Theorem: WittVector.nth_mul_coeff'
File: Mathlib/RingTheory/WittVector/MulCoeff.lean

Proof steps:
state_before: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
⊢ ∃ f,
    ∀ (x y : 𝕎 k),
      f (truncateFun (n + 1) x) (truncateFun (n + 1) y) =
        (x * y).coeff (n + 1) - y.coeff (n + 1) * x.coeff 0 ^ p ^ (n + 1) - x.coeff (n + 1) * y.coeff 0 ^ p ^ (n + 1)
state_after: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
⊢ ∃ f,
    ∀ (x y : 𝕎 k),
      f (truncateFun (n + 1) x) (truncateFun (n + 1) y) =
        peval (polyOfInterest p n) ![fun i => x.coeff i, fun i => y.coeff i]
tactic: simp only [← peval_polyOfInterest']
state_before: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
⊢ ∃ f,
    ∀ (x y : 𝕎 k),
      f (truncateFun (n + 1) x) (truncateFun (n + 1) y) =
        peval (polyOfInterest p n) ![fun i => x.coeff i, fun i => y.coeff i]
state_after: case intro
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
f₀ : ((↑fun a => a ∈ (univ ×ˢ range (n + 1)).val) → k) → k
hf₀ : ∀ (x : Fin 2 × ℕ → k), f₀ (x ∘ Subtype.val) = (aeval x) (polyOfInterest p n)
⊢ ∃ f,
    ∀ (x y : 𝕎 k),
      f (truncateFun (n + 1) x) (truncateFun (n + 1) y) =
        peval (polyOfInterest p n) ![fun i => x.coeff i, fun i => y.coeff i]
tactic: obtain ⟨f₀, hf₀⟩ := exists_restrict_to_vars k (polyOfInterest_vars p n)
state_before: case intro
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
f₀ : ((↑fun a => a ∈ (univ ×ˢ range (n + 1)).val) → k) → k
hf₀ : ∀ (x : Fin 2 × ℕ → k), f₀ (x ∘ Subtype.val) = (aeval x) (polyOfInterest p n)
⊢ ∃ f,
    ∀ (x y : 𝕎 k),
      f (truncateFun (n + 1) x) (truncateFun (n + 1) y) =
        peval (polyOfInterest p n) ![fun i => x.coeff i, fun i => y.coeff i]
state_after: case intro
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
f₀ : ((↑fun a => a ∈ (univ ×ˢ range (n + 1)).val) → k) → k
hf₀ : ∀ (x : Fin 2 × ℕ → k), f₀ (x ∘ Subtype.val) = (aeval x) (polyOfInterest p n)
this : ∀ (a : Multiset (Fin 2)) (b : Multiset ℕ), a ×ˢ b = a.product b
⊢ ∃ f,
    ∀ (x y : 𝕎 k),
      f (truncateFun (n + 1) x) (truncateFun (n + 1) y) =
        peval (polyOfInterest p n) ![fun i => x.coeff i, fun i => y.coeff i]
tactic: have : ∀ (a : Multiset (Fin 2)) (b : Multiset ℕ), a ×ˢ b = a.product b := fun a b => rfl
state_before: case intro
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
f₀ : ((↑fun a => a ∈ (univ ×ˢ range (n + 1)).val) → k) → k
hf₀ : ∀ (x : Fin 2 × ℕ → k), f₀ (x ∘ Subtype.val) = (aeval x) (polyOfInterest p n)
this : ∀ (a : Multiset (Fin 2)) (b : Multiset ℕ), a ×ˢ b = a.product b
⊢ ∃ f,
    ∀ (x y : 𝕎 k),
      f (truncateFun (n + 1) x) (truncateFun (n + 1) y) =
        peval (polyOfInterest p n) ![fun i => x.coeff i, fun i => y.coeff i]
state_after: case intro
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
f₀ : ((↑fun a => a ∈ (univ ×ˢ range (n + 1)).val) → k) → k
hf₀ : ∀ (x : Fin 2 × ℕ → k), f₀ (x ∘ Subtype.val) = (aeval x) (polyOfInterest p n)
this : ∀ (a : Multiset (Fin 2)) (b : Multiset ℕ), a ×ˢ b = a.product b
