awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Riemannian.PathELength (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
H : Type u_2
inst_2 : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst_3 : TopologicalSpace M
inst_4 : ChartedSpace H M
inst_5 : (x : M) → ENorm (TangentSpace I x)
inst_6 : ∀ (x : M), ENormSMulClass.{0, u_1} ℝ (TangentSpace I x)
x_1 y_1 x y : M
r : ℝ≥0∞
hr : riemannianEDist I x y < r
γ : ℝ → M
γ0 : γ 0 = x
γ1 : γ 1 = y
γ_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ
hγ : pathELength I γ 0 1 < r
η : ℝ → ℝ := fun (t : ℝ) ↦ -t
⊢ ContMDiff 𝓘(ℝ, ℝ) I 1 (γ ∘ η)
have apply : ContMDiff 𝓘(ℝ, ℝ) 𝓘(ℝ, ℝ) 1 η
Tactic: apply γ_smooth.comp ?_
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Riemannian.PathELength (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
H : Type u_2
inst_2 : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst_3 : TopologicalSpace M
inst_4 : ChartedSpace H M
inst_5 : (x : M) → ENorm (TangentSpace I x)
inst_6 : ∀ (x : M), ENormSMulClass.{0, u_1} ℝ (TangentSpace I x)
x_1 y_1 x y : M
r : ℝ≥0∞
hr : riemannianEDist I x y < r
γ : ℝ → M
γ0 : γ 0 = x
γ1 : γ 1 = y
γ_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ
hγ : pathELength I γ 0 1 < r
η : ℝ → ℝ := fun (t : ℝ) ↦ -t
⊢ ContMDiff 𝓘(ℝ, ℝ) 𝓘(ℝ, ℝ) 1 η
have simp : ContDiff ℝ 1 η
Tactic: simp only [contMDiff_iff_contDiff]
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Riemannian.PathELength (hypothesis)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
H : Type u_2
inst_2 : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst_3 : TopologicalSpace M
inst_4 : ChartedSpace H M
inst_5 : (x : M) → ENorm (TangentSpace I x)
inst_6 : ∀ (x : M), ENormSMulClass.{0, u_1} ℝ (TangentSpace I x)
x_1 y_1 x y : M
r : ℝ≥0∞
hr : riemannianEDist I x y < r
γ : ℝ → M
γ0 : γ 0 = x
γ1 : γ 1 = y
γ_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ
hγ : pathELength I γ 0 1 < r
η : ℝ → ℝ := fun (t : ℝ) ↦ -t
⊢ riemannianEDist I y x < r
have h_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 (γ ∘ η)
Tactic: have h_smooth : CMDiff 1 (γ ∘ η) := by
    apply γ_smooth.comp ?_
    simp only [contMDiff_iff_contDiff]
    fun_prop
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Riemannian.PathELength (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
H : Type u_2
inst_2 : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst_3 : TopologicalSpace M
inst_4 : ChartedSpace H M
inst_5 : (x : M) → ENorm (TangentSpace I x)
inst_6 : ∀ (x : M), ENormSMulClass.{0, u_1} ℝ (TangentSpace I x)
x_1 y_1 x y : M
r : ℝ≥0∞
hr : riemannianEDist I x y < r
γ : ℝ → M
γ0 : γ 0 = x
γ1 : γ 1 = y
γ_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ
hγ : pathELength I γ 0 1 < r
η : ℝ → ℝ := fun (t : ℝ) ↦ -t
h_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 (γ ∘ η)
⊢ riemannianEDist I y x ≤ pathELength I (γ ∘ η) (η 1) (η 0)
have ha : (γ ∘ η) (η 1) = y
Tactic: apply riemannianEDist_le_pathELength h_smooth.contMDiffOn
