/-- The **Chevalley–Warning theorem**, finitary version.
Let `(f i)` be a finite family of multivariate polynomials
in finitely many variables (`X s`, `s : σ`) over a finite field of characteristic `p`.
Assume that the sum of the total degrees of the `f i` is less than the cardinality of `σ`.
Then the number of common solutions of the `f i` is divisible by `p`. -/
theorem char_dvd_card_solutions_of_sum_lt {s : Finset ι} {f : ι → MvPolynomial σ K}
    (h : (∑ i in s, (f i).totalDegree) < Fintype.card σ) :
    p ∣ Fintype.card { x : σ → K // ∀ i ∈ s, eval x (f i) = 0 } := by
  have hq : 0 < q - 1 := by rw [← Fintype.card_units, Fintype.card_pos_iff]; exact ⟨1⟩
  let S : Finset (σ → K) := { x ∈ univ | ∀ i ∈ s, eval x (f i) = 0 }.toFinset
  have hS : ∀ x : σ → K, x ∈ S ↔ ∀ i : ι, i ∈ s → eval x (f i) = 0 := by
    intro x
    simp only [Set.toFinset_setOf, mem_univ, true_and, mem_filter]
  /- The polynomial `F = ∏ i in s, (1 - (f i)^(q - 1))` has the nice property
    that it takes the value `1` on elements of `{x : σ → K // ∀ i ∈ s, (f i).eval x = 0}`
    while it is `0` outside that locus.
    Hence the sum of its values is equal to the cardinality of
    `{x : σ → K // ∀ i ∈ s, (f i).eval x = 0}` modulo `p`. -/
  let F : MvPolynomial σ K := ∏ i in s, (1 - f i ^ (q - 1))
  have hF : ∀ x, eval x F = if x ∈ S then 1 else 0 := by
    intro x
    calc
      eval x F = ∏ i in s, eval x (1 - f i ^ (q - 1)) := eval_prod s _ x
      _ = if x ∈ S then 1 else 0 := ?_
    simp only [(eval x).map_sub, (eval x).map_pow, (eval x).map_one]
    split_ifs with hx
    · apply Finset.prod_eq_one
      intro i hi
      rw [hS] at hx
      rw [hx i hi, zero_pow hq, sub_zero]
    · obtain ⟨i, hi, hx⟩ : ∃ i : ι, i ∈ s ∧ eval x (f i) ≠ 0 := by
        simpa only [hS, not_forall, not_imp] using hx
      apply Finset.prod_eq_zero hi
      rw [pow_card_sub_one_eq_one (eval x (f i)) hx, sub_self]
  -- In particular, we can now show:
  have key : ∑ x, eval x F = Fintype.card { x : σ → K // ∀ i ∈ s, eval x (f i) = 0 }
  rw [Fintype.card_of_subtype S hS, card_eq_sum_ones, Nat.cast_sum, Nat.cast_one, ←
    Fintype.sum_extend_by_zero S, sum_congr rfl fun x _ => hF x]
  -- With these preparations under our belt, we will approach the main goal.
  show p ∣ Fintype.card { x // ∀ i : ι, i ∈ s → eval x (f i) = 0 }
  rw [← CharP.cast_eq_zero_iff K, ← key]
  show (∑ x, eval x F) = 0
  -- We are now ready to apply the main machine, proven before.
  apply F.sum_eval_eq_zero
  -- It remains to verify the crucial assumption of this machine
  show F.totalDegree < (q - 1) * Fintype.card σ
  calc
    F.totalDegree ≤ ∑ i in s, (1 - f i ^ (q - 1)).totalDegree := totalDegree_finset_prod s _
    _ ≤ ∑ i in s, (q - 1) * (f i).totalDegree := (sum_le_sum fun i _ => ?_)
    -- see ↓
    _ = (q - 1) * ∑ i in s, (f i).totalDegree := mul_sum.symm
    _ < (q - 1) * Fintype.card σ := by rwa [mul_lt_mul_left hq]
  -- Now we prove the remaining step from the preceding calculation
  show (1 - f i ^ (q - 1)).totalDegree ≤ (q - 1) * (f i).totalDegree
  

Proof state:
K : Type u_1
σ : Type u_2
ι : Type u_3
inst✝⁵ : Fintype K
inst✝⁴ : Field K
inst✝³ : Fintype σ
inst✝² : DecidableEq σ
inst✝¹ : DecidableEq K
p : ℕ
inst✝ : CharP K p
s : Finset ι
f : ι → MvPolynomial σ K
h : ∑ i in s, totalDegree (f i) < Fintype.card σ
hq : 0 < q - 1
S : Finset (σ → K) := Set.toFinset {x | x ∈ univ ∧ ∀ i ∈ s, (eval x) (f i) = 0}
hS : ∀ (x : σ → K), x ∈ S ↔ ∀ i ∈ s, (eval x) (f i) = 0
F : MvPolynomial σ K := ∏ i in s, (1 - f i ^ (q - 1))
hF : ∀ (x : σ → K), (eval x) F = if x ∈ S then 1 else 0
key : ∑ x : σ → K, (eval x) F = ↑(Fintype.card { x // ∀ i ∈ s, (eval x) (f i) = 0 })
i : ι
x✝ : i ∈ s
⊢ totalDegree (1 - f i ^ (q - 1)) ≤ (q - 1) * totalDegree (f i)

Next tactic:
calc
    (1 - f i ^ (q - 1)).totalDegree ≤
        max (1 : MvPolynomial σ K).totalDegree (f i ^ (q - 1)).totalDegree := totalDegree_sub _ _
    _ ≤ (f i ^ (q - 1)).totalDegree := by simp
    _ ≤ (q - 1) * (f i).totalDegree := totalDegree_pow _ _