l3lab/ntp-mathlib huggingface dataset:
instance normedAddCommGroup [hp : Fact (1 ≤ p)] : NormedAddCommGroup (lp E p) :=
  AddGroupNorm.toNormedAddCommGroup
    { toFun := norm
      map_zero' := norm_zero
      neg' := norm_neg
      add_le' := fun f g => by
        rcases p.dichotomy with (rfl | hp')
        · cases isEmpty_or_nonempty α
          · -- Porting note: was `simp [lp.eq_zero' f]`
            rw [lp.eq_zero' f]
            simp only [zero_add, norm_zero, le_refl] -- porting note: just `simp` was slow
          refine' (lp.isLUB_norm (f + g)).2 _
          rintro x ⟨i, rfl⟩
          refine' le_trans _ (add_mem_upperBounds_add
            (lp.isLUB_norm f).1 (lp.isLUB_norm g).1 ⟨_, _, ⟨i, rfl⟩, ⟨i, rfl⟩, rfl⟩)
          exact norm_add_le (f i) (g i)
        · 

Proof state:
case inr
α : Type u_1
E : α → Type u_2
p q : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
hp : Fact (1 ≤ p)
f g : ↥(lp E p)
hp' : 1 ≤ ENNReal.toReal p
⊢ ‖f + g‖ ≤ ‖f‖ + ‖g‖

Next tactic:
have hp'' : 0 < p.toReal := zero_lt_one.trans_le hp'
instance normedAddCommGroup [hp : Fact (1 ≤ p)] : NormedAddCommGroup (lp E p) :=
  AddGroupNorm.toNormedAddCommGroup
    { toFun := norm
      map_zero' := norm_zero
      neg' := norm_neg
      add_le' := fun f g => by
        rcases p.dichotomy with (rfl | hp')
        · cases isEmpty_or_nonempty α
          · -- Porting note: was `simp [lp.eq_zero' f]`
            rw [lp.eq_zero' f]
            simp only [zero_add, norm_zero, le_refl] -- porting note: just `simp` was slow
          refine' (lp.isLUB_norm (f + g)).2 _
          rintro x ⟨i, rfl⟩
          refine' le_trans _ (add_mem_upperBounds_add
            (lp.isLUB_norm f).1 (lp.isLUB_norm g).1 ⟨_, _, ⟨i, rfl⟩, ⟨i, rfl⟩, rfl⟩)
          exact norm_add_le (f i) (g i)
        · have hp'' : 0 < p.toReal := zero_lt_one.trans_le hp'
          

Proof state:
case inr
α : Type u_1
E : α → Type u_2
p q : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
hp : Fact (1 ≤ p)
f g : ↥(lp E p)
hp' : 1 ≤ ENNReal.toReal p
hp'' : 0 < ENNReal.toReal p
⊢ ‖f + g‖ ≤ ‖f‖ + ‖g‖

Next tactic:
have hf₁ : ∀ i, 0 ≤ ‖f i‖ := fun i => norm_nonneg _
instance normedAddCommGroup [hp : Fact (1 ≤ p)] : NormedAddCommGroup (lp E p) :=
  AddGroupNorm.toNormedAddCommGroup
    { toFun := norm
      map_zero' := norm_zero
      neg' := norm_neg
      add_le' := fun f g => by
        rcases p.dichotomy with (rfl | hp')
        · cases isEmpty_or_nonempty α
          · -- Porting note: was `simp [lp.eq_zero' f]`
            rw [lp.eq_zero' f]
            simp only [zero_add, norm_zero, le_refl] -- porting note: just `simp` was slow
          refine' (lp.isLUB_norm (f + g)).2 _
          rintro x ⟨i, rfl⟩
          refine' le_trans _ (add_mem_upperBounds_add
            (lp.isLUB_norm f).1 (lp.isLUB_norm g).1 ⟨_, _, ⟨i, rfl⟩, ⟨i, rfl⟩, rfl⟩)
          exact norm_add_le (f i) (g i)
        · have hp'' : 0 < p.toReal := zero_lt_one.trans_le hp'
          have hf₁ : ∀ i, 0 ≤ ‖f i‖ := fun i => norm_nonneg _
          

Proof state:
case inr
α : Type u_1
E : α → Type u_2
p q : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
hp : Fact (1 ≤ p)
f g : ↥(lp E p)
hp' : 1 ≤ ENNReal.toReal p
hp'' : 0 < ENNReal.toReal p
hf₁ : ∀ (i : α), 0 ≤ ‖↑f i‖
⊢ ‖f + g‖ ≤ ‖f‖ + ‖g‖