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Riemannian.PathELength (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
H : Type u_2
inst_2 : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst_3 : TopologicalSpace M
inst_4 : ChartedSpace H M
inst_5 : (x : M) → ENorm (TangentSpace I x)
inst_6 : ∀ (x : M), ENormSMulClass.{0, u_1} ℝ (TangentSpace I x)
x_1 y_1 x y : M
r : ℝ≥0∞
hr : riemannianEDist I x y < r
γ : ℝ → M
γ0 : γ 0 = x
γ1 : γ 1 = y
γ_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ
hγ : pathELength I γ 0 1 < r
η : ℝ → ℝ := fun (t : ℝ) ↦ -t
h_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 (γ ∘ η)
ha : (γ ∘ η) (η 1) = y
⊢ riemannianEDist I y x ≤ pathELength I (γ ∘ η) (η 1) (η 0)
have hb : (γ ∘ η) (η 0) = x
Tactic: apply riemannianEDist_le_pathELength h_smooth.contMDiffOn
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Riemannian.PathELength (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
H : Type u_2
inst_2 : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst_3 : TopologicalSpace M
inst_4 : ChartedSpace H M
inst_5 : (x : M) → ENorm (TangentSpace I x)
inst_6 : ∀ (x : M), ENormSMulClass.{0, u_1} ℝ (TangentSpace I x)
x_1 y_1 x y : M
r : ℝ≥0∞
hr : riemannianEDist I x y < r
γ : ℝ → M
γ0 : γ 0 = x
γ1 : γ 1 = y
γ_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ
hγ : pathELength I γ 0 1 < r
η : ℝ → ℝ := fun (t : ℝ) ↦ -t
h_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 (γ ∘ η)
ha : (γ ∘ η) (η 1) = y
hb : (γ ∘ η) (η 0) = x
⊢ riemannianEDist I y x ≤ pathELength I (γ ∘ η) (η 1) (η 0)
have hab : η 1 ≤ η 0
Tactic: apply riemannianEDist_le_pathELength h_smooth.contMDiffOn
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Riemannian.PathELength (hypothesis)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
H : Type u_2
inst_2 : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst_3 : TopologicalSpace M
inst_4 : ChartedSpace H M
inst_5 : (x : M) → ENorm (TangentSpace I x)
inst_6 : ∀ (x : M), ENormSMulClass.{0, u_1} ℝ (TangentSpace I x)
x_1 y_1 x y : M
r : ℝ≥0∞
hr : riemannianEDist I x y < r
γ : ℝ → M
γ0 : γ 0 = x
γ1 : γ 1 = y
γ_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ
hγ : pathELength I γ 0 1 < r
η : ℝ → ℝ := fun (t : ℝ) ↦ -t
h_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 (γ ∘ η)
⊢ riemannianEDist I y x < r
have this : riemannianEDist I y x ≤ pathELength I (γ ∘ η) (η 1) (η 0)
Tactic: have : riemannianEDist I y x ≤ pathELength I (γ ∘ η) (η 1) (η 0) := by
    apply riemannianEDist_le_pathELength h_smooth.contMDiffOn <;> simp [η, γ0, γ1]
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Riemannian.PathELength (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
H : Type u_2
inst_2 : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst_3 : TopologicalSpace M
inst_4 : ChartedSpace H M
inst_5 : (x : M) → ENorm (TangentSpace I x)
inst_6 : ∀ (x : M), ENormSMulClass.{0, u_1} ℝ (TangentSpace I x)
x_1 y_1 x y : M
r : ℝ≥0∞
hr : riemannianEDist I x y < r
γ : ℝ → M
γ0 : γ 0 = x
γ1 : γ 1 = y
γ_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ
