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

Goal state before:
pp: case pos
p : ℕ
inst✝ : Fact (Nat.Prime p)
hp0' : 0 < ↑p
hp0 : 0 < (↑p)⁻¹
hp1' : 1 < ↑p
hp1 : (↑p)⁻¹ < 1
n : ℕ
hn : Valued.v (↑p ^ n) = exp (-↑n)
x y : WithVal (Rat.padicValuation p)
x' : ℚ := (WithVal.equiv (Rat.padicValuation p)) x
hx : x' = (WithVal.equiv (Rat.padicValuation p)) x
y' : ℚ := (WithVal.equiv (Rat.padicValuation p)) y
hy : y' = (WithVal.equiv (Rat.padicValuation p)) y
h : (if x' - y' = 0 then 0 else exp (-padicValRat p (x' - y'))) < exp (-↑n)
H : x' - y' = 0
⊢ 0 ≤ (↑p ^ n)⁻¹

pp: case neg
p : ℕ
inst✝ : Fact (Nat.Prime p)
hp0' : 0 < ↑p
hp0 : 0 < (↑p)⁻¹
hp1' : 1 < ↑p
hp1 : (↑p)⁻¹ < 1
n : ℕ
hn : Valued.v (↑p ^ n) = exp (-↑n)
x y : WithVal (Rat.padicValuation p)
x' : ℚ := (WithVal.equiv (Rat.padicValuation p)) x
hx : x' = (WithVal.equiv (Rat.padicValuation p)) x
y' : ℚ := (WithVal.equiv (Rat.padicValuation p)) y
hy : y' = (WithVal.equiv (Rat.padicValuation p)) y
h : (if x' - y' = 0 then 0 else exp (-padicValRat p (x' - y'))) < exp (-↑n)
H : ¬x' - y' = 0
⊢ (↑p ^ padicValRat p (x' - y'))⁻¹ ≤ (↑p ^ n)⁻¹

Tactic:
·

Goal state after:
case pos
p : ℕ
inst✝ : Fact (Nat.Prime p)
hp0' : 0 < ↑p
hp0 : 0 < (↑p)⁻¹
hp1' : 1 < ↑p
hp1 : (↑p)⁻¹ < 1
n : ℕ
hn : Valued.v (↑p ^ n) = exp (-↑n)
x y : WithVal (Rat.padicValuation p)
x' : ℚ := (WithVal.equiv (Rat.padicValuation p)) x
hx : x' = (WithVal.equiv (Rat.padicValuation p)) x
y' : ℚ := (WithVal.equiv (Rat.padicValuation p)) y
hy : y' = (WithVal.equiv (Rat.padicValuation p)) y
h : (if x' - y' = 0 then 0 else exp (-padicValRat p (x' - y'))) < exp (-↑n)
H : x' - y' = 0
⊢ 0 ≤ (↑p ^ n)⁻¹
Module: Mathlib.NumberTheory.Padics.WithVal

Goal state before:
pp: case neg
p : ℕ
inst✝ : Fact (Nat.Prime p)
hp0' : 0 < ↑p
hp0 : 0 < (↑p)⁻¹
hp1' : 1 < ↑p
hp1 : (↑p)⁻¹ < 1
n : ℕ
hn : Valued.v (↑p ^ n) = exp (-↑n)
x y : WithVal (Rat.padicValuation p)
x' : ℚ := (WithVal.equiv (Rat.padicValuation p)) x
hx : x' = (WithVal.equiv (Rat.padicValuation p)) x
y' : ℚ := (WithVal.equiv (Rat.padicValuation p)) y
hy : y' = (WithVal.equiv (Rat.padicValuation p)) y
h : (if x' - y' = 0 then 0 else exp (-padicValRat p (x' - y'))) < exp (-↑n)
H : ¬x' - y' = 0
⊢ (↑p ^ padicValRat p (x' - y'))⁻¹ ≤ (↑p ^ n)⁻¹

Tactic:
·

Goal state after:
case neg
p : ℕ
inst✝ : Fact (Nat.Prime p)
hp0' : 0 < ↑p
hp0 : 0 < (↑p)⁻¹
hp1' : 1 < ↑p
hp1 : (↑p)⁻¹ < 1
n : ℕ
hn : Valued.v (↑p ^ n) = exp (-↑n)
x y : WithVal (Rat.padicValuation p)
x' : ℚ := (WithVal.equiv (Rat.padicValuation p)) x
hx : x' = (WithVal.equiv (Rat.padicValuation p)) x
y' : ℚ := (WithVal.equiv (Rat.padicValuation p)) y
hy : y' = (WithVal.equiv (Rat.padicValuation p)) y
h : (if x' - y' = 0 then 0 else exp (-padicValRat p (x' - y'))) < exp (-↑n)
H : ¬x' - y' = 0
⊢ (↑p ^ padicValRat p (x' - y'))⁻¹ ≤ (↑p ^ n)⁻¹
Module: Mathlib.NumberTheory.Padics.WithVal