Next tactic:
have hg₁ : ∀ i, 0 ≤ ‖g i‖ := fun i => norm_nonneg _
instance normedAddCommGroup [hp : Fact (1 ≤ p)] : NormedAddCommGroup (lp E p) :=
  AddGroupNorm.toNormedAddCommGroup
    { toFun := norm
      map_zero' := norm_zero
      neg' := norm_neg
      add_le' := fun f g => by
        rcases p.dichotomy with (rfl | hp')
        · cases isEmpty_or_nonempty α
          · -- Porting note: was `simp [lp.eq_zero' f]`
            rw [lp.eq_zero' f]
            simp only [zero_add, norm_zero, le_refl] -- porting note: just `simp` was slow
          refine' (lp.isLUB_norm (f + g)).2 _
          rintro x ⟨i, rfl⟩
          refine' le_trans _ (add_mem_upperBounds_add
            (lp.isLUB_norm f).1 (lp.isLUB_norm g).1 ⟨_, _, ⟨i, rfl⟩, ⟨i, rfl⟩, rfl⟩)
          exact norm_add_le (f i) (g i)
        · have hp'' : 0 < p.toReal := zero_lt_one.trans_le hp'
          have hf₁ : ∀ i, 0 ≤ ‖f i‖ := fun i => norm_nonneg _
          have hg₁ : ∀ i, 0 ≤ ‖g i‖ := fun i => norm_nonneg _
          

Proof state:
case inr
α : Type u_1
E : α → Type u_2
p q : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
hp : Fact (1 ≤ p)
f g : ↥(lp E p)
hp' : 1 ≤ ENNReal.toReal p
hp'' : 0 < ENNReal.toReal p
hf₁ : ∀ (i : α), 0 ≤ ‖↑f i‖
hg₁ : ∀ (i : α), 0 ≤ ‖↑g i‖
⊢ ‖f + g‖ ≤ ‖f‖ + ‖g‖

Next tactic:
have hf₂ := lp.hasSum_norm hp'' f
instance normedAddCommGroup [hp : Fact (1 ≤ p)] : NormedAddCommGroup (lp E p) :=
  AddGroupNorm.toNormedAddCommGroup
    { toFun := norm
      map_zero' := norm_zero
      neg' := norm_neg
      add_le' := fun f g => by
        rcases p.dichotomy with (rfl | hp')
        · cases isEmpty_or_nonempty α
          · -- Porting note: was `simp [lp.eq_zero' f]`
            rw [lp.eq_zero' f]
            simp only [zero_add, norm_zero, le_refl] -- porting note: just `simp` was slow
          refine' (lp.isLUB_norm (f + g)).2 _
          rintro x ⟨i, rfl⟩
          refine' le_trans _ (add_mem_upperBounds_add
            (lp.isLUB_norm f).1 (lp.isLUB_norm g).1 ⟨_, _, ⟨i, rfl⟩, ⟨i, rfl⟩, rfl⟩)
          exact norm_add_le (f i) (g i)
        · have hp'' : 0 < p.toReal := zero_lt_one.trans_le hp'
          have hf₁ : ∀ i, 0 ≤ ‖f i‖ := fun i => norm_nonneg _
          have hg₁ : ∀ i, 0 ≤ ‖g i‖ := fun i => norm_nonneg _
          have hf₂ := lp.hasSum_norm hp'' f
          

Proof state:
case inr
α : Type u_1
E : α → Type u_2
p q : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
hp : Fact (1 ≤ p)
f g : ↥(lp E p)
hp' : 1 ≤ ENNReal.toReal p
hp'' : 0 < ENNReal.toReal p
hf₁ : ∀ (i : α), 0 ≤ ‖↑f i‖
hg₁ : ∀ (i : α), 0 ≤ ‖↑g i‖
hf₂ : HasSum (fun i => ‖↑f i‖ ^ ENNReal.toReal p) (‖f‖ ^ ENNReal.toReal p)
⊢ ‖f + g‖ ≤ ‖f‖ + ‖g‖

Next tactic:
have hg₂ := lp.hasSum_norm hp'' g
instance normedAddCommGroup [hp : Fact (1 ≤ p)] : NormedAddCommGroup (lp E p) :=
  AddGroupNorm.toNormedAddCommGroup
    { toFun := norm
      map_zero' := norm_zero
      neg' := norm_neg
      add_le' := fun f g => by
        rcases p.dichotomy with (rfl | hp')
        · cases isEmpty_or_nonempty α
          · -- Porting note: was `simp [lp.eq_zero' f]`
            rw [lp.eq_zero' f]
            simp only [zero_add, norm_zero, le_refl] -- porting note: just `simp` was slow
          refine' (lp.isLUB_norm (f + g)).2 _
          rintro x ⟨i, rfl⟩
          refine' le_trans _ (add_mem_upperBounds_add
            (lp.isLUB_norm f).1 (lp.isLUB_norm g).1 ⟨_, _, ⟨i, rfl⟩, ⟨i, rfl⟩, rfl⟩)
          exact norm_add_le (f i) (g i)
        · have hp'' : 0 < p.toReal := zero_lt_one.trans_le hp'
          have hf₁ : ∀ i, 0 ≤ ‖f i‖ := fun i => norm_nonneg _
          have hg₁ : ∀ i, 0 ≤ ‖g i‖ := fun i => norm_nonneg _
          have hf₂ := lp.hasSum_norm hp'' f
          have hg₂ := lp.hasSum_norm hp'' g
          -- apply Minkowski's inequality
          

