-- Mathlib/Analysis/Distribution/SchwartzSpace.lean
theorem eLpNorm_le_seminorm (p : ℝ≥0∞) (μ : Measure E

Proof state:
case intro
𝕜 : Type u_1
E : Type u_4
F : Type u_5
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : MeasurableSpace E
inst✝³ : OpensMeasurableSpace E
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
p : ℝ≥0∞
μ : autoParam (Measure E) _auto✝
hμ : Measure.HasTemperateGrowth μ
k : ℕ
hk : eLpNorm (fun x => (1 + ‖x‖) ^ (-↑k)) p μ < ⊤
f : 𝓢(E, F)
⊢ eLpNorm (⇑f) p μ ≤
    ↑((eLpNorm (fun x => (1 + ‖x‖) ^ (-↑k)) p μ).toNNReal * 2 ^ k) *
      ENNReal.ofReal (((Finset.Iic (k, 0)).sup (schwartzSeminormFamily 𝕜 E F)) f)

Tactic:
have h_one_add (x : E) : 0 < 1 + ‖x‖ := lt_add_of_pos_of_le zero_lt_one (norm_nonneg x)

Result:
case intro
𝕜 : Type u_1
E : Type u_4
F : Type u_5
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : MeasurableSpace E
inst✝³ : OpensMeasurableSpace E
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
p : ℝ≥0∞
μ : autoParam (Measure E) _auto✝
hμ : Measure.HasTemperateGrowth μ
k : ℕ
hk : eLpNorm (fun x => (1 + ‖x‖) ^ (-↑k)) p μ < ⊤
f : 𝓢(E, F)
h_one_add : ∀ (x : E), 0 < 1 + ‖x‖
⊢ eLpNorm (⇑f) p μ ≤
    ↑((eLpNorm (fun x => (1 + ‖x‖) ^ (-↑k)) p μ).toNNReal * 2 ^ k) *
      ENNReal.ofReal (((Finset.Iic (k, 0)).sup (schwartzSeminormFamily 𝕜 E F)) f)

Proof state:
𝕜 : Type u_1
E : Type u_4
F : Type u_5
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : MeasurableSpace E
inst✝³ : OpensMeasurableSpace E
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
p : ℝ≥0∞
μ : autoParam (Measure E) _auto✝
hμ : Measure.HasTemperateGrowth μ
k : ℕ
hk : eLpNorm (fun x => (1 + ‖x‖) ^ (-↑k)) p μ < ⊤
f : 𝓢(E, F)
h_one_add : ∀ (x : E), 0 < 1 + ‖x‖
⊢ eLpNorm (fun x => (1 + ‖x‖) ^ (-↑k)) p μ *
      (2 ^ k * ENNReal.ofReal (((Finset.Iic (k, 0)).sup (schwartzSeminormFamily 𝕜 E F)) f)) =
    eLpNorm (fun x => (1 + ‖x‖) ^ (-↑k)) p μ * ↑(2 ^ k) *
      ENNReal.ofReal (((Finset.Iic (k, 0)).sup (schwartzSeminormFamily 𝕜 E F)) f)

Tactic:
simp only [ENNReal.coe_pow, ENNReal.coe_ofNat]

Result:
𝕜 : Type u_1
E : Type u_4
F : Type u_5
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : MeasurableSpace E
inst✝³ : OpensMeasurableSpace E
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
p : ℝ≥0∞
μ : autoParam (Measure E) _auto✝
hμ : Measure.HasTemperateGrowth μ
k : ℕ
hk : eLpNorm (fun x => (1 + ‖x‖) ^ (-↑k)) p μ < ⊤
f : 𝓢(E, F)
h_one_add : ∀ (x : E), 0 < 1 + ‖x‖
⊢ eLpNorm (fun x => (1 + ‖x‖) ^ (-↑k)) p μ *
      (2 ^ k * ENNReal.ofReal (((Finset.Iic (k, 0)).sup (schwartzSeminormFamily 𝕜 E F)) f)) =
    eLpNorm (fun x => (1 + ‖x‖) ^ (-↑k)) p μ * 2 ^ k *
      ENNReal.ofReal (((Finset.Iic (k, 0)).sup (schwartzSeminormFamily 𝕜 E F)) f)