Goal state before:
pp: case h
p : ℕ
inst✝ : Fact (Nat.Prime p)
hp0' : 0 < ↑p
hp0 : 0 < (↑p)⁻¹
hp1' : 1 < ↑p
hp1 : (↑p)⁻¹ < 1
n : ℕ
hn : Valued.v (↑p ^ n) = exp (-↑n)
x y : WithVal (Rat.padicValuation p)
x' : ℚ := (WithVal.equiv (Rat.padicValuation p)) x
hx : x' = (WithVal.equiv (Rat.padicValuation p)) x
y' : ℚ := (WithVal.equiv (Rat.padicValuation p)) y
hy : y' = (WithVal.equiv (Rat.padicValuation p)) y
h : (Rat.padicValuation p) (x' - y') < exp (-↑n)
⊢ ↑(↑(p ^ n)).num ≠ 0

Tactic:
·

Goal state after:
case h
p : ℕ
inst✝ : Fact (Nat.Prime p)
hp0' : 0 < ↑p
hp0 : 0 < (↑p)⁻¹
hp1' : 1 < ↑p
hp1 : (↑p)⁻¹ < 1
n : ℕ
hn : Valued.v (↑p ^ n) = exp (-↑n)
x y : WithVal (Rat.padicValuation p)
x' : ℚ := (WithVal.equiv (Rat.padicValuation p)) x
hx : x' = (WithVal.equiv (Rat.padicValuation p)) x
y' : ℚ := (WithVal.equiv (Rat.padicValuation p)) y
hy : y' = (WithVal.equiv (Rat.padicValuation p)) y
h : (Rat.padicValuation p) (x' - y') < exp (-↑n)
⊢ ↑(↑(p ^ n)).num ≠ 0
Module: Mathlib.NumberTheory.Padics.MahlerBasis

Goal state before:
pp: case a
p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : Module ℤ_[p] E
inst✝ : IsBoundedSMul ℤ_[p] E
a : E
n : ℕ
⊢ ‖mahlerTerm a n‖ ≤ ‖a‖

pp: case a
p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : Module ℤ_[p] E
inst✝ : IsBoundedSMul ℤ_[p] E
a : E
n : ℕ
⊢ ‖a‖ ≤ ‖mahlerTerm a n‖

Tactic:
·

Goal state after:
case a
p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : Module ℤ_[p] E
inst✝ : IsBoundedSMul ℤ_[p] E
a : E
n : ℕ
⊢ ‖mahlerTerm a n‖ ≤ ‖a‖
Module: Mathlib.NumberTheory.Padics.MahlerBasis

Goal state before:
pp: case a
p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : Module ℤ_[p] E
inst✝ : IsBoundedSMul ℤ_[p] E
a : E
n : ℕ
⊢ ‖a‖ ≤ ‖mahlerTerm a n‖

Tactic:
·

Goal state after:
case a
p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : Module ℤ_[p] E
inst✝ : IsBoundedSMul ℤ_[p] E
a : E
n : ℕ
⊢ ‖a‖ ≤ ‖mahlerTerm a n‖
Module: Mathlib.NumberTheory.Padics.WithVal

Goal state before:
pp: case right
p : ℕ
inst✝ : Fact (Nat.Prime p)
hp0' : 0 < ↑p
hp0 : 0 < (↑p)⁻¹
hp1' : 1 < ↑p
hp1 : (↑p)⁻¹ < 1
⊢ ∀ (j : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ˣ),
    ∃ i,
      ∀ (x y : WithVal (Rat.padicValuation p)),
        ↑(padicNorm p ((WithVal.equiv (Rat.padicValuation p)) (x - y))) ≤ (↑p ^ i)⁻¹ → Valued.v.restrict (y - x) < ↑j

Tactic:
·

Goal state after:
case right
p : ℕ
inst✝ : Fact (Nat.Prime p)
hp0' : 0 < ↑p
hp0 : 0 < (↑p)⁻¹
hp1' : 1 < ↑p
hp1 : (↑p)⁻¹ < 1
⊢ ∀ (j : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ˣ),
    ∃ i,
      ∀ (x y : WithVal (Rat.padicValuation p)),
        ↑(padicNorm p ((WithVal.equiv (Rat.padicValuation p)) (x - y))) ≤ (↑p ^ i)⁻¹ → Valued.v.restrict (y - x) < ↑j
Module: Mathlib.NumberTheory.Padics.MahlerBasis

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : Module ℤ_[p] E
inst✝² : IsBoundedSMul ℤ_[p] E
inst✝¹ : IsUltrametricDist E
inst✝ : CompleteSpace E
a : ℕ → E
ha : Tendsto a atTop (𝓝 0)
m n : ℕ
hmn : m ≤ n
i : ℕ
⊢ m < i + (n + 1)