f : TruncatedWittVector p (n + 1) k → TruncatedWittVector p (n + 1) k → k :=
  fun x y => f₀ fun a => Subtype.casesOn a fun a ha => Function.uncurry ![x, y] (a.1, ⟨a.2, ⋯⟩)
⊢ ∃ f,
    ∀ (x y : 𝕎 k),
      f (truncateFun (n + 1) x) (truncateFun (n + 1) y) =
        peval (polyOfInterest p n) ![fun i => x.coeff i, fun i => y.coeff i]
tactic: let f : TruncatedWittVector p (n + 1) k → TruncatedWittVector p (n + 1) k → k := by
    intro x y
    apply f₀
    rintro ⟨a, ha⟩
    apply Function.uncurry ![x, y]
    simp_rw [product_val, this, Multiset.mem_product, mem_univ_val, true_and_iff, range_val,
      Multiset.range_succ, Multiset.mem_cons, Multiset.mem_range] at ha
    refine ⟨a.fst, ⟨a.snd, ?_⟩⟩
    cases' ha with ha ha <;> omega
state_before: case intro
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
f₀ : ((↑fun a => a ∈ (univ ×ˢ range (n + 1)).val) → k) → k
hf₀ : ∀ (x : Fin 2 × ℕ → k), f₀ (x ∘ Subtype.val) = (aeval x) (polyOfInterest p n)
this : ∀ (a : Multiset (Fin 2)) (b : Multiset ℕ), a ×ˢ b = a.product b
f : TruncatedWittVector p (n + 1) k → TruncatedWittVector p (n + 1) k → k :=
  fun x y => f₀ fun a => Subtype.casesOn a fun a ha => Function.uncurry ![x, y] (a.1, ⟨a.2, ⋯⟩)
⊢ ∃ f,
    ∀ (x y : 𝕎 k),
      f (truncateFun (n + 1) x) (truncateFun (n + 1) y) =
        peval (polyOfInterest p n) ![fun i => x.coeff i, fun i => y.coeff i]
state_after: case h
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
f₀ : ((↑fun a => a ∈ (univ ×ˢ range (n + 1)).val) → k) → k
hf₀ : ∀ (x : Fin 2 × ℕ → k), f₀ (x ∘ Subtype.val) = (aeval x) (polyOfInterest p n)
this : ∀ (a : Multiset (Fin 2)) (b : Multiset ℕ), a ×ˢ b = a.product b
f : TruncatedWittVector p (n + 1) k → TruncatedWittVector p (n + 1) k → k :=
  fun x y => f₀ fun a => Subtype.casesOn a fun a ha => Function.uncurry ![x, y] (a.1, ⟨a.2, ⋯⟩)
⊢ ∀ (x y : 𝕎 k),
    f (truncateFun (n + 1) x) (truncateFun (n + 1) y) =
      peval (polyOfInterest p n) ![fun i => x.coeff i, fun i => y.coeff i]
tactic: use f
state_before: case h
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
f₀ : ((↑fun a => a ∈ (univ ×ˢ range (n + 1)).val) → k) → k
hf₀ : ∀ (x : Fin 2 × ℕ → k), f₀ (x ∘ Subtype.val) = (aeval x) (polyOfInterest p n)
this : ∀ (a : Multiset (Fin 2)) (b : Multiset ℕ), a ×ˢ b = a.product b
f : TruncatedWittVector p (n + 1) k → TruncatedWittVector p (n + 1) k → k :=
  fun x y => f₀ fun a => Subtype.casesOn a fun a ha => Function.uncurry ![x, y] (a.1, ⟨a.2, ⋯⟩)
⊢ ∀ (x y : 𝕎 k),
    f (truncateFun (n + 1) x) (truncateFun (n + 1) y) =
      peval (polyOfInterest p n) ![fun i => x.coeff i, fun i => y.coeff i]
state_after: case h
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
f₀ : ((↑fun a => a ∈ (univ ×ˢ range (n + 1)).val) → k) → k
hf₀ : ∀ (x : Fin 2 × ℕ → k), f₀ (x ∘ Subtype.val) = (aeval x) (polyOfInterest p n)
this : ∀ (a : Multiset (Fin 2)) (b : Multiset ℕ), a ×ˢ b = a.product b
f : TruncatedWittVector p (n + 1) k → TruncatedWittVector p (n + 1) k → k :=
  fun x y => f₀ fun a => Subtype.casesOn a fun a ha => Function.uncurry ![x, y] (a.1, ⟨a.2, ⋯⟩)
