-- Mathlib/Algebra/MvPolynomial/SchwartzZippel.lean
/-- The **Schwartz-Zippel lemma**

For a nonzero multivariable polynomial `p` over an integral domain, the probability that `p`
evaluates to zero at points drawn at random from a product of finite subsets `S i` of the integral
domain is bounded by the supremum of `∑ i, degᵢ s / #(S i)` ranging over monomials `s` of `p`. -/
lemma schwartz_zippel_sup_sum :
    ∀ {n} {p : MvPolynomial (Fin n) R} (hp : p ≠ 0) (S : Fin n → Finset R),
      #{x ∈ S ^^ n | eval x p = 0} / ∏ i, (#(S i) : ℚ≥0) ≤
        p.support.sup fun s ↦ ∑ i, (s i / #(S i) : ℚ≥0)
  | 0, p, hp, S => by
    -- Because `p` is a polynomial over zero variables, it is constant.
    rw [p.eq_C_of_isEmpty] at *
    simp [C_ne_zero.mp hp]
    -- Now, assume that the theorem holds for all polynomials in `n` variables.
  | n + 1, p, hp, S => by
    -- We can consider `p` to be a polynomial over multivariable polynomials in one fewer variables.
    set p' : Polynomial (MvPolynomial (Fin n) R) := finSuccEquiv R n p with hp'
    -- Since `p` is not identically zero, there is some `k` such that `pₖ` is not identically zero.
    -- WLOG `k` is the largest such.
    set k := p'.natDegree with hk
    set pₖ := p'.leadingCoeff with hpₖ
    have hp'₀ : p' ≠ 0 := EmbeddingLike.map_ne_zero_iff.2 hp
    have hpₖ₀ : pₖ ≠ 0

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DecidableEq R
n : ℕ
p : MvPolynomial (Fin (n + 1)) R
hp : p ≠ 0
S : Fin (n + 1) → Finset R
p' : Polynomial (MvPolynomial (Fin n) R) := (finSuccEquiv R n) p
hp' : p' = (finSuccEquiv R n) p
k : ℕ := p'.natDegree
hk : k = p'.natDegree
pₖ : MvPolynomial (Fin n) R := p'.leadingCoeff
hpₖ : pₖ = p'.leadingCoeff
hp'₀ : p' ≠ 0
hpₖ₀ : pₖ ≠ 0
⊢ ∑ xₜ ∈ filter (fun xₜ => (eval xₜ) pₖ ≠ 0) (piFinset fun i => tail S i), degreeOf 0 p ≤
    ∑ _xₜ ∈ piFinset fun i => tail S i, degreeOf 0 p

Tactic:
gcongr

Result:
case h
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DecidableEq R
n : ℕ
p : MvPolynomial (Fin (n + 1)) R
hp : p ≠ 0
S : Fin (n + 1) → Finset R
p' : Polynomial (MvPolynomial (Fin n) R) := (finSuccEquiv R n) p
hp' : p' = (finSuccEquiv R n) p
k : ℕ := p'.natDegree
hk : k = p'.natDegree
pₖ : MvPolynomial (Fin n) R := p'.leadingCoeff
hpₖ : pₖ = p'.leadingCoeff
hp'₀ : p' ≠ 0
hpₖ₀ : pₖ ≠ 0
⊢ filter (fun xₜ => (eval xₜ) pₖ ≠ 0) (piFinset fun i => tail S i) ⊆ piFinset fun i => tail S i

Proof state:
case calc_2
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DecidableEq R
n : ℕ
p : MvPolynomial (Fin (n + 1)) R
hp : p ≠ 0
S : Fin (n + 1) → Finset R
p' : Polynomial (MvPolynomial (Fin n) R) := (finSuccEquiv R n) p
hp' : p' = (finSuccEquiv R n) p
k : ℕ := p'.natDegree
hk : k = p'.natDegree
pₖ : MvPolynomial (Fin n) R := p'.leadingCoeff
hpₖ : pₖ = p'.leadingCoeff
hp'₀ : p' ≠ 0
hpₖ₀ : pₖ ≠ 0
⊢ (pₖ.support.sup fun i => ∑ i_1 : Fin n, ↑(i i_1) / ↑(#(S i_1.succ)) + ↑(degreeOf 0 p) / ↑(#(S 0))) ≤
    p.support.sup fun s => ∑ i : Fin (n + 1), ↑(s i) / ↑(#(S i))

Tactic:
apply Finset.sup_le

Result:
case calc_2.a
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DecidableEq R
n : ℕ
p : MvPolynomial (Fin (n + 1)) R
hp : p ≠ 0
S : Fin (n + 1) → Finset R
p' : Polynomial (MvPolynomial (Fin n) R) := (finSuccEquiv R n) p
hp' : p' = (finSuccEquiv R n) p
k : ℕ := p'.natDegree
hk : k = p'.natDegree
pₖ : MvPolynomial (Fin n) R := p'.leadingCoeff
hpₖ : pₖ = p'.leadingCoeff
hp'₀ : p' ≠ 0
hpₖ₀ : pₖ ≠ 0
⊢ ∀ b ∈ pₖ.support,
    ∑ i : Fin n, ↑(b i) / ↑(#(S i.succ)) + ↑(degreeOf 0 p) / ↑(#(S 0)) ≤
      p.support.sup fun s => ∑ i : Fin (n + 1), ↑(s i) / ↑(#(S i))

