l3lab/ntp-mathlib huggingface dataset:
theorem polarCoord_source_ae_eq_univ : polarCoord.source =ᵐ[volume] univ := by
  have A : polarCoord.sourceᶜ ⊆ LinearMap.ker (LinearMap.snd ℝ ℝ ℝ) := by
    intro x hx
    simp only [polarCoord_source, compl_union, mem_inter_iff, mem_compl_iff, mem_setOf_eq, not_lt,
      Classical.not_not] at hx
    exact hx.2
  have B : volume (LinearMap.ker (LinearMap.snd ℝ ℝ ℝ) : Set (ℝ × ℝ)) = 0 := by
    

Proof state:
A : polarCoord.sourceᶜ ⊆ ↑(LinearMap.ker (LinearMap.snd ℝ ℝ ℝ))
⊢ ↑↑volume ↑(LinearMap.ker (LinearMap.snd ℝ ℝ ℝ)) = 0

Next tactic:
apply Measure.addHaar_submodule
theorem polarCoord_source_ae_eq_univ : polarCoord.source =ᵐ[volume] univ := by
  have A : polarCoord.sourceᶜ ⊆ LinearMap.ker (LinearMap.snd ℝ ℝ ℝ) := by
    intro x hx
    simp only [polarCoord_source, compl_union, mem_inter_iff, mem_compl_iff, mem_setOf_eq, not_lt,
      Classical.not_not] at hx
    exact hx.2
  have B : volume (LinearMap.ker (LinearMap.snd ℝ ℝ ℝ) : Set (ℝ × ℝ)) = 0 := by
    apply Measure.addHaar_submodule
    

Proof state:
case hs
A : polarCoord.sourceᶜ ⊆ ↑(LinearMap.ker (LinearMap.snd ℝ ℝ ℝ))
⊢ LinearMap.ker (LinearMap.snd ℝ ℝ ℝ) ≠ ⊤

Next tactic:
rw [Ne.def, LinearMap.ker_eq_top]
theorem polarCoord_source_ae_eq_univ : polarCoord.source =ᵐ[volume] univ := by
  have A : polarCoord.sourceᶜ ⊆ LinearMap.ker (LinearMap.snd ℝ ℝ ℝ) := by
    intro x hx
    simp only [polarCoord_source, compl_union, mem_inter_iff, mem_compl_iff, mem_setOf_eq, not_lt,
      Classical.not_not] at hx
    exact hx.2
  have B : volume (LinearMap.ker (LinearMap.snd ℝ ℝ ℝ) : Set (ℝ × ℝ)) = 0 := by
    apply Measure.addHaar_submodule
    rw [Ne.def, LinearMap.ker_eq_top]
    

Proof state:
case hs
A : polarCoord.sourceᶜ ⊆ ↑(LinearMap.ker (LinearMap.snd ℝ ℝ ℝ))
⊢ ¬LinearMap.snd ℝ ℝ ℝ = 0

Next tactic:
intro h
theorem polarCoord_source_ae_eq_univ : polarCoord.source =ᵐ[volume] univ := by
  have A : polarCoord.sourceᶜ ⊆ LinearMap.ker (LinearMap.snd ℝ ℝ ℝ) := by
    intro x hx
    simp only [polarCoord_source, compl_union, mem_inter_iff, mem_compl_iff, mem_setOf_eq, not_lt,
      Classical.not_not] at hx
    exact hx.2
  have B : volume (LinearMap.ker (LinearMap.snd ℝ ℝ ℝ) : Set (ℝ × ℝ)) = 0 := by
    apply Measure.addHaar_submodule
    rw [Ne.def, LinearMap.ker_eq_top]
    intro h
    

Proof state:
case hs
A : polarCoord.sourceᶜ ⊆ ↑(LinearMap.ker (LinearMap.snd ℝ ℝ ℝ))
h : LinearMap.snd ℝ ℝ ℝ = 0
⊢ False

