l3lab/ntp-mathlib huggingface dataset:
theorem factorial_smul_hasseDeriv : ⇑(k ! • @hasseDeriv R _ k) = (@derivative R _)^[k] := by
  induction' k with k ih
  · rw [hasseDeriv_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,
    hasseDeriv_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
  

Proof state:
case succ.h.a.e_a.e_a
R : Type u_1
inst✝ : Semiring R
k✝ : ℕ
f✝ : R[X]
k : ℕ
ih : ⇑(k ! • hasseDeriv k) = (⇑derivative)^[k]
f : R[X]
n : ℕ
⊢ (k + 1) * k ! * choose (n + k + 1) (k + 1) = k ! * choose (n + k + 1) (n + 1) * (n + 1)

Next tactic:
rw [mul_comm (k+1) _, mul_assoc, mul_assoc]
theorem factorial_smul_hasseDeriv : ⇑(k ! • @hasseDeriv R _ k) = (@derivative R _)^[k] := by
  induction' k with k ih
  · rw [hasseDeriv_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,
    hasseDeriv_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
  rw [mul_comm (k+1) _, mul_assoc, mul_assoc]
  

Proof state:
case succ.h.a.e_a.e_a
R : Type u_1
inst✝ : Semiring R
k✝ : ℕ
f✝ : R[X]
k : ℕ
ih : ⇑(k ! • hasseDeriv k) = (⇑derivative)^[k]
f : R[X]
n : ℕ
⊢ k ! * ((k + 1) * choose (n + k + 1) (k + 1)) = k ! * (choose (n + k + 1) (n + 1) * (n + 1))

Next tactic:
congr 1
theorem factorial_smul_hasseDeriv : ⇑(k ! • @hasseDeriv R _ k) = (@derivative R _)^[k] := by
  induction' k with k ih
  · rw [hasseDeriv_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,
    hasseDeriv_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
  rw [mul_comm (k+1) _, mul_assoc, mul_assoc]
  congr 1
  

Proof state:
case succ.h.a.e_a.e_a.e_a
R : Type u_1
inst✝ : Semiring R
k✝ : ℕ
f✝ : R[X]
k : ℕ
ih : ⇑(k ! • hasseDeriv k) = (⇑derivative)^[k]
f : R[X]
n : ℕ
⊢ (k + 1) * choose (n + k + 1) (k + 1) = choose (n + k + 1) (n + 1) * (n + 1)

Next tactic:
have : n + k + 1 = n + (k + 1) := by apply add_assoc
theorem factorial_smul_hasseDeriv : ⇑(k ! • @hasseDeriv R _ k) = (@derivative R _)^[k] := by
  induction' k with k ih
  · rw [hasseDeriv_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,
    hasseDeriv_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
  rw [mul_comm (k+1) _, mul_assoc, mul_assoc]
  congr 1
  have : n + k + 1 = n + (k + 1) := by 

Proof state:
R : Type u_1
inst✝ : Semiring R
k✝ : ℕ
f✝ : R[X]
k : ℕ
ih : ⇑(k ! • hasseDeriv k) = (⇑derivative)^[k]
f : R[X]
n : ℕ
⊢ n + k + 1 = n + (k + 1)

Next tactic:
apply add_assoc
theorem factorial_smul_hasseDeriv : ⇑(k ! • @hasseDeriv R _ k) = (@derivative R _)^[k] := by
  induction' k with k ih
  · rw [hasseDeriv_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,
    hasseDeriv_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
  rw [mul_comm (k+1) _, mul_assoc, mul_assoc]
  congr 1
  have : n + k + 1 = n + (k + 1) := by apply add_assoc
  

Proof state:
case succ.h.a.e_a.e_a.e_a
R : Type u_1
inst✝ : Semiring R
k✝ : ℕ
f✝ : R[X]
k : ℕ
ih : ⇑(k ! • hasseDeriv k) = (⇑derivative)^[k]
f : R[X]
n : ℕ
this : n + k + 1 = n + (k + 1)
⊢ (k + 1) * choose (n + k + 1) (k + 1) = choose (n + k + 1) (n + 1) * (n + 1)

