mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith

Goal state before:
pp: x : ℝ
n : ℕ
hn : 0 < n
hn' : 0 < ↑n
this : x < ↑(⌊x * ↑n⌋ + 1) / ↑n
⊢ ↑(⌊x * ↑n⌋ + 1) ≤ x * ↑n + 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith

Goal state before:
pp: x : ℝ
n : ℕ
hn : 0 < n
hn' : 0 < ↑n
this : x < ↑(⌊x * ↑n⌋ + 1) / ↑n
⊢ x * ↑n + 1 < x * ↑n + 2

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith

Goal state before:
pp: x : ℝ
⊢ LiouvilleWith 1 x

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.Residual

Goal state before:
pp: ⊢ {x | Liouville x} = ⋂ n, ⋃ a, ⋃ b, ⋃ (_ : 1 < b), ball (↑a / ↑b) (1 / ↑b ^ n) \ {↑a / ↑b}

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.Residual

Goal state before:
pp: ⊢ IsGδ {x | Liouville x}

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.Residual

Goal state before:
pp: n : ℕ
a b : ℤ
hb : 1 < b
⊢ ↑(↑a / ↑b) = ↑a / ↑b

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.RamificationInertia.Basic

Goal state before:
pp: R : Type u
inst✝¹⁷ : CommRing R
S : Type v
inst✝¹⁶ : CommRing S
inst✝¹⁵ : Algebra R S
inst✝¹⁴ : IsDedekindDomain S
K : Type u_1
L : Type u_2
inst✝¹³ : Field K
inst✝¹² : Field L
inst✝¹¹ : IsDedekindDomain R
inst✝¹⁰ : Algebra R K
inst✝⁹ : IsFractionRing R K
inst✝⁸ : Algebra S L
inst✝⁷ : IsFractionRing S L
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra R L
inst✝⁴ : IsScalarTower R S L
inst✝³ : IsScalarTower R K L
inst✝² : Module.Finite R S
inst✝¹ : NoZeroSMulDivisors R S
p : Ideal R
inst✝ : p.IsMaximal
P : Ideal S
hP₁ : P.IsPrime
hP₂ : P.LiesOver p
hp0 : p ≠ ⊥
⊢ p.inertiaDeg' P ≤ finrank K L

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Lindemann.AnalyticalPart

Goal state before:
pp: case h.right.refine_2
f : ℤ[X]
hf : eval 0 f ≠ 0
c' : ℂ → ℝ
c'0 : ∀ (s : ℂ), c' s ≥ 0
Pp'_le : ∀ (s : ℂ) (p : ℕ), p ≠ 0 → ‖P (map (algebraMap ℤ ℂ) (X ^ (p - 1) * f ^ p)) s‖ ≤ c' s ^ p
p : ℕ
p_gt : p > (eval 0 f).natAbs
prime_p : Nat.Prime p
gp' : ℤ[X]
h' : eval 0 (sumIDeriv (X ^ (p - 1) * f ^ p)) = (p - 1)! • eval 0 (f ^ p) + p ! • eval 0 gp'
gp : ℤ[X]
gp'_le : gp.natDegree ≤ (X ^ (p - 1) * f ^ p).natDegree - p
r : ℂ
h : (aeval r) (sumIDeriv (X ^ (p - 1) * f ^ p)) = ↑p ! * (aeval r) gp
hr : r ∈ f.aroots ℂ
this :
  (↑(eval 0 f ^ p + ↑p * eval 0 gp') * cexp r - ↑p * (aeval r) gp) * ↑(p - 1)! =
    ↑(eval 0 f) ^ p * cexp r * ↑(p - 1)! + ↑(p * (p - 1)!) * (↑(eval 0 gp') * cexp r - (aeval r) gp)
⊢ ‖↑(eval 0 f ^ p + ↑p * eval 0 gp') * cexp r - ↑p * (aeval r) gp‖ ≤
    (if h : (Multiset.map c' (f.aroots ℂ)).toFinset.Nonempty then (Multiset.map c' (f.aroots ℂ)).toFinset.max' h
        else 0) ^
        p /
      ↑(p - 1)!