Proof state:
case inr
α : Type u_1
E : α → Type u_2
p q : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
hp : Fact (1 ≤ p)
f g : ↥(lp E p)
hp' : 1 ≤ ENNReal.toReal p
hp'' : 0 < ENNReal.toReal p
hf₁ : ∀ (i : α), 0 ≤ ‖↑f i‖
hg₁ : ∀ (i : α), 0 ≤ ‖↑g i‖
hf₂ : HasSum (fun i => ‖↑f i‖ ^ ENNReal.toReal p) (‖f‖ ^ ENNReal.toReal p)
hg₂ : HasSum (fun i => ‖↑g i‖ ^ ENNReal.toReal p) (‖g‖ ^ ENNReal.toReal p)
⊢ ‖f + g‖ ≤ ‖f‖ + ‖g‖

Next tactic:
obtain ⟨C, hC₁, hC₂, hCfg⟩ :=
            Real.Lp_add_le_hasSum_of_nonneg hp' hf₁ hg₁ (norm_nonneg' _) (norm_nonneg' _) hf₂ hg₂
instance normedAddCommGroup [hp : Fact (1 ≤ p)] : NormedAddCommGroup (lp E p) :=
  AddGroupNorm.toNormedAddCommGroup
    { toFun := norm
      map_zero' := norm_zero
      neg' := norm_neg
      add_le' := fun f g => by
        rcases p.dichotomy with (rfl | hp')
        · cases isEmpty_or_nonempty α
          · -- Porting note: was `simp [lp.eq_zero' f]`
            rw [lp.eq_zero' f]
            simp only [zero_add, norm_zero, le_refl] -- porting note: just `simp` was slow
          refine' (lp.isLUB_norm (f + g)).2 _
          rintro x ⟨i, rfl⟩
          refine' le_trans _ (add_mem_upperBounds_add
            (lp.isLUB_norm f).1 (lp.isLUB_norm g).1 ⟨_, _, ⟨i, rfl⟩, ⟨i, rfl⟩, rfl⟩)
          exact norm_add_le (f i) (g i)
        · have hp'' : 0 < p.toReal := zero_lt_one.trans_le hp'
          have hf₁ : ∀ i, 0 ≤ ‖f i‖ := fun i => norm_nonneg _
          have hg₁ : ∀ i, 0 ≤ ‖g i‖ := fun i => norm_nonneg _
          have hf₂ := lp.hasSum_norm hp'' f
          have hg₂ := lp.hasSum_norm hp'' g
          -- apply Minkowski's inequality
          obtain ⟨C, hC₁, hC₂, hCfg⟩ :=
            Real.Lp_add_le_hasSum_of_nonneg hp' hf₁ hg₁ (norm_nonneg' _) (norm_nonneg' _) hf₂ hg₂
          

Proof state:
case inr.intro.intro.intro
α : Type u_1
E : α → Type u_2
p q : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
hp : Fact (1 ≤ p)
f g : ↥(lp E p)
hp' : 1 ≤ ENNReal.toReal p
hp'' : 0 < ENNReal.toReal p
hf₁ : ∀ (i : α), 0 ≤ ‖↑f i‖
hg₁ : ∀ (i : α), 0 ≤ ‖↑g i‖
hf₂ : HasSum (fun i => ‖↑f i‖ ^ ENNReal.toReal p) (‖f‖ ^ ENNReal.toReal p)
hg₂ : HasSum (fun i => ‖↑g i‖ ^ ENNReal.toReal p) (‖g‖ ^ ENNReal.toReal p)
C : ℝ
hC₁ : 0 ≤ C
hC₂ : C ≤ ‖f‖ + ‖g‖
hCfg : HasSum (fun i => (‖↑f i‖ + ‖↑g i‖) ^ ENNReal.toReal p) (C ^ ENNReal.toReal p)
⊢ ‖f + g‖ ≤ ‖f‖ + ‖g‖

