l3lab/ntp-mathlib huggingface dataset:
/-- Let `f` be a function which is sufficiently close (in the Lipschitz sense) to a given linear
map `A`. Then it expands the volume of any set by at most `m` for any `m > det A`. -/
theorem addHaar_image_le_mul_of_det_lt (A : E →L[ℝ] E) {m : ℝ≥0}
    (hm : ENNReal.ofReal |A.det| < m) :
    ∀ᶠ δ in 𝓝[>] (0 : ℝ≥0),
      ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → μ (f '' s) ≤ m * μ s := by
  apply nhdsWithin_le_nhds
  let d := ENNReal.ofReal |A.det|
  -- construct a small neighborhood of `A '' (closedBall 0 1)` with measure comparable to
  -- the determinant of `A`.
  obtain ⟨ε, hε, εpos⟩ :
    ∃ ε : ℝ, μ (closedBall 0 ε + A '' closedBall 0 1) < m * μ (closedBall 0 1) ∧ 0 < ε := by
    have HC : IsCompact (A '' closedBall 0 1) :=
      (ProperSpace.isCompact_closedBall _ _).image A.continuous
    have L0 :
      Tendsto (fun ε => μ (cthickening ε (A '' closedBall 0 1))) (𝓝[>] 0)
        (𝓝 (μ (A '' closedBall 0 1))) := by
      apply Tendsto.mono_left _ nhdsWithin_le_nhds
      exact tendsto_measure_cthickening_of_isCompact HC
    have L1 :
      Tendsto (fun ε => μ (closedBall 0 ε + A '' closedBall 0 1)) (𝓝[>] 0)
        (𝓝 (μ (A '' closedBall 0 1))) := by
      apply L0.congr' _
      filter_upwards [self_mem_nhdsWithin] with r hr
      rw [← HC.add_closedBall_zero (le_of_lt hr), add_comm]
    have L2 :
      Tendsto (fun ε => μ (closedBall 0 ε + A '' closedBall 0 1)) (𝓝[>] 0)
        (𝓝 (d * μ (closedBall 0 1))) := by
      convert L1
      exact (addHaar_image_continuousLinearMap _ _ _).symm
    have I : d * μ (closedBall 0 1) < m * μ (closedBall 0 1) :=
      (ENNReal.mul_lt_mul_right (measure_closedBall_pos μ _ zero_lt_one).ne'
            measure_closedBall_lt_top.ne).2
        hm
    have H :
      ∀ᶠ b : ℝ in 𝓝[>] 0, μ (closedBall 0 b + A '' closedBall 0 1) < m * μ (closedBall 0 1) :=
      (tendsto_order.1 L2).2 _ I
    exact (H.and self_mem_nhdsWithin).exists
  have : Iio (⟨ε, εpos.le⟩ : ℝ≥0) ∈ 𝓝 (0 : ℝ≥0) := by apply Iio_mem_nhds; exact εpos
  

Proof state:
case a.intro.intro
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
A : E →L[ℝ] E
m : ℝ≥0
hm : ENNReal.ofReal |ContinuousLinearMap.det A| < ↑m
d : ℝ≥0∞ := ENNReal.ofReal |ContinuousLinearMap.det A|
ε : ℝ
hε : ↑↑μ (closedBall 0 ε + ⇑A '' closedBall 0 1) < ↑m * ↑↑μ (closedBall 0 1)
εpos : 0 < ε
this : Iio { val := ε, property := (_ : 0 ≤ ε) } ∈ 𝓝 0
⊢ {x | (fun δ => ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → ↑↑μ (f '' s) ≤ ↑m * ↑↑μ s) x} ∈ 𝓝 0