Next tactic:
rw [← choose_symm_of_eq_add this, choose_succ_right_eq, mul_comm]
theorem factorial_smul_hasseDeriv : ⇑(k ! • @hasseDeriv R _ k) = (@derivative R _)^[k] := by
  induction' k with k ih
  · rw [hasseDeriv_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,
    hasseDeriv_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
  rw [mul_comm (k+1) _, mul_assoc, mul_assoc]
  congr 1
  have : n + k + 1 = n + (k + 1) := by apply add_assoc
  rw [← choose_symm_of_eq_add this, choose_succ_right_eq, mul_comm]
  

Proof state:
case succ.h.a.e_a.e_a.e_a
R : Type u_1
inst✝ : Semiring R
k✝ : ℕ
f✝ : R[X]
k : ℕ
ih : ⇑(k ! • hasseDeriv k) = (⇑derivative)^[k]
f : R[X]
n : ℕ
this : n + k + 1 = n + (k + 1)
⊢ choose (n + k + 1) n * (k + 1) = choose (n + k + 1) n * (n + k + 1 - n)

Next tactic:
congr
theorem factorial_smul_hasseDeriv : ⇑(k ! • @hasseDeriv R _ k) = (@derivative R _)^[k] := by
  induction' k with k ih
  · rw [hasseDeriv_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,
    hasseDeriv_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
  rw [mul_comm (k+1) _, mul_assoc, mul_assoc]
  congr 1
  have : n + k + 1 = n + (k + 1) := by apply add_assoc
  rw [← choose_symm_of_eq_add this, choose_succ_right_eq, mul_comm]
  congr
  

Proof state:
case succ.h.a.e_a.e_a.e_a.e_a
R : Type u_1
inst✝ : Semiring R
k✝ : ℕ
f✝ : R[X]
k : ℕ
ih : ⇑(k ! • hasseDeriv k) = (⇑derivative)^[k]
f : R[X]
n : ℕ
this : n + k + 1 = n + (k + 1)
⊢ k + 1 = n + k + 1 - n

Next tactic:
rw [add_assoc, add_tsub_cancel_left]
theorem hasseDeriv_comp (k l : ℕ) :
    (@hasseDeriv R _ k).comp (hasseDeriv l) = (k + l).choose k • hasseDeriv (k + l) := by
  

Proof state:
R : Type u_1
inst✝ : Semiring R
k✝ : ℕ
f : R[X]
k l : ℕ
⊢ hasseDeriv k ∘ₗ hasseDeriv l = choose (k + l) k • hasseDeriv (k + l)

Next tactic:
ext i : 2
theorem hasseDeriv_comp (k l : ℕ) :
    (@hasseDeriv R _ k).comp (hasseDeriv l) = (k + l).choose k • hasseDeriv (k + l) := by
  ext i : 2
  

Proof state:
case h.h
R : Type u_1
inst✝ : Semiring R
k✝ : ℕ
f : R[X]
k l i : ℕ
⊢ ((hasseDeriv k ∘ₗ hasseDeriv l) ∘ₗ monomial i) 1 = ((choose (k + l) k • hasseDeriv (k + l)) ∘ₗ monomial i) 1

Next tactic:
simp only [LinearMap.smul_apply, comp_apply, LinearMap.coe_comp, smul_monomial, hasseDeriv_apply,
    mul_one, monomial_eq_zero_iff, sum_monomial_index, mul_zero, ←
    tsub_add_eq_tsub_tsub, add_comm l k]
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_apply, LinearMap.coe_comp, smul_monomial, hasseDeriv_apply,
    mul_one, monomial_eq_zero_iff, sum_monomial_index, mul_zero, ←
    tsub_add_eq_tsub_tsub, add_comm l k]
  

Proof state:
case h.h
R : Type u_1
inst✝ : Semiring R
k✝ : ℕ
f : R[X]
k l i : ℕ
⊢ (monomial (i - (k + l))) (↑(choose (i - l) k) * ↑(choose i l)) =
    (monomial (i - (k + l))) (choose (k + l) k • ↑(choose i (k + l)))

Next tactic:
rw_mod_cast [nsmul_eq_mul]
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_apply, LinearMap.coe_comp, smul_monomial, hasseDeriv_apply,
    mul_one, monomial_eq_zero_iff, sum_monomial_index, mul_zero, ←
    tsub_add_eq_tsub_tsub, add_comm l k]
  rw_mod_cast [nsmul_eq_mul]
  

Proof state:
case h.h
R : Type u_1
inst✝ : Semiring R
k✝ : ℕ
f : R[X]
k l i : ℕ
⊢ (monomial (i - (k + l))) ↑(choose (i - l) k * choose i l) =
    (monomial (i - (k + l))) (↑(choose (k + l) k) * ↑(choose i (k + l)))

