awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.VectorBundle.Riemannian (goal)
Goal:
B : Type u_1
inst : TopologicalSpace B
F : Type u_2
inst_1 : NormedAddCommGroup F
inst_2 : NormedSpace ℝ F
E : B → Type u_3
inst_3 : TopologicalSpace (TotalSpace F E)
inst_4 : (x : B) → NormedAddCommGroup (E x)
inst_5 : (x : B) → InnerProductSpace ℝ (E x)
inst_6 : FiberBundle F E
inst_7 : VectorBundle ℝ F E
h : IsContinuousRiemannianBundle F E
x : B
r : ℝ
hr : 1 < r
r' : ℝ
hr' : 1 < r'
r'r : r' < r
h'x : x ∈ Trivialization.baseSet (trivializationAt F E x)
G : ContinuousLinearEquiv.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F :=
  Trivialization.continuousLinearEquivAt ℝ (E := E) (trivializationAt F E x) x h'x
C : ℝ := HPow.hPow (β := ℕ) ‖(↑G : E x →L[ℝ] F)‖ 2
δ : ℝ
δpos : 0 < δ
hδ : (HPow.hPow (β := ℕ) r' 2)⁻¹ < 1 - δ * C
g : (x : B) → E x →L[ℝ] E x →L[ℝ] ℝ
g_cont :
  Continuous[inst, topologicalSpaceTotalSpace (RingHom.id ℝ) F E (F →L[ℝ] ℝ) fun (x : B) ↦ E x →L[ℝ] ℝ] fun (x : B) ↦
    TotalSpace.mk' (E := fun (x : B) ↦ E x →L[ℝ] E x →L[ℝ] ℝ) (F →L[ℝ] F →L[ℝ] ℝ) x (g x)
hg : ∀ (x : B) (v w : E x), ⟪v, w⟫ = (((g x : (a : E x) → E x →L[ℝ] ℝ) v : E x →L[ℝ] ℝ) : (a : E x) → ℝ) w
g' : B → F →L[ℝ] F →L[ℝ] ℝ := fun (y : B) ↦ inCoordinates F E (F →L[ℝ] ℝ) (fun (x : B) ↦ E x →L[ℝ] ℝ) x y x y (g y)
hg' : ContinuousAt g' x
this_1 : ∀ᶠ (y : B) in 𝓝 x, dist (g' y) (g' x) < δ
y : B
hy : dist (g' y) (g' x) < δ
h'y : y ∈ Trivialization.baseSet (trivializationAt F E x)
this : ‖g' x - g' y‖ ≤ δ
v : E y
w : F :=
  ((Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) y :
        ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F) :
      (a : E y) → F)
    v
⊢ Trivialization.symm (E := E) (trivializationAt F E x) y
      (((Trivialization.linearMapAt ℝ (E := E) (trivializationAt F E x) y :
            LinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F) :
          (a : E y) → F)
        v) =
    v
have h₁ : Eq (α := (a : E y) → F)
  (⇑(Trivialization.linearMapAt ℝ (E := E) (trivializationAt F E x) y :
        LinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F) :
    (a : E y) → F)
  (⇑(Trivialization.continuousLinearEquivAt ℝ (E := E) (trivializationAt F E x) y h'y :
        ContinuousLinearEquiv.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F) :
    (a : E y) → F)
Tactic: convert! ((trivializationAt F E x).continuousLinearEquivAt ℝ _ h'y).symm_apply_apply v
-- Lean 4 proof step from Mathlib.Topology.VectorBundle.Riemannian (hypothesis)
Goal:
B : Type u_1
inst : TopologicalSpace B
F : Type u_2
inst_1 : NormedAddCommGroup F
inst_2 : NormedSpace ℝ F
E : B → Type u_3
inst_3 : TopologicalSpace (TotalSpace F E)
inst_4 : (x : B) → NormedAddCommGroup (E x)
inst_5 : (x : B) → InnerProductSpace ℝ (E x)
inst_6 : FiberBundle F E
inst_7 : VectorBundle ℝ F E
h : IsContinuousRiemannianBundle F E
x : B
r : ℝ
hr : 1 < r
r' : ℝ
hr' : 1 < r'
r'r : r' < r
h'x : x ∈ Trivialization.baseSet (trivializationAt F E x)