Next tactic:
filter_upwards [this]
/-- Let `f` be a function which is sufficiently close (in the Lipschitz sense) to a given linear
map `A`. Then it expands the volume of any set by at most `m` for any `m > det A`. -/
theorem addHaar_image_le_mul_of_det_lt (A : E →L[ℝ] E) {m : ℝ≥0}
    (hm : ENNReal.ofReal |A.det| < m) :
    ∀ᶠ δ in 𝓝[>] (0 : ℝ≥0),
      ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → μ (f '' s) ≤ m * μ s := by
  apply nhdsWithin_le_nhds
  let d := ENNReal.ofReal |A.det|
  -- construct a small neighborhood of `A '' (closedBall 0 1)` with measure comparable to
  -- the determinant of `A`.
  obtain ⟨ε, hε, εpos⟩ :
    ∃ ε : ℝ, μ (closedBall 0 ε + A '' closedBall 0 1) < m * μ (closedBall 0 1) ∧ 0 < ε := by
    have HC : IsCompact (A '' closedBall 0 1) :=
      (ProperSpace.isCompact_closedBall _ _).image A.continuous
    have L0 :
      Tendsto (fun ε => μ (cthickening ε (A '' closedBall 0 1))) (𝓝[>] 0)
        (𝓝 (μ (A '' closedBall 0 1))) := by
      apply Tendsto.mono_left _ nhdsWithin_le_nhds
      exact tendsto_measure_cthickening_of_isCompact HC
    have L1 :
      Tendsto (fun ε => μ (closedBall 0 ε + A '' closedBall 0 1)) (𝓝[>] 0)
        (𝓝 (μ (A '' closedBall 0 1))) := by
      apply L0.congr' _
      filter_upwards [self_mem_nhdsWithin] with r hr
      rw [← HC.add_closedBall_zero (le_of_lt hr), add_comm]
    have L2 :
      Tendsto (fun ε => μ (closedBall 0 ε + A '' closedBall 0 1)) (𝓝[>] 0)
        (𝓝 (d * μ (closedBall 0 1))) := by
      convert L1
      exact (addHaar_image_continuousLinearMap _ _ _).symm
    have I : d * μ (closedBall 0 1) < m * μ (closedBall 0 1) :=
      (ENNReal.mul_lt_mul_right (measure_closedBall_pos μ _ zero_lt_one).ne'
            measure_closedBall_lt_top.ne).2
        hm
    have H :
      ∀ᶠ b : ℝ in 𝓝[>] 0, μ (closedBall 0 b + A '' closedBall 0 1) < m * μ (closedBall 0 1) :=
      (tendsto_order.1 L2).2 _ I
    exact (H.and self_mem_nhdsWithin).exists
  have : Iio (⟨ε, εpos.le⟩ : ℝ≥0) ∈ 𝓝 (0 : ℝ≥0) := by apply Iio_mem_nhds; exact εpos
  filter_upwards [this]
  -- fix a function `f` which is close enough to `A`.
  

Proof state:
case h
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
A : E →L[ℝ] E
m : ℝ≥0
hm : ENNReal.ofReal |ContinuousLinearMap.det A| < ↑m
d : ℝ≥0∞ := ENNReal.ofReal |ContinuousLinearMap.det A|
ε : ℝ
hε : ↑↑μ (closedBall 0 ε + ⇑A '' closedBall 0 1) < ↑m * ↑↑μ (closedBall 0 1)
εpos : 0 < ε
this : Iio { val := ε, property := (_ : 0 ≤ ε) } ∈ 𝓝 0
⊢ ∀ a ∈ Iio { val := ε, property := (_ : 0 ≤ ε) },
    ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s a → ↑↑μ (f '' s) ≤ ↑m * ↑↑μ s