Tactic:
by lia

Goal state after:
Module: Mathlib.NumberTheory.Padics.WithVal

Goal state before:
pp: case h
p : ℕ
inst✝ : Fact (Nat.Prime p)
hp0' : 0 < ↑p
hp0 : 0 < (↑p)⁻¹
hp1' : 1 < ↑p
hp1 : (↑p)⁻¹ < 1
γ : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ˣ
x y : WithVal (Rat.padicValuation p)
h : padicNorm p ((WithVal.equiv (Rat.padicValuation p)) (x - y)) ≤ (↑(p ^ (embedding ↑γ * exp (-1)).log.natAbs))⁻¹
x' : ℚ := (WithVal.equiv (Rat.padicValuation p)) x
hx : x' = (WithVal.equiv (Rat.padicValuation p)) x
y' : ℚ := (WithVal.equiv (Rat.padicValuation p)) y
hy : y' = (WithVal.equiv (Rat.padicValuation p)) y
⊢ (Rat.padicValuation p) (x' - y') < embedding ↑γ

pp: case h
p : ℕ
inst✝ : Fact (Nat.Prime p)
hp0' : 0 < ↑p
hp0 : 0 < (↑p)⁻¹
hp1' : 1 < ↑p
hp1 : (↑p)⁻¹ < 1
γ : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ˣ
x y : WithVal (Rat.padicValuation p)
h : ↑(padicNorm p ((WithVal.equiv (Rat.padicValuation p)) (x - y))) ≤ (↑↑(p ^ (embedding ↑γ * exp (-1)).log.natAbs))⁻¹
x' : ℚ := (WithVal.equiv (Rat.padicValuation p)) x
hx : x' = (WithVal.equiv (Rat.padicValuation p)) x
y' : ℚ := (WithVal.equiv (Rat.padicValuation p)) y
hy : y' = (WithVal.equiv (Rat.padicValuation p)) y
⊢ ↑(↑(p ^ (embedding ↑γ * exp (-1)).log.natAbs)).num ≠ 0

Tactic:
·

Goal state after:
case h
p : ℕ
inst✝ : Fact (Nat.Prime p)
hp0' : 0 < ↑p
hp0 : 0 < (↑p)⁻¹
hp1' : 1 < ↑p
hp1 : (↑p)⁻¹ < 1
γ : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ˣ
x y : WithVal (Rat.padicValuation p)
h : padicNorm p ((WithVal.equiv (Rat.padicValuation p)) (x - y)) ≤ (↑(p ^ (embedding ↑γ * exp (-1)).log.natAbs))⁻¹
x' : ℚ := (WithVal.equiv (Rat.padicValuation p)) x
hx : x' = (WithVal.equiv (Rat.padicValuation p)) x
y' : ℚ := (WithVal.equiv (Rat.padicValuation p)) y
hy : y' = (WithVal.equiv (Rat.padicValuation p)) y
⊢ (Rat.padicValuation p) (x' - y') < embedding ↑γ
Module: Mathlib.NumberTheory.Padics.MahlerBasis

Goal state before:
pp: case a
p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : Module ℤ_[p] E
inst✝² : IsBoundedSMul ℤ_[p] E
a✝ : E
n : ℕ
x : ℤ_[p]
inst✝¹ : IsUltrametricDist E
inst✝ : CompleteSpace E
a : ℕ → E
f : C(ℤ_[p], E)
⊢ ‖({ toFun := fun f ↦ { toFun := fun x ↦ Δ_[1]^[x] (⇑f) 0, continuous_toFun := ⋯, zero_at_infty' := ⋯ }, map_add' := ⋯,
            map_smul' := ⋯ }
          f).toBCF‖ ≤
    ‖f‖

pp: case a
p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : Module ℤ_[p] E
inst✝² : IsBoundedSMul ℤ_[p] E
a✝ : E
n : ℕ
x : ℤ_[p]
inst✝¹ : IsUltrametricDist E
inst✝ : CompleteSpace E
a : ℕ → E
f : C(ℤ_[p], E)
⊢ ‖f‖ ≤
    ‖({ toFun := fun f ↦ { toFun := fun x ↦ Δ_[1]^[x] (⇑f) 0, continuous_toFun := ⋯, zero_at_infty' := ⋯ },
            map_add' := ⋯, map_smul' := ⋯ }
          f).toBCF‖

Tactic:
·

Goal state after:
case a
p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : Module ℤ_[p] E
inst✝² : IsBoundedSMul ℤ_[p] E
a✝ : E
n : ℕ
x : ℤ_[p]
inst✝¹ : IsUltrametricDist E
inst✝ : CompleteSpace E
a : ℕ → E
f : C(ℤ_[p], E)
⊢ ‖({ toFun := fun f ↦ { toFun := fun x ↦ Δ_[1]^[x] (⇑f) 0, continuous_toFun := ⋯, zero_at_infty' := ⋯ }, map_add' := ⋯,
            map_smul' := ⋯ }
          f).toBCF‖ ≤
    ‖f‖
Module: Mathlib.NumberTheory.Padics.WithVal