hγ : pathELength I γ 0 1 < r
η : ℝ → ℝ := fun (t : ℝ) ↦ -t
h_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 (γ ∘ η)
this : riemannianEDist I y x ≤ pathELength I (γ ∘ η) (η 1) (η 0)
⊢ riemannianEDist I y x < r
have rw : ∀ (this : riemannianEDist I y x ≤ pathELength I ((γ ∘ η) ∘ η) 0 1), riemannianEDist I y x < r
Tactic: rw [← pathELength_comp_of_antitoneOn zero_le_one] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Riemannian.PathELength (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
H : Type u_2
inst_2 : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst_3 : TopologicalSpace M
inst_4 : ChartedSpace H M
inst_5 : (x : M) → ENorm (TangentSpace I x)
inst_6 : ∀ (x : M), ENormSMulClass.{0, u_1} ℝ (TangentSpace I x)
x_1 y_1 x y : M
r : ℝ≥0∞
hr : riemannianEDist I x y < r
γ : ℝ → M
γ0 : γ 0 = x
γ1 : γ 1 = y
γ_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ
hγ : pathELength I γ 0 1 < r
η : ℝ → ℝ := fun (t : ℝ) ↦ -t
h_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 (γ ∘ η)
this : riemannianEDist I y x ≤ pathELength I (γ ∘ η) (η 1) (η 0)
rw : ∀ (this : riemannianEDist I y x ≤ pathELength I ((γ ∘ η) ∘ η) 0 1), riemannianEDist I y x < r
⊢ riemannianEDist I y x < r
have hf : AntitoneOn η (Icc 0 1)
Tactic: rw [← pathELength_comp_of_antitoneOn zero_le_one] at this
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Riemannian.PathELength (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
H : Type u_2
inst_2 : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst_3 : TopologicalSpace M
inst_4 : ChartedSpace H M
inst_5 : (x : M) → ENorm (TangentSpace I x)
inst_6 : ∀ (x : M), ENormSMulClass.{0, u_1} ℝ (TangentSpace I x)
x_1 y_1 x y : M
r : ℝ≥0∞
hr : riemannianEDist I x y < r
γ : ℝ → M
γ0 : γ 0 = x
γ1 : γ 1 = y
γ_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ
hγ : pathELength I γ 0 1 < r
η : ℝ → ℝ := fun (t : ℝ) ↦ -t
h_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 (γ ∘ η)
this : riemannianEDist I y x ≤ pathELength I (γ ∘ η) (η 1) (η 0)
rw : ∀ (this : riemannianEDist I y x ≤ pathELength I ((γ ∘ η) ∘ η) 0 1), riemannianEDist I y x < r
hf : AntitoneOn η (Icc 0 1)
⊢ riemannianEDist I y x < r
have h'f : DifferentiableOn ℝ η (Icc 0 1)
Tactic: rw [← pathELength_comp_of_antitoneOn zero_le_one] at this
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Riemannian.PathELength (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
H : Type u_2
inst_2 : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst_3 : TopologicalSpace M
inst_4 : ChartedSpace H M
inst_5 : (x : M) → ENorm (TangentSpace I x)
inst_6 : ∀ (x : M), ENormSMulClass.{0, u_1} ℝ (TangentSpace I x)
x_1 y_1 x y : M
r : ℝ≥0∞
hr : riemannianEDist I x y < r
γ : ℝ → M
γ0 : γ 0 = x
γ1 : γ 1 = y
γ_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ
hγ : pathELength I γ 0 1 < r
η : ℝ → ℝ := fun (t : ℝ) ↦ -t
h_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 (γ ∘ η)
this : riemannianEDist I y x ≤ pathELength I (γ ∘ η) (η 1) (η 0)