Next tactic:
have : (LinearMap.snd ℝ ℝ ℝ) (0, 1) = (0 : ℝ × ℝ →ₗ[ℝ] ℝ) (0, 1) := by rw [h]
theorem polarCoord_source_ae_eq_univ : polarCoord.source =ᵐ[volume] univ := by
  have A : polarCoord.sourceᶜ ⊆ LinearMap.ker (LinearMap.snd ℝ ℝ ℝ) := by
    intro x hx
    simp only [polarCoord_source, compl_union, mem_inter_iff, mem_compl_iff, mem_setOf_eq, not_lt,
      Classical.not_not] at hx
    exact hx.2
  have B : volume (LinearMap.ker (LinearMap.snd ℝ ℝ ℝ) : Set (ℝ × ℝ)) = 0 := by
    apply Measure.addHaar_submodule
    rw [Ne.def, LinearMap.ker_eq_top]
    intro h
    have : (LinearMap.snd ℝ ℝ ℝ) (0, 1) = (0 : ℝ × ℝ →ₗ[ℝ] ℝ) (0, 1) := by 

Proof state:
A : polarCoord.sourceᶜ ⊆ ↑(LinearMap.ker (LinearMap.snd ℝ ℝ ℝ))
h : LinearMap.snd ℝ ℝ ℝ = 0
⊢ (LinearMap.snd ℝ ℝ ℝ) (0, 1) = 0 (0, 1)

Next tactic:
rw [h]
theorem polarCoord_source_ae_eq_univ : polarCoord.source =ᵐ[volume] univ := by
  have A : polarCoord.sourceᶜ ⊆ LinearMap.ker (LinearMap.snd ℝ ℝ ℝ) := by
    intro x hx
    simp only [polarCoord_source, compl_union, mem_inter_iff, mem_compl_iff, mem_setOf_eq, not_lt,
      Classical.not_not] at hx
    exact hx.2
  have B : volume (LinearMap.ker (LinearMap.snd ℝ ℝ ℝ) : Set (ℝ × ℝ)) = 0 := by
    apply Measure.addHaar_submodule
    rw [Ne.def, LinearMap.ker_eq_top]
    intro h
    have : (LinearMap.snd ℝ ℝ ℝ) (0, 1) = (0 : ℝ × ℝ →ₗ[ℝ] ℝ) (0, 1) := by rw [h]
    

Proof state:
case hs
A : polarCoord.sourceᶜ ⊆ ↑(LinearMap.ker (LinearMap.snd ℝ ℝ ℝ))
h : LinearMap.snd ℝ ℝ ℝ = 0
this : (LinearMap.snd ℝ ℝ ℝ) (0, 1) = 0 (0, 1)
⊢ False

Next tactic:
simp at this
theorem polarCoord_source_ae_eq_univ : polarCoord.source =ᵐ[volume] univ := by
  have A : polarCoord.sourceᶜ ⊆ LinearMap.ker (LinearMap.snd ℝ ℝ ℝ) := by
    intro x hx
    simp only [polarCoord_source, compl_union, mem_inter_iff, mem_compl_iff, mem_setOf_eq, not_lt,
      Classical.not_not] at hx
    exact hx.2
  have B : volume (LinearMap.ker (LinearMap.snd ℝ ℝ ℝ) : Set (ℝ × ℝ)) = 0 := by
    apply Measure.addHaar_submodule
    rw [Ne.def, LinearMap.ker_eq_top]
    intro h
    have : (LinearMap.snd ℝ ℝ ℝ) (0, 1) = (0 : ℝ × ℝ →ₗ[ℝ] ℝ) (0, 1) := by rw [h]
    simp at this
  

Proof state:
A : polarCoord.sourceᶜ ⊆ ↑(LinearMap.ker (LinearMap.snd ℝ ℝ ℝ))
B : ↑↑volume ↑(LinearMap.ker (LinearMap.snd ℝ ℝ ℝ)) = 0
⊢ polarCoord.source =ᵐ[volume] univ

Next tactic:
simp only [ae_eq_univ]
theorem polarCoord_source_ae_eq_univ : polarCoord.source =ᵐ[volume] univ := by
  have A : polarCoord.sourceᶜ ⊆ LinearMap.ker (LinearMap.snd ℝ ℝ ℝ) := by
    intro x hx
    simp only [polarCoord_source, compl_union, mem_inter_iff, mem_compl_iff, mem_setOf_eq, not_lt,
      Classical.not_not] at hx
    exact hx.2
  have B : volume (LinearMap.ker (LinearMap.snd ℝ ℝ ℝ) : Set (ℝ × ℝ)) = 0 := by
    apply Measure.addHaar_submodule
    rw [Ne.def, LinearMap.ker_eq_top]
    intro h
    have : (LinearMap.snd ℝ ℝ ℝ) (0, 1) = (0 : ℝ × ℝ →ₗ[ℝ] ℝ) (0, 1) := by rw [h]
    simp at this
  simp only [ae_eq_univ]
  