Next tactic:
refine' le_trans _ hC₂
instance normedAddCommGroup [hp : Fact (1 ≤ p)] : NormedAddCommGroup (lp E p) :=
  AddGroupNorm.toNormedAddCommGroup
    { toFun := norm
      map_zero' := norm_zero
      neg' := norm_neg
      add_le' := fun f g => by
        rcases p.dichotomy with (rfl | hp')
        · cases isEmpty_or_nonempty α
          · -- Porting note: was `simp [lp.eq_zero' f]`
            rw [lp.eq_zero' f]
            simp only [zero_add, norm_zero, le_refl] -- porting note: just `simp` was slow
          refine' (lp.isLUB_norm (f + g)).2 _
          rintro x ⟨i, rfl⟩
          refine' le_trans _ (add_mem_upperBounds_add
            (lp.isLUB_norm f).1 (lp.isLUB_norm g).1 ⟨_, _, ⟨i, rfl⟩, ⟨i, rfl⟩, rfl⟩)
          exact norm_add_le (f i) (g i)
        · have hp'' : 0 < p.toReal := zero_lt_one.trans_le hp'
          have hf₁ : ∀ i, 0 ≤ ‖f i‖ := fun i => norm_nonneg _
          have hg₁ : ∀ i, 0 ≤ ‖g i‖ := fun i => norm_nonneg _
          have hf₂ := lp.hasSum_norm hp'' f
          have hg₂ := lp.hasSum_norm hp'' g
          -- apply Minkowski's inequality
          obtain ⟨C, hC₁, hC₂, hCfg⟩ :=
            Real.Lp_add_le_hasSum_of_nonneg hp' hf₁ hg₁ (norm_nonneg' _) (norm_nonneg' _) hf₂ hg₂
          refine' le_trans _ hC₂
          

Proof state:
case inr.intro.intro.intro
α : Type u_1
E : α → Type u_2
p q : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
hp : Fact (1 ≤ p)
f g : ↥(lp E p)
hp' : 1 ≤ ENNReal.toReal p
hp'' : 0 < ENNReal.toReal p
hf₁ : ∀ (i : α), 0 ≤ ‖↑f i‖
hg₁ : ∀ (i : α), 0 ≤ ‖↑g i‖
hf₂ : HasSum (fun i => ‖↑f i‖ ^ ENNReal.toReal p) (‖f‖ ^ ENNReal.toReal p)
hg₂ : HasSum (fun i => ‖↑g i‖ ^ ENNReal.toReal p) (‖g‖ ^ ENNReal.toReal p)
C : ℝ
hC₁ : 0 ≤ C
hC₂ : C ≤ ‖f‖ + ‖g‖
hCfg : HasSum (fun i => (‖↑f i‖ + ‖↑g i‖) ^ ENNReal.toReal p) (C ^ ENNReal.toReal p)
⊢ ‖f + g‖ ≤ C

Next tactic:
rw [← Real.rpow_le_rpow_iff (norm_nonneg' (f + g)) hC₁ hp'']
instance normedAddCommGroup [hp : Fact (1 ≤ p)] : NormedAddCommGroup (lp E p) :=
  AddGroupNorm.toNormedAddCommGroup
    { toFun := norm
      map_zero' := norm_zero
      neg' := norm_neg
      add_le' := fun f g => by
        rcases p.dichotomy with (rfl | hp')
        · cases isEmpty_or_nonempty α
          · -- Porting note: was `simp [lp.eq_zero' f]`
            rw [lp.eq_zero' f]
            simp only [zero_add, norm_zero, le_refl] -- porting note: just `simp` was slow
          refine' (lp.isLUB_norm (f + g)).2 _
          rintro x ⟨i, rfl⟩
          refine' le_trans _ (add_mem_upperBounds_add
            (lp.isLUB_norm f).1 (lp.isLUB_norm g).1 ⟨_, _, ⟨i, rfl⟩, ⟨i, rfl⟩, rfl⟩)
          exact norm_add_le (f i) (g i)
        · have hp'' : 0 < p.toReal := zero_lt_one.trans_le hp'
          have hf₁ : ∀ i, 0 ≤ ‖f i‖ := fun i => norm_nonneg _
          have hg₁ : ∀ i, 0 ≤ ‖g i‖ := fun i => norm_nonneg _
          have hf₂ := lp.hasSum_norm hp'' f
          have hg₂ := lp.hasSum_norm hp'' g
          -- apply Minkowski's inequality
          obtain ⟨C, hC₁, hC₂, hCfg⟩ :=
            Real.Lp_add_le_hasSum_of_nonneg hp' hf₁ hg₁ (norm_nonneg' _) (norm_nonneg' _) hf₂ hg₂
          refine' le_trans _ hC₂
          rw [← Real.rpow_le_rpow_iff (norm_nonneg' (f + g)) hC₁ hp'']
          

Proof state:
case inr.intro.intro.intro
α : Type u_1
E : α → Type u_2
p q : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
hp : Fact (1 ≤ p)
f g : ↥(lp E p)
hp' : 1 ≤ ENNReal.toReal p
hp'' : 0 < ENNReal.toReal p
hf₁ : ∀ (i : α), 0 ≤ ‖↑f i‖
hg₁ : ∀ (i : α), 0 ≤ ‖↑g i‖
hf₂ : HasSum (fun i => ‖↑f i‖ ^ ENNReal.toReal p) (‖f‖ ^ ENNReal.toReal p)
hg₂ : HasSum (fun i => ‖↑g i‖ ^ ENNReal.toReal p) (‖g‖ ^ ENNReal.toReal p)
C : ℝ
hC₁ : 0 ≤ C
hC₂ : C ≤ ‖f‖ + ‖g‖
hCfg : HasSum (fun i => (‖↑f i‖ + ‖↑g i‖) ^ ENNReal.toReal p) (C ^ ENNReal.toReal p)
⊢ ‖f + g‖ ^ ENNReal.toReal p ≤ C ^ ENNReal.toReal p