Next tactic:
intro δ hδ s f hf
/-- Let `f` be a function which is sufficiently close (in the Lipschitz sense) to a given linear
map `A`. Then it expands the volume of any set by at most `m` for any `m > det A`. -/
theorem addHaar_image_le_mul_of_det_lt (A : E →L[ℝ] E) {m : ℝ≥0}
    (hm : ENNReal.ofReal |A.det| < m) :
    ∀ᶠ δ in 𝓝[>] (0 : ℝ≥0),
      ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → μ (f '' s) ≤ m * μ s := by
  apply nhdsWithin_le_nhds
  let d := ENNReal.ofReal |A.det|
  -- construct a small neighborhood of `A '' (closedBall 0 1)` with measure comparable to
  -- the determinant of `A`.
  obtain ⟨ε, hε, εpos⟩ :
    ∃ ε : ℝ, μ (closedBall 0 ε + A '' closedBall 0 1) < m * μ (closedBall 0 1) ∧ 0 < ε := by
    have HC : IsCompact (A '' closedBall 0 1) :=
      (ProperSpace.isCompact_closedBall _ _).image A.continuous
    have L0 :
      Tendsto (fun ε => μ (cthickening ε (A '' closedBall 0 1))) (𝓝[>] 0)
        (𝓝 (μ (A '' closedBall 0 1))) := by
      apply Tendsto.mono_left _ nhdsWithin_le_nhds
      exact tendsto_measure_cthickening_of_isCompact HC
    have L1 :
      Tendsto (fun ε => μ (closedBall 0 ε + A '' closedBall 0 1)) (𝓝[>] 0)
        (𝓝 (μ (A '' closedBall 0 1))) := by
      apply L0.congr' _
      filter_upwards [self_mem_nhdsWithin] with r hr
      rw [← HC.add_closedBall_zero (le_of_lt hr), add_comm]
    have L2 :
      Tendsto (fun ε => μ (closedBall 0 ε + A '' closedBall 0 1)) (𝓝[>] 0)
        (𝓝 (d * μ (closedBall 0 1))) := by
      convert L1
      exact (addHaar_image_continuousLinearMap _ _ _).symm
    have I : d * μ (closedBall 0 1) < m * μ (closedBall 0 1) :=
      (ENNReal.mul_lt_mul_right (measure_closedBall_pos μ _ zero_lt_one).ne'
            measure_closedBall_lt_top.ne).2
        hm
    have H :
      ∀ᶠ b : ℝ in 𝓝[>] 0, μ (closedBall 0 b + A '' closedBall 0 1) < m * μ (closedBall 0 1) :=
      (tendsto_order.1 L2).2 _ I
    exact (H.and self_mem_nhdsWithin).exists
  have : Iio (⟨ε, εpos.le⟩ : ℝ≥0) ∈ 𝓝 (0 : ℝ≥0) := by apply Iio_mem_nhds; exact εpos
  filter_upwards [this]
  -- fix a function `f` which is close enough to `A`.
  intro δ hδ s f hf
  -- This function expands the volume of any ball by at most `m`
  have I : ∀ x r, x ∈ s → 0 ≤ r → μ (f '' (s ∩ closedBall x r)) ≤ m * μ (closedBall x r) := by
    

Proof state:
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
s✝ : Set E
f✝ : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
A : E →L[ℝ] E
m : ℝ≥0
hm : ENNReal.ofReal |ContinuousLinearMap.det A| < ↑m
d : ℝ≥0∞ := ENNReal.ofReal |ContinuousLinearMap.det A|
ε : ℝ
hε : ↑↑μ (closedBall 0 ε + ⇑A '' closedBall 0 1) < ↑m * ↑↑μ (closedBall 0 1)
εpos : 0 < ε
this : Iio { val := ε, property := (_ : 0 ≤ ε) } ∈ 𝓝 0
δ : ℝ≥0
hδ : δ ∈ Iio { val := ε, property := (_ : 0 ≤ ε) }
s : Set E
f : E → E
hf : ApproximatesLinearOn f A s δ
⊢ ∀ (x : E) (r : ℝ), x ∈ s → 0 ≤ r → ↑↑μ (f '' (s ∩ closedBall x r)) ≤ ↑m * ↑↑μ (closedBall x r)

Next tactic:
intro x r xs r0
/-- Let `f` be a function which is sufficiently close (in the Lipschitz sense) to a given linear
map `A`. Then it expands the volume of any set by at least `m` for any `m < det A`. -/
theorem mul_le_addHaar_image_of_lt_det (A : E →L[ℝ] E) {m : ℝ≥0}
    (hm : (m : ℝ≥0∞) < ENNReal.ofReal |A.det|) :
    ∀ᶠ δ in 𝓝[>] (0 : ℝ≥0),
      ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → (m : ℝ≥0∞) * μ s ≤ μ (f '' s) := by
  