x y : 𝕎 k
⊢ f (truncateFun (n + 1) x) (truncateFun (n + 1) y) =
    peval (polyOfInterest p n) ![fun i => x.coeff i, fun i => y.coeff i]
tactic: intro x y
state_before: case h
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
f₀ : ((↑fun a => a ∈ (univ ×ˢ range (n + 1)).val) → k) → k
hf₀ : ∀ (x : Fin 2 × ℕ → k), f₀ (x ∘ Subtype.val) = (aeval x) (polyOfInterest p n)
this : ∀ (a : Multiset (Fin 2)) (b : Multiset ℕ), a ×ˢ b = a.product b
f : TruncatedWittVector p (n + 1) k → TruncatedWittVector p (n + 1) k → k :=
  fun x y => f₀ fun a => Subtype.casesOn a fun a ha => Function.uncurry ![x, y] (a.1, ⟨a.2, ⋯⟩)
x y : 𝕎 k
⊢ f (truncateFun (n + 1) x) (truncateFun (n + 1) y) =
    peval (polyOfInterest p n) ![fun i => x.coeff i, fun i => y.coeff i]
state_after: case h
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
f₀ : ((↑fun a => a ∈ (univ ×ˢ range (n + 1)).val) → k) → k
hf₀ : ∀ (x : Fin 2 × ℕ → k), f₀ (x ∘ Subtype.val) = (aeval x) (polyOfInterest p n)
this : ∀ (a : Multiset (Fin 2)) (b : Multiset ℕ), a ×ˢ b = a.product b
f : TruncatedWittVector p (n + 1) k → TruncatedWittVector p (n + 1) k → k :=
  fun x y => f₀ fun a => Subtype.casesOn a fun a ha => Function.uncurry ![x, y] (a.1, ⟨a.2, ⋯⟩)
x y : 𝕎 k
⊢ (f₀ fun a => ![truncateFun (n + 1) x, truncateFun (n + 1) y] (↑a).1 ⟨(↑a).2, ⋯⟩) =
    (aeval (Function.uncurry ![fun i => x.coeff i, fun i => y.coeff i])) (polyOfInterest p n)
tactic: dsimp [f, peval]
state_before: case h
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
f₀ : ((↑fun a => a ∈ (univ ×ˢ range (n + 1)).val) → k) → k
hf₀ : ∀ (x : Fin 2 × ℕ → k), f₀ (x ∘ Subtype.val) = (aeval x) (polyOfInterest p n)
this : ∀ (a : Multiset (Fin 2)) (b : Multiset ℕ), a ×ˢ b = a.product b
f : TruncatedWittVector p (n + 1) k → TruncatedWittVector p (n + 1) k → k :=
  fun x y => f₀ fun a => Subtype.casesOn a fun a ha => Function.uncurry ![x, y] (a.1, ⟨a.2, ⋯⟩)
x y : 𝕎 k
⊢ (f₀ fun a => ![truncateFun (n + 1) x, truncateFun (n + 1) y] (↑a).1 ⟨(↑a).2, ⋯⟩) =
    (aeval (Function.uncurry ![fun i => x.coeff i, fun i => y.coeff i])) (polyOfInterest p n)
state_after: case h
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
f₀ : ((↑fun a => a ∈ (univ ×ˢ range (n + 1)).val) → k) → k
hf₀ : ∀ (x : Fin 2 × ℕ → k), f₀ (x ∘ Subtype.val) = (aeval x) (polyOfInterest p n)
this : ∀ (a : Multiset (Fin 2)) (b : Multiset ℕ), a ×ˢ b = a.product b
f : TruncatedWittVector p (n + 1) k → TruncatedWittVector p (n + 1) k → k :=
  fun x y => f₀ fun a => Subtype.casesOn a fun a ha => Function.uncurry ![x, y] (a.1, ⟨a.2, ⋯⟩)
x y : 𝕎 k
⊢ (f₀ fun a => ![truncateFun (n + 1) x, truncateFun (n + 1) y] (↑a).1 ⟨(↑a).2, ⋯⟩) =
    f₀ (Function.uncurry ![fun i => x.coeff i, fun i => y.coeff i] ∘ Subtype.val)
tactic: rw [← hf₀]
state_before: case h
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
f₀ : ((↑fun a => a ∈ (univ ×ˢ range (n + 1)).val) → k) → k
hf₀ : ∀ (x : Fin 2 × ℕ → k), f₀ (x ∘ Subtype.val) = (aeval x) (polyOfInterest p n)
this : ∀ (a : Multiset (Fin 2)) (b : Multiset ℕ), a ×ˢ b = a.product b
