awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
g : SL(2, ℤ)
z : ℍ
cd : Fin 2 → ℤ
hcd : IsCoprime (cd 0) (cd 1)
⊢ ¬HPow.hPow (β := ℕ) (cd 0) 2 + HPow.hPow (β := ℕ) (-cd 1) 2 = 0
have rw : ¬HPow.hPow (β := ℕ) (cd 0) 2 + HPow.hPow (β := ℕ) (cd 1) 2 = 0
Tactic: rw [neg_sq]
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
cd : Fin 2 → ℤ
hcd : IsCoprime (cd 0) (cd 1)
mB : ℝ → Matrix (Fin 2) (Fin 2) ℝ :=
  fun (t : ℝ) ↦ (of : (a : Fin 2 → Fin 2 → ℝ) → Matrix (Fin 2) (Fin 2) ℝ) ![![t, -(↑1 : ℝ)], Int.cast ∘ cd]
⊢ Continuous mB
have refine : ∀ (i j : Fin 2), Continuous (Y := ℝ) fun (a : ℝ) ↦ mB a i j
Tactic: refine continuous_matrix ?_
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (hypothesis)
Goal:
cd : Fin 2 → ℤ
hcd : IsCoprime (cd 0) (cd 1)
mB : ℝ → Matrix (Fin 2) (Fin 2) ℝ :=
  fun (t : ℝ) ↦ (of : (a : Fin 2 → Fin 2 → ℝ) → Matrix (Fin 2) (Fin 2) ℝ) ![![t, -(↑1 : ℝ)], Int.cast ∘ cd]
⊢ Tendsto
    (fun (g : { g : SL(2, ℤ) // (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 = cd }) ↦
      (lcRow0 cd : (a : Matrix (Fin 2) (Fin 2) ℝ) → ℝ)
        (↑((SpecialLinearGroup.map (Int.castRingHom ℝ) : (a : SL(2, ℤ)) → SL(2, ℝ)) (↑g : SL(2, ℤ))) :
          Matrix (Fin 2) (Fin 2) ℝ))
    cofinite (cocompact ℝ)
have hmB : Continuous mB
Tactic: have hmB : Continuous mB := by
    refine continuous_matrix ?_
    simp only [mB, Fin.forall_fin_two, continuous_const, continuous_id', of_apply, cons_val_zero,
      cons_val_one, and_self_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
cd : Fin 2 → ℤ
hcd : IsCoprime (cd 0) (cd 1)
mB : ℝ → Matrix (Fin 2) (Fin 2) ℝ :=
  fun (t : ℝ) ↦ (of : (a : Fin 2 → Fin 2 → ℝ) → Matrix (Fin 2) (Fin 2) ℝ) ![![t, -(↑1 : ℝ)], Int.cast ∘ cd]
hmB : Continuous mB
⊢ Tendsto
    (fun (g : { g : SL(2, ℤ) // (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 = cd }) ↦
      (lcRow0 cd : (a : Matrix (Fin 2) (Fin 2) ℝ) → ℝ)
        (↑((SpecialLinearGroup.map (Int.castRingHom ℝ) : (a : SL(2, ℤ)) → SL(2, ℝ)) (↑g : SL(2, ℤ))) :
          Matrix (Fin 2) (Fin 2) ℝ))
    cofinite (cocompact ℝ)
have refine : Tendsto (α := { g : SL(2, ℤ) // (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 = cd })
  (mB ∘ fun (g : { g : SL(2, ℤ) // (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 = cd }) ↦
    (lcRow0 cd : (a : Matrix (Fin 2) (Fin 2) ℝ) → ℝ)
      (↑((SpecialLinearGroup.map (Int.castRingHom ℝ) : (a : SL(2, ℤ)) → SL(2, ℝ)) (↑g : SL(2, ℤ))) :
        Matrix (Fin 2) (Fin 2) ℝ))
  cofinite (cocompact (Matrix (Fin 2) (Fin 2) ℝ))
Tactic: refine Filter.Tendsto.of_tendsto_comp ?_ (comap_cocompact_le hmB)
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (hypothesis)
Goal:
cd : Fin 2 → ℤ
hcd : IsCoprime (cd 0) (cd 1)
mB : ℝ → Matrix (Fin 2) (Fin 2) ℝ :=
  fun (t : ℝ) ↦ (of : (a : Fin 2 → Fin 2 → ℝ) → Matrix (Fin 2) (Fin 2) ℝ) ![![t, -(↑1 : ℝ)], Int.cast ∘ cd]
hmB : Continuous mB