Proof state:
case calc_2.a
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DecidableEq R
n : ℕ
p : MvPolynomial (Fin (n + 1)) R
hp : p ≠ 0
S : Fin (n + 1) → Finset R
p' : Polynomial (MvPolynomial (Fin n) R) := (finSuccEquiv R n) p
hp' : p' = (finSuccEquiv R n) p
k : ℕ := p'.natDegree
hk : k = p'.natDegree
pₖ : MvPolynomial (Fin n) R := p'.leadingCoeff
hpₖ : pₖ = p'.leadingCoeff
hp'₀ : p' ≠ 0
hpₖ₀ : pₖ ≠ 0
i : Fin n →₀ ℕ
hi : i ∈ pₖ.support
⊢ ∑ i_1 : Fin n, ↑(i i_1) / ↑(#(S i_1.succ)) + ↑(degreeOf 0 p) / ↑(#(S 0)) ≤
    p.support.sup fun s => ∑ i : Fin (n + 1), ↑(s i) / ↑(#(S i))

Tactic:
refine le_sup_of_le (mem_support_coeff_finSuccEquiv.mp hi) ?_

Result:
case calc_2.a
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DecidableEq R
n : ℕ
p : MvPolynomial (Fin (n + 1)) R
hp : p ≠ 0
S : Fin (n + 1) → Finset R
p' : Polynomial (MvPolynomial (Fin n) R) := (finSuccEquiv R n) p
hp' : p' = (finSuccEquiv R n) p
k : ℕ := p'.natDegree
hk : k = p'.natDegree
pₖ : MvPolynomial (Fin n) R := p'.leadingCoeff
hpₖ : pₖ = p'.leadingCoeff
hp'₀ : p' ≠ 0
hpₖ₀ : pₖ ≠ 0
i : Fin n →₀ ℕ
hi : i ∈ pₖ.support
⊢ ∑ i_1 : Fin n, ↑(i i_1) / ↑(#(S i_1.succ)) + ↑(degreeOf 0 p) / ↑(#(S 0)) ≤
    ∑ i_1 : Fin (n + 1), ↑((Finsupp.cons p'.natDegree i) i_1) / ↑(#(S i_1))

Proof state:
case calc_1.h₂.hab.h
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DecidableEq R
n : ℕ
p : MvPolynomial (Fin (n + 1)) R
hp : p ≠ 0
S : Fin (n + 1) → Finset R
p' : Polynomial (MvPolynomial (Fin n) R) := (finSuccEquiv R n) p
hp' : p' = (finSuccEquiv R n) p
k : ℕ := p'.natDegree
hk : k = p'.natDegree
pₖ : MvPolynomial (Fin n) R := p'.leadingCoeff
hpₖ : pₖ = p'.leadingCoeff
hp'₀ : p' ≠ 0
hpₖ₀ : pₖ ≠ 0
⊢ #(filter (fun x => (eval x) p = 0 ∧ (eval (tail x)) pₖ ≠ 0) (piFinset fun i => S i)) ≤
    degreeOf 0 p * ∏ i : Fin n, #(S i.succ)

Tactic:
calc
  #{x ∈ S ^^ (n + 1) | eval x p = 0 ∧ eval (tail x) pₖ ≠ 0}
    = #{x ∈ S ^^ (n + 1) | eval (tail x) pₖ ≠ 0 ∧ eval x p = 0} := by simp_rw [and_comm]
  _ = #({xₜ ∈ tail S ^^ n | eval xₜ pₖ ≠ 0}.biUnion fun xₜ ↦ image (fun x₀ ↦ (x₀, xₜ))
        {x₀ ∈ S 0 | eval (cons x₀ xₜ) p = 0}) := by
    rw [← filter_filter, filter_piFinset_eq_map_consEquiv S (fun r ↦ eval r pₖ ≠ 0),
      filter_map, card_map, product_eq_biUnion_right, filter_biUnion]
    simp [Function.comp_def, filter_image, filter_filter]
    rfl
  _ ≤ ∑ xₜ ∈ tail S ^^ n with eval xₜ pₖ ≠ 0,
        #(image (fun x₀ ↦ (x₀, xₜ)) {x₀ ∈ S 0 | eval (cons x₀ xₜ) p = 0}) :=
    card_biUnion_le
  _ ≤ ∑ xₜ ∈ tail S ^^ n with eval xₜ pₖ ≠ 0, #{x₀ ∈ S 0 | eval (cons x₀ xₜ) p = 0} := by
    gcongr; exact card_image_le
  _ ≤ ∑ xₜ ∈ tail S ^^ n with eval xₜ pₖ ≠ 0, p.degreeOf 0 := ?_
  _ ≤ ∑ _xₜ ∈ tail S ^^ n, p.degreeOf 0 := by gcongr; exact filter_subset ..
  _ = p.degreeOf 0 * ∏ i, #(S (.succ i)) := by simp [mul_comm, tail]

Result:
case calc_1.h₂.hab.h
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DecidableEq R
n : ℕ
p : MvPolynomial (Fin (n + 1)) R
hp : p ≠ 0
S : Fin (n + 1) → Finset R
p' : Polynomial (MvPolynomial (Fin n) R) := (finSuccEquiv R n) p
hp' : p' = (finSuccEquiv R n) p
k : ℕ := p'.natDegree
hk : k = p'.natDegree
pₖ : MvPolynomial (Fin n) R := p'.leadingCoeff
hpₖ : pₖ = p'.leadingCoeff
hp'₀ : p' ≠ 0
hpₖ₀ : pₖ ≠ 0
⊢ ∑ xₜ ∈ filter (fun xₜ => (eval xₜ) pₖ ≠ 0) (piFinset fun i => tail S i),
      #(filter (fun x₀ => (eval (Fin.cons x₀ xₜ)) p = 0) (S 0)) ≤
    ∑ xₜ ∈ filter (fun xₜ => (eval xₜ) pₖ ≠ 0) (piFinset fun i => tail S i), degreeOf 0 p