Proof state:
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
A : E →L[ℝ] E
m : ℝ≥0
hm : ↑m < ENNReal.ofReal |ContinuousLinearMap.det A|
⊢ ∀ᶠ (δ : ℝ≥0) in 𝓝[>] 0, ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → ↑m * ↑↑μ s ≤ ↑↑μ (f '' s)

Next tactic:
apply nhdsWithin_le_nhds
/-- Let `f` be a function which is sufficiently close (in the Lipschitz sense) to a given linear
map `A`. Then it expands the volume of any set by at least `m` for any `m < det A`. -/
theorem mul_le_addHaar_image_of_lt_det (A : E →L[ℝ] E) {m : ℝ≥0}
    (hm : (m : ℝ≥0∞) < ENNReal.ofReal |A.det|) :
    ∀ᶠ δ in 𝓝[>] (0 : ℝ≥0),
      ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → (m : ℝ≥0∞) * μ s ≤ μ (f '' s) := by
  apply nhdsWithin_le_nhds
  -- The assumption `hm` implies that `A` is invertible. If `f` is close enough to `A`, it is also
  -- invertible. One can then pass to the inverses, and deduce the estimate from
  -- `addHaar_image_le_mul_of_det_lt` applied to `f⁻¹` and `A⁻¹`.
  -- exclude first the trivial case where `m = 0`.
  

Proof state:
case a
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
A : E →L[ℝ] E
m : ℝ≥0
hm : ↑m < ENNReal.ofReal |ContinuousLinearMap.det A|
⊢ {x | (fun δ => ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → ↑m * ↑↑μ s ≤ ↑↑μ (f '' s)) x} ∈ 𝓝 0

Next tactic:
rcases eq_or_lt_of_le (zero_le m) with (rfl | mpos)
/-- Let `f` be a function which is sufficiently close (in the Lipschitz sense) to a given linear
map `A`. Then it expands the volume of any set by at least `m` for any `m < det A`. -/
theorem mul_le_addHaar_image_of_lt_det (A : E →L[ℝ] E) {m : ℝ≥0}
    (hm : (m : ℝ≥0∞) < ENNReal.ofReal |A.det|) :
    ∀ᶠ δ in 𝓝[>] (0 : ℝ≥0),
      ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → (m : ℝ≥0∞) * μ s ≤ μ (f '' s) := by
  apply nhdsWithin_le_nhds
  -- The assumption `hm` implies that `A` is invertible. If `f` is close enough to `A`, it is also
  -- invertible. One can then pass to the inverses, and deduce the estimate from
  -- `addHaar_image_le_mul_of_det_lt` applied to `f⁻¹` and `A⁻¹`.
  -- exclude first the trivial case where `m = 0`.
  rcases eq_or_lt_of_le (zero_le m) with (rfl | mpos)
  · 

Proof state:
case a.inl
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
A : E →L[ℝ] E
hm : ↑0 < ENNReal.ofReal |ContinuousLinearMap.det A|
⊢ {x | (fun δ => ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → ↑0 * ↑↑μ s ≤ ↑↑μ (f '' s)) x} ∈ 𝓝 0

Next tactic:
apply eventually_of_forall
/-- Let `f` be a function which is sufficiently close (in the Lipschitz sense) to a given linear
map `A`. Then it expands the volume of any set by at least `m` for any `m < det A`. -/
theorem mul_le_addHaar_image_of_lt_det (A : E →L[ℝ] E) {m : ℝ≥0}
    (hm : (m : ℝ≥0∞) < ENNReal.ofReal |A.det|) :
    ∀ᶠ δ in 𝓝[>] (0 : ℝ≥0),
      ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → (m : ℝ≥0∞) * μ s ≤ μ (f '' s) := by
  apply nhdsWithin_le_nhds
  -- The assumption `hm` implies that `A` is invertible. If `f` is close enough to `A`, it is also
  -- invertible. One can then pass to the inverses, and deduce the estimate from
  -- `addHaar_image_le_mul_of_det_lt` applied to `f⁻¹` and `A⁻¹`.
  -- exclude first the trivial case where `m = 0`.
  rcases eq_or_lt_of_le (zero_le m) with (rfl | mpos)
  · apply eventually_of_forall
    

