/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin

! This file was ported from Lean 3 source module data.polynomial.hasse_deriv
! leanprover-community/mathlib commit 10bf4f825ad729c5653adc039dafa3622e7f93c9
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
import Mathbin.Algebra.Polynomial.BigOperators
import Mathbin.Data.Nat.Choose.Cast
import Mathbin.Data.Nat.Choose.Vandermonde
import Mathbin.Data.Polynomial.Derivative

/-!
# Hasse derivative of polynomials

> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.

The `k`th Hasse derivative of a polynomial `∑ a_i X^i` is `∑ (i.choose k) a_i X^(i-k)`.
It is a variant of the usual derivative, and satisfies `k! * (hasse_deriv k f) = derivative^[k] f`.
The main benefit is that is gives an atomic way of talking about expressions such as
`(derivative^[k] f).eval r / k!`, that occur in Taylor expansions, for example.

## Main declarations

In the following, we write `D k` for the `k`-th Hasse derivative `hasse_deriv k`.

* `polynomial.hasse_deriv`: the `k`-th Hasse derivative of a polynomial
* `polynomial.hasse_deriv_zero`: the `0`th Hasse derivative is the identity
* `polynomial.hasse_deriv_one`: the `1`st Hasse derivative is the usual derivative
* `polynomial.factorial_smul_hasse_deriv`: the identity `k! • (D k f) = derivative^[k] f`
* `polynomial.hasse_deriv_comp`: the identity `(D k).comp (D l) = (k+l).choose k • D (k+l)`
* `polynomial.hasse_deriv_mul`:
  the "Leibniz rule" `D k (f * g) = ∑ ij in antidiagonal k, D ij.1 f * D ij.2 g`

For the identity principle, see `polynomial.eq_zero_of_hasse_deriv_eq_zero`
in `data/polynomial/taylor.lean`.

## Reference

https://math.fontein.de/2009/08/12/the-hasse-derivative/

-/

noncomputable section

namespace Polynomial

open Nat BigOperators Polynomial

open Function

open Nat hiding nsmul_eq_mul

variable {R : Type _} [Semiring R] (k : ℕ) (f : R[X])

#print Polynomial.hasseDeriv /-
/-- The `k`th Hasse derivative of a polynomial `∑ a_i X^i` is `∑ (i.choose k) a_i X^(i-k)`.
It satisfies `k! * (hasse_deriv k f) = derivative^[k] f`. -/
def hasseDeriv (k : ℕ) : R[X] →ₗ[R] R[X] :=
  lsum fun i => monomial (i - k) ∘ₗ DistribMulAction.toLinearMap R R (i.choose k)
#align polynomial.hasse_deriv Polynomial.hasseDeriv
-/

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Case conversion may be inaccurate. Consider using '#align polynomial.hasse_deriv_apply Polynomial.hasseDeriv_applyₓ'. -/
theorem hasseDeriv_apply : hasseDeriv k f = f.Sum fun i r => monomial (i - k) (↑(i.choose k) * r) :=
  by simpa only [← nsmul_eq_mul]
#align polynomial.hasse_deriv_apply Polynomial.hasseDeriv_apply

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Case conversion may be inaccurate. Consider using '#align polynomial.hasse_deriv_coeff Polynomial.hasseDeriv_coeffₓ'. -/
theorem hasseDeriv_coeff (n : ℕ) : (hasseDeriv k f).coeff n = (n + k).choose k * f.coeff (n + k) :=
  by
  rw [hasse_deriv_apply, coeff_sum, sum_def, Finset.sum_eq_single (n + k), coeff_monomial]
  · simp only [if_true, add_tsub_cancel_right, eq_self_iff_true]
  · intro i hi hink
    rw [coeff_monomial]
    by_cases hik : i < k
    · simp only [Nat.choose_eq_zero_of_lt hik, if_t_t, Nat.cast_zero, MulZeroClass.zero_mul]
    · push_neg  at hik
      rw [if_neg]
      contrapose! hink
      exact (tsub_eq_iff_eq_add_of_le hik).mp hink
  · intro h
    simp only [not_mem_support_iff.mp h, monomial_zero_right, MulZeroClass.mul_zero, coeff_zero]
#align polynomial.hasse_deriv_coeff Polynomial.hasseDeriv_coeff