Proof state:
A : polarCoord.sourceᶜ ⊆ ↑(LinearMap.ker (LinearMap.snd ℝ ℝ ℝ))
B : ↑↑volume ↑(LinearMap.ker (LinearMap.snd ℝ ℝ ℝ)) = 0
⊢ ↑↑volume polarCoord.sourceᶜ = 0

Next tactic:
exact le_antisymm ((measure_mono A).trans (le_of_eq B)) bot_le
theorem integral_comp_polarCoord_symm {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    (f : ℝ × ℝ → E) :
    (∫ p in polarCoord.target, p.1 • f (polarCoord.symm p)) = ∫ p, f p := by
  

Proof state:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ × ℝ → E
⊢ ∫ (p : ℝ × ℝ) in polarCoord.target, p.1 • f (↑(PartialHomeomorph.symm polarCoord) p) = ∫ (p : ℝ × ℝ), f p

Next tactic:
set B : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ := fun p =>
    LinearMap.toContinuousLinearMap (Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)
      !![cos p.2, -p.1 * sin p.2; sin p.2, p.1 * cos p.2])
theorem integral_comp_polarCoord_symm {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    (f : ℝ × ℝ → E) :
    (∫ p in polarCoord.target, p.1 • f (polarCoord.symm p)) = ∫ p, f p := by
  set B : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ := fun p =>
    LinearMap.toContinuousLinearMap (Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)
      !![cos p.2, -p.1 * sin p.2; sin p.2, p.1 * cos p.2])
  

Proof state:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ × ℝ → E
B : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ :=
  fun p =>
    LinearMap.toContinuousLinearMap
      ((Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ))
        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))
⊢ ∫ (p : ℝ × ℝ) in polarCoord.target, p.1 • f (↑(PartialHomeomorph.symm polarCoord) p) = ∫ (p : ℝ × ℝ), f p

Next tactic:
have A : ∀ p ∈ polarCoord.symm.source, HasFDerivAt polarCoord.symm (B p) p := fun p _ =>
    hasFDerivAt_polarCoord_symm p
theorem integral_comp_polarCoord_symm {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    (f : ℝ × ℝ → E) :
    (∫ p in polarCoord.target, p.1 • f (polarCoord.symm p)) = ∫ p, f p := by
  set B : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ := fun p =>
    LinearMap.toContinuousLinearMap (Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)
      !![cos p.2, -p.1 * sin p.2; sin p.2, p.1 * cos p.2])
  have A : ∀ p ∈ polarCoord.symm.source, HasFDerivAt polarCoord.symm (B p) p := fun p _ =>
    hasFDerivAt_polarCoord_symm p
  

Proof state:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ × ℝ → E
B : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ :=
  fun p =>
    LinearMap.toContinuousLinearMap
      ((Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ))
        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))
A :
  ∀ p ∈ (PartialHomeomorph.symm polarCoord).toPartialEquiv.source,
    HasFDerivAt (↑(PartialHomeomorph.symm polarCoord)) (B p) p
⊢ ∫ (p : ℝ × ℝ) in polarCoord.target, p.1 • f (↑(PartialHomeomorph.symm polarCoord) p) = ∫ (p : ℝ × ℝ), f p

Next tactic:
have B_det : ∀ p, (B p).det = p.1 := by
    intro p
    conv_rhs => rw [← one_mul p.1, ← cos_sq_add_sin_sq p.2]
    simp only [neg_mul, LinearMap.det_toContinuousLinearMap, LinearMap.det_toLin,
      Matrix.det_fin_two_of, sub_neg_eq_add]
    ring
theorem integral_comp_polarCoord_symm {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    (f : ℝ × ℝ → E) :
    (∫ p in polarCoord.target, p.1 • f (polarCoord.symm p)) = ∫ p, f p := by
  set B : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ := fun p =>
    LinearMap.toContinuousLinearMap (Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)
      !![cos p.2, -p.1 * sin p.2; sin p.2, p.1 * cos p.2])
  have A : ∀ p ∈ polarCoord.symm.source, HasFDerivAt polarCoord.symm (B p) p := fun p _ =>
    hasFDerivAt_polarCoord_symm p
  have B_det : ∀ p, (B p).det = p.1 := by
    