f₁ : SL(2, ℤ) → Matrix (Fin 2) (Fin 2) ℝ := fun (g : SL(2, ℤ)) ↦ (↑g : Matrix (Fin 2) (Fin 2) ℤ).map Int.cast
⊢ Tendsto (α := { g : SL(2, ℤ) // (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 = cd })
    (mB ∘ fun (g : { g : SL(2, ℤ) // (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 = cd }) ↦
      (lcRow0 cd : (a : Matrix (Fin 2) (Fin 2) ℝ) → ℝ)
        (↑((SpecialLinearGroup.map (Int.castRingHom ℝ) : (a : SL(2, ℤ)) → SL(2, ℝ)) (↑g : SL(2, ℤ))) :
          Matrix (Fin 2) (Fin 2) ℝ))
    cofinite (cocompact (Matrix (Fin 2) (Fin 2) ℝ))
have cocompact_ℝ_to_cofinite_ℤ_matrix : Tendsto (β := Matrix (Fin 2) (Fin 2) ℝ) (fun (m : Matrix (Fin 2) (Fin 2) ℤ) ↦ m.map Int.cast) cofinite
  (cocompact (Matrix (Fin 2) (Fin 2) ℝ))
Tactic: have cocompact_ℝ_to_cofinite_ℤ_matrix :
    Tendsto (fun m : Matrix (Fin 2) (Fin 2) ℤ => Matrix.map m ((↑) : ℤ → ℝ)) cofinite
      (cocompact _) := by
    simpa only [coprodᵢ_cofinite, coprodᵢ_cocompact] using
      Tendsto.pi_map_coprodᵢ fun _ : Fin 2 =>
        Tendsto.pi_map_coprodᵢ fun _ : Fin 2 => Int.tendsto_coe_cofinite
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (hypothesis)
Goal:
cd : Fin 2 → ℤ
hcd : IsCoprime (cd 0) (cd 1)
mB : ℝ → Matrix (Fin 2) (Fin 2) ℝ :=
  fun (t : ℝ) ↦ (of : (a : Fin 2 → Fin 2 → ℝ) → Matrix (Fin 2) (Fin 2) ℝ) ![![t, -(↑1 : ℝ)], Int.cast ∘ cd]
hmB : Continuous mB
f₁ : SL(2, ℤ) → Matrix (Fin 2) (Fin 2) ℝ := fun (g : SL(2, ℤ)) ↦ (↑g : Matrix (Fin 2) (Fin 2) ℤ).map Int.cast
cocompact_ℝ_to_cofinite_ℤ_matrix :
  Tendsto (β := Matrix (Fin 2) (Fin 2) ℝ) (fun (m : Matrix (Fin 2) (Fin 2) ℤ) ↦ m.map Int.cast) cofinite
    (cocompact (Matrix (Fin 2) (Fin 2) ℝ))
⊢ Tendsto (α := { g : SL(2, ℤ) // (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 = cd })
    (mB ∘ fun (g : { g : SL(2, ℤ) // (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 = cd }) ↦
      (lcRow0 cd : (a : Matrix (Fin 2) (Fin 2) ℝ) → ℝ)
        (↑((SpecialLinearGroup.map (Int.castRingHom ℝ) : (a : SL(2, ℤ)) → SL(2, ℝ)) (↑g : SL(2, ℤ))) :
          Matrix (Fin 2) (Fin 2) ℝ))
    cofinite (cocompact (Matrix (Fin 2) (Fin 2) ℝ))
have hf₁ : Tendsto f₁ cofinite (cocompact (Matrix (Fin 2) (Fin 2) ℝ))
Tactic: have hf₁ : Tendsto f₁ cofinite (cocompact _) :=
    cocompact_ℝ_to_cofinite_ℤ_matrix.comp Subtype.coe_injective.tendsto_cofinite
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (hypothesis)
Goal:
cd : Fin 2 → ℤ
hcd : IsCoprime (cd 0) (cd 1)
mB : ℝ → Matrix (Fin 2) (Fin 2) ℝ :=
  fun (t : ℝ) ↦ (of : (a : Fin 2 → Fin 2 → ℝ) → Matrix (Fin 2) (Fin 2) ℝ) ![![t, -(↑1 : ℝ)], Int.cast ∘ cd]
hmB : Continuous mB
f₁ : SL(2, ℤ) → Matrix (Fin 2) (Fin 2) ℝ := fun (g : SL(2, ℤ)) ↦ (↑g : Matrix (Fin 2) (Fin 2) ℤ).map Int.cast
cocompact_ℝ_to_cofinite_ℤ_matrix :
  Tendsto (β := Matrix (Fin 2) (Fin 2) ℝ) (fun (m : Matrix (Fin 2) (Fin 2) ℤ) ↦ m.map Int.cast) cofinite
    (cocompact (Matrix (Fin 2) (Fin 2) ℝ))
hf₁ : Tendsto f₁ cofinite (cocompact (Matrix (Fin 2) (Fin 2) ℝ))