f : TruncatedWittVector p (n + 1) k → TruncatedWittVector p (n + 1) k → k :=
  fun x y => f₀ fun a => Subtype.casesOn a fun a ha => Function.uncurry ![x, y] (a.1, ⟨a.2, ⋯⟩)
x y : 𝕎 k
⊢ (f₀ fun a => ![truncateFun (n + 1) x, truncateFun (n + 1) y] (↑a).1 ⟨(↑a).2, ⋯⟩) =
    f₀ (Function.uncurry ![fun i => x.coeff i, fun i => y.coeff i] ∘ Subtype.val)
state_after: case h.e_a
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
f₀ : ((↑fun a => a ∈ (univ ×ˢ range (n + 1)).val) → k) → k
hf₀ : ∀ (x : Fin 2 × ℕ → k), f₀ (x ∘ Subtype.val) = (aeval x) (polyOfInterest p n)
this : ∀ (a : Multiset (Fin 2)) (b : Multiset ℕ), a ×ˢ b = a.product b
f : TruncatedWittVector p (n + 1) k → TruncatedWittVector p (n + 1) k → k :=
  fun x y => f₀ fun a => Subtype.casesOn a fun a ha => Function.uncurry ![x, y] (a.1, ⟨a.2, ⋯⟩)
x y : 𝕎 k
⊢ (fun a => ![truncateFun (n + 1) x, truncateFun (n + 1) y] (↑a).1 ⟨(↑a).2, ⋯⟩) =
    Function.uncurry ![fun i => x.coeff i, fun i => y.coeff i] ∘ Subtype.val
tactic: congr
state_before: case h.e_a
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
f₀ : ((↑fun a => a ∈ (univ ×ˢ range (n + 1)).val) → k) → k
hf₀ : ∀ (x : Fin 2 × ℕ → k), f₀ (x ∘ Subtype.val) = (aeval x) (polyOfInterest p n)
this : ∀ (a : Multiset (Fin 2)) (b : Multiset ℕ), a ×ˢ b = a.product b
f : TruncatedWittVector p (n + 1) k → TruncatedWittVector p (n + 1) k → k :=
  fun x y => f₀ fun a => Subtype.casesOn a fun a ha => Function.uncurry ![x, y] (a.1, ⟨a.2, ⋯⟩)
x y : 𝕎 k
⊢ (fun a => ![truncateFun (n + 1) x, truncateFun (n + 1) y] (↑a).1 ⟨(↑a).2, ⋯⟩) =
    Function.uncurry ![fun i => x.coeff i, fun i => y.coeff i] ∘ Subtype.val
state_after: case h.e_a.h
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
f₀ : ((↑fun a => a ∈ (univ ×ˢ range (n + 1)).val) → k) → k
hf₀ : ∀ (x : Fin 2 × ℕ → k), f₀ (x ∘ Subtype.val) = (aeval x) (polyOfInterest p n)
this : ∀ (a : Multiset (Fin 2)) (b : Multiset ℕ), a ×ˢ b = a.product b
f : TruncatedWittVector p (n + 1) k → TruncatedWittVector p (n + 1) k → k :=
  fun x y => f₀ fun a => Subtype.casesOn a fun a ha => Function.uncurry ![x, y] (a.1, ⟨a.2, ⋯⟩)
x y : 𝕎 k
a : ↑fun a => a ∈ univ.val ×ˢ Multiset.range (n + 1)
⊢ ![truncateFun (n + 1) x, truncateFun (n + 1) y] (↑a).1 ⟨(↑a).2, ⋯⟩ =
    (Function.uncurry ![fun i => x.coeff i, fun i => y.coeff i] ∘ Subtype.val) a
tactic: ext a
state_before: case h.e_a.h
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
f₀ : ((↑fun a => a ∈ (univ ×ˢ range (n + 1)).val) → k) → k
hf₀ : ∀ (x : Fin 2 × ℕ → k), f₀ (x ∘ Subtype.val) = (aeval x) (polyOfInterest p n)
this : ∀ (a : Multiset (Fin 2)) (b : Multiset ℕ), a ×ˢ b = a.product b
f : TruncatedWittVector p (n + 1) k → TruncatedWittVector p (n + 1) k → k :=
  fun x y => f₀ fun a => Subtype.casesOn a fun a ha => Function.uncurry ![x, y] (a.1, ⟨a.2, ⋯⟩)
x y : 𝕎 k
a : ↑fun a => a ∈ univ.val ×ˢ Multiset.range (n + 1)
⊢ ![truncateFun (n + 1) x, truncateFun (n + 1) y] (↑a).1 ⟨(↑a).2, ⋯⟩ =
    (Function.uncurry ![fun i => x.coeff i, fun i => y.coeff i] ∘ Subtype.val) a
