ }))) :
          (a : MvPolynomial ℕ ℤ) → ULift.{u, 0} ℤ)
        (wittPolynomial p ℤ k) :
      ULift.{u, 0} ℤ)
    ((eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ))
            (coeff
              (g (mk p fun (i : ℕ) ↦ { down := x (0, i) : ULift.{u, 0} ℤ })
                (mk p fun (i : ℕ) ↦ { down := x (1, i) : ULift.{u, 0} ℤ }))) :
          (a : MvPolynomial ℕ ℤ) → ULift.{u, 0} ℤ)
        (wittPolynomial p ℤ k) :
      ULift.{u, 0} ℤ)
⊢ Eq.{u + 1} (α := ULift.{u, 0} ℤ)
    ((RingEquiv.symm ULift.ringEquiv : (a : ℤ) → ULift.{u, 0} ℤ)
        ((eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun (i : ℕ) ↦
                (MvPolynomial.eval x : (a : MvPolynomial (Fin 2 × ℕ) ℤ) → ℤ) (φ i) :
              (a : MvPolynomial ℕ ℤ) → ℤ)
            (wittPolynomial p ℤ k) :
          ℤ) :
      ULift.{u, 0} ℤ)
    ((RingEquiv.symm ULift.ringEquiv : (a : ℤ) → ULift.{u, 0} ℤ)
        ((eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun (i : ℕ) ↦
                (MvPolynomial.eval x : (a : MvPolynomial (Fin 2 × ℕ) ℤ) → ℤ) (ψ i) :
              (a : MvPolynomial ℕ ℤ) → ℤ)
            (wittPolynomial p ℤ k) :
          ℤ) :
      ULift.{u, 0} ℤ)
have h₁ : Eq (α := ULift.{u, 0} ℤ)
  ((eval₂Hom
          (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ)
            (RingHom.comp (MvPolynomial.eval x) C))
          fun (i : ℕ) ↦
          ((↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ) : (a : ℤ) → ULift.{u, 0} ℤ)
            ((MvPolynomial.eval x : (a : MvPolynomial (Fin 2 × ℕ) ℤ) → ℤ) (φ i) : ℤ) :
        (a : MvPolynomial ℕ ℤ) → ULift.{u, 0} ℤ)
      (wittPolynomial p ℤ k) :
    ULift.{u, 0} ℤ)
  ((eval₂Hom
          (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ)
            (RingHom.comp (MvPolynomial.eval x) C))
          fun (i : ℕ) ↦
          ((↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ) : (a : ℤ) → ULift.{u, 0} ℤ)
            ((MvPolynomial.eval x : (a : MvPolynomial (Fin 2 × ℕ) ℤ) → ℤ) (ψ i) : ℤ) :
        (a : MvPolynomial ℕ ℤ) → ULift.{u, 0} ℤ)
      (wittPolynomial p ℤ k) :
    ULift.{u, 0} ℤ)
Tactic: simp only [← RingEquiv.coe_toRingHom, map_eval₂Hom]
-- Lean 4 proof step from Mathlib.RingTheory.WittVector.IsPoly (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R
φ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ
hf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun (n : ℕ) ↦ peval (φ n) ![x.coeff, y.coeff]
ψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ
hg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun (n : ℕ) ↦ peval (ψ n) ![x.coeff, y.coeff]
R : Type u
_Rcr : CommRing R
x_1 y : 𝕎 R
n k : ℕ
x : Fin 2 × ℕ → ℤ
h :
  Eq.{u + 1} (α := ULift.{u, 0} ℤ)
    ((eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ))
            (coeff
              (f (mk p fun (i : ℕ) ↦ { down := x (0, i) : ULift.{u, 0} ℤ })
                (mk p fun (i : ℕ) ↦ { down := x (1, i) : ULift.{u, 0} ℤ }))) :
          (a : MvPolynomial ℕ ℤ) → ULift.{u, 0} ℤ)
        (wittPolynomial p ℤ k) :
      ULift.{u, 0} ℤ)
    ((eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ))
            (coeff
              (g (mk p fun (i : ℕ) ↦ { down := x (0, i) : ULift.{u, 0} ℤ })
                (mk p fun (i : ℕ) ↦ { down := x (1, i) : ULift.{u, 0} ℤ }))) :
          (a : MvPolynomial ℕ ℤ) → ULift.{u, 0} ℤ)
        (wittPolynomial p ℤ k) :