/- warning: polynomial.hasse_deriv_zero' -> Polynomial.hasseDeriv_zero' is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.hasse_deriv_zero' Polynomial.hasseDeriv_zero'ₓ'. -/
theorem hasseDeriv_zero' : hasseDeriv 0 f = f := by
  simp only [hasse_deriv_apply, tsub_zero, Nat.choose_zero_right, Nat.cast_one, one_mul,
    sum_monomial_eq]
#align polynomial.hasse_deriv_zero' Polynomial.hasseDeriv_zero'

#print Polynomial.hasseDeriv_zero /-
@[simp]
theorem hasseDeriv_zero : @hasseDeriv R _ 0 = LinearMap.id :=
  LinearMap.ext <| hasseDeriv_zero'
#align polynomial.hasse_deriv_zero Polynomial.hasseDeriv_zero
-/

/- warning: polynomial.hasse_deriv_eq_zero_of_lt_nat_degree -> Polynomial.hasseDeriv_eq_zero_of_lt_natDegree is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.hasse_deriv_eq_zero_of_lt_nat_degree Polynomial.hasseDeriv_eq_zero_of_lt_natDegreeₓ'. -/
theorem hasseDeriv_eq_zero_of_lt_natDegree (p : R[X]) (n : ℕ) (h : p.natDegree < n) :
    hasseDeriv n p = 0 := by
  rw [hasse_deriv_apply, sum_def]
  refine' Finset.sum_eq_zero fun x hx => _
  simp [Nat.choose_eq_zero_of_lt ((le_nat_degree_of_mem_supp _ hx).trans_lt h)]
#align polynomial.hasse_deriv_eq_zero_of_lt_nat_degree Polynomial.hasseDeriv_eq_zero_of_lt_natDegree

/- warning: polynomial.hasse_deriv_one' -> Polynomial.hasseDeriv_one' is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.hasse_deriv_one' Polynomial.hasseDeriv_one'ₓ'. -/
theorem hasseDeriv_one' : hasseDeriv 1 f = derivative f := by
  simp only [hasse_deriv_apply, derivative_apply, ← C_mul_X_pow_eq_monomial, Nat.choose_one_right,
    (Nat.cast_commute _ _).Eq]
#align polynomial.hasse_deriv_one' Polynomial.hasseDeriv_one'

#print Polynomial.hasseDeriv_one /-
@[simp]
theorem hasseDeriv_one : @hasseDeriv R _ 1 = derivative :=
  LinearMap.ext <| hasseDeriv_one'
#align polynomial.hasse_deriv_one Polynomial.hasseDeriv_one
-/

/- warning: polynomial.hasse_deriv_monomial -> Polynomial.hasseDeriv_monomial is a dubious translation:
lean 3 declaration is
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Case conversion may be inaccurate. Consider using '#align polynomial.hasse_deriv_monomial Polynomial.hasseDeriv_monomialₓ'. -/
@[simp]
theorem hasseDeriv_monomial (n : ℕ) (r : R) :
    hasseDeriv k (monomial n r) = monomial (n - k) (↑(n.choose k) * r) :=
  by
  ext i
  simp only [hasse_deriv_coeff, coeff_monomial]
  by_cases hnik : n = i + k
  · rw [if_pos hnik, if_pos, ← hnik]
    apply tsub_eq_of_eq_add_rev
    rwa [add_comm]
  · rw [if_neg hnik, MulZeroClass.mul_zero]
    by_cases hkn : k ≤ n
    · rw [← tsub_eq_iff_eq_add_of_le hkn] at hnik
      rw [if_neg hnik]
    · push_neg  at hkn
      rw [Nat.choose_eq_zero_of_lt hkn, Nat.cast_zero, MulZeroClass.zero_mul, if_t_t]
#align polynomial.hasse_deriv_monomial Polynomial.hasseDeriv_monomial