Proof state:
case a.inl.hp
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
A : E →L[ℝ] E
hm : ↑0 < ENNReal.ofReal |ContinuousLinearMap.det A|
⊢ ∀ (x : ℝ≥0) (s : Set E) (f : E → E), ApproximatesLinearOn f A s x → ↑0 * ↑↑μ s ≤ ↑↑μ (f '' s)

Next tactic:
simp only [forall_const, zero_mul, imp_true_iff, zero_le, ENNReal.coe_zero]
/-- Let `f` be a function which is sufficiently close (in the Lipschitz sense) to a given linear
map `A`. Then it expands the volume of any set by at least `m` for any `m < det A`. -/
theorem mul_le_addHaar_image_of_lt_det (A : E →L[ℝ] E) {m : ℝ≥0}
    (hm : (m : ℝ≥0∞) < ENNReal.ofReal |A.det|) :
    ∀ᶠ δ in 𝓝[>] (0 : ℝ≥0),
      ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → (m : ℝ≥0∞) * μ s ≤ μ (f '' s) := by
  apply nhdsWithin_le_nhds
  -- The assumption `hm` implies that `A` is invertible. If `f` is close enough to `A`, it is also
  -- invertible. One can then pass to the inverses, and deduce the estimate from
  -- `addHaar_image_le_mul_of_det_lt` applied to `f⁻¹` and `A⁻¹`.
  -- exclude first the trivial case where `m = 0`.
  rcases eq_or_lt_of_le (zero_le m) with (rfl | mpos)
  · apply eventually_of_forall
    simp only [forall_const, zero_mul, imp_true_iff, zero_le, ENNReal.coe_zero]
  

Proof state:
case a.inr
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
A : E →L[ℝ] E
m : ℝ≥0
hm : ↑m < ENNReal.ofReal |ContinuousLinearMap.det A|
mpos : 0 < m
⊢ {x | (fun δ => ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → ↑m * ↑↑μ s ≤ ↑↑μ (f '' s)) x} ∈ 𝓝 0

Next tactic:
have hA : A.det ≠ 0 := by
    intro h; simp only [h, ENNReal.not_lt_zero, ENNReal.ofReal_zero, abs_zero] at hm
/-- Let `f` be a function which is sufficiently close (in the Lipschitz sense) to a given linear
map `A`. Then it expands the volume of any set by at least `m` for any `m < det A`. -/
theorem mul_le_addHaar_image_of_lt_det (A : E →L[ℝ] E) {m : ℝ≥0}
    (hm : (m : ℝ≥0∞) < ENNReal.ofReal |A.det|) :
    ∀ᶠ δ in 𝓝[>] (0 : ℝ≥0),
      ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → (m : ℝ≥0∞) * μ s ≤ μ (f '' s) := by
  apply nhdsWithin_le_nhds
  -- The assumption `hm` implies that `A` is invertible. If `f` is close enough to `A`, it is also
  -- invertible. One can then pass to the inverses, and deduce the estimate from
  -- `addHaar_image_le_mul_of_det_lt` applied to `f⁻¹` and `A⁻¹`.
  -- exclude first the trivial case where `m = 0`.
  rcases eq_or_lt_of_le (zero_le m) with (rfl | mpos)
  · apply eventually_of_forall
    simp only [forall_const, zero_mul, imp_true_iff, zero_le, ENNReal.coe_zero]
  have hA : A.det ≠ 0 := by
    

Proof state:
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
A : E →L[ℝ] E
m : ℝ≥0
hm : ↑m < ENNReal.ofReal |ContinuousLinearMap.det A|
mpos : 0 < m
⊢ ContinuousLinearMap.det A ≠ 0

