mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.MaricaSchoenheim

Goal state before:
pp: case calc_3
n : ℕ
f : ℕ → ℕ
hf' : ∀ k < n, Squarefree (f k)
hn : n ≠ 0
hf : StrictMonoOn f (Set.Iio n)
this : ∀ i < n, ∀ j < n, f i < (f i).gcd (f j) * n
𝒜 : Finset (Finset ℕ) := image (fun n ↦ (f n).primeFactors) (Iio n)
hf'' : ∀ i < n, ∀ (j : ℕ), Squarefree (f i / (f i).gcd (f j))
⊢ Set.InjOn (fun s ↦ ∏ p ∈ s, p) ↑(𝒜 \\ 𝒜)

Tactic:
simp only [𝒜, Set.InjOn, mem_coe, forall_mem_diffs, forall_mem_image, mem_Iio]

Goal state after:
case calc_3
n : ℕ
f : ℕ → ℕ
hf' : ∀ k < n, Squarefree (f k)
hn : n ≠ 0
hf : StrictMonoOn f (Set.Iio n)
this : ∀ i < n, ∀ j < n, f i < (f i).gcd (f j) * n
𝒜 : Finset (Finset ℕ) := image (fun n ↦ (f n).primeFactors) (Iio n)
hf'' : ∀ i < n, ∀ (j : ℕ), Squarefree (f i / (f i).gcd (f j))
⊢ ∀ ⦃x : ℕ⦄,
    x < n →
      ∀ ⦃x_1 : ℕ⦄,
        x_1 < n →
          ∀ ⦃x_2 : ℕ⦄,
            x_2 < n →
              ∀ ⦃x_3 : ℕ⦄,
                x_3 < n →
                  ∏ p ∈ (f x).primeFactors \ (f x_1).primeFactors, p =
                      ∏ p ∈ (f x_2).primeFactors \ (f x_3).primeFactors, p →
                    (f x).primeFactors \ (f x_1).primeFactors = (f x_2).primeFactors \ (f x_3).primeFactors
Module: Mathlib.NumberTheory.MahlerMeasure

Goal state before:
pp: p : ℤ[X]
h : (map (castRingHom ℂ) p).mahlerMeasure = 1
z : ℂ
hz : z ∈ p.aroots ℂ
⊢ IsIntegral ℤ z

Tactic:
have : p.leadingCoeff = 1 ∨ p.leadingCoeff = -1 := abs_eq_abs.mp <|
    abs_leadingCoeff_eq_one_of_mahlerMeasure_eq_one h

Goal state after:
p : ℤ[X]
h : (map (castRingHom ℂ) p).mahlerMeasure = 1
z : ℂ
hz : z ∈ p.aroots ℂ
this : p.leadingCoeff = 1 ∨ p.leadingCoeff = -1
⊢ IsIntegral ℤ z
Module: Mathlib.NumberTheory.LucasLehmer

Goal state before:
pp: case h₁
q : ℕ
⊢ ((1 + α) ^ 2).1 = (2 * ω).1

Tactic:
simp [α, ω, sq]

Goal state after:
case h₁
q : ℕ
⊢ 1 + 3 = 2 * 2
Module: Mathlib.NumberTheory.LucasLehmer

Goal state before:
pp: case h₂
q : ℕ
⊢ ((1 + α) ^ 2).2 = (2 * ω).2

Tactic:
simp [α, ω, sq]

Goal state after:
case h₂
q : ℕ
⊢ 1 + 1 = 2
Module: Mathlib.NumberTheory.LucasLehmer

Goal state before:
pp: k : ℕ
inst✝ : Fact (Nat.Prime (2 * k + 1))
leg3 : legendreSym (2 * k + 1) 3 = -1
⊢ (1 + α) ^ (2 * k + 1) = 1 - α

Tactic:
let q := 2 * k + 1

Goal state after:
k : ℕ
inst✝ : Fact (Nat.Prime (2 * k + 1))
leg3 : legendreSym (2 * k + 1) 3 = -1
q : ℕ := 2 * k + 1
⊢ (1 + α) ^ (2 * k + 1) = 1 - α
Module: Mathlib.NumberTheory.MahlerMeasure

Goal state before:
pp: p : ℤ[X]
h : (map (castRingHom ℂ) p).mahlerMeasure = 1
z : ℂ
hz₀ : z ≠ 0
hz : z ∈ p.aroots ℂ
w✝ : ℕ
left✝ : 0 < w✝
hz_pow : z ^ w✝ = 1
⊢ ∃ n, 0 < n ∧ IsPrimitiveRoot z n

Tactic:
exact IsPrimitiveRoot.exists_pos hz_pow (by omega)