G : ContinuousLinearEquiv.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F :=
  Trivialization.continuousLinearEquivAt ℝ (E := E) (trivializationAt F E x) x h'x
C : ℝ := HPow.hPow (β := ℕ) ‖(↑G : E x →L[ℝ] F)‖ 2
δ : ℝ
δpos : 0 < δ
hδ : (HPow.hPow (β := ℕ) r' 2)⁻¹ < 1 - δ * C
g : (x : B) → E x →L[ℝ] E x →L[ℝ] ℝ
g_cont :
  Continuous[inst, topologicalSpaceTotalSpace (RingHom.id ℝ) F E (F →L[ℝ] ℝ) fun (x : B) ↦ E x →L[ℝ] ℝ] fun (x : B) ↦
    TotalSpace.mk' (E := fun (x : B) ↦ E x →L[ℝ] E x →L[ℝ] ℝ) (F →L[ℝ] F →L[ℝ] ℝ) x (g x)
hg : ∀ (x : B) (v w : E x), ⟪v, w⟫ = (((g x : (a : E x) → E x →L[ℝ] ℝ) v : E x →L[ℝ] ℝ) : (a : E x) → ℝ) w
g' : B → F →L[ℝ] F →L[ℝ] ℝ := fun (y : B) ↦ inCoordinates F E (F →L[ℝ] ℝ) (fun (x : B) ↦ E x →L[ℝ] ℝ) x y x y (g y)
hg' : ContinuousAt g' x
this_1 : ∀ᶠ (y : B) in 𝓝 x, dist (g' y) (g' x) < δ
y : B
hy : dist (g' y) (g' x) < δ
h'y : y ∈ Trivialization.baseSet (trivializationAt F E x)
this : ‖g' x - g' y‖ ≤ δ
v : E y
w : F :=
  ((Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) y :
        ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F) :
      (a : E y) → F)
    v
⊢ Eq (α := ℝ) ((((g y : (a : E y) → E y →L[ℝ] ℝ) v : E y →L[ℝ] ℝ) : (a : E y) → ℝ) v : ℝ)
    (((Trivialization.linearMapAt ℝ (E := fun (x : B) ↦ ℝ) (trivializationAt ℝ (fun (x : B) ↦ ℝ) x) y : ℝ →ₗ[ℝ] ℝ) :
          (a : ℝ) → ℝ)
        ((((g y : (a : E y) → E y →L[ℝ] ℝ) (Trivialization.symm (E := E) (trivializationAt F E x) y w) : E y →L[ℝ] ℝ) :
              (a : E y) → ℝ)
            (Trivialization.symm (E := E) (trivializationAt F E x) y w) :
          ℝ) :
      ℝ)
have A : Trivialization.symm (E := E) (trivializationAt F E x) y
    (((Trivialization.linearMapAt ℝ (E := E) (trivializationAt F E x) y :
          LinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F) :
        (a : E y) → F)
      v) =
  v
Tactic: have A : ((trivializationAt F E x).symm y)
       ((trivializationAt F E x).linearMapAt ℝ y v) = v := by
      convert! ((trivializationAt F E x).continuousLinearEquivAt ℝ _ h'y).symm_apply_apply v
      simp [Trivialization.coe_continuousLinearEquivAt_eq _ h'y]
-- Lean 4 proof step from Mathlib.Topology.VectorBundle.Riemannian (hypothesis)
Goal:
B : Type u_1
inst : TopologicalSpace B
F : Type u_2
inst_1 : NormedAddCommGroup F
inst_2 : NormedSpace ℝ F
E : B → Type u_3
inst_3 : TopologicalSpace (TotalSpace F E)
inst_4 : (x : B) → NormedAddCommGroup (E x)
inst_5 : (x : B) → InnerProductSpace ℝ (E x)
inst_6 : FiberBundle F E
inst_7 : VectorBundle ℝ F E
h : IsContinuousRiemannianBundle F E
x : B
r : ℝ
hr : 1 < r
r' : ℝ
hr' : 1 < r'
r'r : r' < r
h'x : x ∈ Trivialization.baseSet (trivializationAt F E x)
G : ContinuousLinearEquiv.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F :=
  Trivialization.continuousLinearEquivAt ℝ (E := E) (trivializationAt F E x) x h'x
C : ℝ := HPow.hPow (β := ℕ) ‖(↑G : E x →L[ℝ] F)‖ 2
δ : ℝ
δpos : 0 < δ
hδ : (HPow.hPow (β := ℕ) r' 2)⁻¹ < 1 - δ * C
g : (x : B) → E x →L[ℝ] E x →L[ℝ] ℝ