⊢ Tendsto (α := { g : SL(2, ℤ) // (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 = cd })
    (mB ∘ fun (g : { g : SL(2, ℤ) // (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 = cd }) ↦
      (lcRow0 cd : (a : Matrix (Fin 2) (Fin 2) ℝ) → ℝ)
        (↑((SpecialLinearGroup.map (Int.castRingHom ℝ) : (a : SL(2, ℤ)) → SL(2, ℝ)) (↑g : SL(2, ℤ))) :
          Matrix (Fin 2) (Fin 2) ℝ))
    cofinite (cocompact (Matrix (Fin 2) (Fin 2) ℝ))
have hf₂ : IsClosedEmbedding (X := Matrix (Fin 2) (Fin 2) ℝ) (Y := Matrix (Fin 2) (Fin 2) ℝ)
  (⇑(lcRow0Extend hcd) : (a : Matrix (Fin 2) (Fin 2) ℝ) → Matrix (Fin 2) (Fin 2) ℝ)
Tactic: have hf₂ : IsClosedEmbedding (lcRow0Extend hcd) :=
    (lcRow0Extend hcd).toContinuousLinearEquiv.toHomeomorph.isClosedEmbedding
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
cd : Fin 2 → ℤ
hcd : IsCoprime (cd 0) (cd 1)
mB : ℝ → Matrix (Fin 2) (Fin 2) ℝ :=
  fun (t : ℝ) ↦ (of : (a : Fin 2 → Fin 2 → ℝ) → Matrix (Fin 2) (Fin 2) ℝ) ![![t, -(↑1 : ℝ)], Int.cast ∘ cd]
hmB : Continuous mB
f₁ : SL(2, ℤ) → Matrix (Fin 2) (Fin 2) ℝ := fun (g : SL(2, ℤ)) ↦ (↑g : Matrix (Fin 2) (Fin 2) ℤ).map Int.cast
cocompact_ℝ_to_cofinite_ℤ_matrix :
  Tendsto (β := Matrix (Fin 2) (Fin 2) ℝ) (fun (m : Matrix (Fin 2) (Fin 2) ℤ) ↦ m.map Int.cast) cofinite
    (cocompact (Matrix (Fin 2) (Fin 2) ℝ))
hf₁ : Tendsto f₁ cofinite (cocompact (Matrix (Fin 2) (Fin 2) ℝ))
hf₂ :
  IsClosedEmbedding (X := Matrix (Fin 2) (Fin 2) ℝ) (Y := Matrix (Fin 2) (Fin 2) ℝ)
    (⇑(lcRow0Extend hcd) : (a : Matrix (Fin 2) (Fin 2) ℝ) → Matrix (Fin 2) (Fin 2) ℝ)
⊢ Tendsto (α := { g : SL(2, ℤ) // (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 = cd })
    (mB ∘ fun (g : { g : SL(2, ℤ) // (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 = cd }) ↦
      (lcRow0 cd : (a : Matrix (Fin 2) (Fin 2) ℝ) → ℝ)
        (↑((SpecialLinearGroup.map (Int.castRingHom ℝ) : (a : SL(2, ℤ)) → SL(2, ℝ)) (↑g : SL(2, ℤ))) :
          Matrix (Fin 2) (Fin 2) ℝ))
    cofinite (cocompact (Matrix (Fin 2) (Fin 2) ℝ))
have h : Eq (α := { g : SL(2, ℤ) // (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 = cd } → Matrix (Fin 2) (Fin 2) ℝ)
  (mB ∘ fun (g : { g : SL(2, ℤ) // (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 = cd }) ↦
    (lcRow0 cd : (a : Matrix (Fin 2) (Fin 2) ℝ) → ℝ)
      (↑((SpecialLinearGroup.map (Int.castRingHom ℝ) : (a : SL(2, ℤ)) → SL(2, ℝ)) (↑g : SL(2, ℤ))) :
        Matrix (Fin 2) (Fin 2) ℝ))
  ((⇑(lcRow0Extend hcd) : (a : Matrix (Fin 2) (Fin 2) ℝ) → Matrix (Fin 2) (Fin 2) ℝ) ∘
    f₁ ∘ fun (a : { g : SL(2, ℤ) // (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 = cd }) ↦ (↑a : SL(2, ℤ)))
Tactic: convert! hf₂.tendsto_cocompact.comp (hf₁.comp Subtype.coe_injective.tendsto_cofinite) using 1
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
cd : Fin 2 → ℤ
hcd : IsCoprime (cd 0) (cd 1)
mB : ℝ → Matrix (Fin 2) (Fin 2) ℝ :=