Proof state:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ × ℝ → E
B : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ :=
  fun p =>
    LinearMap.toContinuousLinearMap
      ((Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ))
        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))
A :
  ∀ p ∈ (PartialHomeomorph.symm polarCoord).toPartialEquiv.source,
    HasFDerivAt (↑(PartialHomeomorph.symm polarCoord)) (B p) p
⊢ ∀ (p : ℝ × ℝ), ContinuousLinearMap.det (B p) = p.1

Next tactic:
intro p
theorem integral_comp_polarCoord_symm {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    (f : ℝ × ℝ → E) :
    (∫ p in polarCoord.target, p.1 • f (polarCoord.symm p)) = ∫ p, f p := by
  set B : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ := fun p =>
    LinearMap.toContinuousLinearMap (Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)
      !![cos p.2, -p.1 * sin p.2; sin p.2, p.1 * cos p.2])
  have A : ∀ p ∈ polarCoord.symm.source, HasFDerivAt polarCoord.symm (B p) p := fun p _ =>
    hasFDerivAt_polarCoord_symm p
  have B_det : ∀ p, (B p).det = p.1 := by
    intro p
    

Proof state:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ × ℝ → E
B : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ :=
  fun p =>
    LinearMap.toContinuousLinearMap
      ((Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ))
        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))
A :
  ∀ p ∈ (PartialHomeomorph.symm polarCoord).toPartialEquiv.source,
    HasFDerivAt (↑(PartialHomeomorph.symm polarCoord)) (B p) p
p : ℝ × ℝ
⊢ ContinuousLinearMap.det (B p) = p.1

Next tactic:
conv_rhs => rw [← one_mul p.1, ← cos_sq_add_sin_sq p.2]
theorem integral_comp_polarCoord_symm {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    (f : ℝ × ℝ → E) :
    (∫ p in polarCoord.target, p.1 • f (polarCoord.symm p)) = ∫ p, f p := by
  set B : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ := fun p =>
    LinearMap.toContinuousLinearMap (Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)
      !![cos p.2, -p.1 * sin p.2; sin p.2, p.1 * cos p.2])
  have A : ∀ p ∈ polarCoord.symm.source, HasFDerivAt polarCoord.symm (B p) p := fun p _ =>
    hasFDerivAt_polarCoord_symm p
  have B_det : ∀ p, (B p).det = p.1 := by
    intro p
    conv_rhs => 

Proof state:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ × ℝ → E
B : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ :=
  fun p =>
    LinearMap.toContinuousLinearMap
      ((Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ))
        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))
A :
  ∀ p ∈ (PartialHomeomorph.symm polarCoord).toPartialEquiv.source,
    HasFDerivAt (↑(PartialHomeomorph.symm polarCoord)) (B p) p
p : ℝ × ℝ
| p.1

Next tactic:
rw [← one_mul p.1, ← cos_sq_add_sin_sq p.2]
theorem integral_comp_polarCoord_symm {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    (f : ℝ × ℝ → E) :
    (∫ p in polarCoord.target, p.1 • f (polarCoord.symm p)) = ∫ p, f p := by
  set B : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ := fun p =>
    LinearMap.toContinuousLinearMap (Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)
      !![cos p.2, -p.1 * sin p.2; sin p.2, p.1 * cos p.2])
  have A : ∀ p ∈ polarCoord.symm.source, HasFDerivAt polarCoord.symm (B p) p := fun p _ =>
    hasFDerivAt_polarCoord_symm p
  have B_det : ∀ p, (B p).det = p.1 := by
    intro p
    conv_rhs => rw [← one_mul p.1, ← cos_sq_add_sin_sq p.2]
    

Proof state:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ × ℝ → E
B : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ :=
  fun p =>
    LinearMap.toContinuousLinearMap
      ((Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ))
        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))
A :
  ∀ p ∈ (PartialHomeomorph.symm polarCoord).toPartialEquiv.source,
    HasFDerivAt (↑(PartialHomeomorph.symm polarCoord)) (B p) p
p : ℝ × ℝ
⊢ ContinuousLinearMap.det (B p) = (cos p.2 ^ 2 + sin p.2 ^ 2) * p.1