/- warning: polynomial.hasse_deriv_C -> Polynomial.hasseDeriv_C is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (k : Nat) (r : R), (LT.lt.{0} Nat Nat.hasLt (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))) k) -> (Eq.{succ u1} (Polynomial.{u1} R _inst_1) (coeFn.{succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) (fun (_x : LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) => (Polynomial.{u1} R _inst_1) -> (Polynomial.{u1} R _inst_1)) (LinearMap.hasCoeToFun.{u1, u1, u1, u1} R R (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.hasseDeriv.{u1} R _inst_1 k) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (Polynomial.C.{u1} R _inst_1) r)) (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (OfNat.mk.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.zero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1)))))
but is expected to have type
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Case conversion may be inaccurate. Consider using '#align polynomial.hasse_deriv_C Polynomial.hasseDeriv_Cₓ'. -/
theorem hasseDeriv_C (r : R) (hk : 0 < k) : hasseDeriv k (C r) = 0 := by
  rw [← monomial_zero_left, hasse_deriv_monomial, Nat.choose_eq_zero_of_lt hk, Nat.cast_zero,
    MulZeroClass.zero_mul, monomial_zero_right]
#align polynomial.hasse_deriv_C Polynomial.hasseDeriv_C

/- warning: polynomial.hasse_deriv_apply_one -> Polynomial.hasseDeriv_apply_one is a dubious translation:
lean 3 declaration is
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Case conversion may be inaccurate. Consider using '#align polynomial.hasse_deriv_apply_one Polynomial.hasseDeriv_apply_oneₓ'. -/
theorem hasseDeriv_apply_one (hk : 0 < k) : hasseDeriv k (1 : R[X]) = 0 := by
  rw [← C_1, hasse_deriv_C k _ hk]
#align polynomial.hasse_deriv_apply_one Polynomial.hasseDeriv_apply_one

/- warning: polynomial.hasse_deriv_X -> Polynomial.hasseDeriv_X is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.hasse_deriv_X Polynomial.hasseDeriv_Xₓ'. -/
theorem hasseDeriv_X (hk : 1 < k) : hasseDeriv k (X : R[X]) = 0 := by
  rw [← monomial_one_one_eq_X, hasse_deriv_monomial, Nat.choose_eq_zero_of_lt hk, Nat.cast_zero,
    MulZeroClass.zero_mul, monomial_zero_right]
#align polynomial.hasse_deriv_X Polynomial.hasseDeriv_X

/- warning: polynomial.factorial_smul_hasse_deriv -> Polynomial.factorial_smul_hasseDeriv is a dubious translation:
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but is expected to have type
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(NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) (Module.End.semiring.{u1, u1} R (Polynomial.{u1} R _inst_1) _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))))))))) (Nat.factorial k) (Polynomial.hasseDeriv.{u1} R _inst_1 k))) (Nat.iterate.{succ u1} (Polynomial.{u1} R _inst_1) (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) (Polynomial.{u1} R _inst_1) (fun (_x : Polynomial.{u1} R _inst_1) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : Polynomial.{u1} R _inst_1) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.derivative.{u1} R _inst_1)) k)
Case conversion may be inaccurate. Consider using '#align polynomial.factorial_smul_hasse_deriv Polynomial.factorial_smul_hasseDerivₓ'. -/
theorem factorial_smul_hasseDeriv : ⇑(k ! • @hasseDeriv R _ k) = @derivative R _^[k] :=
  by
  induction' k with k ih
  · rw [hasse_deriv_zero, factorial_zero, iterate_zero, one_smul, LinearMap.id_coe]
  ext (f n) : 2
  rw [iterate_succ_apply', ← ih]
  simp only [LinearMap.smul_apply, coeff_smul, LinearMap.map_smul_of_tower, coeff_derivative,
    hasse_deriv_coeff, ← @choose_symm_add _ k]
  simp only [nsmul_eq_mul, factorial_succ, mul_assoc, succ_eq_add_one, ← add_assoc,
    add_right_comm n 1 k, ← cast_succ]
  rw [← (cast_commute (n + 1) (f.coeff (n + k + 1))).Eq]
  simp only [← mul_assoc]
  norm_cast
  congr 2
  apply @cast_injective ℚ
  have h1 : n + 1 ≤ n + k + 1 := succ_le_succ le_self_add
  have h2 : k + 1 ≤ n + k + 1 := succ_le_succ le_add_self
  have H : ∀ n : ℕ, (n ! : ℚ) ≠ 0 := by exact_mod_cast factorial_ne_zero
  -- why can't `field_simp` help me here?
  simp only [cast_mul, cast_choose ℚ, h1, h2, -one_div, -mul_eq_zero, succ_sub_succ_eq_sub,
    add_tsub_cancel_right, add_tsub_cancel_left, field_simps]
  rw [eq_div_iff_mul_eq (mul_ne_zero (H _) (H _)), eq_comm, div_mul_eq_mul_div,
    eq_div_iff_mul_eq (mul_ne_zero (H _) (H _))]
  norm_cast
  simp only [factorial_succ, succ_eq_add_one]
  ring
#align polynomial.factorial_smul_hasse_deriv Polynomial.factorial_smul_hasseDeriv