g_cont :
  Continuous[inst, topologicalSpaceTotalSpace (RingHom.id ℝ) F E (F →L[ℝ] ℝ) fun (x : B) ↦ E x →L[ℝ] ℝ] fun (x : B) ↦
    TotalSpace.mk' (E := fun (x : B) ↦ E x →L[ℝ] E x →L[ℝ] ℝ) (F →L[ℝ] F →L[ℝ] ℝ) x (g x)
hg : ∀ (x : B) (v w : E x), ⟪v, w⟫ = (((g x : (a : E x) → E x →L[ℝ] ℝ) v : E x →L[ℝ] ℝ) : (a : E x) → ℝ) w
g' : B → F →L[ℝ] F →L[ℝ] ℝ := fun (y : B) ↦ inCoordinates F E (F →L[ℝ] ℝ) (fun (x : B) ↦ E x →L[ℝ] ℝ) x y x y (g y)
hg' : ContinuousAt g' x
this_1 : ∀ᶠ (y : B) in 𝓝 x, dist (g' y) (g' x) < δ
y : B
hy : dist (g' y) (g' x) < δ
h'y : y ∈ Trivialization.baseSet (trivializationAt F E x)
this : ‖g' x - g' y‖ ≤ δ
v : E y
w : F :=
  ((Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) y :
        ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F) :
      (a : E y) → F)
    v
⊢ (((g x : (a : E x) → E x →L[ℝ] ℝ) (Trivialization.symm (E := E) (trivializationAt F E x) x w) : E x →L[ℝ] ℝ) :
        (a : E x) → ℝ)
      (Trivialization.symm (E := E) (trivializationAt F E x) x w) ≤
    HMul.hMul (β := ℝ) (HPow.hPow (β := ℕ) r' 2)
      ((((g y : (a : E y) → E y →L[ℝ] ℝ) v : E y →L[ℝ] ℝ) : (a : E y) → ℝ) v : ℝ)
have hgy : Eq (α := ℝ) ((((g y : (a : E y) → E y →L[ℝ] ℝ) v : E y →L[ℝ] ℝ) : (a : E y) → ℝ) v : ℝ)
  ((((g' y : (a : F) → F →L[ℝ] ℝ) w : F →L[ℝ] ℝ) : (a : F) → ℝ) w : ℝ)
Tactic: have hgy : g y v v = g' y w w := by
    rw [inCoordinates_apply_eq₂ h'y h'y (Set.mem_univ _)]
    have A : ((trivializationAt F E x).symm y)
       ((trivializationAt F E x).linearMapAt ℝ y v) = v := by
      convert! ((trivializationAt F E x).continuousLinearEquivAt ℝ _ h'y).symm_apply_apply v
      simp [Trivialization.coe_continuousLinearEquivAt_eq _ h'y]
    simp [A, w]
-- Lean 4 proof step from Mathlib.Topology.VectorBundle.Riemannian (goal)
Goal:
B : Type u_1
inst : TopologicalSpace B
F : Type u_2
inst_1 : NormedAddCommGroup F
inst_2 : NormedSpace ℝ F
E : B → Type u_3
inst_3 : TopologicalSpace (TotalSpace F E)
inst_4 : (x : B) → NormedAddCommGroup (E x)
inst_5 : (x : B) → InnerProductSpace ℝ (E x)
inst_6 : FiberBundle F E
inst_7 : VectorBundle ℝ F E
h : IsContinuousRiemannianBundle F E
x : B
r : ℝ
hr : 1 < r
r' : ℝ
hr' : 1 < r'
r'r : r' < r
h'x : x ∈ Trivialization.baseSet (trivializationAt F E x)
G : ContinuousLinearEquiv.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F :=
  Trivialization.continuousLinearEquivAt ℝ (E := E) (trivializationAt F E x) x h'x
C : ℝ := HPow.hPow (β := ℕ) ‖(↑G : E x →L[ℝ] F)‖ 2
δ : ℝ
δpos : 0 < δ
hδ : (HPow.hPow (β := ℕ) r' 2)⁻¹ < 1 - δ * C
g : (x : B) → E x →L[ℝ] E x →L[ℝ] ℝ
g_cont :
  Continuous[inst, topologicalSpaceTotalSpace (RingHom.id ℝ) F E (F →L[ℝ] ℝ) fun (x : B) ↦ E x →L[ℝ] ℝ] fun (x : B) ↦
    TotalSpace.mk' (E := fun (x : B) ↦ E x →L[ℝ] E x →L[ℝ] ℝ) (F →L[ℝ] F →L[ℝ] ℝ) x (g x)
hg : ∀ (x : B) (v w : E x), ⟪v, w⟫ = (((g x : (a : E x) → E x →L[ℝ] ℝ) v : E x →L[ℝ] ℝ) : (a : E x) → ℝ) w
g' : B → F →L[ℝ] F →L[ℝ] ℝ := fun (y : B) ↦ inCoordinates F E (F →L[ℝ] ℝ) (fun (x : B) ↦ E x →L[ℝ] ℝ) x y x y (g y)