Next tactic:
simp only [neg_mul, LinearMap.det_toContinuousLinearMap, LinearMap.det_toLin,
      Matrix.det_fin_two_of, sub_neg_eq_add]
theorem integral_comp_polarCoord_symm {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    (f : ℝ × ℝ → E) :
    (∫ p in polarCoord.target, p.1 • f (polarCoord.symm p)) = ∫ p, f p := by
  set B : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ := fun p =>
    LinearMap.toContinuousLinearMap (Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)
      !![cos p.2, -p.1 * sin p.2; sin p.2, p.1 * cos p.2])
  have A : ∀ p ∈ polarCoord.symm.source, HasFDerivAt polarCoord.symm (B p) p := fun p _ =>
    hasFDerivAt_polarCoord_symm p
  have B_det : ∀ p, (B p).det = p.1 := by
    intro p
    conv_rhs => rw [← one_mul p.1, ← cos_sq_add_sin_sq p.2]
    simp only [neg_mul, LinearMap.det_toContinuousLinearMap, LinearMap.det_toLin,
      Matrix.det_fin_two_of, sub_neg_eq_add]
    

Proof state:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ × ℝ → E
B : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ :=
  fun p =>
    LinearMap.toContinuousLinearMap
      ((Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ))
        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))
A :
  ∀ p ∈ (PartialHomeomorph.symm polarCoord).toPartialEquiv.source,
    HasFDerivAt (↑(PartialHomeomorph.symm polarCoord)) (B p) p
p : ℝ × ℝ
⊢ cos p.2 * (p.1 * cos p.2) + p.1 * sin p.2 * sin p.2 = (cos p.2 ^ 2 + sin p.2 ^ 2) * p.1

Next tactic:
ring
theorem integral_comp_polarCoord_symm {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    (f : ℝ × ℝ → E) :
    (∫ p in polarCoord.target, p.1 • f (polarCoord.symm p)) = ∫ p, f p := by
  set B : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ := fun p =>
    LinearMap.toContinuousLinearMap (Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)
      !![cos p.2, -p.1 * sin p.2; sin p.2, p.1 * cos p.2])
  have A : ∀ p ∈ polarCoord.symm.source, HasFDerivAt polarCoord.symm (B p) p := fun p _ =>
    hasFDerivAt_polarCoord_symm p
  have B_det : ∀ p, (B p).det = p.1 := by
    intro p
    conv_rhs => rw [← one_mul p.1, ← cos_sq_add_sin_sq p.2]
    simp only [neg_mul, LinearMap.det_toContinuousLinearMap, LinearMap.det_toLin,
      Matrix.det_fin_two_of, sub_neg_eq_add]
    ring
  

Proof state:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ × ℝ → E
B : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ :=
  fun p =>
    LinearMap.toContinuousLinearMap
      ((Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ))
        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))
A :
  ∀ p ∈ (PartialHomeomorph.symm polarCoord).toPartialEquiv.source,
    HasFDerivAt (↑(PartialHomeomorph.symm polarCoord)) (B p) p
B_det : ∀ (p : ℝ × ℝ), ContinuousLinearMap.det (B p) = p.1
⊢ ∫ (p : ℝ × ℝ) in polarCoord.target, p.1 • f (↑(PartialHomeomorph.symm polarCoord) p) = ∫ (p : ℝ × ℝ), f p

Next tactic:
symm
theorem integral_comp_polarCoord_symm {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    (f : ℝ × ℝ → E) :
    (∫ p in polarCoord.target, p.1 • f (polarCoord.symm p)) = ∫ p, f p := by
  set B : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ := fun p =>
    LinearMap.toContinuousLinearMap (Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)
      !![cos p.2, -p.1 * sin p.2; sin p.2, p.1 * cos p.2])
  have A : ∀ p ∈ polarCoord.symm.source, HasFDerivAt polarCoord.symm (B p) p := fun p _ =>
    hasFDerivAt_polarCoord_symm p
  have B_det : ∀ p, (B p).det = p.1 := by
    intro p
    conv_rhs => rw [← one_mul p.1, ← cos_sq_add_sin_sq p.2]
    simp only [neg_mul, LinearMap.det_toContinuousLinearMap, LinearMap.det_toLin,
      Matrix.det_fin_two_of, sub_neg_eq_add]
    ring
  symm
  

