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
⊢ ∃ C > 0,
    ∀ᶠ (y : B) in 𝓝 x,
      norm (E := ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F)
          (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) y :
            ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F) <
        C
have refine : ∀ᶠ (y : B) in 𝓝 x,
  norm (E := ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F)
      (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) y :
        ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F) <
    (1 +
        norm (E := ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F)
          (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x :
            ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F)) *
      2
Tactic: refine ⟨(1 + ‖(trivializationAt F E x).continuousLinearMapAt ℝ  x‖) * 2, 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
⊢ ∀ᶠ (y : B) in 𝓝 x,
    norm (E := ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F)
        (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) y :
          ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F) <
      (1 +
          norm (E := ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F)
            (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x :
              ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F)) *
        2
have h₁ : ∀ (y : B)
  (hy :
    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) <
      2),
  norm (E := ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F)
      (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) y :
        ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F) <
    (1 +
        norm (E := ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F)
          (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x :
            ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F)) *
      2
Tactic: filter_upwards [eventually_norm_symmL_trivializationAt_self_comp_lt F E x one_lt_two] with y hy
-- 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 y : B
hy :
  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) <
    2
⊢ Eq (α := F →L[ℝ] F)
    ((Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x :
        ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F) ∘SL
      (Trivialization.symmL ℝ (E := E) (trivializationAt F E x) x :
        ContinuousLinearMap.{0, 0, u_2, u_3} (RingHom.id ℝ) F (E x)))
    (ContinuousLinearMap.id ℝ F)
have h₁ : ∀ (v : F),
  Eq (α := F)
    (((Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x :
              ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F) ∘SL
            (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) → F)
        v :
      F)
    ((ContinuousLinearMap.id ℝ F : (a : F) → F) v : F)
Tactic: ext 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 y : B
hy :
  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) <
    2
v : F
⊢ Eq (α := F)
    (((Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x :
              ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F) ∘SL
            (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) → F)
        v :
      F)
    ((ContinuousLinearMap.id ℝ F : (a : F) → F) v : F)
have h'x : x ∈ Trivialization.baseSet (trivializationAt F E x)
Tactic: have h'x : x ∈ (trivializationAt F E x).baseSet := FiberBundle.mem_baseSet_trivializationAt' x
-- 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 y : B
hy :
  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) <
    2
v : F
h'x : x ∈ Trivialization.baseSet (trivializationAt F E x)
⊢ Eq (α := F)
    (((Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x :
              ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F) ∘SL
            (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) → F)
        v :
      F)
    ((ContinuousLinearMap.id ℝ F : (a : F) → F) v : F)
have h₁ : ((Trivialization.linearMapAt ℝ (E := E) (trivializationAt F E x) x :
        LinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F) :
      (a : E x) → F)
    (Trivialization.symm (E := E) (trivializationAt F E x) x v) =
  v
Tactic: simp only [coe_comp', Trivialization.continuousLinearMapAt_apply, Trivialization.symmL_apply,
      Function.comp_apply, coe_id', id_eq]
-- 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 y : B
hy :
  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) <
    2
v : F
h'x : x ∈ Trivialization.baseSet (trivializationAt F E x)
⊢ ((Trivialization.linearMapAt ℝ (E := E) (trivializationAt F E x) x :
          LinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F) :
        (a : E x) → F)
      (Trivialization.symm (E := E) (trivializationAt F E x) x v) =
    v
have h₁ : Eq (α := (a : E x) → F)
  (⇑(Trivialization.linearMapAt ℝ (E := E) (trivializationAt F E x) x :
        LinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F) :
    (a : E x) → F)
  (⇑(Trivialization.continuousLinearEquivAt ℝ (E := E) (trivializationAt F E x) x h'x :
        ContinuousLinearEquiv.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F) :
    (a : E x) → F)
Tactic: convert! ((trivializationAt F E x).continuousLinearEquivAt ℝ _ h'x).apply_symm_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 y : B
hy :
  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) <
    2
⊢ norm (E := ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F)
      (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) y :
        ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F) <
    (1 +
        norm (E := ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F)
          (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x :
            ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F)) *
      2
have A : Eq (α := F →L[ℝ] F)
  ((Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x :
      ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F) ∘SL
    (Trivialization.symmL ℝ (E := E) (trivializationAt F E x) x :
      ContinuousLinearMap.{0, 0, u_2, u_3} (RingHom.id ℝ) F (E x)))
  (ContinuousLinearMap.id ℝ F)
Tactic: have A : ((trivializationAt F E x).continuousLinearMapAt ℝ x) ∘L
      ((trivializationAt F E x).symmL ℝ x) = ContinuousLinearMap.id _ _ := by
    ext v
    have h'x : x ∈ (trivializationAt F E x).baseSet := FiberBundle.mem_baseSet_trivializationAt' x
    simp only [coe_comp', Trivialization.continuousLinearMapAt_apply, Trivialization.symmL_apply,
      Function.comp_apply, coe_id', id_eq]
    convert! ((trivializationAt F E x).continuousLinearEquivAt ℝ _ h'x).apply_symm_apply v
    simp [Trivialization.coe_continuousLinearEquivAt_eq _ h'x]
-- 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 y : B
hy :
  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) <
    2
A :
  Eq (α := F →L[ℝ] F)
    ((Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x :
        ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F) ∘SL
      (Trivialization.symmL ℝ (E := E) (trivializationAt F E x) x :
        ContinuousLinearMap.{0, 0, u_2, u_3} (RingHom.id ℝ) F (E x)))
    (ContinuousLinearMap.id ℝ F)
⊢ norm (E := ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F)
      (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) y :
        ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F) <
    (1 +
        norm (E := ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F)
          (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x :
            ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F)) *
      2
have this : Eq (α := ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F)
  (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) y :
    ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F)
  (comp (σ₁₂ := RingHom.id ℝ) (ContinuousLinearMap.id ℝ F)
      (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) y) :
    ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F)
Tactic: have : (trivializationAt F E x).continuousLinearMapAt ℝ y =
    (ContinuousLinearMap.id _ _) ∘L ((trivializationAt F E x).continuousLinearMapAt ℝ y) := by 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 y : B
hy :
  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) <
    2
A :
  Eq (α := F →L[ℝ] F)
    ((Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x :
        ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F) ∘SL
      (Trivialization.symmL ℝ (E := E) (trivializationAt F E x) x :
        ContinuousLinearMap.{0, 0, u_2, u_3} (RingHom.id ℝ) F (E x)))
    (ContinuousLinearMap.id ℝ F)
this :
  Eq (α := ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F)
    (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) y :
      ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F)
    (comp (σ₁₂ := RingHom.id ℝ) (ContinuousLinearMap.id ℝ F)
        (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) y) :
      ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F)
⊢ norm (E := ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F)
      (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) y :
        ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F) <
    (1 +
        norm (E := ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F)
          (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x :
            ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F)) *
      2
have h₁ : norm (E := ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F)
      (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x :
        ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F) *
    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) <
  (1 +
      norm (E := ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F)
        (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x :
          ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F)) *
    2
Tactic: grw [this, ← A, comp_assoc, opNorm_comp_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 y : B
hy :
  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) <
    2
A :
  Eq (α := F →L[ℝ] F)
    ((Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x :
        ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F) ∘SL
      (Trivialization.symmL ℝ (E := E) (trivializationAt F E x) x :
        ContinuousLinearMap.{0, 0, u_2, u_3} (RingHom.id ℝ) F (E x)))
    (ContinuousLinearMap.id ℝ F)
this :
  Eq (α := ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F)
    (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) y :
      ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F)
    (comp (σ₁₂ := RingHom.id ℝ) (ContinuousLinearMap.id ℝ F)
        (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) y) :
      ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F)
⊢ norm (E := ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F)
        (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x :
          ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F) *
      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) <
    (1 +
        norm (E := ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F)
          (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x :
            ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F)) *
      2
have h₁ : norm (E := ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F)
    (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x :
      ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F) <
  1 +
    norm (E := ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F)
      (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x :
        ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F)
Tactic: gcongr
-- 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 y : B
hy :
  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) <
    2
A :
  Eq (α := F →L[ℝ] F)
    ((Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x :
        ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F) ∘SL
      (Trivialization.symmL ℝ (E := E) (trivializationAt F E x) x :
        ContinuousLinearMap.{0, 0, u_2, u_3} (RingHom.id ℝ) F (E x)))
    (ContinuousLinearMap.id ℝ F)
this :
  Eq (α := ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F)
    (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) y :
      ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F)
    (comp (σ₁₂ := RingHom.id ℝ) (ContinuousLinearMap.id ℝ F)
        (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) y) :
      ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E y) F)
⊢ norm (E := ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F)
      (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x :
        ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F) <
    1 +
      norm (E := ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F)
        (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x :
          ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F)
have linarith : 1 +
      norm (E := ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F)
        (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x :
          ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F) ≤
    norm (E := ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F)
      (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x :
        ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F) →
  -1 +
      (1 +
          norm (E := ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F)
            (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x :
              ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F) -
        norm (E := ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F)
          (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x :
            ContinuousLinearMap.{0, 0, u_3, u_2} (RingHom.id ℝ) (E x) F)) =
    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
⊢ ∀ᶠ (y : B) in 𝓝 x,
    norm (E := E x →L[ℝ] E y)
        (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) y)
            (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x) :
          E x →L[ℝ] E y) <
      r
have obtain : ∀ (r' : ℝ) (hr' : 1 < r') (r'r : r' < r),
  ∀ᶠ (y : B) in 𝓝 x,
    norm (E := E x →L[ℝ] E y)
        (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) y)
            (Trivialization.continuousLinearMapAt ℝ (E := E) (trivializationAt F E x) x) :
          E x →L[ℝ] E y) <
      r
Tactic: obtain ⟨r', hr', r'r⟩ : ∃ r', 1 < r' ∧ r' < r := exists_between hr