#print Polynomial.hasseDeriv_comp /-
theorem hasseDeriv_comp (k l : ℕ) :
    (@hasseDeriv R _ k).comp (hasseDeriv l) = (k + l).choose k • hasseDeriv (k + l) :=
  by
  ext i : 2
  simp only [LinearMap.smul_apply, comp_app, LinearMap.coe_comp, smul_monomial, hasse_deriv_apply,
    mul_one, monomial_eq_zero_iff, sum_monomial_index, MulZeroClass.mul_zero, ←
    tsub_add_eq_tsub_tsub, add_comm l k]
  rw_mod_cast [nsmul_eq_mul]
  congr 2
  by_cases hikl : i < k + l
  · rw [choose_eq_zero_of_lt hikl, MulZeroClass.mul_zero]
    by_cases hil : i < l
    · rw [choose_eq_zero_of_lt hil, MulZeroClass.mul_zero]
    · push_neg  at hil
      rw [← tsub_lt_iff_right hil] at hikl
      rw [choose_eq_zero_of_lt hikl, MulZeroClass.zero_mul]
  push_neg  at hikl
  apply @cast_injective ℚ
  have h1 : l ≤ i := le_of_add_le_right hikl
  have h2 : k ≤ i - l := le_tsub_of_add_le_right hikl
  have h3 : k ≤ k + l := le_self_add
  have H : ∀ n : ℕ, (n ! : ℚ) ≠ 0 := by exact_mod_cast factorial_ne_zero
  -- why can't `field_simp` help me here?
  simp only [cast_mul, cast_choose ℚ, h1, h2, h3, hikl, -one_div, -mul_eq_zero,
    succ_sub_succ_eq_sub, add_tsub_cancel_right, add_tsub_cancel_left, field_simps]
  rw [eq_div_iff_mul_eq, eq_comm, div_mul_eq_mul_div, eq_div_iff_mul_eq, ← tsub_add_eq_tsub_tsub,
    add_comm l k]
  · ring
  all_goals apply_rules [mul_ne_zero, H]
#align polynomial.hasse_deriv_comp Polynomial.hasseDeriv_comp
-/

/- warning: polynomial.nat_degree_hasse_deriv_le -> Polynomial.natDegree_hasseDeriv_le is a dubious translation:
lean 3 declaration is
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but is expected to have type
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (p : Polynomial.{u1} R _inst_1) (n : Nat), LE.le.{0} Nat instLENat (Polynomial.natDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) (Polynomial.{u1} R _inst_1) (fun (_x : Polynomial.{u1} R _inst_1) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : Polynomial.{u1} R _inst_1) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.hasseDeriv.{u1} R _inst_1 n) p)) (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat instSubNat) (Polynomial.natDegree.{u1} R _inst_1 p) n)
Case conversion may be inaccurate. Consider using '#align polynomial.nat_degree_hasse_deriv_le Polynomial.natDegree_hasseDeriv_leₓ'. -/
theorem natDegree_hasseDeriv_le (p : R[X]) (n : ℕ) : natDegree (hasseDeriv n p) ≤ natDegree p - n :=
  by
  classical
    rw [hasse_deriv_apply, sum_def]
    refine' (nat_degree_sum_le _ _).trans _
    simp_rw [Function.comp, nat_degree_monomial]
    rw [Finset.fold_ite, Finset.fold_const]
    · simp only [if_t_t, max_eq_right, zero_le', Finset.fold_max_le, true_and_iff, and_imp,
        tsub_le_iff_right, mem_support_iff, Ne.def, Finset.mem_filter]
      intro x hx hx'
      have hxp : x ≤ p.nat_degree := le_nat_degree_of_ne_zero hx
      have hxn : n ≤ x := by
        contrapose! hx'
        simp [Nat.choose_eq_zero_of_lt hx']
      rwa [tsub_add_cancel_of_le (hxn.trans hxp)]
    · simp
#align polynomial.nat_degree_hasse_deriv_le Polynomial.natDegree_hasseDeriv_le