Next tactic:
rw [← Nat.cast_mul]
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_apply, LinearMap.coe_comp, smul_monomial, hasseDeriv_apply,
    mul_one, monomial_eq_zero_iff, sum_monomial_index, mul_zero, ←
    tsub_add_eq_tsub_tsub, add_comm l k]
  rw_mod_cast [nsmul_eq_mul]
  rw [← Nat.cast_mul]
  

Proof state:
case h.h
R : Type u_1
inst✝ : Semiring R
k✝ : ℕ
f : R[X]
k l i : ℕ
⊢ (monomial (i - (k + l))) ↑(choose (i - l) k * choose i l) =
    (monomial (i - (k + l))) ↑(choose (k + l) k * choose i (k + l))

Next tactic:
congr 2
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_apply, LinearMap.coe_comp, smul_monomial, hasseDeriv_apply,
    mul_one, monomial_eq_zero_iff, sum_monomial_index, mul_zero, ←
    tsub_add_eq_tsub_tsub, add_comm l k]
  rw_mod_cast [nsmul_eq_mul]
  rw [← Nat.cast_mul]
  congr 2
  

Proof state:
case h.h.h.e_6.h.e_a
R : Type u_1
inst✝ : Semiring R
k✝ : ℕ
f : R[X]
k l i : ℕ
⊢ choose (i - l) k * choose i l = choose (k + l) k * choose i (k + l)

Next tactic:
by_cases hikl : i < k + l
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_apply, LinearMap.coe_comp, smul_monomial, hasseDeriv_apply,
    mul_one, monomial_eq_zero_iff, sum_monomial_index, mul_zero, ←
    tsub_add_eq_tsub_tsub, add_comm l k]
  rw_mod_cast [nsmul_eq_mul]
  rw [← Nat.cast_mul]
  congr 2
  by_cases hikl : i < k + l
  · 

Proof state:
case pos
R : Type u_1
inst✝ : Semiring R
k✝ : ℕ
f : R[X]
k l i : ℕ
hikl : i < k + l
⊢ choose (i - l) k * choose i l = choose (k + l) k * choose i (k + l)

Next tactic:
rw [choose_eq_zero_of_lt hikl, mul_zero]
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_apply, LinearMap.coe_comp, smul_monomial, hasseDeriv_apply,
    mul_one, monomial_eq_zero_iff, sum_monomial_index, mul_zero, ←
    tsub_add_eq_tsub_tsub, add_comm l k]
  rw_mod_cast [nsmul_eq_mul]
  rw [← Nat.cast_mul]
  congr 2
  by_cases hikl : i < k + l
  · rw [choose_eq_zero_of_lt hikl, mul_zero]
    

Proof state:
case pos
R : Type u_1
inst✝ : Semiring R
k✝ : ℕ
f : R[X]
k l i : ℕ
hikl : i < k + l
⊢ choose (i - l) k * choose i l = 0

Next tactic:
by_cases hil : i < l
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_apply, LinearMap.coe_comp, smul_monomial, hasseDeriv_apply,
    mul_one, monomial_eq_zero_iff, sum_monomial_index, mul_zero, ←
    tsub_add_eq_tsub_tsub, add_comm l k]
  rw_mod_cast [nsmul_eq_mul]
  rw [← Nat.cast_mul]
  congr 2
  by_cases hikl : i < k + l
  · rw [choose_eq_zero_of_lt hikl, mul_zero]
    by_cases hil : i < l
    · 

Proof state:
case pos
R : Type u_1
inst✝ : Semiring R
k✝ : ℕ
f : R[X]
k l i : ℕ
hikl : i < k + l
hil : i < l
⊢ choose (i - l) k * choose i l = 0

Next tactic:
rw [choose_eq_zero_of_lt hil, mul_zero]
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_apply, LinearMap.coe_comp, smul_monomial, hasseDeriv_apply,
    mul_one, monomial_eq_zero_iff, sum_monomial_index, mul_zero, ←
    tsub_add_eq_tsub_tsub, add_comm l k]
  rw_mod_cast [nsmul_eq_mul]
  rw [← Nat.cast_mul]
  congr 2
  by_cases hikl : i < k + l
  · rw [choose_eq_zero_of_lt hikl, mul_zero]
    by_cases hil : i < l
    · rw [choose_eq_zero_of_lt hil, mul_zero]
    · 