Next tactic:
intro h
/-- Let `f` be a function which is sufficiently close (in the Lipschitz sense) to a given linear
map `A`. Then it expands the volume of any set by at least `m` for any `m < det A`. -/
theorem mul_le_addHaar_image_of_lt_det (A : E →L[ℝ] E) {m : ℝ≥0}
    (hm : (m : ℝ≥0∞) < ENNReal.ofReal |A.det|) :
    ∀ᶠ δ in 𝓝[>] (0 : ℝ≥0),
      ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → (m : ℝ≥0∞) * μ s ≤ μ (f '' s) := by
  apply nhdsWithin_le_nhds
  -- The assumption `hm` implies that `A` is invertible. If `f` is close enough to `A`, it is also
  -- invertible. One can then pass to the inverses, and deduce the estimate from
  -- `addHaar_image_le_mul_of_det_lt` applied to `f⁻¹` and `A⁻¹`.
  -- exclude first the trivial case where `m = 0`.
  rcases eq_or_lt_of_le (zero_le m) with (rfl | mpos)
  · apply eventually_of_forall
    simp only [forall_const, zero_mul, imp_true_iff, zero_le, ENNReal.coe_zero]
  have hA : A.det ≠ 0 := by
    intro h; 

Proof state:
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
A : E →L[ℝ] E
m : ℝ≥0
hm : ↑m < ENNReal.ofReal |ContinuousLinearMap.det A|
mpos : 0 < m
h : ContinuousLinearMap.det A = 0
⊢ False

Next tactic:
simp only [h, ENNReal.not_lt_zero, ENNReal.ofReal_zero, abs_zero] at hm
/-- Let `f` be a function which is sufficiently close (in the Lipschitz sense) to a given linear
map `A`. Then it expands the volume of any set by at least `m` for any `m < det A`. -/
theorem mul_le_addHaar_image_of_lt_det (A : E →L[ℝ] E) {m : ℝ≥0}
    (hm : (m : ℝ≥0∞) < ENNReal.ofReal |A.det|) :
    ∀ᶠ δ in 𝓝[>] (0 : ℝ≥0),
      ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → (m : ℝ≥0∞) * μ s ≤ μ (f '' s) := by
  apply nhdsWithin_le_nhds
  -- The assumption `hm` implies that `A` is invertible. If `f` is close enough to `A`, it is also
  -- invertible. One can then pass to the inverses, and deduce the estimate from
  -- `addHaar_image_le_mul_of_det_lt` applied to `f⁻¹` and `A⁻¹`.
  -- exclude first the trivial case where `m = 0`.
  rcases eq_or_lt_of_le (zero_le m) with (rfl | mpos)
  · apply eventually_of_forall
    simp only [forall_const, zero_mul, imp_true_iff, zero_le, ENNReal.coe_zero]
  have hA : A.det ≠ 0 := by
    intro h; simp only [h, ENNReal.not_lt_zero, ENNReal.ofReal_zero, abs_zero] at hm
  -- let `B` be the continuous linear equiv version of `A`.
  

Proof state:
case a.inr
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
A : E →L[ℝ] E
m : ℝ≥0
hm : ↑m < ENNReal.ofReal |ContinuousLinearMap.det A|
mpos : 0 < m
hA : ContinuousLinearMap.det A ≠ 0
⊢ {x | (fun δ => ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → ↑m * ↑↑μ s ≤ ↑↑μ (f '' s)) x} ∈ 𝓝 0

Next tactic:
let B := A.toContinuousLinearEquivOfDetNeZero hA
/-- Let `f` be a function which is sufficiently close (in the Lipschitz sense) to a given linear
map `A`. Then it expands the volume of any set by at least `m` for any `m < det A`. -/
theorem mul_le_addHaar_image_of_lt_det (A : E →L[ℝ] E) {m : ℝ≥0}
    (hm : (m : ℝ≥0∞) < ENNReal.ofReal |A.det|) :
    ∀ᶠ δ in 𝓝[>] (0 : ℝ≥0),
      ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → (m : ℝ≥0∞) * μ s ≤ μ (f '' s) := by
  apply nhdsWithin_le_nhds
  -- The assumption `hm` implies that `A` is invertible. If `f` is close enough to `A`, it is also
  -- invertible. One can then pass to the inverses, and deduce the estimate from
  -- `addHaar_image_le_mul_of_det_lt` applied to `f⁻¹` and `A⁻¹`.
  -- exclude first the trivial case where `m = 0`.
  rcases eq_or_lt_of_le (zero_le m) with (rfl | mpos)
  · apply eventually_of_forall
    simp only [forall_const, zero_mul, imp_true_iff, zero_le, ENNReal.coe_zero]
  have hA : A.det ≠ 0 := by
    intro h; simp only [h, ENNReal.not_lt_zero, ENNReal.ofReal_zero, abs_zero] at hm
  -- let `B` be the continuous linear equiv version of `A`.
  let B := A.toContinuousLinearEquivOfDetNeZero hA
  -- the determinant of `B.symm` is bounded by `m⁻¹`
  