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.CongruenceSubgroups

Goal state before:
pp: Γ : Subgroup SL(2, ℤ)
h : IsCongruenceSubgroup Γ
⊢ Γ.FiniteIndex

Tactic:
obtain ⟨N, hN⟩ := h

Goal state after:
Γ : Subgroup SL(2, ℤ)
N : ℕ
hN : N ≠ 0 ∧ Γ(N) ≤ Γ
⊢ Γ.FiniteIndex
Module: Mathlib.NumberTheory.ModularForms.CongruenceSubgroups

Goal state before:
pp: g : GL (Fin 2) ℚ
M : ℕ
inst✝ : NeZero M
A₁ : Matrix (Fin 2) (Fin 2) ℚ := ↑g
A₂ : Matrix (Fin 2) (Fin 2) ℚ := ↑g⁻¹
hA₁₂ : A₁ * A₂ = 1
⊢ ∃ N, N ≠ 0 ∧ ∀ x ∈ Γ(N), g * (mapGL ℚ) x * g⁻¹ ∈ Subgroup.map (mapGL ℚ) Γ(M)

Tactic:
let a₁ := A₁.den

Goal state after:
g : GL (Fin 2) ℚ
M : ℕ
inst✝ : NeZero M
A₁ : Matrix (Fin 2) (Fin 2) ℚ := ↑g
A₂ : Matrix (Fin 2) (Fin 2) ℚ := ↑g⁻¹
hA₁₂ : A₁ * A₂ = 1
a₁ : ℕ := A₁.den
⊢ ∃ N, N ≠ 0 ∧ ∀ x ∈ Γ(N), g * (mapGL ℚ) x * g⁻¹ ∈ Subgroup.map (mapGL ℚ) Γ(M)
Module: Mathlib.NumberTheory.ModularForms.SlashActions

Goal state before:
pp: case e'_1.e'_2
k : ℤ
f : ℍ → ℂ
γ : SL(2, ℤ)
z : ℍ
⊢ f (γ • z) * denom (Matrix.SpecialLinearGroup.toGL ((Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) γ)) ↑z ^ (-k) =
    ((↑(↑γ 1 0) * ↑z + ↑(↑γ 1 1)) ^ k)⁻¹ * f (γ • z)

Tactic:
simp only [mul_comm (f _), denom, zpow_neg]

Goal state after:
case e'_1.e'_2
k : ℤ
f : ℍ → ℂ
γ : SL(2, ℤ)
z : ℍ
⊢ ((↑(↑(Matrix.SpecialLinearGroup.toGL ((Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) γ)) 1 0) * ↑z +
            ↑(↑(Matrix.SpecialLinearGroup.toGL ((Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) γ)) 1 1)) ^
          k)⁻¹ *
      f (γ • z) =
    ((↑(↑γ 1 0) * ↑z + ↑(↑γ 1 1)) ^ k)⁻¹ * f (γ • z)
Module: Mathlib.NumberTheory.ModularForms.Cusps

Goal state before:
pp: case mp
c : OnePoint ℝ
⊢ IsCusp c (mapGL ℝ).range → c ∈ Set.range (OnePoint.map Rat.cast)

Tactic:
rintro ⟨-, ⟨g, rfl⟩, hgp, hgc⟩

Goal state after:
case mp
c : OnePoint ℝ
g : SL(2, ℤ)
hgp : ((mapGL ℝ) g).IsParabolic
hgc : (mapGL ℝ) g • c = c
⊢ c ∈ Set.range (OnePoint.map Rat.cast)
Module: Mathlib.NumberTheory.ModularForms.Cusps

Goal state before:
pp: case mpr
c : OnePoint ℝ
⊢ c ∈ Set.range (OnePoint.map Rat.cast) → IsCusp c (mapGL ℝ).range

Tactic:
rintro ⟨c, rfl⟩

Goal state after:
case mpr
c : OnePoint ℚ
⊢ IsCusp (OnePoint.map Rat.cast c) (mapGL ℝ).range
Module: Mathlib.NumberTheory.ModularForms.Cusps

Goal state before:
pp: case mp
c : OnePoint ℝ
⊢ c ∈ Set.range (OnePoint.map Rat.cast) → ∃ g, c = (mapGL ℝ) g • ∞