Proof state:
case neg
R : Type u_1
inst✝ : Semiring R
k✝ : ℕ
f : R[X]
k l i : ℕ
hikl : i < k + l
hil : ¬i < l
⊢ choose (i - l) k * choose i l = 0

Next tactic:
push_neg at hil
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_apply, LinearMap.coe_comp, smul_monomial, hasseDeriv_apply,
    mul_one, monomial_eq_zero_iff, sum_monomial_index, mul_zero, ←
    tsub_add_eq_tsub_tsub, add_comm l k]
  rw_mod_cast [nsmul_eq_mul]
  rw [← Nat.cast_mul]
  congr 2
  by_cases hikl : i < k + l
  · rw [choose_eq_zero_of_lt hikl, mul_zero]
    by_cases hil : i < l
    · rw [choose_eq_zero_of_lt hil, mul_zero]
    · push_neg at hil
      

Proof state:
case neg
R : Type u_1
inst✝ : Semiring R
k✝ : ℕ
f : R[X]
k l i : ℕ
hikl : i < k + l
hil : l ≤ i
⊢ choose (i - l) k * choose i l = 0

Next tactic:
rw [← tsub_lt_iff_right hil] at hikl
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_apply, LinearMap.coe_comp, smul_monomial, hasseDeriv_apply,
    mul_one, monomial_eq_zero_iff, sum_monomial_index, mul_zero, ←
    tsub_add_eq_tsub_tsub, add_comm l k]
  rw_mod_cast [nsmul_eq_mul]
  rw [← Nat.cast_mul]
  congr 2
  by_cases hikl : i < k + l
  · rw [choose_eq_zero_of_lt hikl, mul_zero]
    by_cases hil : i < l
    · rw [choose_eq_zero_of_lt hil, mul_zero]
    · push_neg at hil
      rw [← tsub_lt_iff_right hil] at hikl
      

Proof state:
case neg
R : Type u_1
inst✝ : Semiring R
k✝ : ℕ
f : R[X]
k l i : ℕ
hikl : i - l < k
hil : l ≤ i
⊢ choose (i - l) k * choose i l = 0

Next tactic:
rw [choose_eq_zero_of_lt hikl, zero_mul]
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_apply, LinearMap.coe_comp, smul_monomial, hasseDeriv_apply,
    mul_one, monomial_eq_zero_iff, sum_monomial_index, mul_zero, ←
    tsub_add_eq_tsub_tsub, add_comm l k]
  rw_mod_cast [nsmul_eq_mul]
  rw [← Nat.cast_mul]
  congr 2
  by_cases hikl : i < k + l
  · rw [choose_eq_zero_of_lt hikl, mul_zero]
    by_cases hil : i < l
    · rw [choose_eq_zero_of_lt hil, mul_zero]
    · push_neg at hil
      rw [← tsub_lt_iff_right hil] at hikl
      rw [choose_eq_zero_of_lt hikl, zero_mul]
  

Proof state:
case neg
R : Type u_1
inst✝ : Semiring R
k✝ : ℕ
f : R[X]
k l i : ℕ
hikl : ¬i < k + l
⊢ choose (i - l) k * choose i l = choose (k + l) k * choose i (k + l)

Next tactic:
push_neg at hikl
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_apply, LinearMap.coe_comp, smul_monomial, hasseDeriv_apply,
    mul_one, monomial_eq_zero_iff, sum_monomial_index, mul_zero, ←
    tsub_add_eq_tsub_tsub, add_comm l k]
  rw_mod_cast [nsmul_eq_mul]
  rw [← Nat.cast_mul]
  congr 2
  by_cases hikl : i < k + l
  · rw [choose_eq_zero_of_lt hikl, mul_zero]
    by_cases hil : i < l
    · rw [choose_eq_zero_of_lt hil, mul_zero]
    · push_neg at hil
      rw [← tsub_lt_iff_right hil] at hikl
      rw [choose_eq_zero_of_lt hikl, zero_mul]
  push_neg at hikl
  

Proof state:
case neg
R : Type u_1
inst✝ : Semiring R
k✝ : ℕ
f : R[X]
k l i : ℕ
hikl : k + l ≤ i
⊢ choose (i - l) k * choose i l = choose (k + l) k * choose i (k + l)