Tactic:
le_div_iff₀ (Nat.cast_pos.mpr (Nat.factorial_pos _) : (0 : ℝ) < _),

Goal state after:
case h.right.refine_2
f : ℤ[X]
hf : eval 0 f ≠ 0
c' : ℂ → ℝ
c'0 : ∀ (s : ℂ), c' s ≥ 0
Pp'_le : ∀ (s : ℂ) (p : ℕ), p ≠ 0 → ‖P (map (algebraMap ℤ ℂ) (X ^ (p - 1) * f ^ p)) s‖ ≤ c' s ^ p
p : ℕ
p_gt : p > (eval 0 f).natAbs
prime_p : Nat.Prime p
gp' : ℤ[X]
h' : eval 0 (sumIDeriv (X ^ (p - 1) * f ^ p)) = (p - 1)! • eval 0 (f ^ p) + p ! • eval 0 gp'
gp : ℤ[X]
gp'_le : gp.natDegree ≤ (X ^ (p - 1) * f ^ p).natDegree - p
r : ℂ
h : (aeval r) (sumIDeriv (X ^ (p - 1) * f ^ p)) = ↑p ! * (aeval r) gp
hr : r ∈ f.aroots ℂ
this :
  (↑(eval 0 f ^ p + ↑p * eval 0 gp') * cexp r - ↑p * (aeval r) gp) * ↑(p - 1)! =
    ↑(eval 0 f) ^ p * cexp r * ↑(p - 1)! + ↑(p * (p - 1)!) * (↑(eval 0 gp') * cexp r - (aeval r) gp)
⊢ ‖↑(eval 0 f ^ p + ↑p * eval 0 gp') * cexp r - ↑p * (aeval r) gp‖ * ↑(p - 1)! ≤
    (if h : (Multiset.map c' (f.aroots ℂ)).toFinset.Nonempty then (Multiset.map c' (f.aroots ℂ)).toFinset.max' h
      else 0) ^
      p
Module: Mathlib.NumberTheory.Transcendental.Liouville.Residual

Goal state before:
pp: ⊢ {x | Liouville x} = {x | Irrational x} ∩ ⋂ n, ⋃ a, ⋃ b, ⋃ (_ : 1 < b), ball (↑a / ↑b) (1 / ↑b ^ n)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith

Goal state before:
pp: p x : ℝ
h : LiouvilleWith p x
⊢ ∃ C, ∃ (_ : 0 < C), ∃ᶠ (n : ℕ) in atTop, 1 ≤ n ∧ ∃ m, x ≠ ↑m / ↑n ∧ |x - ↑m / ↑n| < C / ↑n ^ p

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.RamificationInertia.Basic

Goal state before:
pp: R : Type u
inst✝¹⁷ : CommRing R
S : Type v
inst✝¹⁶ : CommRing S
inst✝¹⁵ : Algebra R S
inst✝¹⁴ : IsDedekindDomain S
K : Type u_1
L : Type u_2
inst✝¹³ : Field K
inst✝¹² : Field L
inst✝¹¹ : IsDedekindDomain R
inst✝¹⁰ : Algebra R K
inst✝⁹ : IsFractionRing R K
inst✝⁸ : Algebra S L
inst✝⁷ : IsFractionRing S L
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra R L
inst✝⁴ : IsScalarTower R S L
inst✝³ : IsScalarTower R K L
inst✝² : Module.Finite R S
inst✝¹ : NoZeroSMulDivisors R S
p : Ideal R
inst✝ : p.IsMaximal
P : Ideal S
hP₁ : P.IsPrime
hP₂ : P.LiesOver p
⊢ p.ramificationIdx' P ≤ finrank K L

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.Residual

Goal state before:
pp: ⊢ ∀ᶠ (x : ℝ) in residual ℝ, Liouville x

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith

Goal state before:
pp: p q x : ℝ
h : LiouvilleWith p x
hle : q ≤ p
⊢ LiouvilleWith q x

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith

Goal state before:
pp: p q x : ℝ
h : LiouvilleWith p x
hlt : q < p
C : ℝ
_hC₀ : 0 < C
hC : ∃ᶠ (n : ℕ) in atTop, 1 ≤ n ∧ ∃ m, x ≠ ↑m / ↑n ∧ |x - ↑m / ↑n| < C / ↑n ^ p
⊢ ∀ᶠ (n : ℕ) in atTop, C < ↑n ^ (p - q)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.RamificationInertia.Basic

Goal state before:
pp: R : Type u
inst✝¹⁷ : CommRing R
S : Type v
inst✝¹⁶ : CommRing S
inst✝¹⁵ : Algebra R S
inst✝¹⁴ : IsDedekindDomain S
K : Type u_1
L : Type u_2
inst✝¹³ : Field K
inst✝¹² : Field L
inst✝¹¹ : IsDedekindDomain R
inst✝¹⁰ : Algebra R K
inst✝⁹ : IsFractionRing R K
inst✝⁸ : Algebra S L
inst✝⁷ : IsFractionRing S L
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra R L
inst✝⁴ : IsScalarTower R S L
inst✝³ : IsScalarTower R K L
inst✝² : Module.Finite R S
inst✝¹ : NoZeroSMulDivisors R S
p : Ideal R
inst✝ : p.IsMaximal
hp0 : p ≠ ⊥
⊢ (IsDedekindDomain.primesOverFinset p S).card ≤ finrank K L