Next tactic:
refine' hasSum_le _ (lp.hasSum_norm hp'' (f + g)) hCfg
instance normedAddCommGroup [hp : Fact (1 ≤ p)] : NormedAddCommGroup (lp E p) :=
  AddGroupNorm.toNormedAddCommGroup
    { toFun := norm
      map_zero' := norm_zero
      neg' := norm_neg
      add_le' := fun f g => by
        rcases p.dichotomy with (rfl | hp')
        · cases isEmpty_or_nonempty α
          · -- Porting note: was `simp [lp.eq_zero' f]`
            rw [lp.eq_zero' f]
            simp only [zero_add, norm_zero, le_refl] -- porting note: just `simp` was slow
          refine' (lp.isLUB_norm (f + g)).2 _
          rintro x ⟨i, rfl⟩
          refine' le_trans _ (add_mem_upperBounds_add
            (lp.isLUB_norm f).1 (lp.isLUB_norm g).1 ⟨_, _, ⟨i, rfl⟩, ⟨i, rfl⟩, rfl⟩)
          exact norm_add_le (f i) (g i)
        · have hp'' : 0 < p.toReal := zero_lt_one.trans_le hp'
          have hf₁ : ∀ i, 0 ≤ ‖f i‖ := fun i => norm_nonneg _
          have hg₁ : ∀ i, 0 ≤ ‖g i‖ := fun i => norm_nonneg _
          have hf₂ := lp.hasSum_norm hp'' f
          have hg₂ := lp.hasSum_norm hp'' g
          -- apply Minkowski's inequality
          obtain ⟨C, hC₁, hC₂, hCfg⟩ :=
            Real.Lp_add_le_hasSum_of_nonneg hp' hf₁ hg₁ (norm_nonneg' _) (norm_nonneg' _) hf₂ hg₂
          refine' le_trans _ hC₂
          rw [← Real.rpow_le_rpow_iff (norm_nonneg' (f + g)) hC₁ hp'']
          refine' hasSum_le _ (lp.hasSum_norm hp'' (f + g)) hCfg
          

Proof state:
case inr.intro.intro.intro
α : Type u_1
E : α → Type u_2
p q : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
hp : Fact (1 ≤ p)
f g : ↥(lp E p)
hp' : 1 ≤ ENNReal.toReal p
hp'' : 0 < ENNReal.toReal p
hf₁ : ∀ (i : α), 0 ≤ ‖↑f i‖
hg₁ : ∀ (i : α), 0 ≤ ‖↑g i‖
hf₂ : HasSum (fun i => ‖↑f i‖ ^ ENNReal.toReal p) (‖f‖ ^ ENNReal.toReal p)
hg₂ : HasSum (fun i => ‖↑g i‖ ^ ENNReal.toReal p) (‖g‖ ^ ENNReal.toReal p)
C : ℝ
hC₁ : 0 ≤ C
hC₂ : C ≤ ‖f‖ + ‖g‖
hCfg : HasSum (fun i => (‖↑f i‖ + ‖↑g i‖) ^ ENNReal.toReal p) (C ^ ENNReal.toReal p)
⊢ ∀ (i : α), ‖↑(f + g) i‖ ^ ENNReal.toReal p ≤ (‖↑f i‖ + ‖↑g i‖) ^ ENNReal.toReal p

Next tactic:
intro i
instance normedAddCommGroup [hp : Fact (1 ≤ p)] : NormedAddCommGroup (lp E p) :=
  AddGroupNorm.toNormedAddCommGroup
    { toFun := norm
      map_zero' := norm_zero
      neg' := norm_neg
      add_le' := fun f g => by
        rcases p.dichotomy with (rfl | hp')
        · cases isEmpty_or_nonempty α
          · -- Porting note: was `simp [lp.eq_zero' f]`
            rw [lp.eq_zero' f]
            simp only [zero_add, norm_zero, le_refl] -- porting note: just `simp` was slow
          refine' (lp.isLUB_norm (f + g)).2 _
          rintro x ⟨i, rfl⟩
          refine' le_trans _ (add_mem_upperBounds_add
            (lp.isLUB_norm f).1 (lp.isLUB_norm g).1 ⟨_, _, ⟨i, rfl⟩, ⟨i, rfl⟩, rfl⟩)
          exact norm_add_le (f i) (g i)
        · have hp'' : 0 < p.toReal := zero_lt_one.trans_le hp'
          have hf₁ : ∀ i, 0 ≤ ‖f i‖ := fun i => norm_nonneg _
          have hg₁ : ∀ i, 0 ≤ ‖g i‖ := fun i => norm_nonneg _
          have hf₂ := lp.hasSum_norm hp'' f
          have hg₂ := lp.hasSum_norm hp'' g
          -- apply Minkowski's inequality
          obtain ⟨C, hC₁, hC₂, hCfg⟩ :=
            Real.Lp_add_le_hasSum_of_nonneg hp' hf₁ hg₁ (norm_nonneg' _) (norm_nonneg' _) hf₂ hg₂
          refine' le_trans _ hC₂
          rw [← Real.rpow_le_rpow_iff (norm_nonneg' (f + g)) hC₁ hp'']
          refine' hasSum_le _ (lp.hasSum_norm hp'' (f + g)) hCfg
          intro i
          