Tactic:
rintro ⟨c, rfl⟩

Goal state after:
case mp
c : OnePoint ℚ
⊢ ∃ g, OnePoint.map Rat.cast c = (mapGL ℝ) g • ∞
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: z : ℍ
this : Module ℝ (Fin 2 → ℝ) := Pi.normedSpace.toModule
π₀ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 0
π₁ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 1
f : (Fin 2 → ℝ) →ₗ[ℝ] ℂ := π₀.smulRight ↑z + π₁.smulRight 1
f_def : ⇑f = fun p ↦ ↑(p 0) * ↑z + ↑(p 1)
x✝ : Fin 2 → ℤ
⊢ normSq (↑(x✝ 0) * ↑z + ↑(x✝ 1)) = (⇑normSq ∘ ⇑f ∘ fun p ↦ Int.cast ∘ p) x✝

Tactic:
f_def

Goal state after:
z : ℍ
this : Module ℝ (Fin 2 → ℝ) := Pi.normedSpace.toModule
π₀ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 0
π₁ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 1
f : (Fin 2 → ℝ) →ₗ[ℝ] ℂ := π₀.smulRight ↑z + π₁.smulRight 1
f_def : ⇑f = fun p ↦ ↑(p 0) * ↑z + ↑(p 1)
x✝ : Fin 2 → ℤ
⊢ normSq (↑(x✝ 0) * ↑z + ↑(x✝ 1)) = (⇑normSq ∘ (fun p ↦ ↑(p 0) * ↑z + ↑(p 1)) ∘ fun p ↦ Int.cast ∘ p) x✝
Module: Mathlib.NumberTheory.ModularForms.Cusps

Goal state before:
pp: Γ : Subgroup SL(2, ℤ)
hΓ : ModularGroup.T ∈ Γ
m : ℤ
i j : Fin 2
⊢ (algebraMap ℤ ℝ).mapMatrix (↑(ModularGroup.T ^ m)) i j = ↑(upperRightHom ((fun x ↦ x • 1) m)) i j

Tactic:
fin_cases i <;> fin_cases j

Goal state after:
case «0».«0»
Γ : Subgroup SL(2, ℤ)
hΓ : ModularGroup.T ∈ Γ
m : ℤ
⊢ (algebraMap ℤ ℝ).mapMatrix (↑(ModularGroup.T ^ m)) ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩) =
    ↑(upperRightHom ((fun x ↦ x • 1) m)) ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)

case «0».«1»
Γ : Subgroup SL(2, ℤ)
hΓ : ModularGroup.T ∈ Γ
m : ℤ
⊢ (algebraMap ℤ ℝ).mapMatrix (↑(ModularGroup.T ^ m)) ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩) =
    ↑(upperRightHom ((fun x ↦ x • 1) m)) ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩)

case «1».«0»
Γ : Subgroup SL(2, ℤ)
hΓ : ModularGroup.T ∈ Γ
m : ℤ
⊢ (algebraMap ℤ ℝ).mapMatrix (↑(ModularGroup.T ^ m)) ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩) =
    ↑(upperRightHom ((fun x ↦ x • 1) m)) ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)

case «1».«1»
Γ : Subgroup SL(2, ℤ)
hΓ : ModularGroup.T ∈ Γ
m : ℤ
⊢ (algebraMap ℤ ℝ).mapMatrix (↑(ModularGroup.T ^ m)) ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩) =
    ↑(upperRightHom ((fun x ↦ x • 1) m)) ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩)
Module: Mathlib.NumberTheory.ModularForms.Cusps

Goal state before:
pp: case mpr
N : ℕ
x : ℝ
this : AddSubgroup.zmultiples ↑N = AddSubgroup.map (Int.castAddHom ℝ) (AddSubgroup.zmultiples ↑N)
a : ℤ
ha : (Int.castAddHom ℝ) (a • ↑N) = x
i j : Fin 2
⊢ (algebraMap ℤ ℝ).mapMatrix (↑(ModularGroup.T ^ (a * ↑N))) i j = ↑(upperRightHom x) i j

Tactic:
fin_cases i <;> fin_cases j

Goal state after:
case mpr.«0».«0»
N : ℕ
x : ℝ
this : AddSubgroup.zmultiples ↑N = AddSubgroup.map (Int.castAddHom ℝ) (AddSubgroup.zmultiples ↑N)
a : ℤ
ha : (Int.castAddHom ℝ) (a • ↑N) = x
⊢ (algebraMap ℤ ℝ).mapMatrix (↑(ModularGroup.T ^ (a * ↑N))) ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩) =
    ↑(upperRightHom x) ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)

case mpr.«0».«1»
N : ℕ
x : ℝ
this : AddSubgroup.zmultiples ↑N = AddSubgroup.map (Int.castAddHom ℝ) (AddSubgroup.zmultiples ↑N)
a : ℤ
ha : (Int.castAddHom ℝ) (a • ↑N) = x
⊢ (algebraMap ℤ ℝ).mapMatrix (↑(ModularGroup.T ^ (a * ↑N))) ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩) =
    ↑(upperRightHom x) ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩)