Proof state:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ × ℝ → E
B : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ :=
  fun p =>
    LinearMap.toContinuousLinearMap
      ((Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ))
        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))
A :
  ∀ p ∈ (PartialHomeomorph.symm polarCoord).toPartialEquiv.source,
    HasFDerivAt (↑(PartialHomeomorph.symm polarCoord)) (B p) p
B_det : ∀ (p : ℝ × ℝ), ContinuousLinearMap.det (B p) = p.1
⊢ ∫ (p : ℝ × ℝ), f p = ∫ (p : ℝ × ℝ) in polarCoord.target, p.1 • f (↑(PartialHomeomorph.symm polarCoord) p)

Next tactic:
calc
    ∫ p, f p = ∫ p in polarCoord.source, f p := by
      rw [← integral_univ]
      apply set_integral_congr_set_ae
      exact polarCoord_source_ae_eq_univ.symm
    _ = ∫ p in polarCoord.target, abs (B p).det • f (polarCoord.symm p) := by
      apply integral_target_eq_integral_abs_det_fderiv_smul volume A
    _ = ∫ p in polarCoord.target, p.1 • f (polarCoord.symm p) := by
      apply set_integral_congr polarCoord.open_target.measurableSet fun x hx => ?_
      rw [B_det, abs_of_pos]
      exact hx.1
theorem integral_comp_polarCoord_symm {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    (f : ℝ × ℝ → E) :
    (∫ p in polarCoord.target, p.1 • f (polarCoord.symm p)) = ∫ p, f p := by
  set B : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ := fun p =>
    LinearMap.toContinuousLinearMap (Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)
      !![cos p.2, -p.1 * sin p.2; sin p.2, p.1 * cos p.2])
  have A : ∀ p ∈ polarCoord.symm.source, HasFDerivAt polarCoord.symm (B p) p := fun p _ =>
    hasFDerivAt_polarCoord_symm p
  have B_det : ∀ p, (B p).det = p.1 := by
    intro p
    conv_rhs => rw [← one_mul p.1, ← cos_sq_add_sin_sq p.2]
    simp only [neg_mul, LinearMap.det_toContinuousLinearMap, LinearMap.det_toLin,
      Matrix.det_fin_two_of, sub_neg_eq_add]
    ring
  symm
  calc
    ∫ p, f p = ∫ p in polarCoord.source, f p := by
      

Proof state:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ × ℝ → E
B : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ :=
  fun p =>
    LinearMap.toContinuousLinearMap
      ((Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ))
        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))
A :
  ∀ p ∈ (PartialHomeomorph.symm polarCoord).toPartialEquiv.source,
    HasFDerivAt (↑(PartialHomeomorph.symm polarCoord)) (B p) p
B_det : ∀ (p : ℝ × ℝ), ContinuousLinearMap.det (B p) = p.1
⊢ ∫ (p : ℝ × ℝ), f p = ∫ (p : ℝ × ℝ) in polarCoord.source, f p

Next tactic:
rw [← integral_univ]
theorem integral_comp_polarCoord_symm {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    (f : ℝ × ℝ → E) :
    (∫ p in polarCoord.target, p.1 • f (polarCoord.symm p)) = ∫ p, f p := by
  set B : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ := fun p =>
    LinearMap.toContinuousLinearMap (Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)
      !![cos p.2, -p.1 * sin p.2; sin p.2, p.1 * cos p.2])
  have A : ∀ p ∈ polarCoord.symm.source, HasFDerivAt polarCoord.symm (B p) p := fun p _ =>
    hasFDerivAt_polarCoord_symm p
  have B_det : ∀ p, (B p).det = p.1 := by
    intro p
    conv_rhs => rw [← one_mul p.1, ← cos_sq_add_sin_sq p.2]
    simp only [neg_mul, LinearMap.det_toContinuousLinearMap, LinearMap.det_toLin,
      Matrix.det_fin_two_of, sub_neg_eq_add]
    ring
  symm
  calc
    ∫ p, f p = ∫ p in polarCoord.source, f p := by
      rw [← integral_univ]
      