rw : ∀ (this : riemannianEDist I y x ≤ pathELength I ((γ ∘ η) ∘ η) 0 1), riemannianEDist I y x < r
hf : AntitoneOn η (Icc 0 1)
h'f : DifferentiableOn ℝ η (Icc 0 1)
⊢ riemannianEDist I y x < r
have hγ₁ : MDiff[Icc (η 1) (η 0)] (γ ∘ η)
Tactic: rw [← pathELength_comp_of_antitoneOn zero_le_one] at this
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Riemannian.PathELength (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
H : Type u_2
inst_2 : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst_3 : TopologicalSpace M
inst_4 : ChartedSpace H M
inst_5 : (x : M) → ENorm (TangentSpace I x)
inst_6 : ∀ (x : M), ENormSMulClass.{0, u_1} ℝ (TangentSpace I x)
x_1 y_1 x y : M
r : ℝ≥0∞
hr : riemannianEDist I x y < r
γ : ℝ → M
γ0 : γ 0 = x
γ1 : γ 1 = y
γ_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ
hγ : pathELength I γ 0 1 < r
η : ℝ → ℝ := fun (t : ℝ) ↦ -t
h_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 (γ ∘ η)
this : riemannianEDist I y x ≤ pathELength I ((γ ∘ η) ∘ η) 0 1
⊢ riemannianEDist I y x < r
have apply : pathELength I ((γ ∘ η) ∘ η) 0 1 < r
Tactic: apply this.trans_lt
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Riemannian.PathELength (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
H : Type u_2
inst_2 : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst_3 : TopologicalSpace M
inst_4 : ChartedSpace H M
inst_5 : (x : M) → ENorm (TangentSpace I x)
inst_6 : ∀ (x : M), ENormSMulClass.{0, u_1} ℝ (TangentSpace I x)
x_1 y_1 x y : M
r : ℝ≥0∞
hr : riemannianEDist I x y < r
γ : ℝ → M
γ0 : γ 0 = x
γ1 : γ 1 = y
γ_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ
hγ : pathELength I γ 0 1 < r
η : ℝ → ℝ := fun (t : ℝ) ↦ -t
h_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 (γ ∘ η)
this : riemannianEDist I y x ≤ pathELength I ((γ ∘ η) ∘ η) 0 1
⊢ pathELength I ((γ ∘ η) ∘ η) 0 1 < r
have h : (γ ∘ η) ∘ η = γ
Tactic: convert! hγ
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Riemannian.PathELength (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
H : Type u_2
inst_2 : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst_3 : TopologicalSpace M
inst_4 : ChartedSpace H M
inst_5 : (x : M) → ENorm (TangentSpace I x)
inst_6 : ∀ (x : M), ENormSMulClass.{0, u_1} ℝ (TangentSpace I x)
x_1 y_1 x y : M
r : ℝ≥0∞
hr : riemannianEDist I x y < r
γ : ℝ → M
γ0 : γ 0 = x
γ1 : γ 1 = y
γ_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ
hγ : pathELength I γ 0 1 < r
η : ℝ → ℝ := fun (t : ℝ) ↦ -t
h_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 (γ ∘ η)
this : riemannianEDist I y x ≤ pathELength I ((γ ∘ η) ∘ η) 0 1
⊢ (γ ∘ η) ∘ η = γ
have h : ∀ (t : ℝ), ((γ ∘ η) ∘ η) t = γ t
Tactic: ext t
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Riemannian.PathELength (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
H : Type u_2
inst_2 : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst_3 : TopologicalSpace M
inst_4 : ChartedSpace H M
inst_5 : (x : M) → ENorm (TangentSpace I x)
inst_6 : ∀ (x : M), ENormSMulClass.{0, u_1} ℝ (TangentSpace I x)