case mpr.«1».«0»
N : ℕ
x : ℝ
this : AddSubgroup.zmultiples ↑N = AddSubgroup.map (Int.castAddHom ℝ) (AddSubgroup.zmultiples ↑N)
a : ℤ
ha : (Int.castAddHom ℝ) (a • ↑N) = x
⊢ (algebraMap ℤ ℝ).mapMatrix (↑(ModularGroup.T ^ (a * ↑N))) ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩) =
    ↑(upperRightHom x) ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)

case mpr.«1».«1»
N : ℕ
x : ℝ
this : AddSubgroup.zmultiples ↑N = AddSubgroup.map (Int.castAddHom ℝ) (AddSubgroup.zmultiples ↑N)
a : ℤ
ha : (Int.castAddHom ℝ) (a • ↑N) = x
⊢ (algebraMap ℤ ℝ).mapMatrix (↑(ModularGroup.T ^ (a * ↑N))) ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩) =
    ↑(upperRightHom x) ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩)
Module: Mathlib.NumberTheory.ModularForms.Bounds

Goal state before:
pp: case inl
E✝ : Type u_1
inst✝ : SeminormedAddCommGroup E✝
f : ℍ → E✝
hf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
t : ℝ
hf_infinity : f =O[atImInfty] fun z ↦ z.im ^ t
D : ℝ
hD : D > 0
y : ℝ
hy : ∀ (b : ℝ), y ≤ b → ∀ (a : ℍ), a.im = b → ‖f a‖ ≤ D * a.im ^ t
hfm : ContinuousOn (fun τ ↦ ‖f τ‖ / τ.im ^ t) (truncatedFundamentalDomain y)
E : ℝ
hE : ∀ τ ∈ truncatedFundamentalDomain y, ‖f τ‖ / τ.im ^ t ≤ E
τ : ℍ
hτ : τ ∈ 𝒟
hτ' : y ≤ τ.im
⊢ ‖f τ‖ ≤ max D E * τ.im ^ t

pp: case inr
E✝ : Type u_1
inst✝ : SeminormedAddCommGroup E✝
f : ℍ → E✝
hf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
t : ℝ
hf_infinity : f =O[atImInfty] fun z ↦ z.im ^ t
D : ℝ
hD : D > 0
y : ℝ
hy : ∀ (b : ℝ), y ≤ b → ∀ (a : ℍ), a.im = b → ‖f a‖ ≤ D * a.im ^ t
hfm : ContinuousOn (fun τ ↦ ‖f τ‖ / τ.im ^ t) (truncatedFundamentalDomain y)
E : ℝ
hE : ∀ τ ∈ truncatedFundamentalDomain y, ‖f τ‖ / τ.im ^ t ≤ E
τ : ℍ
hτ : τ ∈ 𝒟
hτ' : τ.im ≤ y
⊢ ‖f τ‖ ≤ max D E * τ.im ^ t

Tactic:
· exact (hy _ hτ' _ rfl).trans <| mul_le_mul_of_nonneg_right (le_max_left ..) (by positivity)

Goal state after:
case inr
E✝ : Type u_1
inst✝ : SeminormedAddCommGroup E✝
f : ℍ → E✝
hf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
t : ℝ
hf_infinity : f =O[atImInfty] fun z ↦ z.im ^ t
D : ℝ
hD : D > 0
y : ℝ
hy : ∀ (b : ℝ), y ≤ b → ∀ (a : ℍ), a.im = b → ‖f a‖ ≤ D * a.im ^ t
hfm : ContinuousOn (fun τ ↦ ‖f τ‖ / τ.im ^ t) (truncatedFundamentalDomain y)
E : ℝ
hE : ∀ τ ∈ truncatedFundamentalDomain y, ‖f τ‖ / τ.im ^ t ≤ E
τ : ℍ
hτ : τ ∈ 𝒟
hτ' : τ.im ≤ y
⊢ ‖f τ‖ ≤ max D E * τ.im ^ t
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Defs

Goal state before:
pp: r : ℕ
inst✝ : NeZero r
v : Fin 2 → ℤ
hv : finGcdMap v = r
⊢ gcd (v 0) (v 1) ≠ 0

Tactic:
have := NeZero.ne r

Goal state after:
r : ℕ
inst✝ : NeZero r
v : Fin 2 → ℤ
hv : finGcdMap v = r
this : r ≠ 0
⊢ gcd (v 0) (v 1) ≠ 0
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: case h
g : SL(2, ℤ)
hc : ↑g 1 0 = 0
had : ↑g 0 0 * ↑g 1 1 = 1
ha : ↑g 0 0 = 1
hd : ↑g 1 1 = 1
i j : Fin 2
⊢ ↑g i j = ↑(T ^ ↑g 0 1) i j