Proof state:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ × ℝ → E
B : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ :=
  fun p =>
    LinearMap.toContinuousLinearMap
      ((Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ))
        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))
A :
  ∀ p ∈ (PartialHomeomorph.symm polarCoord).toPartialEquiv.source,
    HasFDerivAt (↑(PartialHomeomorph.symm polarCoord)) (B p) p
B_det : ∀ (p : ℝ × ℝ), ContinuousLinearMap.det (B p) = p.1
⊢ ∫ (x : ℝ × ℝ) in univ, f x = ∫ (p : ℝ × ℝ) in polarCoord.source, f p

Next tactic:
apply set_integral_congr_set_ae
theorem integral_comp_polarCoord_symm {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    (f : ℝ × ℝ → E) :
    (∫ p in polarCoord.target, p.1 • f (polarCoord.symm p)) = ∫ p, f p := by
  set B : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ := fun p =>
    LinearMap.toContinuousLinearMap (Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)
      !![cos p.2, -p.1 * sin p.2; sin p.2, p.1 * cos p.2])
  have A : ∀ p ∈ polarCoord.symm.source, HasFDerivAt polarCoord.symm (B p) p := fun p _ =>
    hasFDerivAt_polarCoord_symm p
  have B_det : ∀ p, (B p).det = p.1 := by
    intro p
    conv_rhs => rw [← one_mul p.1, ← cos_sq_add_sin_sq p.2]
    simp only [neg_mul, LinearMap.det_toContinuousLinearMap, LinearMap.det_toLin,
      Matrix.det_fin_two_of, sub_neg_eq_add]
    ring
  symm
  calc
    ∫ p, f p = ∫ p in polarCoord.source, f p := by
      rw [← integral_univ]
      apply set_integral_congr_set_ae
      

Proof state:
case hst
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ × ℝ → E
B : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ :=
  fun p =>
    LinearMap.toContinuousLinearMap
      ((Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ))
        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))
A :
  ∀ p ∈ (PartialHomeomorph.symm polarCoord).toPartialEquiv.source,
    HasFDerivAt (↑(PartialHomeomorph.symm polarCoord)) (B p) p
B_det : ∀ (p : ℝ × ℝ), ContinuousLinearMap.det (B p) = p.1
⊢ univ =ᵐ[volume] polarCoord.source

Next tactic:
exact polarCoord_source_ae_eq_univ.symm
theorem integral_comp_polarCoord_symm {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    (f : ℝ × ℝ → E) :
    (∫ p in polarCoord.target, p.1 • f (polarCoord.symm p)) = ∫ p, f p := by
  set B : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ := fun p =>
    LinearMap.toContinuousLinearMap (Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)
      !![cos p.2, -p.1 * sin p.2; sin p.2, p.1 * cos p.2])
  have A : ∀ p ∈ polarCoord.symm.source, HasFDerivAt polarCoord.symm (B p) p := fun p _ =>
    hasFDerivAt_polarCoord_symm p
  have B_det : ∀ p, (B p).det = p.1 := by
    intro p
    conv_rhs => rw [← one_mul p.1, ← cos_sq_add_sin_sq p.2]
    simp only [neg_mul, LinearMap.det_toContinuousLinearMap, LinearMap.det_toLin,
      Matrix.det_fin_two_of, sub_neg_eq_add]
    ring
  symm
  calc
    ∫ p, f p = ∫ p in polarCoord.source, f p := by
      rw [← integral_univ]
      apply set_integral_congr_set_ae
      exact polarCoord_source_ae_eq_univ.symm
    _ = ∫ p in polarCoord.target, abs (B p).det • f (polarCoord.symm p) := by
      

Proof state:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ × ℝ → E
B : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ :=
  fun p =>
    LinearMap.toContinuousLinearMap
      ((Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ))
        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))
A :
  ∀ p ∈ (PartialHomeomorph.symm polarCoord).toPartialEquiv.source,
    HasFDerivAt (↑(PartialHomeomorph.symm polarCoord)) (B p) p
B_det : ∀ (p : ℝ × ℝ), ContinuousLinearMap.det (B p) = p.1
⊢ ∫ (p : ℝ × ℝ) in polarCoord.source, f p =
    ∫ (p : ℝ × ℝ) in polarCoord.target, |ContinuousLinearMap.det (B p)| • f (↑(PartialHomeomorph.symm polarCoord) p)

Next tactic:
apply integral_target_eq_integral_abs_det_fderiv_smul volume A