hg' : ContinuousAt g' x
this_1 : ∀ᶠ (y : B) in 𝓝 x, dist (g' y) (g' x) < δ
y : B
hy : dist (g' y) (g' x) < δ
h'y : y ∈ Trivialization.baseSet (trivializationAt F E x)
this : ‖g' x - g' y‖ ≤ δ
v : E y
w : F :=
  ((Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) y :
        ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F) :
      (a : E y) → F)
    v
hgy :
  Eq (α := ℝ) ((((g y : (a : E y) → E y →L[ℝ] ℝ) v : E y →L[ℝ] ℝ) : (a : E y) → ℝ) v : ℝ)
    ((((g' y : (a : F) → F →L[ℝ] ℝ) w : F →L[ℝ] ℝ) : (a : F) → ℝ) w : ℝ)
⊢ Eq (α := ℝ)
    ((((g x : (a : E x) → E x →L[ℝ] ℝ) (Trivialization.symm (E := E) (trivializationAt F E x) x w) : E x →L[ℝ] ℝ) :
          (a : E x) → ℝ)
        (Trivialization.symm (E := E) (trivializationAt F E x) x w) :
      ℝ)
    ((((g' x : (a : F) → F →L[ℝ] ℝ) w : F →L[ℝ] ℝ) : (a : F) → ℝ) w : ℝ)
have rw : Eq (α := ℝ)
  ((((g x : (a : E x) → E x →L[ℝ] ℝ) (Trivialization.symm (E := E) (trivializationAt F E x) x w) : E x →L[ℝ] ℝ) :
        (a : E x) → ℝ)
      (Trivialization.symm (E := E) (trivializationAt F E x) x w) :
    ℝ)
  (((Trivialization.linearMapAt ℝ (E := fun (x : B) ↦ ℝ) (trivializationAt ℝ (fun (x : B) ↦ ℝ) x) x : ℝ →ₗ[ℝ] ℝ) :
        (a : ℝ) → ℝ)
      ((((g x : (a : E x) → E x →L[ℝ] ℝ) (Trivialization.symm (E := E) (trivializationAt F E x) x w) : E x →L[ℝ] ℝ) :
            (a : E x) → ℝ)
          (Trivialization.symm (E := E) (trivializationAt F E x) x w) :
        ℝ) :
    ℝ)
Tactic: rw [inCoordinates_apply_eq₂ h'x h'x (Set.mem_univ _)]
-- Lean 4 proof step from Mathlib.Topology.VectorBundle.Riemannian (hypothesis)
Goal:
B : Type u_1
inst : TopologicalSpace B
F : Type u_2
inst_1 : NormedAddCommGroup F
inst_2 : NormedSpace ℝ F
E : B → Type u_3
inst_3 : TopologicalSpace (TotalSpace F E)
inst_4 : (x : B) → NormedAddCommGroup (E x)
inst_5 : (x : B) → InnerProductSpace ℝ (E x)
inst_6 : FiberBundle F E
inst_7 : VectorBundle ℝ F E
h : IsContinuousRiemannianBundle F E
x : B
r : ℝ
hr : 1 < r
r' : ℝ
hr' : 1 < r'
r'r : r' < r
h'x : x ∈ Trivialization.baseSet (trivializationAt F E x)
G : ContinuousLinearEquiv.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F :=
  Trivialization.continuousLinearEquivAt ℝ (E := E) (trivializationAt F E x) x h'x
C : ℝ := HPow.hPow (β := ℕ) ‖(↑G : E x →L[ℝ] F)‖ 2
δ : ℝ
δpos : 0 < δ
hδ : (HPow.hPow (β := ℕ) r' 2)⁻¹ < 1 - δ * C
g : (x : B) → E x →L[ℝ] E x →L[ℝ] ℝ
g_cont :
  Continuous[inst, topologicalSpaceTotalSpace (RingHom.id ℝ) F E (F →L[ℝ] ℝ) fun (x : B) ↦ E x →L[ℝ] ℝ] fun (x : B) ↦
    TotalSpace.mk' (E := fun (x : B) ↦ E x →L[ℝ] E x →L[ℝ] ℝ) (F →L[ℝ] F →L[ℝ] ℝ) x (g x)
hg : ∀ (x : B) (v w : E x), ⟪v, w⟫ = (((g x : (a : E x) → E x →L[ℝ] ℝ) v : E x →L[ℝ] ℝ) : (a : E x) → ℝ) w
g' : B → F →L[ℝ] F →L[ℝ] ℝ := fun (y : B) ↦ inCoordinates F E (F →L[ℝ] ℝ) (fun (x : B) ↦ E x →L[ℝ] ℝ) x y x y (g y)
hg' : ContinuousAt g' x
this_1 : ∀ᶠ (y : B) in 𝓝 x, dist (g' y) (g' x) < δ
y : B
hy : dist (g' y) (g' x) < δ