x y z : M
⊢ riemannianEDist I x z ≤ riemannianEDist I x y + riemannianEDist I y z
have apply : ∀ (r : ℝ≥0∞) (hr : riemannianEDist I x y + riemannianEDist I y z < r), riemannianEDist I x z < r
Tactic: apply le_of_forall_gt (fun r hr ↦ ?_)
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Riemannian.PathELength (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
H : Type u_2
inst_2 : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst_3 : TopologicalSpace M
inst_4 : ChartedSpace H M
inst_5 : (x : M) → ENorm (TangentSpace I x)
inst_6 : ∀ (x : M), ENormSMulClass.{0, u_1} ℝ (TangentSpace I x)
x y z : M
r : ℝ≥0∞
hr : riemannianEDist I x y + riemannianEDist I y z < r
⊢ riemannianEDist I x z < r
have rcases : ∀ (u : ℝ≥0∞) (hu : u > riemannianEDist I x y) (v : ℝ≥0∞) (hv : v > riemannianEDist I y z) (huv : u + v < r),
  riemannianEDist I x z < r
Tactic: rcases ENNReal.exists_add_lt_of_add_lt hr with ⟨u, hu, v, hv, huv⟩
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Riemannian.PathELength (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
H : Type u_2
inst_2 : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst_3 : TopologicalSpace M
inst_4 : ChartedSpace H M
inst_5 : (x : M) → ENorm (TangentSpace I x)
inst_6 : ∀ (x : M), ENormSMulClass.{0, u_1} ℝ (TangentSpace I x)
x y z : M
r : ℝ≥0∞
hr : riemannianEDist I x y + riemannianEDist I y z < r
u : ℝ≥0∞
hu : u > riemannianEDist I x y
v : ℝ≥0∞
hv : v > riemannianEDist I y z
huv : u + v < r
⊢ riemannianEDist I x z < r
have rcases : ∀ (γ₁ : ℝ → M) (hγ₁0 : γ₁ 0 = x) (hγ₁1 : γ₁ 1 = y) (hγ₁_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ₁)
  (hγ₁ : pathELength I γ₁ 0 1 < u) (hγ₁_const : γ₁ =ᶠ[𝓝 1] fun (x : ℝ) ↦ y), riemannianEDist I x z < r
Tactic: rcases exists_lt_locally_constant_of_riemannianEDist_lt hu zero_lt_one with
    ⟨γ₁, hγ₁0, hγ₁1, hγ₁_smooth, hγ₁, -, hγ₁_const⟩
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Riemannian.PathELength (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
H : Type u_2
inst_2 : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst_3 : TopologicalSpace M
inst_4 : ChartedSpace H M
inst_5 : (x : M) → ENorm (TangentSpace I x)
inst_6 : ∀ (x : M), ENormSMulClass.{0, u_1} ℝ (TangentSpace I x)
x y z : M
r : ℝ≥0∞
hr : riemannianEDist I x y + riemannianEDist I y z < r
u : ℝ≥0∞
hu : u > riemannianEDist I x y
v : ℝ≥0∞
hv : v > riemannianEDist I y z
huv : u + v < r
γ₁ : ℝ → M
hγ₁0 : γ₁ 0 = x
hγ₁1 : γ₁ 1 = y
hγ₁_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ₁
hγ₁ : pathELength I γ₁ 0 1 < u
hγ₁_const : γ₁ =ᶠ[𝓝 1] fun (x : ℝ) ↦ y
⊢ riemannianEDist I x z < r
have rcases : ∀ (γ₂ : ℝ → M) (hγ₂1 : γ₂ 1 = y) (hγ₂2 : γ₂ 2 = z) (hγ₂_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ₂)
  (hγ₂ : pathELength I γ₂ 1 2 < v) (hγ₂_const : γ₂ =ᶠ[𝓝 1] fun (x : ℝ) ↦ y), riemannianEDist I x z < r
Tactic: rcases exists_lt_locally_constant_of_riemannianEDist_lt hv one_lt_two with
    ⟨γ₂, hγ₂1, hγ₂2, hγ₂_smooth, hγ₂, hγ₂_const, -⟩