      ULift.{u, 0} ℤ)
⊢ Eq (α := ULift.{u, 0} ℤ)
    ((eval₂Hom
            (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ)
              (RingHom.comp (MvPolynomial.eval x) C))
            fun (i : ℕ) ↦
            ((↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ) : (a : ℤ) → ULift.{u, 0} ℤ)
              ((MvPolynomial.eval x : (a : MvPolynomial (Fin 2 × ℕ) ℤ) → ℤ) (φ i) : ℤ) :
          (a : MvPolynomial ℕ ℤ) → ULift.{u, 0} ℤ)
        (wittPolynomial p ℤ k) :
      ULift.{u, 0} ℤ)
    ((eval₂Hom
            (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ)
              (RingHom.comp (MvPolynomial.eval x) C))
            fun (i : ℕ) ↦
            ((↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ) : (a : ℤ) → ULift.{u, 0} ℤ)
              ((MvPolynomial.eval x : (a : MvPolynomial (Fin 2 × ℕ) ℤ) → ℤ) (ψ i) : ℤ) :
          (a : MvPolynomial ℕ ℤ) → ULift.{u, 0} ℤ)
        (wittPolynomial p ℤ k) :
      ULift.{u, 0} ℤ)
have h₁ : Eq (α := ULift.{u, 0} ℤ)
  ((eval₂Hom
          (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ)
            (RingHom.comp (MvPolynomial.eval x) C))
          fun (i : ℕ) ↦
          ((↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ) : (a : ℤ) → ULift.{u, 0} ℤ)
            ((MvPolynomial.eval x : (a : MvPolynomial (Fin 2 × ℕ) ℤ) → ℤ) (φ i) : ℤ) :
        (a : MvPolynomial ℕ ℤ) → ULift.{u, 0} ℤ)
      (wittPolynomial p ℤ k) :
    ULift.{u, 0} ℤ)
  ((eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ))
          (coeff
            (f (mk p fun (i : ℕ) ↦ { down := x (0, i) : ULift.{u, 0} ℤ })
              (mk p fun (i : ℕ) ↦ { down := x (1, i) : ULift.{u, 0} ℤ }))) :
        (a : MvPolynomial ℕ ℤ) → ULift.{u, 0} ℤ)
      (wittPolynomial p ℤ k) :
    ULift.{u, 0} ℤ)
Tactic: convert! h using 1
-- Lean 4 proof step from Mathlib.RingTheory.WittVector.IsPoly (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R
φ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ
hf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun (n : ℕ) ↦ peval (φ n) ![x.coeff, y.coeff]
ψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ
hg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun (n : ℕ) ↦ peval (ψ n) ![x.coeff, y.coeff]
R : Type u
_Rcr : CommRing R
x_1 y : 𝕎 R
n k : ℕ
x : Fin 2 × ℕ → ℤ
h :
  Eq.{u + 1} (α := ULift.{u, 0} ℤ)
    ((eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ))
            (coeff
              (f (mk p fun (i : ℕ) ↦ { down := x (0, i) : ULift.{u, 0} ℤ })
                (mk p fun (i : ℕ) ↦ { down := x (1, i) : ULift.{u, 0} ℤ }))) :
          (a : MvPolynomial ℕ ℤ) → ULift.{u, 0} ℤ)
        (wittPolynomial p ℤ k) :
      ULift.{u, 0} ℤ)
    ((eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ))
            (coeff
              (g (mk p fun (i : ℕ) ↦ { down := x (0, i) : ULift.{u, 0} ℤ })
                (mk p fun (i : ℕ) ↦ { down := x (1, i) : ULift.{u, 0} ℤ }))) :
          (a : MvPolynomial ℕ ℤ) → ULift.{u, 0} ℤ)
        (wittPolynomial p ℤ k) :
      ULift.{u, 0} ℤ)
⊢ Eq (α := ULift.{u, 0} ℤ)
    ((eval₂Hom
            (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ)
              (RingHom.comp (MvPolynomial.eval x) C))
            fun (i : ℕ) ↦
            ((↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ) : (a : ℤ) → ULift.{u, 0} ℤ)
              ((MvPolynomial.eval x : (a : MvPolynomial (Fin 2 × ℕ) ℤ) → ℤ) (φ i) : ℤ) :
          (a : MvPolynomial ℕ ℤ) → ULift.{u, 0} ℤ)
        (wittPolynomial p ℤ k) :
      ULift.{u, 0} ℤ)
    ((eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ))
            (coeff
              (f (mk p fun (i : ℕ) ↦ { down := x (0, i) : ULift.{u, 0} ℤ })
                (mk p fun (i : ℕ) ↦ { down := x (1, i) : ULift.{u, 0} ℤ }))) :
          (a : MvPolynomial ℕ ℤ) → ULift.{u, 0} ℤ)
        (wittPolynomial p ℤ k) :
      ULift.{u, 0} ℤ)
have h₁ : Eq (α := ULift.{u, 0} ℤ)
  ((eval₂Hom
          (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ) ((eval₂Hom (RingHom.id ℤ) x).comp C))
          fun (i : ℕ) ↦
          (eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ) (RingHom.id ℤ))
                fun (i : Fin 2 × ℕ) ↦
                ((↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ) : (a : ℤ) → ULift.{u, 0} ℤ) (x i) :
              (a : MvPolynomial (Fin 2 × ℕ) ℤ) → ULift.{u, 0} ℤ)
            (φ i) :
        (a : MvPolynomial ℕ ℤ) → ULift.{u, 0} ℤ)
      (wittPolynomial p ℤ k) :
    ULift.{u, 0} ℤ)
  ((eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ)) fun (n : ℕ) ↦
          peval (φ n)
            ![coeff (mk p fun (i : ℕ) ↦ { down := x (0, i) : ULift.{u, 0} ℤ }),
              coeff (mk p fun (i : ℕ) ↦ { down := x (1, i) : ULift.{u, 0} ℤ })] :
        (a : MvPolynomial ℕ ℤ) → ULift.{u, 0} ℤ)
      (wittPolynomial p ℤ k) :
    ULift.{u, 0} ℤ)
Tactic: simp only [hf, hg, MvPolynomial.eval, map_eval₂Hom]
-- Lean 4 proof step from Mathlib.RingTheory.WittVector.IsPoly (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R
φ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ
hf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun (n : ℕ) ↦ peval (φ n) ![x.coeff, y.coeff]
ψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ
hg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun (n : ℕ) ↦ peval (ψ n) ![x.coeff, y.coeff]
R : Type u
_Rcr : CommRing R
x_1 y : 𝕎 R
n k : ℕ
x : Fin 2 × ℕ → ℤ
h :
  Eq.{u + 1} (α := ULift.{u, 0} ℤ)
    ((eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ))
            (coeff
              (f (mk p fun (i : ℕ) ↦ { down := x (0, i) : ULift.{u, 0} ℤ })
                (mk p fun (i : ℕ) ↦ { down := x (1, i) : ULift.{u, 0} ℤ }))) :
          (a : MvPolynomial ℕ ℤ) → ULift.{u, 0} ℤ)
        (wittPolynomial p ℤ k) :
      ULift.{u, 0} ℤ)
    ((eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ))
            (coeff
              (g (mk p fun (i : ℕ) ↦ { down := x (0, i) : ULift.{u, 0} ℤ })
                (mk p fun (i : ℕ) ↦ { down := x (1, i) : ULift.{u, 0} ℤ }))) :
          (a : MvPolynomial ℕ ℤ) → ULift.{u, 0} ℤ)
        (wittPolynomial p ℤ k) :
      ULift.{u, 0} ℤ)
⊢ Eq (α := ULift.{u, 0} ℤ)
    ((eval₂Hom
            (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ)
              ((eval₂Hom (RingHom.id ℤ) x).comp C))
            fun (i : ℕ) ↦
            (eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ) (RingHom.id ℤ))
                  fun (i : Fin 2 × ℕ) ↦
                  ((↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ) : (a : ℤ) → ULift.{u, 0} ℤ) (x i) :
                (a : MvPolynomial (Fin 2 × ℕ) ℤ) → ULift.{u, 0} ℤ)
              (φ i) :
          (a : MvPolynomial ℕ ℤ) → ULift.{u, 0} ℤ)
        (wittPolynomial p ℤ k) :