/- warning: polynomial.nat_degree_hasse_deriv -> Polynomial.natDegree_hasseDeriv is a dubious translation:
lean 3 declaration is
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  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] [_inst_2 : NoZeroSMulDivisors.{0, u1} Nat R (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero) (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (AddMonoid.SMul.{u1} R (AddMonoidWithOne.toAddMonoid.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))))] (p : Polynomial.{u1} R _inst_1) (n : Nat), Eq.{1} Nat (Polynomial.natDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) (Polynomial.{u1} R _inst_1) (fun (_x : Polynomial.{u1} R _inst_1) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : Polynomial.{u1} R _inst_1) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.hasseDeriv.{u1} R _inst_1 n) p)) (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat instSubNat) (Polynomial.natDegree.{u1} R _inst_1 p) n)
Case conversion may be inaccurate. Consider using '#align polynomial.nat_degree_hasse_deriv Polynomial.natDegree_hasseDerivₓ'. -/
theorem natDegree_hasseDeriv [NoZeroSMulDivisors ℕ R] (p : R[X]) (n : ℕ) :
    natDegree (hasseDeriv n p) = natDegree p - n :=
  by
  cases' lt_or_le p.nat_degree n with hn hn
  · simpa [hasse_deriv_eq_zero_of_lt_nat_degree, hn] using (tsub_eq_zero_of_le hn.le).symm
  · refine' map_nat_degree_eq_sub _ _
    · exact fun h => hasse_deriv_eq_zero_of_lt_nat_degree _ _
    ·
      classical
        simp only [ite_eq_right_iff, Ne.def, nat_degree_monomial, hasse_deriv_monomial]
        intro k c c0 hh
        -- this is where we use the `smul_eq_zero` from `no_zero_smul_divisors`
        rw [← nsmul_eq_mul, smul_eq_zero, Nat.choose_eq_zero_iff] at hh
        exact (tsub_eq_zero_of_le (Or.resolve_right hh c0).le).symm
#align polynomial.nat_degree_hasse_deriv Polynomial.natDegree_hasseDeriv