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Lindemann.AnalyticalPart

Goal state before:
pp: f : ℤ[X]
hf : eval 0 f ≠ 0
c' : ℂ → ℝ
c'0 : ∀ (s : ℂ), c' s ≥ 0
Pp'_le : ∀ (s : ℂ) (p : ℕ), p ≠ 0 → ‖P (map (algebraMap ℤ ℂ) (X ^ (p - 1) * f ^ p)) s‖ ≤ c' s ^ p
p : ℕ
p_gt : p > (eval 0 f).natAbs
prime_p : Nat.Prime p
gp' : ℤ[X]
h' : eval 0 (sumIDeriv (X ^ (p - 1) * f ^ p)) = (p - 1)! • eval 0 (f ^ p) + p ! • eval 0 gp'
gp : ℤ[X]
gp'_le : gp.natDegree ≤ (X ^ (p - 1) * f ^ p).natDegree - p
r : ℂ
h : (aeval r) (sumIDeriv (X ^ (p - 1) * f ^ p)) = ↑p ! * (aeval r) gp
hr : r ∈ f.aroots ℂ
this :
  (↑(eval 0 f ^ p + ↑p * eval 0 gp') * cexp r - ↑p * (aeval r) gp) * ↑(p - 1)! =
    ↑(eval 0 f) ^ p * cexp r * ↑(p - 1)! + ↑(p * (p - 1)!) * (↑(eval 0 gp') * cexp r - (aeval r) gp)
⊢ ↑(eval 0 f) ^ p * cexp r * ↑(p - 1)! +
      (↑p ! * (↑(eval 0 gp') * cexp r) - (aeval r) (sumIDeriv (X ^ (p - 1) * f ^ p))) =
    ↑((p - 1)! • eval 0 (f ^ p) + p ! • eval 0 gp') * cexp r - (aeval r) (sumIDeriv (X ^ (p - 1) * f ^ p))

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.RamificationInertia.Basic

Goal state before:
pp: R : Type u
inst✝¹⁷ : CommRing R
S : Type v
inst✝¹⁶ : CommRing S
inst✝¹⁵ : Algebra R S
inst✝¹⁴ : IsDedekindDomain S
K : Type u_1
L : Type u_2
inst✝¹³ : Field K
inst✝¹² : Field L
inst✝¹¹ : IsDedekindDomain R
inst✝¹⁰ : Algebra R K
inst✝⁹ : IsFractionRing R K
inst✝⁸ : Algebra S L
inst✝⁷ : IsFractionRing S L
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra R L
inst✝⁴ : IsScalarTower R S L
inst✝³ : IsScalarTower R K L
inst✝² : Module.Finite R S
inst✝¹ : IsLocalRing S
p : Ideal R
inst✝ : p.IsMaximal
hp0 : p ≠ ⊥
⊢ p.ramificationIdx' (IsLocalRing.maximalIdeal S) * p.inertiaDeg' (IsLocalRing.maximalIdeal S) = finrank K L

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith

Goal state before:
pp: p q x : ℝ
h : LiouvilleWith p x
hlt : q < p
⊢ ∃ᶠ (n : ℕ) in atTop, ∃ m, x ≠ ↑m / ↑n ∧ |x - ↑m / ↑n| < ↑n ^ (-q)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith

Goal state before:
pp: p x : ℝ
r : ℚ
h : LiouvilleWith p x
hr : r ≠ 0
C : ℝ
_hC₀ : 0 < C
hC : ∃ᶠ (n : ℕ) in atTop, 1 ≤ n ∧ ∃ m, x ≠ ↑m / ↑n ∧ |x - ↑m / ↑n| < C / ↑n ^ p
n : ℕ
_hn : 1 ≤ n
m : ℤ
hne : x ≠ ↑m / ↑n
hlt : |x - ↑m / ↑n| < C / ↑n ^ p
⊢ ↑(r.num * m) / ↑(r.den • id n) = ↑m / ↑n * ↑r

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith

Goal state before:
pp: p x : ℝ
r : ℚ
h : LiouvilleWith p x
hr : r ≠ 0
C : ℝ
_hC₀ : 0 < C
hC : ∃ᶠ (n : ℕ) in atTop, 1 ≤ n ∧ ∃ m, x ≠ ↑m / ↑n ∧ |x - ↑m / ↑n| < C / ↑n ^ p
n : ℕ
_hn : 1 ≤ n
m : ℤ
hne : x ≠ ↑m / ↑n
hlt : |x - ↑m / ↑n| < C / ↑n ^ p
A : ↑(r.num * m) / ↑(r.den • id n) = ↑m / ↑n * ↑r
⊢ |x - ↑m / ↑n| * |↑r| < C / ↑n ^ p * |↑r|