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Riemannian.PathELength (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
H : Type u_2
inst_2 : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst_3 : TopologicalSpace M
inst_4 : ChartedSpace H M
inst_5 : (x : M) → ENorm (TangentSpace I x)
inst_6 : ∀ (x : M), ENormSMulClass.{0, u_1} ℝ (TangentSpace I x)
x y z : M
r : ℝ≥0∞
hr : riemannianEDist I x y + riemannianEDist I y z < r
u : ℝ≥0∞
hu : u > riemannianEDist I x y
v : ℝ≥0∞
hv : v > riemannianEDist I y z
huv : u + v < r
γ₁ : ℝ → M
hγ₁0 : γ₁ 0 = x
hγ₁1 : γ₁ 1 = y
hγ₁_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ₁
hγ₁ : pathELength I γ₁ 0 1 < u
hγ₁_const : γ₁ =ᶠ[𝓝 1] fun (x : ℝ) ↦ y
γ₂ : ℝ → M
hγ₂1 : γ₂ 1 = y
hγ₂2 : γ₂ 2 = z
hγ₂_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ₂
hγ₂ : pathELength I γ₂ 1 2 < v
hγ₂_const : γ₂ =ᶠ[𝓝 1] fun (x : ℝ) ↦ y
γ : (i : ℝ) → M := piecewise (β := fun (i : ℝ) ↦ M) (Iic 1) γ₁ γ₂
⊢ riemannianEDist I x z ≤ pathELength I γ 0 2
have hγ : ContMDiffOn 𝓘(ℝ, ℝ) I 1 γ (Icc 0 2)
Tactic: apply riemannianEDist_le_pathELength
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Riemannian.PathELength (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
H : Type u_2
inst_2 : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst_3 : TopologicalSpace M
inst_4 : ChartedSpace H M
inst_5 : (x : M) → ENorm (TangentSpace I x)
inst_6 : ∀ (x : M), ENormSMulClass.{0, u_1} ℝ (TangentSpace I x)
x y z : M
r : ℝ≥0∞
hr : riemannianEDist I x y + riemannianEDist I y z < r
u : ℝ≥0∞
hu : u > riemannianEDist I x y
v : ℝ≥0∞
hv : v > riemannianEDist I y z
huv : u + v < r
γ₁ : ℝ → M
hγ₁0 : γ₁ 0 = x
hγ₁1 : γ₁ 1 = y
hγ₁_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ₁
hγ₁ : pathELength I γ₁ 0 1 < u
hγ₁_const : γ₁ =ᶠ[𝓝 1] fun (x : ℝ) ↦ y
γ₂ : ℝ → M
hγ₂1 : γ₂ 1 = y
hγ₂2 : γ₂ 2 = z
hγ₂_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ₂
hγ₂ : pathELength I γ₂ 1 2 < v
hγ₂_const : γ₂ =ᶠ[𝓝 1] fun (x : ℝ) ↦ y
γ : (i : ℝ) → M := piecewise (β := fun (i : ℝ) ↦ M) (Iic 1) γ₁ γ₂
hγ : ContMDiffOn 𝓘(ℝ, ℝ) I 1 γ (Icc 0 2)
⊢ riemannianEDist I x z ≤ pathELength I γ 0 2
have ha : γ 0 = x
Tactic: apply riemannianEDist_le_pathELength
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Riemannian.PathELength (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
H : Type u_2
inst_2 : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst_3 : TopologicalSpace M
inst_4 : ChartedSpace H M
inst_5 : (x : M) → ENorm (TangentSpace I x)
inst_6 : ∀ (x : M), ENormSMulClass.{0, u_1} ℝ (TangentSpace I x)
x y z : M
r : ℝ≥0∞
hr : riemannianEDist I x y + riemannianEDist I y z < r
u : ℝ≥0∞
hu : u > riemannianEDist I x y
v : ℝ≥0∞
hv : v > riemannianEDist I y z
huv : u + v < r
γ₁ : ℝ → M
hγ₁0 : γ₁ 0 = x
hγ₁1 : γ₁ 1 = y
hγ₁_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ₁
hγ₁ : pathELength I γ₁ 0 1 < u
hγ₁_const : γ₁ =ᶠ[𝓝 1] fun (x : ℝ) ↦ y
γ₂ : ℝ → M
hγ₂1 : γ₂ 1 = y
hγ₂2 : γ₂ 2 = z
hγ₂_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ₂
hγ₂ : pathELength I γ₂ 1 2 < v
hγ₂_const : γ₂ =ᶠ[𝓝 1] fun (x : ℝ) ↦ y
