awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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)
W :
  ContinuousAt (Y := TotalSpace.{u_1, u_2, u_3} (F →L[ℝ] F →L[ℝ] ℝ) fun (x : B) ↦ E x →L[ℝ] 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)) x
⊢ ContinuousAt g' x
have W₁ : ContinuousAt (Y := B) (fun (x : B) ↦ x) x ∧
  ContinuousAt.{u_1, u_2} (Y := F →L[ℝ] F →L[ℝ] ℝ)
    (fun (x_1 : B) ↦ inCoordinates F E (F →L[ℝ] ℝ) (fun (x : B) ↦ E x →L[ℝ] ℝ) x x_1 x x_1 (g x_1)) x
Tactic: simp only [continuousAt_hom_bundle] at W
Cleared hypotheses: W
-- 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)
⊢ ∀ᶠ (y : B) in 𝓝 x,
    norm (E := E y →L[ℝ] E x)
        (comp.{0, 0, 0, u_3, u_2, u_3} (R₂ := ℝ) (σ₁₂ := RingHom.id ℝ) (σ₂₃ := RingHom.id ℝ) (M₂ := F)
            (Trivialization.symmL ℝ (E := E) (trivializationAt F E x) x)
            (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) y) :
          E y →L[ℝ] E x) <
      r
have hg' : ContinuousAt g' x
Tactic: have hg' : ContinuousAt g' x := by
    have W := g_cont.continuousAt (x := x)
    simp only [continuousAt_hom_bundle] at W
    exact W.2
-- 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
⊢ ∀ᶠ (y : B) in 𝓝 x,
    norm (E := E y →L[ℝ] E x)
        (comp.{0, 0, 0, u_3, u_2, u_3} (R₂ := ℝ) (σ₁₂ := RingHom.id ℝ) (σ₂₃ := RingHom.id ℝ) (M₂ := F)
            (Trivialization.symmL ℝ (E := E) (trivializationAt F E x) x)
            (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) y) :
          E y →L[ℝ] E x) <
      r
have this : ∀ᶠ (y : B) in 𝓝 x, dist (g' y) (g' x) < δ
Tactic: have : ∀ᶠ y in 𝓝 x, dist (g' y) (g' x) < δ := by
    rw [Metric.continuousAt_iff'] at hg'
    apply hg' _ δpos
-- 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 : ∀ᶠ (y : B) in 𝓝 x, dist (g' y) (g' x) < δ
⊢ ∀ᶠ (y : B) in 𝓝 x,
    norm (E := E y →L[ℝ] E x)
        (comp.{0, 0, 0, u_3, u_2, u_3} (R₂ := ℝ) (σ₁₂ := RingHom.id ℝ) (σ₂₃ := RingHom.id ℝ) (M₂ := F)
            (Trivialization.symmL ℝ (E := E) (trivializationAt F E x) x)
            (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) y) :
          E y →L[ℝ] E x) <
      r
have h₁ : ∀ (y : B) (hy : dist (g' y) (g' x) < δ) (h'y : y ∈ Trivialization.baseSet (trivializationAt F E x)),
  norm (E := E y →L[ℝ] E x)
      (comp.{0, 0, 0, u_3, u_2, u_3} (R₂ := ℝ) (σ₁₂ := RingHom.id ℝ) (σ₂₃ := RingHom.id ℝ) (M₂ := F)
          (Trivialization.symmL ℝ (E := E) (trivializationAt F E x) x)
          (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) y) :
        E y →L[ℝ] E x) <
    r
Tactic: filter_upwards [this, (trivializationAt F E x).open_baseSet.mem_nhds h'x] with y hy 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 : ∀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)
⊢ norm (E := E y →L[ℝ] E x)
      (comp.{0, 0, 0, u_3, u_2, u_3} (R₂ := ℝ) (σ₁₂ := RingHom.id ℝ) (σ₂₃ := RingHom.id ℝ) (M₂ := F)
          (Trivialization.symmL ℝ (E := E) (trivializationAt F E x) x)
          (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) y) :