Goal state before:
pp: case pos
p : ℕ
inst✝ : Fact (Nat.Prime p)
hp0' : 0 < ↑p
hp0 : 0 < (↑p)⁻¹
hp1' : 1 < ↑p
hp1 : (↑p)⁻¹ < 1
γ : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ˣ
x y : WithVal (Rat.padicValuation p)
x' : ℚ := (WithVal.equiv (Rat.padicValuation p)) x
hx : x' = (WithVal.equiv (Rat.padicValuation p)) x
y' : ℚ := (WithVal.equiv (Rat.padicValuation p)) y
hy : y' = (WithVal.equiv (Rat.padicValuation p)) y
h : (if x' - y' = 0 then 0 else (↑p ^ padicValRat p (x' - y'))⁻¹) ≤ (↑p ^ (embedding ↑γ * exp (-1)).log.natAbs)⁻¹
H : x' - y' = 0
⊢ 0 < embedding ↑γ

pp: case neg
p : ℕ
inst✝ : Fact (Nat.Prime p)
hp0' : 0 < ↑p
hp0 : 0 < (↑p)⁻¹
hp1' : 1 < ↑p
hp1 : (↑p)⁻¹ < 1
γ : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ˣ
x y : WithVal (Rat.padicValuation p)
x' : ℚ := (WithVal.equiv (Rat.padicValuation p)) x
hx : x' = (WithVal.equiv (Rat.padicValuation p)) x
y' : ℚ := (WithVal.equiv (Rat.padicValuation p)) y
hy : y' = (WithVal.equiv (Rat.padicValuation p)) y
h : (if x' - y' = 0 then 0 else (↑p ^ padicValRat p (x' - y'))⁻¹) ≤ (↑p ^ (embedding ↑γ * exp (-1)).log.natAbs)⁻¹
H : ¬x' - y' = 0
⊢ exp (-padicValRat p (x' - y')) < embedding ↑γ

Tactic:
·

Goal state after:
case pos
p : ℕ
inst✝ : Fact (Nat.Prime p)
hp0' : 0 < ↑p
hp0 : 0 < (↑p)⁻¹
hp1' : 1 < ↑p
hp1 : (↑p)⁻¹ < 1
γ : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ˣ
x y : WithVal (Rat.padicValuation p)
x' : ℚ := (WithVal.equiv (Rat.padicValuation p)) x
hx : x' = (WithVal.equiv (Rat.padicValuation p)) x
y' : ℚ := (WithVal.equiv (Rat.padicValuation p)) y
hy : y' = (WithVal.equiv (Rat.padicValuation p)) y
h : (if x' - y' = 0 then 0 else (↑p ^ padicValRat p (x' - y'))⁻¹) ≤ (↑p ^ (embedding ↑γ * exp (-1)).log.natAbs)⁻¹
H : x' - y' = 0
⊢ 0 < embedding ↑γ
Module: Mathlib.NumberTheory.Padics.MahlerBasis

Goal state before:
pp: case a
p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : Module ℤ_[p] E
inst✝² : IsBoundedSMul ℤ_[p] E
a✝ : E
n : ℕ
x : ℤ_[p]
inst✝¹ : IsUltrametricDist E
inst✝ : CompleteSpace E
a : ℕ → E
f : C(ℤ_[p], E)
⊢ ‖f‖ ≤
    ‖({ toFun := fun f ↦ { toFun := fun x ↦ Δ_[1]^[x] (⇑f) 0, continuous_toFun := ⋯, zero_at_infty' := ⋯ },
            map_add' := ⋯, map_smul' := ⋯ }
          f).toBCF‖

Tactic:
·

Goal state after:
case a
p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : Module ℤ_[p] E
inst✝² : IsBoundedSMul ℤ_[p] E
a✝ : E
n : ℕ
x : ℤ_[p]
inst✝¹ : IsUltrametricDist E
inst✝ : CompleteSpace E
a : ℕ → E
f : C(ℤ_[p], E)
⊢ ‖f‖ ≤
    ‖({ toFun := fun f ↦ { toFun := fun x ↦ Δ_[1]^[x] (⇑f) 0, continuous_toFun := ⋯, zero_at_infty' := ⋯ },
            map_add' := ⋯, map_smul' := ⋯ }
          f).toBCF‖
Module: Mathlib.NumberTheory.Padics.WithVal