state_after: case h.e_a.h.mk
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
f₀ : ((↑fun a => a ∈ (univ ×ˢ range (n + 1)).val) → k) → k
hf₀ : ∀ (x : Fin 2 × ℕ → k), f₀ (x ∘ Subtype.val) = (aeval x) (polyOfInterest p n)
this : ∀ (a : Multiset (Fin 2)) (b : Multiset ℕ), a ×ˢ b = a.product b
f : TruncatedWittVector p (n + 1) k → TruncatedWittVector p (n + 1) k → k :=
  fun x y => f₀ fun a => Subtype.casesOn a fun a ha => Function.uncurry ![x, y] (a.1, ⟨a.2, ⋯⟩)
x y : 𝕎 k
a : Fin 2 × ℕ
ha : a ∈ fun a => a ∈ univ.val ×ˢ Multiset.range (n + 1)
⊢ ![truncateFun (n + 1) x, truncateFun (n + 1) y] (↑⟨a, ha⟩).1 ⟨(↑⟨a, ha⟩).2, ⋯⟩ =
    (Function.uncurry ![fun i => x.coeff i, fun i => y.coeff i] ∘ Subtype.val) ⟨a, ha⟩
tactic: cases' a with a ha
state_before: case h.e_a.h.mk
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
f₀ : ((↑fun a => a ∈ (univ ×ˢ range (n + 1)).val) → k) → k
hf₀ : ∀ (x : Fin 2 × ℕ → k), f₀ (x ∘ Subtype.val) = (aeval x) (polyOfInterest p n)
this : ∀ (a : Multiset (Fin 2)) (b : Multiset ℕ), a ×ˢ b = a.product b
f : TruncatedWittVector p (n + 1) k → TruncatedWittVector p (n + 1) k → k :=
  fun x y => f₀ fun a => Subtype.casesOn a fun a ha => Function.uncurry ![x, y] (a.1, ⟨a.2, ⋯⟩)
x y : 𝕎 k
a : Fin 2 × ℕ
ha : a ∈ fun a => a ∈ univ.val ×ˢ Multiset.range (n + 1)
⊢ ![truncateFun (n + 1) x, truncateFun (n + 1) y] (↑⟨a, ha⟩).1 ⟨(↑⟨a, ha⟩).2, ⋯⟩ =
    (Function.uncurry ![fun i => x.coeff i, fun i => y.coeff i] ∘ Subtype.val) ⟨a, ha⟩
state_after: case h.e_a.h.mk.mk
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
f₀ : ((↑fun a => a ∈ (univ ×ˢ range (n + 1)).val) → k) → k
hf₀ : ∀ (x : Fin 2 × ℕ → k), f₀ (x ∘ Subtype.val) = (aeval x) (polyOfInterest p n)
this : ∀ (a : Multiset (Fin 2)) (b : Multiset ℕ), a ×ˢ b = a.product b
f : TruncatedWittVector p (n + 1) k → TruncatedWittVector p (n + 1) k → k :=
  fun x y => f₀ fun a => Subtype.casesOn a fun a ha => Function.uncurry ![x, y] (a.1, ⟨a.2, ⋯⟩)
x y : 𝕎 k
i : Fin 2
m : ℕ
ha : (i, m) ∈ fun a => a ∈ univ.val ×ˢ Multiset.range (n + 1)
⊢ ![truncateFun (n + 1) x, truncateFun (n + 1) y] (↑⟨(i, m), ha⟩).1 ⟨(↑⟨(i, m), ha⟩).2, ⋯⟩ =
    (Function.uncurry ![fun i => x.coeff i, fun i => y.coeff i] ∘ Subtype.val) ⟨(i, m), ha⟩
tactic: cases' a with i m
state_before: case h.e_a.h.mk.mk
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
f₀ : ((↑fun a => a ∈ (univ ×ˢ range (n + 1)).val) → k) → k
hf₀ : ∀ (x : Fin 2 × ℕ → k), f₀ (x ∘ Subtype.val) = (aeval x) (polyOfInterest p n)
this : ∀ (a : Multiset (Fin 2)) (b : Multiset ℕ), a ×ˢ b = a.product b
f : TruncatedWittVector p (n + 1) k → TruncatedWittVector p (n + 1) k → k :=
  fun x y => f₀ fun a => Subtype.casesOn a fun a ha => Function.uncurry ![x, y] (a.1, ⟨a.2, ⋯⟩)
x y : 𝕎 k
i : Fin 2
m : ℕ
ha : (i, m) ∈ fun a => a ∈ univ.val ×ˢ Multiset.range (n + 1)
⊢ ![truncateFun (n + 1) x, truncateFun (n + 1) y] (↑⟨(i, m), ha⟩).1 ⟨(↑⟨(i, m), ha⟩).2, ⋯⟩ =
    (Function.uncurry ![fun i => x.coeff i, fun i => y.coeff i] ∘ Subtype.val) ⟨(i, m), ha⟩
state_after: no goals
tactic: fin_cases i <;> rfl