Tactic:
fin_cases i <;> fin_cases j

Goal state after:
case h.«0».«0»
g : SL(2, ℤ)
hc : ↑g 1 0 = 0
had : ↑g 0 0 * ↑g 1 1 = 1
ha : ↑g 0 0 = 1
hd : ↑g 1 1 = 1
⊢ ↑g ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩) = ↑(T ^ ↑g 0 1) ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)

case h.«0».«1»
g : SL(2, ℤ)
hc : ↑g 1 0 = 0
had : ↑g 0 0 * ↑g 1 1 = 1
ha : ↑g 0 0 = 1
hd : ↑g 1 1 = 1
⊢ ↑g ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩) = ↑(T ^ ↑g 0 1) ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩)

case h.«1».«0»
g : SL(2, ℤ)
hc : ↑g 1 0 = 0
had : ↑g 0 0 * ↑g 1 1 = 1
ha : ↑g 0 0 = 1
hd : ↑g 1 1 = 1
⊢ ↑g ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩) = ↑(T ^ ↑g 0 1) ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)

case h.«1».«1»
g : SL(2, ℤ)
hc : ↑g 1 0 = 0
had : ↑g 0 0 * ↑g 1 1 = 1
ha : ↑g 0 0 = 1
hd : ↑g 1 1 = 1
⊢ ↑g ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩) = ↑(T ^ ↑g 0 1) ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩)
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: case h
g : SL(2, ℤ)
hc : ↑g 1 0 = 0
had : ↑g 0 0 * ↑g 1 1 = 1
ha : ↑g 0 0 = -1
hd : ↑g 1 1 = -1
i j : Fin 2
⊢ ↑g i j = ↑(-T ^ (-↑g 0 1)) i j

Tactic:
fin_cases i <;> fin_cases j

Goal state after:
case h.«0».«0»
g : SL(2, ℤ)
hc : ↑g 1 0 = 0
had : ↑g 0 0 * ↑g 1 1 = 1
ha : ↑g 0 0 = -1
hd : ↑g 1 1 = -1
⊢ ↑g ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩) = ↑(-T ^ (-↑g 0 1)) ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)

case h.«0».«1»
g : SL(2, ℤ)
hc : ↑g 1 0 = 0
had : ↑g 0 0 * ↑g 1 1 = 1
ha : ↑g 0 0 = -1
hd : ↑g 1 1 = -1
⊢ ↑g ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩) = ↑(-T ^ (-↑g 0 1)) ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩)

case h.«1».«0»
g : SL(2, ℤ)
hc : ↑g 1 0 = 0
had : ↑g 0 0 * ↑g 1 1 = 1
ha : ↑g 0 0 = -1
hd : ↑g 1 1 = -1
⊢ ↑g ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩) = ↑(-T ^ (-↑g 0 1)) ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)

case h.«1».«1»
g : SL(2, ℤ)
hc : ↑g 1 0 = 0
had : ↑g 0 0 * ↑g 1 1 = 1
ha : ↑g 0 0 = -1
hd : ↑g 1 1 = -1
⊢ ↑g ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩) = ↑(-T ^ (-↑g 0 1)) ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩)
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: z : ℍ
h : 1 < normSq ↑z
⊢ normSq
      ((↑((algebraMap ℤ ℝ) (↑S 0 0)) * ↑z + ↑((algebraMap ℤ ℝ) (↑S 0 1))) /
        (↑((algebraMap ℤ ℝ) (↑S 1 0)) * ↑z + ↑((algebraMap ℤ ℝ) (↑S 1 1)))) <
    1

Tactic:
simpa [coe_S, num, denom] using (inv_lt_inv₀ z.normSq_pos zero_lt_one).mpr h

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.Bounds

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
f : ℍ → E
hf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
t : ℝ
ht : 0 ≤ t
hf_infinity : ∀ (g : SL(2, ℤ)), (fun τ ↦ f (g • τ)) =O[atImInfty] fun z ↦ z.im ^ t
Γ : Subgroup SL(2, ℤ)
inst✝ : Γ.FiniteIndex
hf_inv : ∀ g ∈ Γ, ∀ (τ : ℍ), f (g • τ) = f τ
f' : ℍ → SL(2, ℤ) ⧸ Γ → E := fun τ ↦ Quotient.lift (fun g ↦ f (g⁻¹ • τ)) ⋯
hf'_cont : ∀ (γ : SL(2, ℤ) ⧸ Γ), Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦ f' x γ
hf'_inv : ∀ (τ : ℍ) (g : SL(2, ℤ)) (γ : SL(2, ℤ) ⧸ Γ), f' (g • τ) (g • γ) = f' τ γ
hf'_infty : ∀ (γ : SL(2, ℤ) ⧸ Γ), (fun x ↦ f' x γ) =O[atImInfty] fun z ↦ z.im ^ t
⊢ ∃ C, ∀ (τ : ℍ), ‖f τ‖ ≤ C * max τ.im (1 / τ.im) ^ t