γ : (i : ℝ) → M := piecewise (β := fun (i : ℝ) ↦ M) (Iic 1) γ₁ γ₂
hγ : ContMDiffOn 𝓘(ℝ, ℝ) I 1 γ (Icc 0 2)
ha : γ 0 = x
⊢ riemannianEDist I x z ≤ pathELength I γ 0 2
have hb : γ 2 = z
Tactic: apply riemannianEDist_le_pathELength
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Riemannian.PathELength (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
H : Type u_2
inst_2 : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst_3 : TopologicalSpace M
inst_4 : ChartedSpace H M
inst_5 : (x : M) → ENorm (TangentSpace I x)
inst_6 : ∀ (x : M), ENormSMulClass.{0, u_1} ℝ (TangentSpace I x)
x y z : M
r : ℝ≥0∞
hr : riemannianEDist I x y + riemannianEDist I y z < r
u : ℝ≥0∞
hu : u > riemannianEDist I x y
v : ℝ≥0∞
hv : v > riemannianEDist I y z
huv : u + v < r
γ₁ : ℝ → M
hγ₁0 : γ₁ 0 = x
hγ₁1 : γ₁ 1 = y
hγ₁_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ₁
hγ₁ : pathELength I γ₁ 0 1 < u
hγ₁_const : γ₁ =ᶠ[𝓝 1] fun (x : ℝ) ↦ y
γ₂ : ℝ → M
hγ₂1 : γ₂ 1 = y
hγ₂2 : γ₂ 2 = z
hγ₂_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ₂
hγ₂ : pathELength I γ₂ 1 2 < v
hγ₂_const : γ₂ =ᶠ[𝓝 1] fun (x : ℝ) ↦ y
γ : (i : ℝ) → M := piecewise (β := fun (i : ℝ) ↦ M) (Iic 1) γ₁ γ₂
hγ : ContMDiffOn 𝓘(ℝ, ℝ) I 1 γ (Icc 0 2)
ha : γ 0 = x
hb : γ 2 = z
⊢ riemannianEDist I x z ≤ pathELength I γ 0 2
have hab : LE.le (α := ℝ) 0 2
Tactic: apply riemannianEDist_le_pathELength
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Riemannian.PathELength (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
H : Type u_2
inst_2 : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst_3 : TopologicalSpace M
inst_4 : ChartedSpace H M
inst_5 : (x : M) → ENorm (TangentSpace I x)
inst_6 : ∀ (x : M), ENormSMulClass.{0, u_1} ℝ (TangentSpace I x)
x y z : M
r : ℝ≥0∞
hr : riemannianEDist I x y + riemannianEDist I y z < r
u : ℝ≥0∞
hu : u > riemannianEDist I x y
v : ℝ≥0∞
hv : v > riemannianEDist I y z
huv : u + v < r
γ₁ : ℝ → M
hγ₁0 : γ₁ 0 = x
hγ₁1 : γ₁ 1 = y
hγ₁_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ₁
hγ₁ : pathELength I γ₁ 0 1 < u
hγ₁_const : γ₁ =ᶠ[𝓝 1] fun (x : ℝ) ↦ y
γ₂ : ℝ → M
hγ₂1 : γ₂ 1 = y
hγ₂2 : γ₂ 2 = z
hγ₂_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ₂
hγ₂ : pathELength I γ₂ 1 2 < v
hγ₂_const : γ₂ =ᶠ[𝓝 1] fun (x : ℝ) ↦ y
γ : (i : ℝ) → M := piecewise (β := fun (i : ℝ) ↦ M) (Iic 1) γ₁ γ₂
⊢ ContMDiffOn 𝓘(ℝ, ℝ) I 1 γ (Icc 0 2)
have hγ : ContMDiff 𝓘(ℝ, ℝ) I 1 γ
Tactic: apply ContMDiff.contMDiffOn
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Riemannian.PathELength (hypothesis)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
H : Type u_2
inst_2 : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst_3 : TopologicalSpace M
inst_4 : ChartedSpace H M
inst_5 : (x : M) → ENorm (TangentSpace I x)
inst_6 : ∀ (x : M), ENormSMulClass.{0, u_1} ℝ (TangentSpace I x)
x y z : M
r : ℝ≥0∞
hr : riemannianEDist I x y + riemannianEDist I y z < r
u : ℝ≥0∞
hu : u > riemannianEDist I x y
v : ℝ≥0∞
hv : v > riemannianEDist I y z
huv : u + v < r
γ₁ : ℝ → M
hγ₁0 : γ₁ 0 = x
hγ₁1 : γ₁ 1 = y