Proof state:
case bc
𝕜 : Type u_1
E : Type u_4
F : Type u_5
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : MeasurableSpace E
inst✝³ : OpensMeasurableSpace E
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
p : ℝ≥0∞
μ : autoParam (Measure E) _auto✝
hμ : Measure.HasTemperateGrowth μ
k : ℕ
hk : eLpNorm (fun x => (1 + ‖x‖) ^ (-↑k)) p μ < ⊤
f : 𝓢(E, F)
h_one_add : ∀ (x : E), 0 < 1 + ‖x‖
x : E
⊢ ‖(1 + ‖x‖) ^ k • f x‖₊ ≤ ↑2 ^ k * (((Finset.Iic (k, 0)).sup (schwartzSeminormFamily 𝕜 E F)) f).toNNReal

Tactic:
rw [← norm_toNNReal, Real.toNNReal_le_iff_le_coe]

Result:
case bc
𝕜 : Type u_1
E : Type u_4
F : Type u_5
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : MeasurableSpace E
inst✝³ : OpensMeasurableSpace E
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
p : ℝ≥0∞
μ : autoParam (Measure E) _auto✝
hμ : Measure.HasTemperateGrowth μ
k : ℕ
hk : eLpNorm (fun x => (1 + ‖x‖) ^ (-↑k)) p μ < ⊤
f : 𝓢(E, F)
h_one_add : ∀ (x : E), 0 < 1 + ‖x‖
x : E
⊢ ‖(1 + ‖x‖) ^ k • f x‖ ≤ ↑(↑2 ^ k * (((Finset.Iic (k, 0)).sup (schwartzSeminormFamily 𝕜 E F)) f).toNNReal)

Proof state:
𝕜 : Type u_1
E : Type u_4
F : Type u_5
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : MeasurableSpace E
inst✝³ : OpensMeasurableSpace E
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
p : ℝ≥0∞
μ : autoParam (Measure E) _auto✝
hμ : Measure.HasTemperateGrowth μ
k : ℕ
hk : eLpNorm (fun x => (1 + ‖x‖) ^ (-↑k)) p μ < ⊤
f : 𝓢(E, F)
h_one_add : ∀ (x : E), 0 < 1 + ‖x‖
⊢ AEStronglyMeasurable (fun x => (1 + ‖x‖) ^ (-↑k)) μ

Tactic:
refine Continuous.aestronglyMeasurable ?_

Result:
𝕜 : Type u_1
E : Type u_4
F : Type u_5
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : MeasurableSpace E
inst✝³ : OpensMeasurableSpace E
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
p : ℝ≥0∞
μ : autoParam (Measure E) _auto✝
hμ : Measure.HasTemperateGrowth μ
k : ℕ
hk : eLpNorm (fun x => (1 + ‖x‖) ^ (-↑k)) p μ < ⊤
f : 𝓢(E, F)
h_one_add : ∀ (x : E), 0 < 1 + ‖x‖
⊢ Continuous fun x => (1 + ‖x‖) ^ (-↑k)

Proof state:
case intro
𝕜 : Type u_1
E : Type u_4
F : Type u_5
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : MeasurableSpace E
inst✝³ : OpensMeasurableSpace E
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
p : ℝ≥0∞
μ : autoParam (Measure E) _auto✝
hμ : Measure.HasTemperateGrowth μ
k : ℕ
hk : eLpNorm (fun x => (1 + ‖x‖) ^ (-↑k)) p μ < ⊤
⊢ ∃ k C,
    ∀ (f : 𝓢(E, F)), eLpNorm (⇑f) p μ ≤ ↑C * ENNReal.ofReal (((Finset.Iic (k, 0)).sup (schwartzSeminormFamily 𝕜 E F)) f)

Tactic:
refine ⟨k, (eLpNorm (fun x ↦ (1 + ‖x‖) ^ (-k : ℝ)) p μ).toNNReal * 2 ^ k, fun f ↦ ?_⟩

Result:
case intro
𝕜 : Type u_1
E : Type u_4
F : Type u_5
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : MeasurableSpace E
inst✝³ : OpensMeasurableSpace E
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
p : ℝ≥0∞
μ : autoParam (Measure E) _auto✝
hμ : Measure.HasTemperateGrowth μ
k : ℕ
hk : eLpNorm (fun x => (1 + ‖x‖) ^ (-↑k)) p μ < ⊤
f : 𝓢(E, F)
⊢ eLpNorm (⇑f) p μ ≤
    ↑((eLpNorm (fun x => (1 + ‖x‖) ^ (-↑k)) p μ).toNNReal * 2 ^ k) *
      ENNReal.ofReal (((Finset.Iic (k, 0)).sup (schwartzSeminormFamily 𝕜 E F)) f)