Next tactic:
apply @cast_injective ℚ
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_apply, LinearMap.coe_comp, smul_monomial, hasseDeriv_apply,
    mul_one, monomial_eq_zero_iff, sum_monomial_index, mul_zero, ←
    tsub_add_eq_tsub_tsub, add_comm l k]
  rw_mod_cast [nsmul_eq_mul]
  rw [← Nat.cast_mul]
  congr 2
  by_cases hikl : i < k + l
  · rw [choose_eq_zero_of_lt hikl, mul_zero]
    by_cases hil : i < l
    · rw [choose_eq_zero_of_lt hil, mul_zero]
    · push_neg at hil
      rw [← tsub_lt_iff_right hil] at hikl
      rw [choose_eq_zero_of_lt hikl, zero_mul]
  push_neg at hikl
  apply @cast_injective ℚ
  

Proof state:
case neg.a
R : Type u_1
inst✝ : Semiring R
k✝ : ℕ
f : R[X]
k l i : ℕ
hikl : k + l ≤ i
⊢ ↑(choose (i - l) k * choose i l) = ↑(choose (k + l) k * choose i (k + l))

Next tactic:
have h1 : l ≤ i := le_of_add_le_right hikl
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_apply, LinearMap.coe_comp, smul_monomial, hasseDeriv_apply,
    mul_one, monomial_eq_zero_iff, sum_monomial_index, mul_zero, ←
    tsub_add_eq_tsub_tsub, add_comm l k]
  rw_mod_cast [nsmul_eq_mul]
  rw [← Nat.cast_mul]
  congr 2
  by_cases hikl : i < k + l
  · rw [choose_eq_zero_of_lt hikl, mul_zero]
    by_cases hil : i < l
    · rw [choose_eq_zero_of_lt hil, mul_zero]
    · push_neg at hil
      rw [← tsub_lt_iff_right hil] at hikl
      rw [choose_eq_zero_of_lt hikl, zero_mul]
  push_neg at hikl
  apply @cast_injective ℚ
  have h1 : l ≤ i := le_of_add_le_right hikl
  

Proof state:
case neg.a
R : Type u_1
inst✝ : Semiring R
k✝ : ℕ
f : R[X]
k l i : ℕ
hikl : k + l ≤ i
h1 : l ≤ i
⊢ ↑(choose (i - l) k * choose i l) = ↑(choose (k + l) k * choose i (k + l))

Next tactic:
have h2 : k ≤ i - l := le_tsub_of_add_le_right hikl
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_apply, LinearMap.coe_comp, smul_monomial, hasseDeriv_apply,
    mul_one, monomial_eq_zero_iff, sum_monomial_index, mul_zero, ←
    tsub_add_eq_tsub_tsub, add_comm l k]
  rw_mod_cast [nsmul_eq_mul]
  rw [← Nat.cast_mul]
  congr 2
  by_cases hikl : i < k + l
  · rw [choose_eq_zero_of_lt hikl, mul_zero]
    by_cases hil : i < l
    · rw [choose_eq_zero_of_lt hil, mul_zero]
    · push_neg at hil
      rw [← tsub_lt_iff_right hil] at hikl
      rw [choose_eq_zero_of_lt hikl, 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
  

Proof state:
case neg.a
R : Type u_1
inst✝ : Semiring R
k✝ : ℕ
f : R[X]
k l i : ℕ
hikl : k + l ≤ i
h1 : l ≤ i
h2 : k ≤ i - l
⊢ ↑(choose (i - l) k * choose i l) = ↑(choose (k + l) k * choose i (k + l))

Next tactic:
have h3 : k ≤ k + l := le_self_add
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_apply, LinearMap.coe_comp, smul_monomial, hasseDeriv_apply,
    mul_one, monomial_eq_zero_iff, sum_monomial_index, mul_zero, ←
    tsub_add_eq_tsub_tsub, add_comm l k]
  rw_mod_cast [nsmul_eq_mul]
  rw [← Nat.cast_mul]
  congr 2
  by_cases hikl : i < k + l
  · rw [choose_eq_zero_of_lt hikl, mul_zero]
    by_cases hil : i < l
    · rw [choose_eq_zero_of_lt hil, mul_zero]
    · push_neg at hil
      rw [← tsub_lt_iff_right hil] at hikl
      rw [choose_eq_zero_of_lt hikl, 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
  