Proof state:
case inr.intro.intro.intro
α : Type u_1
E : α → Type u_2
p q : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
hp : Fact (1 ≤ p)
f g : ↥(lp E p)
hp' : 1 ≤ ENNReal.toReal p
hp'' : 0 < ENNReal.toReal p
hf₁ : ∀ (i : α), 0 ≤ ‖↑f i‖
hg₁ : ∀ (i : α), 0 ≤ ‖↑g i‖
hf₂ : HasSum (fun i => ‖↑f i‖ ^ ENNReal.toReal p) (‖f‖ ^ ENNReal.toReal p)
hg₂ : HasSum (fun i => ‖↑g i‖ ^ ENNReal.toReal p) (‖g‖ ^ ENNReal.toReal p)
C : ℝ
hC₁ : 0 ≤ C
hC₂ : C ≤ ‖f‖ + ‖g‖
hCfg : HasSum (fun i => (‖↑f i‖ + ‖↑g i‖) ^ ENNReal.toReal p) (C ^ ENNReal.toReal p)
i : α
⊢ ‖↑(f + g) i‖ ^ ENNReal.toReal p ≤ (‖↑f i‖ + ‖↑g i‖) ^ ENNReal.toReal p

Next tactic:
gcongr
instance normedAddCommGroup [hp : Fact (1 ≤ p)] : NormedAddCommGroup (lp E p) :=
  AddGroupNorm.toNormedAddCommGroup
    { toFun := norm
      map_zero' := norm_zero
      neg' := norm_neg
      add_le' := fun f g => by
        rcases p.dichotomy with (rfl | hp')
        · cases isEmpty_or_nonempty α
          · -- Porting note: was `simp [lp.eq_zero' f]`
            rw [lp.eq_zero' f]
            simp only [zero_add, norm_zero, le_refl] -- porting note: just `simp` was slow
          refine' (lp.isLUB_norm (f + g)).2 _
          rintro x ⟨i, rfl⟩
          refine' le_trans _ (add_mem_upperBounds_add
            (lp.isLUB_norm f).1 (lp.isLUB_norm g).1 ⟨_, _, ⟨i, rfl⟩, ⟨i, rfl⟩, rfl⟩)
          exact norm_add_le (f i) (g i)
        · have hp'' : 0 < p.toReal := zero_lt_one.trans_le hp'
          have hf₁ : ∀ i, 0 ≤ ‖f i‖ := fun i => norm_nonneg _
          have hg₁ : ∀ i, 0 ≤ ‖g i‖ := fun i => norm_nonneg _
          have hf₂ := lp.hasSum_norm hp'' f
          have hg₂ := lp.hasSum_norm hp'' g
          -- apply Minkowski's inequality
          obtain ⟨C, hC₁, hC₂, hCfg⟩ :=
            Real.Lp_add_le_hasSum_of_nonneg hp' hf₁ hg₁ (norm_nonneg' _) (norm_nonneg' _) hf₂ hg₂
          refine' le_trans _ hC₂
          rw [← Real.rpow_le_rpow_iff (norm_nonneg' (f + g)) hC₁ hp'']
          refine' hasSum_le _ (lp.hasSum_norm hp'' (f + g)) hCfg
          intro i
          gcongr
          

Proof state:
case inr.intro.intro.intro.h₁
α : Type u_1
E : α → Type u_2
p q : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
hp : Fact (1 ≤ p)
f g : ↥(lp E p)
hp' : 1 ≤ ENNReal.toReal p
hp'' : 0 < ENNReal.toReal p
hf₁ : ∀ (i : α), 0 ≤ ‖↑f i‖
hg₁ : ∀ (i : α), 0 ≤ ‖↑g i‖
hf₂ : HasSum (fun i => ‖↑f i‖ ^ ENNReal.toReal p) (‖f‖ ^ ENNReal.toReal p)
hg₂ : HasSum (fun i => ‖↑g i‖ ^ ENNReal.toReal p) (‖g‖ ^ ENNReal.toReal p)
C : ℝ
hC₁ : 0 ≤ C
hC₂ : C ≤ ‖f‖ + ‖g‖
hCfg : HasSum (fun i => (‖↑f i‖ + ‖↑g i‖) ^ ENNReal.toReal p) (C ^ ENNReal.toReal p)
i : α
⊢ ‖↑(f + g) i‖ ≤ ‖↑f i‖ + ‖↑g i‖