        E y →L[ℝ] E x) <
    r
have this₁ : ‖g' x - g' y‖ ≤ δ
Tactic: have : ‖g' x - g' y‖ ≤ δ := by rw [← dist_eq_norm']; exact hy.le
-- 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
⊢ 0 ≤ r'
have linarith : r' < 0 → HMul.hMul (α := ℝ) 1 1 = 1
Tactic: linarith
-- 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
linarith : r' < 0 → HMul.hMul (α := ℝ) 1 1 = 1
⊢ 0 ≤ r'
have linarith₁ : r' < 0 → -1 + (1 - r') + (r' - 0) = 0
Tactic: linarith
-- 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
⊢ norm (E := E y →L[ℝ] E x)
      (comp.{0, 0, 0, u_3, u_2, u_3} (R₂ := ℝ) (σ₁₂ := RingHom.id ℝ) (σ₂₃ := RingHom.id ℝ) (M₂ := F)
          (Trivialization.symmL ℝ (E := E) (trivializationAt F E x) x)
          (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) y) :
        E y →L[ℝ] E x) <
    r
have apply : ∀ (v : E y),
  norm (E := E x)
      (((comp.{0, 0, 0, u_3, u_2, u_3} (R₂ := ℝ) (σ₁₂ := RingHom.id ℝ) (σ₂₃ := RingHom.id ℝ) (M₂ := F)
                (Trivialization.symmL ℝ (E := E) (trivializationAt F E x) x)
                (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) y) :
              E y →L[ℝ] E x) :
            (a : E y) → E x)
          v :
        E x) ≤
    r' * ‖v‖
Tactic: apply (opNorm_le_bound _ (by linarith) (fun v ↦ ?_)).trans_lt r'r
-- 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
⊢ norm (E := E x)
      (((comp.{0, 0, 0, u_3, u_2, u_3} (R₂ := ℝ) (σ₁₂ := RingHom.id ℝ) (σ₂₃ := RingHom.id ℝ) (M₂ := F)
                (Trivialization.symmL ℝ (E := E) (trivializationAt F E x) x)
                (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) y) :
              E y →L[ℝ] E x) :
            (a : E y) → E x)
          v :
        E x) ≤
    r' * ‖v‖
have suffices : HPow.hPow (β := ℕ)
    (norm.{u_3} (E := E x)
      (((Trivialization.symmL ℝ (E := E) (trivializationAt F E x) x :
              ContinuousLinearMap.{0, 0, u_2, u_3} (RingHom.id ℝ) F (E x)) :
            (a : F) → E x)
          w :
        E x))
    2 ≤
  HPow.hPow (β := ℕ) r' 2 * HPow.hPow (β := ℕ) ‖v‖ 2
Tactic: suffices ‖((trivializationAt F E x).symmL ℝ x) w‖ ^ 2 ≤ r' ^ 2 * ‖v‖ ^ 2 from
    le_of_sq_le_sq (by simpa [mul_pow]) (by positivity)
-- 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
⊢ HPow.hPow (β := ℕ)
      (norm.{u_3} (E := E x)
        (((Trivialization.symmL ℝ (E := E) (trivializationAt F E x) x :
                ContinuousLinearMap.{0, 0, u_2, u_3} (RingHom.id ℝ) F (E x)) :
              (a : F) → E x)
            w :
          E x))
      2 ≤
    HPow.hPow (β := ℕ) r' 2 * HPow.hPow (β := ℕ) ‖v‖ 2
have simp : (((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)
Tactic: simp only [Trivialization.symmL_apply, ← real_inner_self_eq_norm_sq, hg]
-- 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
⊢ 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)
have rw : 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)
Tactic: rw [inCoordinates_apply_eq₂ h'y h'y (Set.mem_univ _)]