h'y : y ∈ Trivialization.baseSet (trivializationAt F E x)
this : ‖g' x - g' y‖ ≤ δ
v : E y
w : F :=
  ((Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) y :
        ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F) :
      (a : E y) → F)
    v
hgy :
  Eq (α := ℝ) ((((g y : (a : E y) → E y →L[ℝ] ℝ) v : E y →L[ℝ] ℝ) : (a : E y) → ℝ) v : ℝ)
    ((((g' y : (a : F) → F →L[ℝ] ℝ) w : F →L[ℝ] ℝ) : (a : F) → ℝ) w : ℝ)
⊢ (((g x : (a : E x) → E x →L[ℝ] ℝ) (Trivialization.symm (E := E) (trivializationAt F E x) x w) : E x →L[ℝ] ℝ) :
        (a : E x) → ℝ)
      (Trivialization.symm (E := E) (trivializationAt F E x) x w) ≤
    HMul.hMul (β := ℝ) (HPow.hPow (β := ℕ) r' 2)
      ((((g y : (a : E y) → E y →L[ℝ] ℝ) v : E y →L[ℝ] ℝ) : (a : E y) → ℝ) v : ℝ)
have hgx : Eq (α := ℝ)
  ((((g x : (a : E x) → E x →L[ℝ] ℝ) (Trivialization.symm (E := E) (trivializationAt F E x) x w) : E x →L[ℝ] ℝ) :
        (a : E x) → ℝ)
      (Trivialization.symm (E := E) (trivializationAt F E x) x w) :
    ℝ)
  ((((g' x : (a : F) → F →L[ℝ] ℝ) w : F →L[ℝ] ℝ) : (a : F) → ℝ) w : ℝ)
Tactic: have hgx : g x ((trivializationAt F E x).symm x w) ((trivializationAt F E x).symm x w) =
      g' x w w := by
    rw [inCoordinates_apply_eq₂ h'x h'x (Set.mem_univ _)]
    simp
-- Lean 4 proof step from Mathlib.Topology.VectorBundle.Riemannian (goal)
Goal:
B : Type u_1
inst : TopologicalSpace B
F : Type u_2
inst_1 : NormedAddCommGroup F
inst_2 : NormedSpace ℝ F
E : B → Type u_3
inst_3 : TopologicalSpace (TotalSpace F E)
inst_4 : (x : B) → NormedAddCommGroup (E x)
inst_5 : (x : B) → InnerProductSpace ℝ (E x)
inst_6 : FiberBundle F E
inst_7 : VectorBundle ℝ F E
h : IsContinuousRiemannianBundle F E
x : B
r : ℝ
hr : 1 < r
r' : ℝ
hr' : 1 < r'
r'r : r' < r
h'x : x ∈ Trivialization.baseSet (trivializationAt F E x)
G : ContinuousLinearEquiv.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F :=
  Trivialization.continuousLinearEquivAt ℝ (E := E) (trivializationAt F E x) x h'x
C : ℝ := HPow.hPow (β := ℕ) ‖(↑G : E x →L[ℝ] F)‖ 2
δ : ℝ
δpos : 0 < δ
hδ : (HPow.hPow (β := ℕ) r' 2)⁻¹ < 1 - δ * C
g : (x : B) → E x →L[ℝ] E x →L[ℝ] ℝ
g_cont :
  Continuous[inst, topologicalSpaceTotalSpace (RingHom.id ℝ) F E (F →L[ℝ] ℝ) fun (x : B) ↦ E x →L[ℝ] ℝ] fun (x : B) ↦
    TotalSpace.mk' (E := fun (x : B) ↦ E x →L[ℝ] E x →L[ℝ] ℝ) (F →L[ℝ] F →L[ℝ] ℝ) x (g x)
hg : ∀ (x : B) (v w : E x), ⟪v, w⟫ = (((g x : (a : E x) → E x →L[ℝ] ℝ) v : E x →L[ℝ] ℝ) : (a : E x) → ℝ) w
g' : B → F →L[ℝ] F →L[ℝ] ℝ := fun (y : B) ↦ inCoordinates F E (F →L[ℝ] ℝ) (fun (x : B) ↦ E x →L[ℝ] ℝ) x y x y (g y)
hg' : ContinuousAt g' x
this_1 : ∀ᶠ (y : B) in 𝓝 x, dist (g' y) (g' x) < δ
y : B
hy : dist (g' y) (g' x) < δ
h'y : y ∈ Trivialization.baseSet (trivializationAt F E x)
this : ‖g' x - g' y‖ ≤ δ
v : E y
w : F :=
  ((Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) y :
        ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F) :
      (a : E y) → F)
    v
hgy :
  Eq (α := ℝ) ((((g y : (a : E y) → E y →L[ℝ] ℝ) v : E y →L[ℝ] ℝ) : (a : E y) → ℝ) v : ℝ)