  fun (t : ℝ) ↦ (of : (a : Fin 2 → Fin 2 → ℝ) → Matrix (Fin 2) (Fin 2) ℝ) ![![t, -(↑1 : ℝ)], Int.cast ∘ cd]
hmB : Continuous mB
f₁ : SL(2, ℤ) → Matrix (Fin 2) (Fin 2) ℝ := fun (g : SL(2, ℤ)) ↦ (↑g : Matrix (Fin 2) (Fin 2) ℤ).map Int.cast
cocompact_ℝ_to_cofinite_ℤ_matrix :
  Tendsto (β := Matrix (Fin 2) (Fin 2) ℝ) (fun (m : Matrix (Fin 2) (Fin 2) ℤ) ↦ m.map Int.cast) cofinite
    (cocompact (Matrix (Fin 2) (Fin 2) ℝ))
hf₁ : Tendsto f₁ cofinite (cocompact (Matrix (Fin 2) (Fin 2) ℝ))
hf₂ :
  IsClosedEmbedding (X := Matrix (Fin 2) (Fin 2) ℝ) (Y := Matrix (Fin 2) (Fin 2) ℝ)
    (⇑(lcRow0Extend hcd) : (a : Matrix (Fin 2) (Fin 2) ℝ) → Matrix (Fin 2) (Fin 2) ℝ)
⊢ Eq (α := { g : SL(2, ℤ) // (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 = cd } → Matrix (Fin 2) (Fin 2) ℝ)
    (mB ∘ fun (g : { g : SL(2, ℤ) // (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 = cd }) ↦
      (lcRow0 cd : (a : Matrix (Fin 2) (Fin 2) ℝ) → ℝ)
        (↑((SpecialLinearGroup.map (Int.castRingHom ℝ) : (a : SL(2, ℤ)) → SL(2, ℝ)) (↑g : SL(2, ℤ))) :
          Matrix (Fin 2) (Fin 2) ℝ))
    ((⇑(lcRow0Extend hcd) : (a : Matrix (Fin 2) (Fin 2) ℝ) → Matrix (Fin 2) (Fin 2) ℝ) ∘
      f₁ ∘ fun (a : { g : SL(2, ℤ) // (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 = cd }) ↦ (↑a : SL(2, ℤ)))
have h : ∀ (g : SL(2, ℤ)) (hcd : IsCoprime ((↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 0) ((↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 1)),
  let mB : ℝ → Matrix (Fin 2) (Fin 2) ℝ := fun (t : ℝ) ↦
    (of : (a : Fin 2 → Fin 2 → ℝ) → Matrix (Fin 2) (Fin 2) ℝ)
      ![![t, -(↑1 : ℝ)], Int.cast ∘ (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1];
  ∀ (hmB : Continuous mB)
    (hf₂ :
      IsClosedEmbedding (X := Matrix (Fin 2) (Fin 2) ℝ) (Y := Matrix (Fin 2) (Fin 2) ℝ)
        (⇑(lcRow0Extend hcd) : (a : Matrix (Fin 2) (Fin 2) ℝ) → Matrix (Fin 2) (Fin 2) ℝ))
    (i j : Fin 2),
    (mB ∘ fun (g_1 : { g_1 : SL(2, ℤ) // (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 }) ↦
          (lcRow0 ((↑g : Matrix (Fin 2) (Fin 2) ℤ) 1) : (a : Matrix (Fin 2) (Fin 2) ℝ) → ℝ)
            (↑((SpecialLinearGroup.map (Int.castRingHom ℝ) : (a : SL(2, ℤ)) → SL(2, ℝ)) (↑g_1 : SL(2, ℤ))) :
              Matrix (Fin 2) (Fin 2) ℝ))
        (Subtype.mk (p := fun (g_1 : SL(2, ℤ)) ↦
          (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1) g sorry)
        i j =
      Function.comp (α := { g_1 : SL(2, ℤ) // (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 })
        (⇑(lcRow0Extend hcd) : (a : Matrix (Fin 2) (Fin 2) ℝ) → Matrix (Fin 2) (Fin 2) ℝ)
        (f₁ ∘ fun (a : { g_1 : SL(2, ℤ) // (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 }) ↦
          (↑a : SL(2, ℤ)))
        (Subtype.mk (p := fun (g_1 : SL(2, ℤ)) ↦
          (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1) g sorry)
        i j
Tactic: ext ⟨g, rfl⟩ i j : 3
Cleared hypotheses: cd, hcd, mB, hmB, hf₂