Next tactic:
apply norm_add_le
/-- Hölder inequality -/
protected theorem tsum_mul_le_mul_norm {p q : ℝ≥0∞} (hpq : p.toReal.IsConjugateExponent q.toReal)
    (f : lp E p) (g : lp E q) :
    (Summable fun i => ‖f i‖ * ‖g i‖) ∧ ∑' i, ‖f i‖ * ‖g i‖ ≤ ‖f‖ * ‖g‖ := by
  

Proof state:
α : Type u_1
E : α → Type u_2
p✝ q✝ : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
p q : ℝ≥0∞
hpq : Real.IsConjugateExponent (ENNReal.toReal p) (ENNReal.toReal q)
f : ↥(lp E p)
g : ↥(lp E q)
⊢ (Summable fun i => ‖↑f i‖ * ‖↑g i‖) ∧ ∑' (i : α), ‖↑f i‖ * ‖↑g i‖ ≤ ‖f‖ * ‖g‖

Next tactic:
have hf₁ : ∀ i, 0 ≤ ‖f i‖ := fun i => norm_nonneg _
/-- Hölder inequality -/
protected theorem tsum_mul_le_mul_norm {p q : ℝ≥0∞} (hpq : p.toReal.IsConjugateExponent q.toReal)
    (f : lp E p) (g : lp E q) :
    (Summable fun i => ‖f i‖ * ‖g i‖) ∧ ∑' i, ‖f i‖ * ‖g i‖ ≤ ‖f‖ * ‖g‖ := by
  have hf₁ : ∀ i, 0 ≤ ‖f i‖ := fun i => norm_nonneg _
  

Proof state:
α : Type u_1
E : α → Type u_2
p✝ q✝ : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
p q : ℝ≥0∞
hpq : Real.IsConjugateExponent (ENNReal.toReal p) (ENNReal.toReal q)
f : ↥(lp E p)
g : ↥(lp E q)
hf₁ : ∀ (i : α), 0 ≤ ‖↑f i‖
⊢ (Summable fun i => ‖↑f i‖ * ‖↑g i‖) ∧ ∑' (i : α), ‖↑f i‖ * ‖↑g i‖ ≤ ‖f‖ * ‖g‖

Next tactic:
have hg₁ : ∀ i, 0 ≤ ‖g i‖ := fun i => norm_nonneg _
/-- Hölder inequality -/
protected theorem tsum_mul_le_mul_norm {p q : ℝ≥0∞} (hpq : p.toReal.IsConjugateExponent q.toReal)
    (f : lp E p) (g : lp E q) :
    (Summable fun i => ‖f i‖ * ‖g i‖) ∧ ∑' i, ‖f i‖ * ‖g i‖ ≤ ‖f‖ * ‖g‖ := by
  have hf₁ : ∀ i, 0 ≤ ‖f i‖ := fun i => norm_nonneg _
  have hg₁ : ∀ i, 0 ≤ ‖g i‖ := fun i => norm_nonneg _
  

Proof state:
α : Type u_1
E : α → Type u_2
p✝ q✝ : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
p q : ℝ≥0∞
hpq : Real.IsConjugateExponent (ENNReal.toReal p) (ENNReal.toReal q)
f : ↥(lp E p)
g : ↥(lp E q)
hf₁ : ∀ (i : α), 0 ≤ ‖↑f i‖
hg₁ : ∀ (i : α), 0 ≤ ‖↑g i‖
⊢ (Summable fun i => ‖↑f i‖ * ‖↑g i‖) ∧ ∑' (i : α), ‖↑f i‖ * ‖↑g i‖ ≤ ‖f‖ * ‖g‖

Next tactic:
have hf₂ := lp.hasSum_norm hpq.pos f
/-- Hölder inequality -/
protected theorem tsum_mul_le_mul_norm {p q : ℝ≥0∞} (hpq : p.toReal.IsConjugateExponent q.toReal)
    (f : lp E p) (g : lp E q) :
    (Summable fun i => ‖f i‖ * ‖g i‖) ∧ ∑' i, ‖f i‖ * ‖g i‖ ≤ ‖f‖ * ‖g‖ := by
  have hf₁ : ∀ i, 0 ≤ ‖f i‖ := fun i => norm_nonneg _
  have hg₁ : ∀ i, 0 ≤ ‖g i‖ := fun i => norm_nonneg _
  have hf₂ := lp.hasSum_norm hpq.pos f
  