Proof state:
case neg.a
R : Type u_1
inst✝ : Semiring R
k✝ : ℕ
f : R[X]
k l i : ℕ
hikl : k + l ≤ i
h1 : l ≤ i
h2 : k ≤ i - l
h3 : k ≤ k + l
⊢ ↑(choose (i - l) k * choose i l) = ↑(choose (k + l) k * choose i (k + l))

Next tactic:
push_cast
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_apply, LinearMap.coe_comp, smul_monomial, hasseDeriv_apply,
    mul_one, monomial_eq_zero_iff, sum_monomial_index, mul_zero, ←
    tsub_add_eq_tsub_tsub, add_comm l k]
  rw_mod_cast [nsmul_eq_mul]
  rw [← Nat.cast_mul]
  congr 2
  by_cases hikl : i < k + l
  · rw [choose_eq_zero_of_lt hikl, mul_zero]
    by_cases hil : i < l
    · rw [choose_eq_zero_of_lt hil, mul_zero]
    · push_neg at hil
      rw [← tsub_lt_iff_right hil] at hikl
      rw [choose_eq_zero_of_lt hikl, 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
  push_cast
  

Proof state:
case neg.a
R : Type u_1
inst✝ : Semiring R
k✝ : ℕ
f : R[X]
k l i : ℕ
hikl : k + l ≤ i
h1 : l ≤ i
h2 : k ≤ i - l
h3 : k ≤ k + l
⊢ ↑(choose (i - l) k) * ↑(choose i l) = ↑(choose (k + l) k) * ↑(choose i (k + l))

Next tactic:
rw [cast_choose ℚ h1, cast_choose ℚ h2, cast_choose ℚ h3, cast_choose ℚ hikl]
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_apply, LinearMap.coe_comp, smul_monomial, hasseDeriv_apply,
    mul_one, monomial_eq_zero_iff, sum_monomial_index, mul_zero, ←
    tsub_add_eq_tsub_tsub, add_comm l k]
  rw_mod_cast [nsmul_eq_mul]
  rw [← Nat.cast_mul]
  congr 2
  by_cases hikl : i < k + l
  · rw [choose_eq_zero_of_lt hikl, mul_zero]
    by_cases hil : i < l
    · rw [choose_eq_zero_of_lt hil, mul_zero]
    · push_neg at hil
      rw [← tsub_lt_iff_right hil] at hikl
      rw [choose_eq_zero_of_lt hikl, 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
  push_cast
  rw [cast_choose ℚ h1, cast_choose ℚ h2, cast_choose ℚ h3, cast_choose ℚ hikl]
  

Proof state:
case neg.a
R : Type u_1
inst✝ : Semiring R
k✝ : ℕ
f : R[X]
k l i : ℕ
hikl : k + l ≤ i
h1 : l ≤ i
h2 : k ≤ i - l
h3 : k ≤ k + l
⊢ ↑(i - l)! / (↑k ! * ↑(i - l - k)!) * (↑i ! / (↑l ! * ↑(i - l)!)) =
    ↑(k + l)! / (↑k ! * ↑(k + l - k)!) * (↑i ! / (↑(k + l)! * ↑(i - (k + l))!))

Next tactic:
rw [show i - (k + l) = i - l - k by rw [add_comm]; apply tsub_add_eq_tsub_tsub]
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_apply, LinearMap.coe_comp, smul_monomial, hasseDeriv_apply,
    mul_one, monomial_eq_zero_iff, sum_monomial_index, mul_zero, ←
    tsub_add_eq_tsub_tsub, add_comm l k]
  rw_mod_cast [nsmul_eq_mul]
  rw [← Nat.cast_mul]
  congr 2
  by_cases hikl : i < k + l
  · rw [choose_eq_zero_of_lt hikl, mul_zero]
    by_cases hil : i < l
    · rw [choose_eq_zero_of_lt hil, mul_zero]
    · push_neg at hil
      rw [← tsub_lt_iff_right hil] at hikl
      rw [choose_eq_zero_of_lt hikl, 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
  push_cast
  rw [cast_choose ℚ h1, cast_choose ℚ h2, cast_choose ℚ h3, cast_choose ℚ hikl]
  rw [show i - (k + l) = i - l - k by 

Proof state:
R : Type u_1
inst✝ : Semiring R
k✝ : ℕ
f : R[X]
k l i : ℕ
hikl : k + l ≤ i
h1 : l ≤ i
h2 : k ≤ i - l
h3 : k ≤ k + l
⊢ i - (k + l) = i - l - k

Next tactic:
rw [add_comm]