section

open AddMonoidHom Finset.Nat

/- warning: polynomial.hasse_deriv_mul -> Polynomial.hasseDeriv_mul is a dubious translation:
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(Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.hasseDeriv.{u1} R _inst_1 (Prod.snd.{0, 0} Nat Nat ij)) g)))
but is expected to have type
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (k : Nat) (f : Polynomial.{u1} R _inst_1) (g : Polynomial.{u1} R _inst_1), Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : Polynomial.{u1} R _inst_1) => Polynomial.{u1} R _inst_1) (HMul.hMul.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHMul.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.mul'.{u1} R _inst_1)) f g)) (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) (Polynomial.{u1} R _inst_1) (fun (_x : Polynomial.{u1} R _inst_1) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : Polynomial.{u1} R _inst_1) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.hasseDeriv.{u1} R _inst_1 k) (HMul.hMul.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHMul.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.mul'.{u1} R _inst_1)) f g)) (Finset.sum.{u1, 0} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : Polynomial.{u1} R _inst_1) => Polynomial.{u1} R _inst_1) f) (Prod.{0, 0} Nat Nat) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : Polynomial.{u1} R _inst_1) => Polynomial.{u1} R _inst_1) f) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : Polynomial.{u1} R _inst_1) => Polynomial.{u1} R _inst_1) f) (Semiring.toNonAssocSemiring.{u1} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : Polynomial.{u1} R _inst_1) => Polynomial.{u1} R _inst_1) f) (Polynomial.semiring.{u1} R _inst_1)))) (Finset.Nat.antidiagonal k) (fun (ij : Prod.{0, 0} Nat Nat) => HMul.hMul.{u1, u1, u1} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : Polynomial.{u1} R _inst_1) => Polynomial.{u1} R _inst_1) f) ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : Polynomial.{u1} R _inst_1) => Polynomial.{u1} R _inst_1) g) ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : Polynomial.{u1} R _inst_1) => Polynomial.{u1} R _inst_1) f) (instHMul.{u1} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : Polynomial.{u1} R _inst_1) => Polynomial.{u1} R _inst_1) f) (Polynomial.mul'.{u1} R _inst_1)) (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) (Polynomial.{u1} R _inst_1) (fun (_x : Polynomial.{u1} R _inst_1) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : Polynomial.{u1} R _inst_1) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.hasseDeriv.{u1} R _inst_1 (Prod.fst.{0, 0} Nat Nat ij)) f) (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) (Polynomial.{u1} R _inst_1) (fun (_x : Polynomial.{u1} R _inst_1) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : Polynomial.{u1} R _inst_1) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.hasseDeriv.{u1} R _inst_1 (Prod.snd.{0, 0} Nat Nat ij)) g)))
Case conversion may be inaccurate. Consider using '#align polynomial.hasse_deriv_mul Polynomial.hasseDeriv_mulₓ'. -/
theorem hasseDeriv_mul (f g : R[X]) :
    hasseDeriv k (f * g) = ∑ ij in antidiagonal k, hasseDeriv ij.1 f * hasseDeriv ij.2 g :=
  by
  let D k := (@hasse_deriv R _ k).toAddMonoidHom
  let Φ := @AddMonoidHom.mul R[X] _
  show
    (comp_hom (D k)).comp Φ f g =
      ∑ ij : ℕ × ℕ in antidiagonal k, ((comp_hom.comp ((comp_hom Φ) (D ij.1))).flip (D ij.2) f) g
  simp only [← finset_sum_apply]
  congr 2
  clear f g
  ext (m r n s) : 4
  simp only [finset_sum_apply, coe_mul_left, coe_comp, flip_apply, comp_app, hasse_deriv_monomial,
    LinearMap.toAddMonoidHom_coe, comp_hom_apply_apply, coe_mul, monomial_mul_monomial]
  have aux :
    ∀ x : ℕ × ℕ,
      x ∈ antidiagonal k →
        monomial (m - x.1 + (n - x.2)) (↑(m.choose x.1) * r * (↑(n.choose x.2) * s)) =
          monomial (m + n - k) (↑(m.choose x.1) * ↑(n.choose x.2) * (r * s)) :=
    by
    intro x hx
    rw [Finset.Nat.mem_antidiagonal] at hx
    subst hx
    by_cases hm : m < x.1
    ·
      simp only [Nat.choose_eq_zero_of_lt hm, Nat.cast_zero, MulZeroClass.zero_mul,
        monomial_zero_right]
    by_cases hn : n < x.2
    ·
      simp only [Nat.choose_eq_zero_of_lt hn, Nat.cast_zero, MulZeroClass.zero_mul,
        MulZeroClass.mul_zero, monomial_zero_right]
    push_neg  at hm hn
    rw [tsub_add_eq_add_tsub hm, ← add_tsub_assoc_of_le hn, ← tsub_add_eq_tsub_tsub,
      add_comm x.2 x.1, mul_assoc, ← mul_assoc r, ← (Nat.cast_commute _ r).Eq, mul_assoc, mul_assoc]
  conv_rhs =>
    apply_congr
    skip
    rw [aux _ H]
  rw_mod_cast [← LinearMap.map_sum, ← Finset.sum_mul, ← Nat.add_choose_eq]
#align polynomial.hasse_deriv_mul Polynomial.hasseDeriv_mul

end

end Polynomial