    ((((g' y : (a : F) → F →L[ℝ] ℝ) w : F →L[ℝ] ℝ) : (a : F) → ℝ) w : ℝ)
hgx :
  Eq (α := ℝ)
    ((((g x : (a : E x) → E x →L[ℝ] ℝ) (Trivialization.symm (E := E) (trivializationAt F E x) x w) : E x →L[ℝ] ℝ) :
          (a : E x) → ℝ)
        (Trivialization.symm (E := E) (trivializationAt F E x) x w) :
      ℝ)
    ((((g' x : (a : F) → F →L[ℝ] ℝ) w : F →L[ℝ] ℝ) : (a : F) → ℝ) w : ℝ)
⊢ (((g x : (a : E x) → E x →L[ℝ] ℝ) (Trivialization.symm (E := E) (trivializationAt F E x) x w) : E x →L[ℝ] ℝ) :
        (a : E x) → ℝ)
      (Trivialization.symm (E := E) (trivializationAt F E x) x w) ≤
    HMul.hMul (β := ℝ) (HPow.hPow (β := ℕ) r' 2)
      ((((g y : (a : E y) → E y →L[ℝ] ℝ) v : E y →L[ℝ] ℝ) : (a : E y) → ℝ) v : ℝ)
have rw : (((g' x : (a : F) → F →L[ℝ] ℝ) w : F →L[ℝ] ℝ) : (a : F) → ℝ) w ≤
  HMul.hMul (β := ℝ) (HPow.hPow (β := ℕ) r' 2) ((((g' y : (a : F) → F →L[ℝ] ℝ) w : F →L[ℝ] ℝ) : (a : F) → ℝ) w : ℝ)
Tactic: rw [hgx, hgy]
-- Lean 4 proof step from Mathlib.Topology.VectorBundle.Riemannian (goal)
Goal:
B : Type u_1
inst : TopologicalSpace B
F : Type u_2
inst_1 : NormedAddCommGroup F
inst_2 : NormedSpace ℝ F
E : B → Type u_3
inst_3 : TopologicalSpace (TotalSpace F E)
inst_4 : (x : B) → NormedAddCommGroup (E x)
inst_5 : (x : B) → InnerProductSpace ℝ (E x)
inst_6 : FiberBundle F E
inst_7 : VectorBundle ℝ F E
h : IsContinuousRiemannianBundle F E
x : B
r : ℝ
hr : 1 < r
r' : ℝ
hr' : 1 < r'
r'r : r' < r
h'x : x ∈ Trivialization.baseSet (trivializationAt F E x)
G : ContinuousLinearEquiv.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F :=
  Trivialization.continuousLinearEquivAt ℝ (E := E) (trivializationAt F E x) x h'x
C : ℝ := HPow.hPow (β := ℕ) ‖(↑G : E x →L[ℝ] F)‖ 2
δ : ℝ
δpos : 0 < δ
hδ : (HPow.hPow (β := ℕ) r' 2)⁻¹ < 1 - δ * C
g : (x : B) → E x →L[ℝ] E x →L[ℝ] ℝ
g_cont :
  Continuous[inst, topologicalSpaceTotalSpace (RingHom.id ℝ) F E (F →L[ℝ] ℝ) fun (x : B) ↦ E x →L[ℝ] ℝ] fun (x : B) ↦
    TotalSpace.mk' (E := fun (x : B) ↦ E x →L[ℝ] E x →L[ℝ] ℝ) (F →L[ℝ] F →L[ℝ] ℝ) x (g x)
hg : ∀ (x : B) (v w : E x), ⟪v, w⟫ = (((g x : (a : E x) → E x →L[ℝ] ℝ) v : E x →L[ℝ] ℝ) : (a : E x) → ℝ) w
g' : B → F →L[ℝ] F →L[ℝ] ℝ := fun (y : B) ↦ inCoordinates F E (F →L[ℝ] ℝ) (fun (x : B) ↦ E x →L[ℝ] ℝ) x y x y (g y)
hg' : ContinuousAt g' x
this_1 : ∀ᶠ (y : B) in 𝓝 x, dist (g' y) (g' x) < δ
y : B
hy : dist (g' y) (g' x) < δ
h'y : y ∈ Trivialization.baseSet (trivializationAt F E x)
this : ‖g' x - g' y‖ ≤ δ
v : E y
w : F :=
  ((Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) y :
        ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F) :
      (a : E y) → F)
    v
hgy :
  Eq (α := ℝ) ((((g y : (a : E y) → E y →L[ℝ] ℝ) v : E y →L[ℝ] ℝ) : (a : E y) → ℝ) v : ℝ)
    ((((g' y : (a : F) → F →L[ℝ] ℝ) w : F →L[ℝ] ℝ) : (a : F) → ℝ) w : ℝ)
hgx :
  Eq (α := ℝ)
    ((((g x : (a : E x) → E x →L[ℝ] ℝ) (Trivialization.symm (E := E) (trivializationAt F E x) x w) : E x →L[ℝ] ℝ) :
          (a : E x) → ℝ)
