/-- Composition with a function on the right is a continuous linear map on Schwartz space
provided that the function is temperate and growths polynomially near infinity. -/
def compCLM {g : D → E} (hg : g.HasTemperateGrowth)
    (hg_upper : ∃ (k : ℕ) (C : ℝ), ∀ x, ‖x‖ ≤ C * (1 + ‖g x‖) ^ k) : 𝓢(E, F) →L[𝕜] 𝓢(D, F) :=
  mkCLM (fun f x => f (g x))
    (fun _ _ _ => by simp only [add_left_inj, Pi.add_apply, eq_self_iff_true]) (fun _ _ _ => rfl)
    (fun f => f.smooth'.comp hg.1)
    (by
      rintro ⟨k, n⟩
      rcases hg.norm_iteratedFDeriv_le_uniform_aux n with ⟨l, C, hC, hgrowth⟩
      rcases hg_upper with ⟨kg, Cg, hg_upper'⟩
      have hCg : 1 ≤ 1 + Cg := by
        refine' le_add_of_nonneg_right _
        specialize hg_upper' 0
        rw [norm_zero] at hg_upper'
        refine' nonneg_of_mul_nonneg_left hg_upper' (by positivity)
      let k' := kg * (k + l * n)
      use Finset.Iic (k', n), (1 + Cg) ^ (k + l * n) * ((C + 1) ^ n * n ! * 2 ^ k'), by positivity
      intro f x
      let seminorm_f := ((Finset.Iic (k', n)).sup (schwartzSeminormFamily 𝕜 _ _)) f
      have hg_upper'' : (1 + ‖x‖) ^ (k + l * n) ≤ (1 + Cg) ^ (k + l * n) * (1 + ‖g x‖) ^ k' := by
        rw [pow_mul, ← mul_pow]
        refine' pow_le_pow_left (by positivity) _ _
        rw [add_mul]
        refine' add_le_add _ (hg_upper' x)
        nth_rw 1 [← one_mul (1 : ℝ)]
        refine' mul_le_mul (le_refl _) (one_le_pow_of_one_le _ _) zero_le_one zero_le_one
        simp only [le_add_iff_nonneg_right, norm_nonneg]
      have hbound :
        ∀ i, i ≤ n → ‖iteratedFDeriv ℝ i f (g x)‖ ≤ 2 ^ k' * seminorm_f / (1 + ‖g x‖) ^ k' := by
        intro i hi
        have hpos : 0 < (1 + ‖g x‖) ^ k' := by positivity
        rw [le_div_iff' hpos]
        change i ≤ (k', n).snd at hi
        exact one_add_le_sup_seminorm_apply le_rfl hi _ _
      have hgrowth' : ∀ N : ℕ, 1 ≤ N → N ≤ n →
          ‖iteratedFDeriv ℝ N g x‖ ≤ ((C + 1) * (1 + ‖x‖) ^ l) ^ N := by
        intro N hN₁ hN₂
        refine' (hgrowth N hN₂ x).trans _
        rw [mul_pow]
        have hN₁' := (lt_of_lt_of_le zero_lt_one hN₁).ne'
        refine' mul_le_mul _ _ (by positivity) (by positivity)
        · exact le_trans (by simp [hC]) (le_self_pow (by simp [hC]) hN₁')
        · refine' le_self_pow (one_le_pow_of_one_le _ l) hN₁'
          simp only [le_add_iff_nonneg_right, norm_nonneg]
      have := norm_iteratedFDeriv_comp_le f.smooth' hg.1 le_top x hbound hgrowth'
      have hxk : ‖x‖ ^ k ≤ (1 + ‖x‖) ^ k :=
        pow_le_pow_left (norm_nonneg _) (by simp only [zero_le_one, le_add_iff_nonneg_left]) _
      refine' le_trans (mul_le_mul hxk this (by positivity) (by positivity)) _
      

Proof state:
case right
𝕜 : Type u_1
𝕜' : Type u_2
D : Type u_3
E : Type u_4
F : Type u_5
G : Type u_6
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace ℝ F
inst✝⁸ : IsROrC 𝕜
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace ℝ D
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace ℝ G
inst✝³ : NormedSpace 𝕜 F
inst✝² : SMulCommClass ℝ 𝕜 F
inst✝¹ : NormedSpace 𝕜 G
inst✝ : SMulCommClass ℝ 𝕜 G
g : D → E
hg : Function.HasTemperateGrowth g
k n l : ℕ
C : ℝ
hC : 0 ≤ C
hgrowth : ∀ N ≤ n, ∀ (x : D), ‖iteratedFDeriv ℝ N g x‖ ≤ C * (1 + ‖x‖) ^ l
kg : ℕ
Cg : ℝ
hg_upper' : ∀ (x : D), ‖x‖ ≤ Cg * (1 + ‖g x‖) ^ kg
hCg : 1 ≤ 1 + Cg
k' : ℕ := kg * (k + l * n)
f : 𝓢(E, F)
x : D
seminorm_f : ℝ := (Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily 𝕜 E F)) f
hg_upper'' : (1 + ‖x‖) ^ (k + l * n) ≤ (1 + Cg) ^ (k + l * n) * (1 + ‖g x‖) ^ k'
hbound : ∀ i ≤ n, ‖iteratedFDeriv ℝ i (⇑f) (g x)‖ ≤ 2 ^ k' * seminorm_f / (1 + ‖g x‖) ^ k'
hgrowth' : ∀ (N : ℕ), 1 ≤ N → N ≤ n → ‖iteratedFDeriv ℝ N g x‖ ≤ ((C + 1) * (1 + ‖x‖) ^ l) ^ N
this :
  ‖iteratedFDeriv ℝ n (f.toFun ∘ g) x‖ ≤ ↑n ! * (2 ^ k' * seminorm_f / (1 + ‖g x‖) ^ k') * ((C + 1) * (1 + ‖x‖) ^ l) ^ n
hxk : ‖x‖ ^ k ≤ (1 + ‖x‖) ^ k
⊢ (1 + ‖x‖) ^ k * (↑n ! * (2 ^ k' * seminorm_f / (1 + ‖g x‖) ^ k') * ((C + 1) * (1 + ‖x‖) ^ l) ^ n) ≤
    (1 + Cg) ^ (k + l * n) * ((C + 1) ^ n * ↑n ! * 2 ^ k') *
      (Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily 𝕜 E F)) f

Next tactic:
have rearrange :
        (1 + ‖x‖) ^ k *
            (n ! * (2 ^ k' * seminorm_f / (1 + ‖g x‖) ^ k') * ((C + 1) * (1 + ‖x‖) ^ l) ^ n) =
          (1 + ‖x‖) ^ (k + l * n) / (1 + ‖g x‖) ^ k' *
            ((C + 1) ^ n * n ! * 2 ^ k' * seminorm_f) := by
        rw [mul_pow, pow_add, ← pow_mul]
        ring