Goal state before:
pp: case neg
p : ℕ
inst✝ : Fact (Nat.Prime p)
hp0' : 0 < ↑p
hp0 : 0 < (↑p)⁻¹
hp1' : 1 < ↑p
hp1 : (↑p)⁻¹ < 1
γ : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ˣ
x y : WithVal (Rat.padicValuation p)
x' : ℚ := (WithVal.equiv (Rat.padicValuation p)) x
hx : x' = (WithVal.equiv (Rat.padicValuation p)) x
y' : ℚ := (WithVal.equiv (Rat.padicValuation p)) y
hy : y' = (WithVal.equiv (Rat.padicValuation p)) y
h : (if x' - y' = 0 then 0 else (↑p ^ padicValRat p (x' - y'))⁻¹) ≤ (↑p ^ (embedding ↑γ * exp (-1)).log.natAbs)⁻¹
H : ¬x' - y' = 0
⊢ exp (-padicValRat p (x' - y')) < embedding ↑γ

Tactic:
·

Goal state after:
case neg
p : ℕ
inst✝ : Fact (Nat.Prime p)
hp0' : 0 < ↑p
hp0 : 0 < (↑p)⁻¹
hp1' : 1 < ↑p
hp1 : (↑p)⁻¹ < 1
γ : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ˣ
x y : WithVal (Rat.padicValuation p)
x' : ℚ := (WithVal.equiv (Rat.padicValuation p)) x
hx : x' = (WithVal.equiv (Rat.padicValuation p)) x
y' : ℚ := (WithVal.equiv (Rat.padicValuation p)) y
hy : y' = (WithVal.equiv (Rat.padicValuation p)) y
h : (if x' - y' = 0 then 0 else (↑p ^ padicValRat p (x' - y'))⁻¹) ≤ (↑p ^ (embedding ↑γ * exp (-1)).log.natAbs)⁻¹
H : ¬x' - y' = 0
⊢ exp (-padicValRat p (x' - y')) < embedding ↑γ
Module: Mathlib.NumberTheory.Padics.WithVal

Goal state before:
pp: case h
p : ℕ
inst✝ : Fact (Nat.Prime p)
hp0' : 0 < ↑p
hp0 : 0 < (↑p)⁻¹
hp1' : 1 < ↑p
hp1 : (↑p)⁻¹ < 1
γ : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ˣ
x y : WithVal (Rat.padicValuation p)
h : ↑(padicNorm p ((WithVal.equiv (Rat.padicValuation p)) (x - y))) ≤ (↑↑(p ^ (embedding ↑γ * exp (-1)).log.natAbs))⁻¹
x' : ℚ := (WithVal.equiv (Rat.padicValuation p)) x
hx : x' = (WithVal.equiv (Rat.padicValuation p)) x
y' : ℚ := (WithVal.equiv (Rat.padicValuation p)) y
hy : y' = (WithVal.equiv (Rat.padicValuation p)) y
⊢ ↑(↑(p ^ (embedding ↑γ * exp (-1)).log.natAbs)).num ≠ 0

Tactic:
·

Goal state after:
case h
p : ℕ
inst✝ : Fact (Nat.Prime p)
hp0' : 0 < ↑p
hp0 : 0 < (↑p)⁻¹
hp1' : 1 < ↑p
hp1 : (↑p)⁻¹ < 1
γ : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ˣ
x y : WithVal (Rat.padicValuation p)
h : ↑(padicNorm p ((WithVal.equiv (Rat.padicValuation p)) (x - y))) ≤ (↑↑(p ^ (embedding ↑γ * exp (-1)).log.natAbs))⁻¹
x' : ℚ := (WithVal.equiv (Rat.padicValuation p)) x
hx : x' = (WithVal.equiv (Rat.padicValuation p)) x
y' : ℚ := (WithVal.equiv (Rat.padicValuation p)) y
hy : y' = (WithVal.equiv (Rat.padicValuation p)) y
⊢ ↑(↑(p ^ (embedding ↑γ * exp (-1)).log.natAbs)).num ≠ 0
Module: Mathlib.Topology.Separation.DisjointCover

Goal state before:
pp: case refine_1
ι : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TotallyDisconnectedSpace X
inst✝¹ : T2Space X
inst✝ : CompactSpace X
U : ι → Opens X
hU : IsOpenCover U
n : ℕ
V : Fin n → Clopens X
hVle : ∀ (j : Fin n), ∃ i, ↑(V j) ⊆ ↑(U i)
hVun : univ ⊆ ⋃ j, ↑(V j)
W : Fin n → Clopens X
hWle : W ≤ V
hWd : Pairwise (Disjoint on W)
hWun : ⋃ i, ↑(W i) = ⋃ i, ↑(V i)
t : Finset (Fin n) := {j | W j ≠ ⊥}
k : Fin #t
⊢ (fun k ↦ W ↑(t.equivFin.symm k)) k ≠ ⊥