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
f₁ : SL(2, ℤ) → Matrix (Fin 2) (Fin 2) ℝ := fun (g : SL(2, ℤ)) ↦ (↑g : Matrix (Fin 2) (Fin 2) ℤ).map Int.cast
cocompact_ℝ_to_cofinite_ℤ_matrix :
  Tendsto (β := Matrix (Fin 2) (Fin 2) ℝ) (fun (m : Matrix (Fin 2) (Fin 2) ℤ) ↦ m.map Int.cast) cofinite
    (cocompact (Matrix (Fin 2) (Fin 2) ℝ))
hf₁ : Tendsto f₁ cofinite (cocompact (Matrix (Fin 2) (Fin 2) ℝ))
g : SL(2, ℤ)
hcd : IsCoprime ((↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 0) ((↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 1)
mB : ℝ → Matrix (Fin 2) (Fin 2) ℝ :=
  fun (t : ℝ) ↦
    (of : (a : Fin 2 → Fin 2 → ℝ) → Matrix (Fin 2) (Fin 2) ℝ)
      ![![t, -(↑1 : ℝ)], Int.cast ∘ (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1]
hmB : Continuous mB
hf₂ :
  IsClosedEmbedding (X := Matrix (Fin 2) (Fin 2) ℝ) (Y := Matrix (Fin 2) (Fin 2) ℝ)
    (⇑(lcRow0Extend hcd) : (a : Matrix (Fin 2) (Fin 2) ℝ) → Matrix (Fin 2) (Fin 2) ℝ)
i j : Fin 2
⊢ (mB ∘ fun (g_1 : { g_1 : SL(2, ℤ) // (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 }) ↦
        (lcRow0 ((↑g : Matrix (Fin 2) (Fin 2) ℤ) 1) : (a : Matrix (Fin 2) (Fin 2) ℝ) → ℝ)
          (↑((SpecialLinearGroup.map (Int.castRingHom ℝ) : (a : SL(2, ℤ)) → SL(2, ℝ)) (↑g_1 : SL(2, ℤ))) :
            Matrix (Fin 2) (Fin 2) ℝ))
      (Subtype.mk (p := fun (g_1 : SL(2, ℤ)) ↦ (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1)
        g ⋯)
      i j =
    Function.comp (α := { g_1 : SL(2, ℤ) // (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 })
      (⇑(lcRow0Extend hcd) : (a : Matrix (Fin 2) (Fin 2) ℝ) → Matrix (Fin 2) (Fin 2) ℝ)
      (f₁ ∘ fun (a : { g_1 : SL(2, ℤ) // (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 }) ↦
        (↑a : SL(2, ℤ)))
      (Subtype.mk (p := fun (g_1 : SL(2, ℤ)) ↦ (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1)
        g ⋯)
      i j
have h : (mB ∘ fun (g_1 : { g_1 : SL(2, ℤ) // (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 }) ↦
      (lcRow0 ((↑g : Matrix (Fin 2) (Fin 2) ℤ) 1) : (a : Matrix (Fin 2) (Fin 2) ℝ) → ℝ)
        (↑((SpecialLinearGroup.map (Int.castRingHom ℝ) : (a : SL(2, ℤ)) → SL(2, ℝ)) (↑g_1 : SL(2, ℤ))) :
          Matrix (Fin 2) (Fin 2) ℝ))
    (Subtype.mk (p := fun (g_1 : SL(2, ℤ)) ↦ (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1) g
      sorry)
    ((fun (i : Fin 2) ↦ i) (⟨0, sorry⟩ : Fin 2)) j =
  Function.comp (α := { g_1 : SL(2, ℤ) // (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 })
    (⇑(lcRow0Extend hcd) : (a : Matrix (Fin 2) (Fin 2) ℝ) → Matrix (Fin 2) (Fin 2) ℝ)
    (f₁ ∘ fun (a : { g_1 : SL(2, ℤ) // (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 }) ↦
      (↑a : SL(2, ℤ)))
    (Subtype.mk (p := fun (g_1 : SL(2, ℤ)) ↦ (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1) g
      sorry)
    ((fun (i : Fin 2) ↦ i) ⟨0, sorry⟩) j
Tactic: fin_cases i
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