Tactic:
have : Fintype (SL(2, ℤ) ⧸ Γ) := Subgroup.fintypeQuotientOfFiniteIndex

Goal state after:
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
f : ℍ → E
hf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
t : ℝ
ht : 0 ≤ t
hf_infinity : ∀ (g : SL(2, ℤ)), (fun τ ↦ f (g • τ)) =O[atImInfty] fun z ↦ z.im ^ t
Γ : Subgroup SL(2, ℤ)
inst✝ : Γ.FiniteIndex
hf_inv : ∀ g ∈ Γ, ∀ (τ : ℍ), f (g • τ) = f τ
f' : ℍ → SL(2, ℤ) ⧸ Γ → E := fun τ ↦ Quotient.lift (fun g ↦ f (g⁻¹ • τ)) ⋯
hf'_cont : ∀ (γ : SL(2, ℤ) ⧸ Γ), Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦ f' x γ
hf'_inv : ∀ (τ : ℍ) (g : SL(2, ℤ)) (γ : SL(2, ℤ) ⧸ Γ), f' (g • τ) (g • γ) = f' τ γ
hf'_infty : ∀ (γ : SL(2, ℤ) ⧸ Γ), (fun x ↦ f' x γ) =O[atImInfty] fun z ↦ z.im ^ t
this : Fintype (SL(2, ℤ) ⧸ Γ)
⊢ ∃ C, ∀ (τ : ℍ), ‖f τ‖ ≤ C * max τ.im (1 / τ.im) ^ t
Module: Mathlib.NumberTheory.ModularForms.QExpansion

Goal state before:
pp: case succ
k : ℤ
Γ : Subgroup (GL (Fin 2) ℝ)
h : ℝ
inst✝ : Γ.HasDetPlusMinusOne
hh : 0 < h
hΓ : h ∈ Γ.strictPeriods
f : ModularForm Γ k
n : ℕ
ih : qExpansion h ⇑(f.pow n) = qExpansion h ⇑f ^ n
⊢ qExpansion h (⇑(f.pow n) * ⇑f) = qExpansion h ⇑f ^ (n + 1)

Tactic:
← coe_mul,

Goal state after:
case succ
k : ℤ
Γ : Subgroup (GL (Fin 2) ℝ)
h : ℝ
inst✝ : Γ.HasDetPlusMinusOne
hh : 0 < h
hΓ : h ∈ Γ.strictPeriods
f : ModularForm Γ k
n : ℕ
ih : qExpansion h ⇑(f.pow n) = qExpansion h ⇑f ^ n
⊢ qExpansion h ⇑((f.pow n).mul f) = qExpansion h ⇑f ^ (n + 1)
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: g✝ : SL(2, ℤ)
z✝ : ℍ
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
hc : ↑g 1 0 = 0
hd' : ↑g 1 1 = 1 ∨ ↑g 1 1 = -1
hd : ↑g 1 1 = 1
ha : ↑g 0 0 = 1
b : ℤ := ↑g 0 1
i j : Fin 2
⊢ ↑g i j = !![1, b; 0, 1] i j

Tactic:
fin_cases i <;> fin_cases j

Goal state after:
case «0».«0»
g✝ : SL(2, ℤ)
z✝ : ℍ
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
hc : ↑g 1 0 = 0
hd' : ↑g 1 1 = 1 ∨ ↑g 1 1 = -1
hd : ↑g 1 1 = 1
ha : ↑g 0 0 = 1
b : ℤ := ↑g 0 1
⊢ ↑g ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩) = !![1, b; 0, 1] ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)

case «0».«1»
g✝ : SL(2, ℤ)
z✝ : ℍ
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
hc : ↑g 1 0 = 0
hd' : ↑g 1 1 = 1 ∨ ↑g 1 1 = -1
hd : ↑g 1 1 = 1
ha : ↑g 0 0 = 1
b : ℤ := ↑g 0 1
⊢ ↑g ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩) = !![1, b; 0, 1] ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩)