        (Trivialization.symm (E := E) (trivializationAt F E x) x w) :
      ℝ)
    ((((g' x : (a : F) → F →L[ℝ] ℝ) w : F →L[ℝ] ℝ) : (a : F) → ℝ) w : ℝ)
⊢ HAdd.hAdd (α := ℝ) (β := ℝ) ((((g' x - g' y : (a : F) → F →L[ℝ] ℝ) w : F →L[ℝ] ℝ) : (a : F) → ℝ) w : ℝ)
      ((((g' y : (a : F) → F →L[ℝ] ℝ) w : F →L[ℝ] ℝ) : (a : F) → ℝ) w : ℝ) ≤
    HAdd.hAdd (β := ℝ) (‖g' x - g' y‖ * ‖w‖ * ‖w‖) ((((g' y : (a : F) → F →L[ℝ] ℝ) w : F →L[ℝ] ℝ) : (a : F) → ℝ) w : ℝ)
have grw : HAdd.hAdd (α := ℝ) (β := ℝ) ((((g' x - g' y : (a : F) → F →L[ℝ] ℝ) w : F →L[ℝ] ℝ) : (a : F) → ℝ) w : ℝ)
    ((((g' y : (a : F) → F →L[ℝ] ℝ) w : F →L[ℝ] ℝ) : (a : F) → ℝ) w : ℝ) ≤
  HAdd.hAdd (β := ℝ) (norm (E := ℝ) ((((g' x - g' y : (a : F) → F →L[ℝ] ℝ) w : F →L[ℝ] ℝ) : (a : F) → ℝ) w : ℝ))
    ((((g' y : (a : F) → F →L[ℝ] ℝ) w : F →L[ℝ] ℝ) : (a : F) → ℝ) w : ℝ)
Tactic: grw [← le_opNorm₂, ← Real.le_norm_self]
-- Lean 4 proof step from Mathlib.Topology.VectorBundle.Riemannian (goal)
Goal:
B : Type u_1
inst : TopologicalSpace B
F : Type u_2
inst_1 : NormedAddCommGroup F
inst_2 : NormedSpace ℝ F
E : B → Type u_3
inst_3 : TopologicalSpace (TotalSpace F E)
inst_4 : (x : B) → NormedAddCommGroup (E x)
inst_5 : (x : B) → InnerProductSpace ℝ (E x)
inst_6 : FiberBundle F E
inst_7 : VectorBundle ℝ F E
h : IsContinuousRiemannianBundle F E
x : B
r : ℝ
hr : 1 < r
r' : ℝ
hr' : 1 < r'
r'r : r' < r
h'x : x ∈ Trivialization.baseSet (trivializationAt F E x)
G : ContinuousLinearEquiv.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F :=
  Trivialization.continuousLinearEquivAt ℝ (E := E) (trivializationAt F E x) x h'x
C : ℝ := HPow.hPow (β := ℕ) ‖(↑G : E x →L[ℝ] F)‖ 2
δ : ℝ
δpos : 0 < δ
hδ : (HPow.hPow (β := ℕ) r' 2)⁻¹ < 1 - δ * C
g : (x : B) → E x →L[ℝ] E x →L[ℝ] ℝ
g_cont :
  Continuous[inst, topologicalSpaceTotalSpace (RingHom.id ℝ) F E (F →L[ℝ] ℝ) fun (x : B) ↦ E x →L[ℝ] ℝ] fun (x : B) ↦
    TotalSpace.mk' (E := fun (x : B) ↦ E x →L[ℝ] E x →L[ℝ] ℝ) (F →L[ℝ] F →L[ℝ] ℝ) x (g x)
hg : ∀ (x : B) (v w : E x), ⟪v, w⟫ = (((g x : (a : E x) → E x →L[ℝ] ℝ) v : E x →L[ℝ] ℝ) : (a : E x) → ℝ) w
g' : B → F →L[ℝ] F →L[ℝ] ℝ := fun (y : B) ↦ inCoordinates F E (F →L[ℝ] ℝ) (fun (x : B) ↦ E x →L[ℝ] ℝ) x y x y (g y)
hg' : ContinuousAt g' x
this_1 : ∀ᶠ (y : B) in 𝓝 x, dist (g' y) (g' x) < δ
y : B
hy : dist (g' y) (g' x) < δ
h'y : y ∈ Trivialization.baseSet (trivializationAt F E x)
this : ‖g' x - g' y‖ ≤ δ
v : E y
w : F :=
  ((Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) y :
        ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F) :
      (a : E y) → F)
    v
hgy :
  Eq (α := ℝ) ((((g y : (a : E y) → E y →L[ℝ] ℝ) v : E y →L[ℝ] ℝ) : (a : E y) → ℝ) v : ℝ)
    ((((g' y : (a : F) → F →L[ℝ] ℝ) w : F →L[ℝ] ℝ) : (a : F) → ℝ) w : ℝ)
hgx :
  Eq (α := ℝ)
    ((((g x : (a : E x) → E x →L[ℝ] ℝ) (Trivialization.symm (E := E) (trivializationAt F E x) x w) : E x →L[ℝ] ℝ) :
          (a : E x) → ℝ)
        (Trivialization.symm (E := E) (trivializationAt F E x) x w) :
      ℝ)
    ((((g' x : (a : F) → F →L[ℝ] ℝ) w : F →L[ℝ] ℝ) : (a : F) → ℝ) w : ℝ)