f₁ : SL(2, ℤ) → Matrix (Fin 2) (Fin 2) ℝ := fun (g : SL(2, ℤ)) ↦ (↑g : Matrix (Fin 2) (Fin 2) ℤ).map Int.cast
cocompact_ℝ_to_cofinite_ℤ_matrix :
  Tendsto (β := Matrix (Fin 2) (Fin 2) ℝ) (fun (m : Matrix (Fin 2) (Fin 2) ℤ) ↦ m.map Int.cast) cofinite
    (cocompact (Matrix (Fin 2) (Fin 2) ℝ))
hf₁ : Tendsto f₁ cofinite (cocompact (Matrix (Fin 2) (Fin 2) ℝ))
g : SL(2, ℤ)
hcd : IsCoprime ((↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 0) ((↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 1)
mB : ℝ → Matrix (Fin 2) (Fin 2) ℝ :=
  fun (t : ℝ) ↦
    (of : (a : Fin 2 → Fin 2 → ℝ) → Matrix (Fin 2) (Fin 2) ℝ)
      ![![t, -(↑1 : ℝ)], Int.cast ∘ (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1]
hmB : Continuous mB
hf₂ :
  IsClosedEmbedding (X := Matrix (Fin 2) (Fin 2) ℝ) (Y := Matrix (Fin 2) (Fin 2) ℝ)
    (⇑(lcRow0Extend hcd) : (a : Matrix (Fin 2) (Fin 2) ℝ) → Matrix (Fin 2) (Fin 2) ℝ)
j : Fin 2
⊢ (mB ∘ fun (g_1 : { g_1 : SL(2, ℤ) // (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 }) ↦
        (lcRow0 ((↑g : Matrix (Fin 2) (Fin 2) ℤ) 1) : (a : Matrix (Fin 2) (Fin 2) ℝ) → ℝ)
          (↑((SpecialLinearGroup.map (Int.castRingHom ℝ) : (a : SL(2, ℤ)) → SL(2, ℝ)) (↑g_1 : SL(2, ℤ))) :
            Matrix (Fin 2) (Fin 2) ℝ))
      (Subtype.mk (p := fun (g_1 : SL(2, ℤ)) ↦ (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1)
        g ⋯)
      ((fun (i : Fin 2) ↦ i) (⟨0, ⋯⟩ : Fin 2)) j =
    Function.comp (α := { g_1 : SL(2, ℤ) // (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 })
      (⇑(lcRow0Extend hcd) : (a : Matrix (Fin 2) (Fin 2) ℝ) → Matrix (Fin 2) (Fin 2) ℝ)
      (f₁ ∘ fun (a : { g_1 : SL(2, ℤ) // (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 }) ↦
        (↑a : SL(2, ℤ)))
      (Subtype.mk (p := fun (g_1 : SL(2, ℤ)) ↦ (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1)
        g ⋯)
      ((fun (i : Fin 2) ↦ i) ⟨0, ⋯⟩) j
have h : (mB ∘ fun (g_1 : { g_1 : SL(2, ℤ) // (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 }) ↦
      (lcRow0 ((↑g : Matrix (Fin 2) (Fin 2) ℤ) 1) : (a : Matrix (Fin 2) (Fin 2) ℝ) → ℝ)
        (↑((SpecialLinearGroup.map (Int.castRingHom ℝ) : (a : SL(2, ℤ)) → SL(2, ℝ)) (↑g_1 : SL(2, ℤ))) :
          Matrix (Fin 2) (Fin 2) ℝ))
    (Subtype.mk (p := fun (g_1 : SL(2, ℤ)) ↦ (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1) g
      sorry)
    ((fun (i : Fin 2) ↦ i) (⟨0, sorry⟩ : Fin 2)) ((fun (i : Fin 2) ↦ i) (⟨0, sorry⟩ : Fin 2)) =
  Function.comp (α := { g_1 : SL(2, ℤ) // (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 })
    (⇑(lcRow0Extend hcd) : (a : Matrix (Fin 2) (Fin 2) ℝ) → Matrix (Fin 2) (Fin 2) ℝ)
    (f₁ ∘ fun (a : { g_1 : SL(2, ℤ) // (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 }) ↦
      (↑a : SL(2, ℤ)))
    (Subtype.mk (p := fun (g_1 : SL(2, ℤ)) ↦ (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1) g
      sorry)
    ((fun (i : Fin 2) ↦ i) ⟨0, sorry⟩) ((fun (i : Fin 2) ↦ i) ⟨0, sorry⟩)
Tactic: fin_cases j
Cleared hypotheses: j
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
f₁ : SL(2, ℤ) → Matrix (Fin 2) (Fin 2) ℝ := fun (g : SL(2, ℤ)) ↦ (↑g : Matrix (Fin 2) (Fin 2) ℤ).map Int.cast
cocompact_ℝ_to_cofinite_ℤ_matrix :
  Tendsto (β := Matrix (Fin 2) (Fin 2) ℝ) (fun (m : Matrix (Fin 2) (Fin 2) ℤ) ↦ m.map Int.cast) cofinite
    (cocompact (Matrix (Fin 2) (Fin 2) ℝ))
hf₁ : Tendsto f₁ cofinite (cocompact (Matrix (Fin 2) (Fin 2) ℝ))
g : SL(2, ℤ)
hcd : IsCoprime ((↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 0) ((↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 1)
mB : ℝ → Matrix (Fin 2) (Fin 2) ℝ :=
  fun (t : ℝ) ↦
    (of : (a : Fin 2 → Fin 2 → ℝ) → Matrix (Fin 2) (Fin 2) ℝ)
      ![![t, -(↑1 : ℝ)], Int.cast ∘ (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1]
hmB : Continuous mB
hf₂ :
  IsClosedEmbedding (X := Matrix (Fin 2) (Fin 2) ℝ) (Y := Matrix (Fin 2) (Fin 2) ℝ)
    (⇑(lcRow0Extend hcd) : (a : Matrix (Fin 2) (Fin 2) ℝ) → Matrix (Fin 2) (Fin 2) ℝ)
i j : Fin 2
⊢ (mB ∘ fun (g_1 : { g_1 : SL(2, ℤ) // (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 }) ↦
        (lcRow0 ((↑g : Matrix (Fin 2) (Fin 2) ℤ) 1) : (a : Matrix (Fin 2) (Fin 2) ℝ) → ℝ)
          (↑((SpecialLinearGroup.map (Int.castRingHom ℝ) : (a : SL(2, ℤ)) → SL(2, ℝ)) (↑g_1 : SL(2, ℤ))) :
            Matrix (Fin 2) (Fin 2) ℝ))
      (Subtype.mk (p := fun (g_1 : SL(2, ℤ)) ↦ (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1)
        g ⋯)
      i j =
    Function.comp (α := { g_1 : SL(2, ℤ) // (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 })
      (⇑(lcRow0Extend hcd) : (a : Matrix (Fin 2) (Fin 2) ℝ) → Matrix (Fin 2) (Fin 2) ℝ)
      (f₁ ∘ fun (a : { g_1 : SL(2, ℤ) // (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 }) ↦
        (↑a : SL(2, ℤ)))
      (Subtype.mk (p := fun (g_1 : SL(2, ℤ)) ↦ (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1)
        g ⋯)
      i j
have h : (mB ∘ fun (g_1 : { g_1 : SL(2, ℤ) // (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 }) ↦
      (lcRow0 ((↑g : Matrix (Fin 2) (Fin 2) ℤ) 1) : (a : Matrix (Fin 2) (Fin 2) ℝ) → ℝ)
        (↑((SpecialLinearGroup.map (Int.castRingHom ℝ) : (a : SL(2, ℤ)) → SL(2, ℝ)) (↑g_1 : SL(2, ℤ))) :
          Matrix (Fin 2) (Fin 2) ℝ))
    (Subtype.mk (p := fun (g_1 : SL(2, ℤ)) ↦ (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1) g
      sorry)
    ((fun (i : Fin 2) ↦ i) (⟨0, sorry⟩ : Fin 2)) ((fun (i : Fin 2) ↦ i) (⟨0, sorry⟩ : Fin 2)) =
  Function.comp (α := { g_1 : SL(2, ℤ) // (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 })
    (⇑(lcRow0Extend hcd) : (a : Matrix (Fin 2) (Fin 2) ℝ) → Matrix (Fin 2) (Fin 2) ℝ)
    (f₁ ∘ fun (a : { g_1 : SL(2, ℤ) // (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 }) ↦
      (↑a : SL(2, ℤ)))
    (Subtype.mk (p := fun (g_1 : SL(2, ℤ)) ↦ (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1) g
      sorry)
    ((fun (i : Fin 2) ↦ i) ⟨0, sorry⟩) ((fun (i : Fin 2) ↦ i) ⟨0, sorry⟩)
Tactic: fin_cases i <;> [fin_cases j; skip]
Cleared hypotheses: i, j
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
f₁ : SL(2, ℤ) → Matrix (Fin 2) (Fin 2) ℝ := fun (g : SL(2, ℤ)) ↦ (↑g : Matrix (Fin 2) (Fin 2) ℤ).map Int.cast
cocompact_ℝ_to_cofinite_ℤ_matrix :
  Tendsto (β := Matrix (Fin 2) (Fin 2) ℝ) (fun (m : Matrix (Fin 2) (Fin 2) ℤ) ↦ m.map Int.cast) cofinite
    (cocompact (Matrix (Fin 2) (Fin 2) ℝ))
hf₁ : Tendsto f₁ cofinite (cocompact (Matrix (Fin 2) (Fin 2) ℝ))
g : SL(2, ℤ)
hcd : IsCoprime ((↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 0) ((↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 1)
mB : ℝ → Matrix (Fin 2) (Fin 2) ℝ :=
  fun (t : ℝ) ↦
    (of : (a : Fin 2 → Fin 2 → ℝ) → Matrix (Fin 2) (Fin 2) ℝ)
      ![![t, -(↑1 : ℝ)], Int.cast ∘ (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1]
hmB : Continuous mB
hf₂ :
  IsClosedEmbedding (X := Matrix (Fin 2) (Fin 2) ℝ) (Y := Matrix (Fin 2) (Fin 2) ℝ)
    (⇑(lcRow0Extend hcd) : (a : Matrix (Fin 2) (Fin 2) ℝ) → Matrix (Fin 2) (Fin 2) ℝ)
⊢ (mB ∘ fun (g_1 : { g_1 : SL(2, ℤ) // (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 }) ↦
        (lcRow0 ((↑g : Matrix (Fin 2) (Fin 2) ℤ) 1) : (a : Matrix (Fin 2) (Fin 2) ℝ) → ℝ)
          (↑((SpecialLinearGroup.map (Int.castRingHom ℝ) : (a : SL(2, ℤ)) → SL(2, ℝ)) (↑g_1 : SL(2, ℤ))) :
            Matrix (Fin 2) (Fin 2) ℝ))
      (Subtype.mk (p := fun (g_1 : SL(2, ℤ)) ↦ (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1)
        g ⋯)
      ((fun (i : Fin 2) ↦ i) (⟨0, ⋯⟩ : Fin 2)) ((fun (i : Fin 2) ↦ i) (⟨1, ⋯⟩ : Fin 2)) =
    Function.comp (α := { g_1 : SL(2, ℤ) // (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 })
      (⇑(lcRow0Extend hcd) : (a : Matrix (Fin 2) (Fin 2) ℝ) → Matrix (Fin 2) (Fin 2) ℝ)
      (f₁ ∘ fun (a : { g_1 : SL(2, ℤ) // (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 }) ↦
        (↑a : SL(2, ℤ)))
      (Subtype.mk (p := fun (g_1 : SL(2, ℤ)) ↦ (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1)
        g ⋯)
      ((fun (i : Fin 2) ↦ i) ⟨0, ⋯⟩) ((fun (i : Fin 2) ↦ i) ⟨1, ⋯⟩)
have h : Eq (α := ℝ)
  (Function.comp (α := { g_1 : SL(2, ℤ) // (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 })
    (⇑(lcRow0Extend hcd) : (a : Matrix (Fin 2) (Fin 2) ℝ) → Matrix (Fin 2) (Fin 2) ℝ)
    (f₁ ∘ fun (a : { g_1 : SL(2, ℤ) // (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 }) ↦
      (↑a : SL(2, ℤ)))
    (Subtype.mk (p := fun (g_1 : SL(2, ℤ)) ↦ (↑g_1 : Matrix (Fin 2) (Fin 2) ℤ) 1 = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1) g
      sorry)
    ((fun (i : Fin 2) ↦ i) ⟨0, sorry⟩) ((fun (i : Fin 2) ↦ i) ⟨1, sorry⟩))
  (-(↑(↑g : Matrix (Fin 2) (Fin 2) ℤ).det : ℝ))
Tactic: convert! congr_arg (fun n : ℤ => (-n : ℝ)) g.det_coe.symm using 1