pp: case refine_2
ι : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TotallyDisconnectedSpace X
inst✝¹ : T2Space X
inst✝ : CompactSpace X
U : ι → Opens X
hU : IsOpenCover U
n : ℕ
V : Fin n → Clopens X
hVle : ∀ (j : Fin n), ∃ i, ↑(V j) ⊆ ↑(U i)
hVun : univ ⊆ ⋃ j, ↑(V j)
W : Fin n → Clopens X
hWle : W ≤ V
hWd : Pairwise (Disjoint on W)
hWun : ⋃ i, ↑(W i) = ⋃ i, ↑(V i)
t : Finset (Fin n) := {j | W j ≠ ⊥}
k : Fin #t
⊢ ∃ i, ↑((fun k ↦ W ↑(t.equivFin.symm k)) k) ⊆ ↑(U i)

pp: case refine_3
ι : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TotallyDisconnectedSpace X
inst✝¹ : T2Space X
inst✝ : CompactSpace X
U : ι → Opens X
hU : IsOpenCover U
n : ℕ
V : Fin n → Clopens X
hVle : ∀ (j : Fin n), ∃ i, ↑(V j) ⊆ ↑(U i)
hVun : univ ⊆ ⋃ j, ↑(V j)
W : Fin n → Clopens X
hWle : W ≤ V
hWd : Pairwise (Disjoint on W)
hWun : ⋃ i, ↑(W i) = ⋃ i, ↑(V i)
t : Finset (Fin n) := {j | W j ≠ ⊥}
x : X
hx : x ∈ univ
⊢ x ∈ ⋃ j, ↑((fun k ↦ W ↑(t.equivFin.symm k)) j)

pp: case refine_4
ι : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TotallyDisconnectedSpace X
inst✝¹ : T2Space X
inst✝ : CompactSpace X
U : ι → Opens X
hU : IsOpenCover U
n : ℕ
V : Fin n → Clopens X
hVle : ∀ (j : Fin n), ∃ i, ↑(V j) ⊆ ↑(U i)
hVun : univ ⊆ ⋃ j, ↑(V j)
W : Fin n → Clopens X
hWle : W ≤ V
hWd : Pairwise (Disjoint on W)
hWun : ⋃ i, ↑(W i) = ⋃ i, ↑(V i)
t : Finset (Fin n) := {j | W j ≠ ⊥}
⊢ Pairwise (Disjoint on fun k ↦ W ↑(t.equivFin.symm k))

Tactic:
·

Goal state after:
case refine_1
ι : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TotallyDisconnectedSpace X
inst✝¹ : T2Space X
inst✝ : CompactSpace X
U : ι → Opens X
hU : IsOpenCover U
n : ℕ
V : Fin n → Clopens X
hVle : ∀ (j : Fin n), ∃ i, ↑(V j) ⊆ ↑(U i)
hVun : univ ⊆ ⋃ j, ↑(V j)
W : Fin n → Clopens X
hWle : W ≤ V
hWd : Pairwise (Disjoint on W)
hWun : ⋃ i, ↑(W i) = ⋃ i, ↑(V i)
t : Finset (Fin n) := {j | W j ≠ ⊥}
k : Fin #t
⊢ (fun k ↦ W ↑(t.equivFin.symm k)) k ≠ ⊥
Module: Mathlib.NumberTheory.Padics.WithVal

Goal state before:
pp: case hp
p : ℕ
inst✝ : Fact (Nat.Prime p)
⊢ IsClosed
    {a |
      ⋯.extend coe' ((extensionHom ((Rat.castHom ℚ_[p]).comp (WithVal.equiv (Rat.padicValuation p)).toRingHom) ⋯) a) =
        a}

pp: case ih
p : ℕ
inst✝ : Fact (Nat.Prime p)
a✝ : WithVal (Rat.padicValuation p)
⊢ ⋯.extend coe' ((extensionHom ((Rat.castHom ℚ_[p]).comp (WithVal.equiv (Rat.padicValuation p)).toRingHom) ⋯) ↑a✝) = ↑a✝

Tactic:
·

Goal state after:
case hp
p : ℕ
inst✝ : Fact (Nat.Prime p)
⊢ IsClosed
    {a |
      ⋯.extend coe' ((extensionHom ((Rat.castHom ℚ_[p]).comp (WithVal.equiv (Rat.padicValuation p)).toRingHom) ⋯) a) =
        a}
Module: Mathlib.NumberTheory.Padics.WithVal

Goal state before:
pp: case ih
p : ℕ
inst✝ : Fact (Nat.Prime p)
a✝ : WithVal (Rat.padicValuation p)
⊢ ⋯.extend coe' ((extensionHom ((Rat.castHom ℚ_[p]).comp (WithVal.equiv (Rat.padicValuation p)).toRingHom) ⋯) ↑a✝) = ↑a✝