case «1».«0»
g✝ : SL(2, ℤ)
z✝ : ℍ
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
hc : ↑g 1 0 = 0
hd' : ↑g 1 1 = 1 ∨ ↑g 1 1 = -1
hd : ↑g 1 1 = 1
ha : ↑g 0 0 = 1
b : ℤ := ↑g 0 1
⊢ ↑g ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩) = !![1, b; 0, 1] ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)

case «1».«1»
g✝ : SL(2, ℤ)
z✝ : ℍ
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
hc : ↑g 1 0 = 0
hd' : ↑g 1 1 = 1 ∨ ↑g 1 1 = -1
hd : ↑g 1 1 = 1
ha : ↑g 0 0 = 1
b : ℤ := ↑g 0 1
⊢ ↑g ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩) = !![1, b; 0, 1] ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩)
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
hg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1
hc : ↑g 1 0 = 1
hd : ↑g 1 1 = 0
hb : ↑g 0 1 = -1
hz' : ‖↑z‖ = 1
i j : Fin 2
⊢ ↑g i j = !![↑g 0 0, -1; 1, 0] i j

Tactic:
fin_cases i <;> fin_cases j

Goal state after:
case «0».«0»
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
hg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1
hc : ↑g 1 0 = 1
hd : ↑g 1 1 = 0
hb : ↑g 0 1 = -1
hz' : ‖↑z‖ = 1
⊢ ↑g ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩) = !![↑g 0 0, -1; 1, 0] ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)

case «0».«1»
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
hg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1
hc : ↑g 1 0 = 1
hd : ↑g 1 1 = 0
hb : ↑g 0 1 = -1
hz' : ‖↑z‖ = 1
⊢ ↑g ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩) = !![↑g 0 0, -1; 1, 0] ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩)

case «1».«0»
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
hg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1
hc : ↑g 1 0 = 1
hd : ↑g 1 1 = 0
hb : ↑g 0 1 = -1
hz' : ‖↑z‖ = 1
⊢ ↑g ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩) = !![↑g 0 0, -1; 1, 0] ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)

case «1».«1»
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
hg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1
hc : ↑g 1 0 = 1
hd : ↑g 1 1 = 0
hb : ↑g 0 1 = -1
hz' : ‖↑z‖ = 1
⊢ ↑g ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩) = !![↑g 0 0, -1; 1, 0] ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩)
Module: Mathlib.Topology.Algebra.InfiniteSum.ConditionalInt

Goal state before:
pp: F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormSMulClass ℤ F
f : ℤ → F
hs : Function.Even f
L : F
hL : Tendsto (fun x ↦ ∑ b ∈ Icc (-x) x, f b) atTop (𝓝 L)
this✝ : Tendsto (fun x ↦ ‖∑ b ∈ Icc (-x) x, f b - ∑ b ∈ Icc (-(x - 1)) (x - 1), f b‖) atTop (𝓝 0)
x : ℤ
hx : 1 ≤ x
this : Icc (-x) x = Icc (-(x - 1)) (x - 1) ∪ {-x, x}
⊢ 2⁻¹ * ‖f x + f x + ∑ x ∈ Icc (-(x - 1)) (x - 1), f x - ∑ b ∈ Icc (-(x - 1)) (x - 1), f b‖ = ‖f x‖

pp: F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormSMulClass ℤ F
f : ℤ → F
hs : Function.Even f
L : F
hL : Tendsto (fun x ↦ ∑ b ∈ Icc (-x) x, f b) atTop (𝓝 L)
this✝ : Tendsto (fun x ↦ ‖∑ b ∈ Icc (-x) x, f b - ∑ b ∈ Icc (-(x - 1)) (x - 1), f b‖) atTop (𝓝 0)
x : ℤ
hx : 1 ≤ x
this : Icc (-x) x = Icc (-(x - 1)) (x - 1) ∪ {-x, x}
⊢ -x ∉ {x}

pp: F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormSMulClass ℤ F
f : ℤ → F
hs : Function.Even f
L : F
hL : Tendsto (fun x ↦ ∑ b ∈ Icc (-x) x, f b) atTop (𝓝 L)
this✝ : Tendsto (fun x ↦ ‖∑ b ∈ Icc (-x) x, f b - ∑ b ∈ Icc (-(x - 1)) (x - 1), f b‖) atTop (𝓝 0)
x : ℤ
hx : 1 ≤ x
this : Icc (-x) x = Icc (-(x - 1)) (x - 1) ∪ {-x, x}
⊢ Disjoint (Icc (-(x - 1)) (x - 1)) {-x, x}