Proof state:
α : Type u_1
E : α → Type u_2
p✝ q✝ : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
p q : ℝ≥0∞
hpq : Real.IsConjugateExponent (ENNReal.toReal p) (ENNReal.toReal q)
f : ↥(lp E p)
g : ↥(lp E q)
hf₁ : ∀ (i : α), 0 ≤ ‖↑f i‖
hg₁ : ∀ (i : α), 0 ≤ ‖↑g i‖
hf₂ : HasSum (fun i => ‖↑f i‖ ^ ENNReal.toReal p) (‖f‖ ^ ENNReal.toReal p)
⊢ (Summable fun i => ‖↑f i‖ * ‖↑g i‖) ∧ ∑' (i : α), ‖↑f i‖ * ‖↑g i‖ ≤ ‖f‖ * ‖g‖

Next tactic:
have hg₂ := lp.hasSum_norm hpq.symm.pos g
/-- Hölder inequality -/
protected theorem tsum_mul_le_mul_norm {p q : ℝ≥0∞} (hpq : p.toReal.IsConjugateExponent q.toReal)
    (f : lp E p) (g : lp E q) :
    (Summable fun i => ‖f i‖ * ‖g i‖) ∧ ∑' i, ‖f i‖ * ‖g i‖ ≤ ‖f‖ * ‖g‖ := by
  have hf₁ : ∀ i, 0 ≤ ‖f i‖ := fun i => norm_nonneg _
  have hg₁ : ∀ i, 0 ≤ ‖g i‖ := fun i => norm_nonneg _
  have hf₂ := lp.hasSum_norm hpq.pos f
  have hg₂ := lp.hasSum_norm hpq.symm.pos g
  

Proof state:
α : Type u_1
E : α → Type u_2
p✝ q✝ : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
p q : ℝ≥0∞
hpq : Real.IsConjugateExponent (ENNReal.toReal p) (ENNReal.toReal q)
f : ↥(lp E p)
g : ↥(lp E q)
hf₁ : ∀ (i : α), 0 ≤ ‖↑f i‖
hg₁ : ∀ (i : α), 0 ≤ ‖↑g i‖
hf₂ : HasSum (fun i => ‖↑f i‖ ^ ENNReal.toReal p) (‖f‖ ^ ENNReal.toReal p)
hg₂ : HasSum (fun i => ‖↑g i‖ ^ ENNReal.toReal q) (‖g‖ ^ ENNReal.toReal q)
⊢ (Summable fun i => ‖↑f i‖ * ‖↑g i‖) ∧ ∑' (i : α), ‖↑f i‖ * ‖↑g i‖ ≤ ‖f‖ * ‖g‖

Next tactic:
obtain ⟨C, -, hC', hC⟩ :=
    Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg hpq (norm_nonneg' _) (norm_nonneg' _) hf₁ hg₁ hf₂ hg₂
/-- Hölder inequality -/
protected theorem tsum_mul_le_mul_norm {p q : ℝ≥0∞} (hpq : p.toReal.IsConjugateExponent q.toReal)
    (f : lp E p) (g : lp E q) :
    (Summable fun i => ‖f i‖ * ‖g i‖) ∧ ∑' i, ‖f i‖ * ‖g i‖ ≤ ‖f‖ * ‖g‖ := by
  have hf₁ : ∀ i, 0 ≤ ‖f i‖ := fun i => norm_nonneg _
  have hg₁ : ∀ i, 0 ≤ ‖g i‖ := fun i => norm_nonneg _
  have hf₂ := lp.hasSum_norm hpq.pos f
  have hg₂ := lp.hasSum_norm hpq.symm.pos g
  obtain ⟨C, -, hC', hC⟩ :=
    Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg hpq (norm_nonneg' _) (norm_nonneg' _) hf₁ hg₁ hf₂ hg₂
  

Proof state:
case intro.intro.intro
α : Type u_1
E : α → Type u_2
p✝ q✝ : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
p q : ℝ≥0∞
hpq : Real.IsConjugateExponent (ENNReal.toReal p) (ENNReal.toReal q)
f : ↥(lp E p)
g : ↥(lp E q)
hf₁ : ∀ (i : α), 0 ≤ ‖↑f i‖
hg₁ : ∀ (i : α), 0 ≤ ‖↑g i‖
hf₂ : HasSum (fun i => ‖↑f i‖ ^ ENNReal.toReal p) (‖f‖ ^ ENNReal.toReal p)
hg₂ : HasSum (fun i => ‖↑g i‖ ^ ENNReal.toReal q) (‖g‖ ^ ENNReal.toReal q)
C : ℝ
hC' : C ≤ ‖f‖ * ‖g‖
hC : HasSum (fun i => ‖↑f i‖ * ‖↑g i‖) C
⊢ (Summable fun i => ‖↑f i‖ * ‖↑g i‖) ∧ ∑' (i : α), ‖↑f i‖ * ‖↑g i‖ ≤ ‖f‖ * ‖g‖

Next tactic:
rw [← hC.tsum_eq] at hC'