Tactic:
·

Goal state after:
case ih
p : ℕ
inst✝ : Fact (Nat.Prime p)
a✝ : WithVal (Rat.padicValuation p)
⊢ ⋯.extend coe' ((extensionHom ((Rat.castHom ℚ_[p]).comp (WithVal.equiv (Rat.padicValuation p)).toRingHom) ⋯) ↑a✝) = ↑a✝
Module: Mathlib.Topology.Separation.DisjointCover

Goal state before:
pp: case refine_2
ι : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TotallyDisconnectedSpace X
inst✝¹ : T2Space X
inst✝ : CompactSpace X
U : ι → Opens X
hU : IsOpenCover U
n : ℕ
V : Fin n → Clopens X
hVle : ∀ (j : Fin n), ∃ i, ↑(V j) ⊆ ↑(U i)
hVun : univ ⊆ ⋃ j, ↑(V j)
W : Fin n → Clopens X
hWle : W ≤ V
hWd : Pairwise (Disjoint on W)
hWun : ⋃ i, ↑(W i) = ⋃ i, ↑(V i)
t : Finset (Fin n) := {j | W j ≠ ⊥}
k : Fin #t
⊢ ∃ i, ↑((fun k ↦ W ↑(t.equivFin.symm k)) k) ⊆ ↑(U i)

pp: case refine_3
ι : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TotallyDisconnectedSpace X
inst✝¹ : T2Space X
inst✝ : CompactSpace X
U : ι → Opens X
hU : IsOpenCover U
n : ℕ
V : Fin n → Clopens X
hVle : ∀ (j : Fin n), ∃ i, ↑(V j) ⊆ ↑(U i)
hVun : univ ⊆ ⋃ j, ↑(V j)
W : Fin n → Clopens X
hWle : W ≤ V
hWd : Pairwise (Disjoint on W)
hWun : ⋃ i, ↑(W i) = ⋃ i, ↑(V i)
t : Finset (Fin n) := {j | W j ≠ ⊥}
x : X
hx : x ∈ univ
⊢ x ∈ ⋃ j, ↑((fun k ↦ W ↑(t.equivFin.symm k)) j)

pp: case refine_4
ι : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TotallyDisconnectedSpace X
inst✝¹ : T2Space X
inst✝ : CompactSpace X
U : ι → Opens X
hU : IsOpenCover U
n : ℕ
V : Fin n → Clopens X
hVle : ∀ (j : Fin n), ∃ i, ↑(V j) ⊆ ↑(U i)
hVun : univ ⊆ ⋃ j, ↑(V j)
W : Fin n → Clopens X
hWle : W ≤ V
hWd : Pairwise (Disjoint on W)
hWun : ⋃ i, ↑(W i) = ⋃ i, ↑(V i)
t : Finset (Fin n) := {j | W j ≠ ⊥}
⊢ Pairwise (Disjoint on fun k ↦ W ↑(t.equivFin.symm k))

Tactic:
·

Goal state after:
case refine_2
ι : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TotallyDisconnectedSpace X
inst✝¹ : T2Space X
inst✝ : CompactSpace X
U : ι → Opens X
hU : IsOpenCover U
n : ℕ
V : Fin n → Clopens X
hVle : ∀ (j : Fin n), ∃ i, ↑(V j) ⊆ ↑(U i)
hVun : univ ⊆ ⋃ j, ↑(V j)
W : Fin n → Clopens X
hWle : W ≤ V
hWd : Pairwise (Disjoint on W)
hWun : ⋃ i, ↑(W i) = ⋃ i, ↑(V i)
t : Finset (Fin n) := {j | W j ≠ ⊥}
k : Fin #t
⊢ ∃ i, ↑((fun k ↦ W ↑(t.equivFin.symm k)) k) ⊆ ↑(U i)
Module: Mathlib.NumberTheory.Padics.WithVal

Goal state before:
pp: case hp
p : ℕ
inst✝ : Fact (Nat.Prime p)
⊢ IsClosed
    {x |
      (extensionHom ((Rat.castHom ℚ_[p]).comp (WithVal.equiv (Rat.padicValuation p)).toRingHom) ⋯) (⋯.extend coe' x) =
        x}

pp: case h
p : ℕ
inst✝ : Fact (Nat.Prime p)
a✝ : ℚ
⊢ (extensionHom ((Rat.castHom ℚ_[p]).comp (WithVal.equiv (Rat.padicValuation p)).toRingHom) ⋯) (⋯.extend coe' ↑a✝) = ↑a✝

Tactic:
·

Goal state after:
case hp
p : ℕ
inst✝ : Fact (Nat.Prime p)
⊢ IsClosed
    {x |
      (extensionHom ((Rat.castHom ℚ_[p]).comp (WithVal.equiv (Rat.padicValuation p)).toRingHom) ⋯) (⋯.extend coe' x) =
        x}
Module: Mathlib.NumberTheory.Padics.WithVal