Tactic:
add_sub_cancel_right,

Goal state after:
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormSMulClass ℤ F
f : ℤ → F
hs : Function.Even f
L : F
hL : Tendsto (fun x ↦ ∑ b ∈ Icc (-x) x, f b) atTop (𝓝 L)
this✝ : Tendsto (fun x ↦ ‖∑ b ∈ Icc (-x) x, f b - ∑ b ∈ Icc (-(x - 1)) (x - 1), f b‖) atTop (𝓝 0)
x : ℤ
hx : 1 ≤ x
this : Icc (-x) x = Icc (-(x - 1)) (x - 1) ∪ {-x, x}
⊢ 2⁻¹ * ‖f x + f x‖ = ‖f x‖

F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormSMulClass ℤ F
f : ℤ → F
hs : Function.Even f
L : F
hL : Tendsto (fun x ↦ ∑ b ∈ Icc (-x) x, f b) atTop (𝓝 L)
this✝ : Tendsto (fun x ↦ ‖∑ b ∈ Icc (-x) x, f b - ∑ b ∈ Icc (-(x - 1)) (x - 1), f b‖) atTop (𝓝 0)
x : ℤ
hx : 1 ≤ x
this : Icc (-x) x = Icc (-(x - 1)) (x - 1) ∪ {-x, x}
⊢ -x ∉ {x}

F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormSMulClass ℤ F
f : ℤ → F
hs : Function.Even f
L : F
hL : Tendsto (fun x ↦ ∑ b ∈ Icc (-x) x, f b) atTop (𝓝 L)
this✝ : Tendsto (fun x ↦ ‖∑ b ∈ Icc (-x) x, f b - ∑ b ∈ Icc (-(x - 1)) (x - 1), f b‖) atTop (𝓝 0)
x : ℤ
hx : 1 ≤ x
this : Icc (-x) x = Icc (-(x - 1)) (x - 1) ∪ {-x, x}
⊢ Disjoint (Icc (-(x - 1)) (x - 1)) {-x, x}
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
hg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1
hc : ↑g 1 0 = 1
hd : ↑g 1 1 = 1
i j : Fin 2
⊢ ↑g i j = !![↑g 0 0, -1 + ↑g 0 0; 1, 1] i j

Tactic:
fin_cases i <;> fin_cases j

Goal state after:
case «0».«0»
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
hg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1
hc : ↑g 1 0 = 1
hd : ↑g 1 1 = 1
⊢ ↑g ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩) = !![↑g 0 0, -1 + ↑g 0 0; 1, 1] ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)

case «0».«1»
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
hg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1
hc : ↑g 1 0 = 1
hd : ↑g 1 1 = 1
⊢ ↑g ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩) = !![↑g 0 0, -1 + ↑g 0 0; 1, 1] ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩)

case «1».«0»
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
hg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1
hc : ↑g 1 0 = 1
hd : ↑g 1 1 = 1
⊢ ↑g ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩) = !![↑g 0 0, -1 + ↑g 0 0; 1, 1] ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)

case «1».«1»
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
hg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1
hc : ↑g 1 0 = 1
hd : ↑g 1 1 = 1
⊢ ↑g ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩) = !![↑g 0 0, -1 + ↑g 0 0; 1, 1] ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩)
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
hg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1
hc : ↑g 1 0 = 1
hd : ↑g 1 1 = -1
this : ↑g 0 1 = -↑g 0 0 - 1
i j : Fin 2
⊢ ↑g i j = !![↑g 0 0, -1 - ↑g 0 0; 1, -1] i j

Tactic:
fin_cases i <;> fin_cases j

Goal state after:
case «0».«0»
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
hg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1
hc : ↑g 1 0 = 1
hd : ↑g 1 1 = -1
this : ↑g 0 1 = -↑g 0 0 - 1
⊢ ↑g ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩) =
    !![↑g 0 0, -1 - ↑g 0 0; 1, -1] ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)

case «0».«1»
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
hg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1
hc : ↑g 1 0 = 1
hd : ↑g 1 1 = -1
this : ↑g 0 1 = -↑g 0 0 - 1
⊢ ↑g ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩) =
    !![↑g 0 0, -1 - ↑g 0 0; 1, -1] ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩)

case «1».«0»
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
hg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1
hc : ↑g 1 0 = 1
hd : ↑g 1 1 = -1
this : ↑g 0 1 = -↑g 0 0 - 1
⊢ ↑g ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩) =
    !![↑g 0 0, -1 - ↑g 0 0; 1, -1] ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)

case «1».«1»
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
hg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1
hc : ↑g 1 0 = 1
hd : ↑g 1 1 = -1
this : ↑g 0 1 = -↑g 0 0 - 1
⊢ ↑g ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩) =
    !![↑g 0 0, -1 - ↑g 0 0; 1, -1] ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩)