awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.WittVector.IsPoly
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)
x_2 : ℕ
apply :
  Eq.{max 1 (u + 1)} (α := Fin 2 × ℕ → ULift.{u, 0} ℤ)
    (fun (i : Fin 2 × ℕ) ↦
      ((↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ) : (a : ℤ) → ULift.{u, 0} ℤ) (x i))
    (uncurry
      ![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 })])
⊢ 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:
apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl

Premises used: MvPolynomial.eval₂Hom_congr, RingHom.ext_int, rfl
-- Module: Mathlib.RingTheory.WittVector.IsPoly
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)
x_2 : ℕ
h₁ :
  ∀ (b : Fin 2) (snd : ℕ),
    Eq (α := ℤ)
      (ULift.down.{u, 0}
        (((↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ) : (a : ℤ) → ULift.{u, 0} ℤ) (x (b, snd))))
      (ULift.down.{u, 0}
        (uncurry
          ![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 })]
          (b, snd)))
⊢ Eq.{max 1 (u + 1)} (α := Fin 2 × ℕ → ULift.{u, 0} ℤ)
    (fun (i : Fin 2 × ℕ) ↦
      ((↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ) : (a : ℤ) → ULift.{u, 0} ℤ) (x i))
    (uncurry
      ![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:
ext ⟨b, _⟩
-- Module: Mathlib.RingTheory.WittVector.IsPoly
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)
x_2 : ℕ
b : Fin 2
snd : ℕ
h₁ :
  Eq (α := ℤ)
    (ULift.down.{u, 0}
      (((↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ) : (a : ℤ) → ULift.{u, 0} ℤ)
        (x ((fun (i : Fin 2) ↦ i) (⟨0, ⋯⟩ : Fin 2), snd))))
    (ULift.down.{u, 0}
      (uncurry
        ![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 })]
        ((fun (i : Fin 2) ↦ i) ⟨0, ⋯⟩, snd)))
h₂ :
  Eq (α := ℤ)
    (ULift.down.{u, 0}
      (((↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ) : (a : ℤ) → ULift.{u, 0} ℤ)
        (x ((fun (i : Fin 2) ↦ i) (⟨1, ⋯⟩ : Fin 2), snd))))
    (ULift.down.{u, 0}
      (uncurry
        ![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 })]
        ((fun (i : Fin 2) ↦ i) ⟨1, ⋯⟩, snd)))
⊢ Eq (α := ℤ)
    (ULift.down.{u, 0}
      (((↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ) : (a : ℤ) → ULift.{u, 0} ℤ) (x (b, snd))))
    (ULift.down.{u, 0}
      (uncurry
        ![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 })]
        (b, snd)))

Tactic:
fin_cases b
-- Module: Mathlib.RingTheory.WittVector.IsPoly
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)
x_2 snd : ℕ
h₁ :
  Eq (α := ℤ)
    (ULift.down.{u, 0}
      (((↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ) : (a : ℤ) → ULift.{u, 0} ℤ)
        (x ((⟨0, ⋯⟩ : Fin 2), snd))))
    (ULift.down.{u, 0}
      (![fun (i : ℕ) ↦ { down := x (0, i) }, fun (i : ℕ) ↦ { down := x (1, i) : ULift.{u, 0} ℤ }] ⟨0, ⋯⟩ snd))
⊢ Eq (α := ℤ)
    (ULift.down.{u, 0}
      (((↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ) : (a : ℤ) → ULift.{u, 0} ℤ)
        (x ((fun (i : Fin 2) ↦ i) (⟨0, ⋯⟩ : Fin 2), snd))))
    (ULift.down.{u, 0}
      (uncurry
        ![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 })]
        ((fun (i : Fin 2) ↦ i) ⟨0, ⋯⟩, snd)))

Tactic:
simp only [coeff_mk, uncurry]

Premises used: WittVector.coeff_mk, Function.uncurry
-- Module: Mathlib.RingTheory.WittVector.IsPoly
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)
x_2 snd : ℕ
h₁ :
  Eq (α := ℤ)
    (ULift.down.{u, 0}
      (((↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ) : (a : ℤ) → ULift.{u, 0} ℤ)
        (x ((⟨1, ⋯⟩ : Fin 2), snd))))
    (ULift.down.{u, 0}
      (![fun (i : ℕ) ↦ { down := x (0, i) }, fun (i : ℕ) ↦ { down := x (1, i) : ULift.{u, 0} ℤ }] ⟨1, ⋯⟩ snd))
⊢ Eq (α := ℤ)
    (ULift.down.{u, 0}
      (((↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ) : (a : ℤ) → ULift.{u, 0} ℤ)
        (x ((fun (i : Fin 2) ↦ i) (⟨1, ⋯⟩ : Fin 2), snd))))
    (ULift.down.{u, 0}
      (uncurry
        ![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 })]
        ((fun (i : Fin 2) ↦ i) ⟨1, ⋯⟩, snd)))

Tactic:
simp only [coeff_mk, uncurry]

Premises used: WittVector.coeff_mk, Function.uncurry
-- Module: Mathlib.RingTheory.WittVector.IsPoly
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)
x_2 snd : ℕ
⊢ Eq (α := ℤ)
    (ULift.down.{u, 0}
      (((↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ) : (a : ℤ) → ULift.{u, 0} ℤ)
        (x ((⟨0, ⋯⟩ : Fin 2), snd))))
    (ULift.down.{u, 0}
      (![fun (i : ℕ) ↦ { down := x (0, i) }, fun (i : ℕ) ↦ { down := x (1, i) : ULift.{u, 0} ℤ }] ⟨0, ⋯⟩ snd))

Tactic:
rfl
-- Module: Mathlib.RingTheory.WittVector.IsPoly
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)
x_2 snd : ℕ
⊢ Eq (α := ℤ)
    (ULift.down.{u, 0}
      (((↑(RingEquiv.symm ULift.ringEquiv) : ℤ →+* ULift.{u, 0} ℤ) : (a : ℤ) → ULift.{u, 0} ℤ)
        (x ((⟨1, ⋯⟩ : Fin 2), snd))))
    (ULift.down.{u, 0}
      (![fun (i : ℕ) ↦ { down := x (0, i) }, fun (i : ℕ) ↦ { down := x (1, i) : ULift.{u, 0} ℤ }] ⟨1, ⋯⟩ snd))

Tactic:
rfl
-- Module: Mathlib.RingTheory.WittVector.IsPoly
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)
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)
              (ψ 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 (ψ 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)
⊢ 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) (ψ 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
              (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)

Tactic:
simp only [hf, hg, MvPolynomial.eval, map_eval₂Hom]

Premises used: MvPolynomial.map_eval₂Hom, MvPolynomial.eval
-- Module: Mathlib.RingTheory.WittVector.IsPoly
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)
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)
        (ψ i))
    fun (n : ℕ) ↦
    peval (ψ 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 })]
⊢ 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)
              (ψ 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 (ψ 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:
apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl

Premises used: MvPolynomial.eval₂Hom_congr, RingHom.ext_int, rfl
-- Module: Mathlib.RingTheory.WittVector.IsPoly
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)
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)
          (ψ x_2) :
        ULift.{u, 0} ℤ)
      (peval (ψ 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)
⊢ 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)
        (ψ i))
    fun (n : ℕ) ↦
    peval (ψ 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:
ext1