Goal state before:
pp: case h
p : ℕ
inst✝ : Fact (Nat.Prime p)
a✝ : ℚ
⊢ (extensionHom ((Rat.castHom ℚ_[p]).comp (WithVal.equiv (Rat.padicValuation p)).toRingHom) ⋯) (⋯.extend coe' ↑a✝) = ↑a✝

Tactic:
·

Goal state after:
case h
p : ℕ
inst✝ : Fact (Nat.Prime p)
a✝ : ℚ
⊢ (extensionHom ((Rat.castHom ℚ_[p]).comp (WithVal.equiv (Rat.padicValuation p)).toRingHom) ⋯) (⋯.extend coe' ↑a✝) = ↑a✝
Module: Mathlib.Topology.Separation.DisjointCover

Goal state before:
pp: case refine_3
ι : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TotallyDisconnectedSpace X
inst✝¹ : T2Space X
inst✝ : CompactSpace X
U : ι → Opens X
hU : IsOpenCover U
n : ℕ
V : Fin n → Clopens X
hVle : ∀ (j : Fin n), ∃ i, ↑(V j) ⊆ ↑(U i)
hVun : univ ⊆ ⋃ j, ↑(V j)
W : Fin n → Clopens X
hWle : W ≤ V
hWd : Pairwise (Disjoint on W)
hWun : ⋃ i, ↑(W i) = ⋃ i, ↑(V i)
t : Finset (Fin n) := {j | W j ≠ ⊥}
x : X
hx : x ∈ univ
⊢ x ∈ ⋃ j, ↑((fun k ↦ W ↑(t.equivFin.symm k)) j)

pp: case refine_4
ι : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TotallyDisconnectedSpace X
inst✝¹ : T2Space X
inst✝ : CompactSpace X
U : ι → Opens X
hU : IsOpenCover U
n : ℕ
V : Fin n → Clopens X
hVle : ∀ (j : Fin n), ∃ i, ↑(V j) ⊆ ↑(U i)
hVun : univ ⊆ ⋃ j, ↑(V j)
W : Fin n → Clopens X
hWle : W ≤ V
hWd : Pairwise (Disjoint on W)
hWun : ⋃ i, ↑(W i) = ⋃ i, ↑(V i)
t : Finset (Fin n) := {j | W j ≠ ⊥}
⊢ Pairwise (Disjoint on fun k ↦ W ↑(t.equivFin.symm k))

Tactic:
·

Goal state after:
case refine_3
ι : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TotallyDisconnectedSpace X
inst✝¹ : T2Space X
inst✝ : CompactSpace X
U : ι → Opens X
hU : IsOpenCover U
n : ℕ
V : Fin n → Clopens X
hVle : ∀ (j : Fin n), ∃ i, ↑(V j) ⊆ ↑(U i)
hVun : univ ⊆ ⋃ j, ↑(V j)
W : Fin n → Clopens X
hWle : W ≤ V
hWd : Pairwise (Disjoint on W)
hWun : ⋃ i, ↑(W i) = ⋃ i, ↑(V i)
t : Finset (Fin n) := {j | W j ≠ ⊥}
x : X
hx : x ∈ univ
⊢ x ∈ ⋃ j, ↑((fun k ↦ W ↑(t.equivFin.symm k)) j)
Module: Mathlib.Topology.Separation.DisjointCover

Goal state before:
pp: case refine_4
ι : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TotallyDisconnectedSpace X
inst✝¹ : T2Space X
inst✝ : CompactSpace X
U : ι → Opens X
hU : IsOpenCover U
n : ℕ
V : Fin n → Clopens X
hVle : ∀ (j : Fin n), ∃ i, ↑(V j) ⊆ ↑(U i)
hVun : univ ⊆ ⋃ j, ↑(V j)
W : Fin n → Clopens X
hWle : W ≤ V
hWd : Pairwise (Disjoint on W)
hWun : ⋃ i, ↑(W i) = ⋃ i, ↑(V i)
t : Finset (Fin n) := {j | W j ≠ ⊥}
⊢ Pairwise (Disjoint on fun k ↦ W ↑(t.equivFin.symm k))

Tactic:
·

Goal state after:
case refine_4
ι : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TotallyDisconnectedSpace X
inst✝¹ : T2Space X
inst✝ : CompactSpace X
U : ι → Opens X
hU : IsOpenCover U
n : ℕ
V : Fin n → Clopens X
hVle : ∀ (j : Fin n), ∃ i, ↑(V j) ⊆ ↑(U i)
hVun : univ ⊆ ⋃ j, ↑(V j)
W : Fin n → Clopens X
hWle : W ≤ V
hWd : Pairwise (Disjoint on W)
hWun : ⋃ i, ↑(W i) = ⋃ i, ↑(V i)
t : Finset (Fin n) := {j | W j ≠ ⊥}
⊢ Pairwise (Disjoint on fun k ↦ W ↑(t.equivFin.symm k))