hγ₁_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ₁
hγ₁ : pathELength I γ₁ 0 1 < u
hγ₁_const : γ₁ =ᶠ[𝓝 1] fun (x : ℝ) ↦ y
γ₂ : ℝ → M
hγ₂1 : γ₂ 1 = y
hγ₂2 : γ₂ 2 = z
hγ₂_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ₂
hγ₂ : pathELength I γ₂ 1 2 < v
hγ₂_const : γ₂ =ᶠ[𝓝 1] fun (x : ℝ) ↦ y
γ : (i : ℝ) → M := piecewise (β := fun (i : ℝ) ↦ M) (Iic 1) γ₁ γ₂
⊢ riemannianEDist I x z < r
have this : riemannianEDist I x z ≤ pathELength I γ 0 2
Tactic: have : riemannianEDist I x z ≤ pathELength I γ 0 2 := by
    apply riemannianEDist_le_pathELength
    · apply ContMDiff.contMDiffOn
      exact ContMDiff.piecewise_Iic hγ₁_smooth hγ₂_smooth (hγ₁_const.trans hγ₂_const.symm)
    · simp [γ, hγ₁0]
    · simp [γ, hγ₂2]
    · exact zero_le_two
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Riemannian.PathELength (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
H : Type u_2
inst_2 : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst_3 : TopologicalSpace M
inst_4 : ChartedSpace H M
inst_5 : (x : M) → ENorm (TangentSpace I x)
inst_6 : ∀ (x : M), ENormSMulClass.{0, u_1} ℝ (TangentSpace I x)
x y z : M
r : ℝ≥0∞
hr : riemannianEDist I x y + riemannianEDist I y z < r
u : ℝ≥0∞
hu : u > riemannianEDist I x y
v : ℝ≥0∞
hv : v > riemannianEDist I y z
huv : u + v < r
γ₁ : ℝ → M
hγ₁0 : γ₁ 0 = x
hγ₁1 : γ₁ 1 = y
hγ₁_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ₁
hγ₁ : pathELength I γ₁ 0 1 < u
hγ₁_const : γ₁ =ᶠ[𝓝 1] fun (x : ℝ) ↦ y
γ₂ : ℝ → M
hγ₂1 : γ₂ 1 = y
hγ₂2 : γ₂ 2 = z
hγ₂_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ₂
hγ₂ : pathELength I γ₂ 1 2 < v
hγ₂_const : γ₂ =ᶠ[𝓝 1] fun (x : ℝ) ↦ y
γ : (i : ℝ) → M := piecewise (β := fun (i : ℝ) ↦ M) (Iic 1) γ₁ γ₂
this : riemannianEDist I x z ≤ pathELength I γ 0 2
⊢ riemannianEDist I x z < r
have apply : pathELength I γ 0 2 < u + v
Tactic: apply this.trans_lt (lt_trans ?_ huv)
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Riemannian.PathELength (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
H : Type u_2
inst_2 : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst_3 : TopologicalSpace M
inst_4 : ChartedSpace H M
inst_5 : (x : M) → ENorm (TangentSpace I x)
inst_6 : ∀ (x : M), ENormSMulClass.{0, u_1} ℝ (TangentSpace I x)
x y z : M
r : ℝ≥0∞
hr : riemannianEDist I x y + riemannianEDist I y z < r
u : ℝ≥0∞
hu : u > riemannianEDist I x y
v : ℝ≥0∞
hv : v > riemannianEDist I y z
huv : u + v < r
γ₁ : ℝ → M
hγ₁0 : γ₁ 0 = x
hγ₁1 : γ₁ 1 = y
hγ₁_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ₁
hγ₁ : pathELength I γ₁ 0 1 < u
hγ₁_const : γ₁ =ᶠ[𝓝 1] fun (x : ℝ) ↦ y
γ₂ : ℝ → M
hγ₂1 : γ₂ 1 = y
hγ₂2 : γ₂ 2 = z
hγ₂_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ₂
hγ₂ : pathELength I γ₂ 1 2 < v
hγ₂_const : γ₂ =ᶠ[𝓝 1] fun (x : ℝ) ↦ y
γ : (i : ℝ) → M := piecewise (β := fun (i : ℝ) ↦ M) (Iic 1) γ₁ γ₂
this : riemannianEDist I x z ≤ pathELength I γ 0 2
⊢ pathELength I γ 0 2 < u + v
have rw : pathELength I γ 0 1 + pathELength I γ 1 2 < u + v
Tactic: rw [← pathELength_add zero_le_one one_le_two]