      ULift.{u, 0} ℤ)
    ((eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ)) fun (n : ℕ) ↦
            peval (φ n)
              ![coeff (mk p fun (i : ℕ) ↦ { down := x (0, i) : ULift.{u, 0} ℤ }),
                coeff (mk p fun (i : ℕ) ↦ { down := x (1, i) : ULift.{u, 0} ℤ })] :
          (a : MvPolynomial ℕ ℤ) → ULift.{u, 0} ℤ)
        (wittPolynomial p ℤ k) :
      ULift.{u, 0} ℤ)
have apply : Eq.{max 1 (u + 1)} (α := ℕ → ULift.{u, 0} ℤ)
  (fun (i : ℕ) ↦
    (eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ) (RingHom.id ℤ))
          fun (i : Fin 2 × ℕ) ↦
          ((↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ) : (a : ℤ) → ULift.{u, 0} ℤ) (x i) :
        (a : MvPolynomial (Fin 2 × ℕ) ℤ) → ULift.{u, 0} ℤ)
      (φ i))
  fun (n : ℕ) ↦
  peval (φ n)
    ![coeff (mk p fun (i : ℕ) ↦ { down := x (0, i) : ULift.{u, 0} ℤ }),
      coeff (mk p fun (i : ℕ) ↦ { down := x (1, i) : ULift.{u, 0} ℤ })]
Tactic: apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl
-- Lean 4 proof step from Mathlib.RingTheory.WittVector.IsPoly (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R
φ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ
hf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun (n : ℕ) ↦ peval (φ n) ![x.coeff, y.coeff]
ψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ
hg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun (n : ℕ) ↦ peval (ψ n) ![x.coeff, y.coeff]
R : Type u
_Rcr : CommRing R
x_1 y : 𝕎 R
n k : ℕ
x : Fin 2 × ℕ → ℤ
h :
  Eq.{u + 1} (α := ULift.{u, 0} ℤ)
    ((eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ))
            (coeff
              (f (mk p fun (i : ℕ) ↦ { down := x (0, i) : ULift.{u, 0} ℤ })
                (mk p fun (i : ℕ) ↦ { down := x (1, i) : ULift.{u, 0} ℤ }))) :
          (a : MvPolynomial ℕ ℤ) → ULift.{u, 0} ℤ)
        (wittPolynomial p ℤ k) :
      ULift.{u, 0} ℤ)
    ((eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ))
            (coeff
              (g (mk p fun (i : ℕ) ↦ { down := x (0, i) : ULift.{u, 0} ℤ })
                (mk p fun (i : ℕ) ↦ { down := x (1, i) : ULift.{u, 0} ℤ }))) :
          (a : MvPolynomial ℕ ℤ) → ULift.{u, 0} ℤ)
        (wittPolynomial p ℤ k) :
      ULift.{u, 0} ℤ)
⊢ Eq.{max 1 (u + 1)} (α := ℕ → ULift.{u, 0} ℤ)
    (fun (i : ℕ) ↦
      (eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ) (RingHom.id ℤ))
            fun (i : Fin 2 × ℕ) ↦
            ((↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ) : (a : ℤ) → ULift.{u, 0} ℤ) (x i) :
          (a : MvPolynomial (Fin 2 × ℕ) ℤ) → ULift.{u, 0} ℤ)
        (φ i))
    fun (n : ℕ) ↦
    peval (φ n)
      ![coeff (mk p fun (i : ℕ) ↦ { down := x (0, i) : ULift.{u, 0} ℤ }),
        coeff (mk p fun (i : ℕ) ↦ { down := x (1, i) : ULift.{u, 0} ℤ })]
have h₁ : ∀ (x_2 : ℕ),
  Eq (α := ULift.{u, 0} ℤ)
    ((eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ) (RingHom.id ℤ))
            fun (i : Fin 2 × ℕ) ↦
            ((↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ) : (a : ℤ) → ULift.{u, 0} ℤ) (x i) :
          (a : MvPolynomial (Fin 2 × ℕ) ℤ) → ULift.{u, 0} ℤ)
        (φ x_2) :
      ULift.{u, 0} ℤ)
    (peval (φ x_2)
        ![coeff (mk p fun (i : ℕ) ↦ { down := x (0, i) : ULift.{u, 0} ℤ }),
          coeff (mk p fun (i : ℕ) ↦ { down := x (1, i) : ULift.{u, 0} ℤ })] :
      ULift.{u, 0} ℤ)
Tactic: ext1