-- Mathlib/RingTheory/WittVector/IsPoly.lean
theorem ext [Fact p.Prime] {f g} (hf : IsPoly₂ p f) (hg : IsPoly₂ p g)
    (h : ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R) (n : ℕ),
        ghostComponent n (f x y) = ghostComponent n (g x y)) :
    ∀ (R) [_Rcr : CommRing R] (x y : 𝕎 R), f x y = g x y

Proof state:
case h.e'_3
p : ℕ
inst✝ : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [inst : 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 => 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 => peval (ψ n) ![x.coeff, y.coeff]
R✝ : Type u
_Rcr✝ : CommRing R✝
x✝ y✝ : 𝕎 R✝
n k : ℕ
x : Fin 2 × ℕ → ℤ
h :
  (eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ))
        (f (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff)
      (wittPolynomial p ℤ k) =
    (eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ))
        (g (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff)
      (wittPolynomial p ℤ k)
⊢ (eval₂Hom ((↑ULift.ringEquiv.symm).comp ((MvPolynomial.eval x).comp C)) fun i =>
        ↑ULift.ringEquiv.symm ((MvPolynomial.eval x) (ψ i)))
      (wittPolynomial p ℤ k) =
    (eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ))
        (g (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff)
      (wittPolynomial p ℤ k)

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

Result:
case h.e'_3
p : ℕ
inst✝ : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [inst : 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 => 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 => peval (ψ n) ![x.coeff, y.coeff]
R✝ : Type u
_Rcr✝ : CommRing R✝
x✝ y✝ : 𝕎 R✝
n k : ℕ
x : Fin 2 × ℕ → ℤ
h :
  (eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ))
        (f (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff)
      (wittPolynomial p ℤ k) =
    (eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ))
        (g (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff)
      (wittPolynomial p ℤ k)
⊢ (eval₂Hom ((↑ULift.ringEquiv.symm).comp ((eval₂Hom (RingHom.id ℤ) x).comp C)) fun i =>
        (eval₂Hom ((↑ULift.ringEquiv.symm).comp (RingHom.id ℤ)) fun i => ↑ULift.ringEquiv.symm (x i)) (ψ i))
      (wittPolynomial p ℤ k) =
    (eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ)) fun n =>
        peval (ψ n) ![(mk p fun i => { down := x (0, i) }).coeff, (mk p fun i => { down := x (1, i) }).coeff])
      (wittPolynomial p ℤ k)

Proof state:
case mk'.intro.mk'.intro.h.h.a
p : ℕ
inst✝ : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [inst : 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 => 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 => peval (ψ n) ![x.coeff, y.coeff]
R✝ : Type u
_Rcr✝ : CommRing R✝
x✝ y✝ : 𝕎 R✝
n k : ℕ
x : Fin 2 × ℕ → ℤ
h :
  (eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ))
        (f (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff)
      (wittPolynomial p ℤ k) =
    (eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ))
        (g (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff)
      (wittPolynomial p ℤ k)
⊢ ULift.ringEquiv.symm
      ((eval₂Hom ((MvPolynomial.eval x).comp C) fun i => (MvPolynomial.eval x) (φ i)) (wittPolynomial p ℤ k)) =
    ULift.ringEquiv.symm
      ((eval₂Hom ((MvPolynomial.eval x).comp C) fun i => (MvPolynomial.eval x) (ψ i)) (wittPolynomial p ℤ k))

Tactic:
simp only [← RingEquiv.coe_toRingHom, map_eval₂Hom]

Result:
case mk'.intro.mk'.intro.h.h.a
p : ℕ
inst✝ : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [inst : 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 => 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 => peval (ψ n) ![x.coeff, y.coeff]
R✝ : Type u
_Rcr✝ : CommRing R✝
x✝ y✝ : 𝕎 R✝
n k : ℕ
x : Fin 2 × ℕ → ℤ
h :
  (eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ))
        (f (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff)
      (wittPolynomial p ℤ k) =
    (eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ))
        (g (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff)
      (wittPolynomial p ℤ k)
⊢ (eval₂Hom ((↑ULift.ringEquiv.symm).comp ((MvPolynomial.eval x).comp C)) fun i =>
        ↑ULift.ringEquiv.symm ((MvPolynomial.eval x) (φ i)))
      (wittPolynomial p ℤ k) =
    (eval₂Hom ((↑ULift.ringEquiv.symm).comp ((MvPolynomial.eval x).comp C)) fun i =>
        ↑ULift.ringEquiv.symm ((MvPolynomial.eval x) (ψ i)))
      (wittPolynomial p ℤ k)

Proof state:
case mk'.intro.mk'.intro
p : ℕ
inst✝ : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R
h : ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R) (n : ℕ), (ghostComponent n) (f x y) = (ghostComponent n) (g x y)
φ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ
hf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun 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 => peval (ψ n) ![x.coeff, y.coeff]
R✝ : Type u
_Rcr✝ : CommRing R✝
x✝ y✝ : 𝕎 R✝
⊢ f x✝ y✝ = g x✝ y✝

Tactic:
ext n

Result:
case mk'.intro.mk'.intro.h
p : ℕ
inst✝ : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R
h : ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R) (n : ℕ), (ghostComponent n) (f x y) = (ghostComponent n) (g x y)
φ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ
hf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun 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 => 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

Proof state:
case mk'.intro.mk'.intro.h.h
p : ℕ
inst✝ : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R
h : ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R) (n : ℕ), (ghostComponent n) (f x y) = (ghostComponent n) (g x y)
φ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ
hf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun 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 => peval (ψ n) ![x.coeff, y.coeff]
R✝ : Type u
_Rcr✝ : CommRing R✝
x✝ y✝ : 𝕎 R✝
n k : ℕ
⊢ ∀ (x : Fin 2 × ℕ → ℤ),
    (MvPolynomial.eval x) ((bind₁ φ) (wittPolynomial p ℤ k)) = (MvPolynomial.eval x) ((bind₁ ψ) (wittPolynomial p ℤ k))

Tactic:
intro x

Result:
case mk'.intro.mk'.intro.h.h
p : ℕ
inst✝ : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R
h : ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R) (n : ℕ), (ghostComponent n) (f x y) = (ghostComponent n) (g x y)
φ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ
hf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun 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 => peval (ψ n) ![x.coeff, y.coeff]
R✝ : Type u
_Rcr✝ : CommRing R✝
x✝ y✝ : 𝕎 R✝
n k : ℕ
x : Fin 2 × ℕ → ℤ
⊢ (MvPolynomial.eval x) ((bind₁ φ) (wittPolynomial p ℤ k)) = (MvPolynomial.eval x) ((bind₁ ψ) (wittPolynomial p ℤ k))