⊢ HAdd.hAdd (β := ℝ) (‖g' x - g' y‖ * ‖w‖ * ‖w‖) ((((g' y : (a : F) → F →L[ℝ] ℝ) w : F →L[ℝ] ℝ) : (a : F) → ℝ) w : ℝ) ≤
    HAdd.hAdd (β := ℝ) (δ * HPow.hPow (β := ℕ) ‖w‖ 2)
      ((((g' y : (a : F) → F →L[ℝ] ℝ) w : F →L[ℝ] ℝ) : (a : F) → ℝ) w : ℝ)
have rw : HAdd.hAdd (β := ℝ) (‖g' x - g' y‖ * (‖w‖ * ‖w‖)) ((((g' y : (a : F) → F →L[ℝ] ℝ) w : F →L[ℝ] ℝ) : (a : F) → ℝ) w : ℝ) ≤
  HAdd.hAdd (β := ℝ) (δ * (‖w‖ * ‖w‖)) ((((g' y : (a : F) → F →L[ℝ] ℝ) w : F →L[ℝ] ℝ) : (a : F) → ℝ) w : ℝ)
Tactic: rw [pow_two, mul_assoc]
-- Lean 4 proof step from Mathlib.Topology.VectorBundle.Riemannian (goal)
Goal:
B : Type u_1
inst : TopologicalSpace B
F : Type u_2
inst_1 : NormedAddCommGroup F
inst_2 : NormedSpace ℝ F
E : B → Type u_3
inst_3 : TopologicalSpace (TotalSpace F E)
inst_4 : (x : B) → NormedAddCommGroup (E x)
inst_5 : (x : B) → InnerProductSpace ℝ (E x)
inst_6 : FiberBundle F E
inst_7 : VectorBundle ℝ F E
h : IsContinuousRiemannianBundle F E
x : B
r : ℝ
hr : 1 < r
r' : ℝ
hr' : 1 < r'
r'r : r' < r
h'x : x ∈ Trivialization.baseSet (trivializationAt F E x)
G : ContinuousLinearEquiv.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F :=
  Trivialization.continuousLinearEquivAt ℝ (E := E) (trivializationAt F E x) x h'x
C : ℝ := HPow.hPow (β := ℕ) ‖(↑G : E x →L[ℝ] F)‖ 2
δ : ℝ
δpos : 0 < δ
hδ : (HPow.hPow (β := ℕ) r' 2)⁻¹ < 1 - δ * C
g : (x : B) → E x →L[ℝ] E x →L[ℝ] ℝ
g_cont :
  Continuous[inst, topologicalSpaceTotalSpace (RingHom.id ℝ) F E (F →L[ℝ] ℝ) fun (x : B) ↦ E x →L[ℝ] ℝ] fun (x : B) ↦
    TotalSpace.mk' (E := fun (x : B) ↦ E x →L[ℝ] E x →L[ℝ] ℝ) (F →L[ℝ] F →L[ℝ] ℝ) x (g x)
hg : ∀ (x : B) (v w : E x), ⟪v, w⟫ = (((g x : (a : E x) → E x →L[ℝ] ℝ) v : E x →L[ℝ] ℝ) : (a : E x) → ℝ) w
g' : B → F →L[ℝ] F →L[ℝ] ℝ := fun (y : B) ↦ inCoordinates F E (F →L[ℝ] ℝ) (fun (x : B) ↦ E x →L[ℝ] ℝ) x y x y (g y)
hg' : ContinuousAt g' x
this_1 : ∀ᶠ (y : B) in 𝓝 x, dist (g' y) (g' x) < δ
y : B
hy : dist (g' y) (g' x) < δ
h'y : y ∈ Trivialization.baseSet (trivializationAt F E x)
this : ‖g' x - g' y‖ ≤ δ
v : E y
w : F :=
  ((Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) y :
        ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F) :
      (a : E y) → F)
    v
hgy :
  Eq (α := ℝ) ((((g y : (a : E y) → E y →L[ℝ] ℝ) v : E y →L[ℝ] ℝ) : (a : E y) → ℝ) v : ℝ)
    ((((g' y : (a : F) → F →L[ℝ] ℝ) w : F →L[ℝ] ℝ) : (a : F) → ℝ) w : ℝ)
hgx :
  Eq (α := ℝ)
    ((((g x : (a : E x) → E x →L[ℝ] ℝ) (Trivialization.symm (E := E) (trivializationAt F E x) x w) : E x →L[ℝ] ℝ) :
          (a : E x) → ℝ)
        (Trivialization.symm (E := E) (trivializationAt F E x) x w) :
      ℝ)
    ((((g' x : (a : F) → F →L[ℝ] ℝ) w : F →L[ℝ] ℝ) : (a : F) → ℝ) w : ℝ)
⊢ HAdd.hAdd (β := ℝ) (‖g' x - g' y‖ * (‖w‖ * ‖w‖))
      ((((g' y : (a : F) → F →L[ℝ] ℝ) w : F →L[ℝ] ℝ) : (a : F) → ℝ) w : ℝ) ≤
    HAdd.hAdd (β := ℝ) (δ * (‖w‖ * ‖w‖)) ((((g' y : (a : F) → F →L[ℝ] ℝ) w : F →L[ℝ] ℝ) : (a : F) → ℝ) w : ℝ)
have bc : 0 ≤ ‖w‖ * ‖w‖
Tactic: gcongr