Proof state:
case a.inr
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
A : E →L[ℝ] E
m : ℝ≥0
hm : ↑m < ENNReal.ofReal |ContinuousLinearMap.det A|
mpos : 0 < m
hA : ContinuousLinearMap.det A ≠ 0
B : E ≃L[ℝ] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA
⊢ {x | (fun δ => ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → ↑m * ↑↑μ s ≤ ↑↑μ (f '' s)) x} ∈ 𝓝 0

Next tactic:
have I : ENNReal.ofReal |(B.symm : E →L[ℝ] E).det| < (m⁻¹ : ℝ≥0) := by
    simp only [ENNReal.ofReal, abs_inv, Real.toNNReal_inv, ContinuousLinearEquiv.det_coe_symm,
      ContinuousLinearMap.coe_toContinuousLinearEquivOfDetNeZero, ENNReal.coe_lt_coe] at hm ⊢
    exact NNReal.inv_lt_inv mpos.ne' hm
/-- Let `f` be a function which is sufficiently close (in the Lipschitz sense) to a given linear
map `A`. Then it expands the volume of any set by at least `m` for any `m < det A`. -/
theorem mul_le_addHaar_image_of_lt_det (A : E →L[ℝ] E) {m : ℝ≥0}
    (hm : (m : ℝ≥0∞) < ENNReal.ofReal |A.det|) :
    ∀ᶠ δ in 𝓝[>] (0 : ℝ≥0),
      ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → (m : ℝ≥0∞) * μ s ≤ μ (f '' s) := by
  apply nhdsWithin_le_nhds
  -- The assumption `hm` implies that `A` is invertible. If `f` is close enough to `A`, it is also
  -- invertible. One can then pass to the inverses, and deduce the estimate from
  -- `addHaar_image_le_mul_of_det_lt` applied to `f⁻¹` and `A⁻¹`.
  -- exclude first the trivial case where `m = 0`.
  rcases eq_or_lt_of_le (zero_le m) with (rfl | mpos)
  · apply eventually_of_forall
    simp only [forall_const, zero_mul, imp_true_iff, zero_le, ENNReal.coe_zero]
  have hA : A.det ≠ 0 := by
    intro h; simp only [h, ENNReal.not_lt_zero, ENNReal.ofReal_zero, abs_zero] at hm
  -- let `B` be the continuous linear equiv version of `A`.
  let B := A.toContinuousLinearEquivOfDetNeZero hA
  -- the determinant of `B.symm` is bounded by `m⁻¹`
  have I : ENNReal.ofReal |(B.symm : E →L[ℝ] E).det| < (m⁻¹ : ℝ≥0) := by
    

Proof state:
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
A : E →L[ℝ] E
m : ℝ≥0
hm : ↑m < ENNReal.ofReal |ContinuousLinearMap.det A|
mpos : 0 < m
hA : ContinuousLinearMap.det A ≠ 0
B : E ≃L[ℝ] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA
⊢ ENNReal.ofReal |ContinuousLinearMap.det ↑(ContinuousLinearEquiv.symm B)| < ↑m⁻¹

Next tactic:
simp only [ENNReal.ofReal, abs_inv, Real.toNNReal_inv, ContinuousLinearEquiv.det_coe_symm,
      ContinuousLinearMap.coe_toContinuousLinearEquivOfDetNeZero, ENNReal.coe_lt_coe] at hm ⊢