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Lindemann.AnalyticalPart

Goal state before:
pp: f : ℤ[X]
hf : eval 0 f ≠ 0
c' : ℂ → ℝ
c'0 : ∀ (s : ℂ), c' s ≥ 0
Pp'_le : ∀ (s : ℂ) (p : ℕ), p ≠ 0 → ‖P (map (algebraMap ℤ ℂ) (X ^ (p - 1) * f ^ p)) s‖ ≤ c' s ^ p
p : ℕ
p_gt : p > (eval 0 f).natAbs
prime_p : Nat.Prime p
gp' : ℤ[X]
h' : eval 0 (sumIDeriv (X ^ (p - 1) * f ^ p)) = (p - 1)! • eval 0 (f ^ p) + p ! • eval 0 gp'
gp : ℤ[X]
gp'_le : gp.natDegree ≤ (X ^ (p - 1) * f ^ p).natDegree - p
r : ℂ
h : (aeval r) (sumIDeriv (X ^ (p - 1) * f ^ p)) = ↑p ! * (aeval r) gp
hr : r ∈ f.aroots ℂ
this✝ :
  (↑(eval 0 f ^ p + ↑p * eval 0 gp') * cexp r - ↑p * (aeval r) gp) * ↑(p - 1)! =
    ↑(eval 0 f) ^ p * cexp r * ↑(p - 1)! + ↑(p * (p - 1)!) * (↑(eval 0 gp') * cexp r - (aeval r) gp)
this :
  ↑(eval 0 f) ^ p * cexp r * ↑(p - 1)! +
      (↑p ! * (↑(eval 0 gp') * cexp r) - (aeval r) (sumIDeriv (X ^ (p - 1) * f ^ p))) =
    ↑((p - 1)! • eval 0 (f ^ p) + p ! • eval 0 gp') * cexp r - (aeval r) (sumIDeriv (X ^ (p - 1) * f ^ p))
⊢ c' r ∈ (Multiset.map c' (f.aroots ℂ)).toFinset

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Lindemann.AnalyticalPart

Goal state before:
pp: f : ℤ[X]
hf : eval 0 f ≠ 0
⊢ ∃ c,
    ∀ p > (eval 0 f).natAbs,
      Nat.Prime p →
        ∃ n,
          ¬↑p ∣ n ∧
            ∃ gp,
              gp.natDegree ≤ p * f.natDegree - 1 ∧
                ∀ {r : ℂ}, r ∈ f.aroots ℂ → ‖n • cexp r - p • (aeval r) gp‖ ≤ c ^ p / ↑(p - 1)!

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith

Goal state before:
pp: p x : ℝ
r : ℚ
h : LiouvilleWith p x
hr : r ≠ 0
⊢ LiouvilleWith p (x * ↑r)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.Measure

Goal state before:
pp: x p : ℝ
hp : p > 2
hxp : LiouvilleWith p x
n : ℕ
hn : 2 + 1 / (↑n + 1) < p
this :
  ∀ (y : ℝ),
    LiouvilleWith p y →
      y ∈ Ico 0 1 → ∃ᶠ (b : ℕ) in atTop, ∃ a ∈ Finset.Icc 0 ↑b, |y - ↑a / ↑b| < 1 / ↑b ^ (2 + 1 / ↑(n + 1))
⊢ x + ↑(-⌊x⌋) ∈ Ico 0 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith

Goal state before:
pp: p x : ℝ
r : ℚ
hr : r ≠ 0
h : LiouvilleWith p (x * ↑r)
⊢ LiouvilleWith p x

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith

Goal state before:
pp: p x : ℝ
r : ℚ
hr : r ≠ 0
⊢ LiouvilleWith p (↑r * x) ↔ LiouvilleWith p x

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith

Goal state before:
pp: p x : ℝ
m : ℤ
hm : m ≠ 0
⊢ LiouvilleWith p (x * ↑m) ↔ LiouvilleWith p x

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith

Goal state before:
pp: p x : ℝ
m : ℤ
hm : m ≠ 0
⊢ LiouvilleWith p (↑m * x) ↔ LiouvilleWith p x

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith

Goal state before:
pp: p x : ℝ
n : ℕ
hn : n ≠ 0
⊢ LiouvilleWith p (x * ↑n) ↔ LiouvilleWith p x

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.Measure

Goal state before:
pp: x p : ℝ
hp : p > 2
hxp : LiouvilleWith p x
n : ℕ
hn : 2 + 1 / (↑n + 1) < p
this :
  ∀ (y : ℝ),
    LiouvilleWith p y →
      y ∈ Ico 0 1 → ∃ᶠ (b : ℕ) in atTop, ∃ a ∈ Finset.Icc 0 ↑b, |y - ↑a / ↑b| < 1 / ↑b ^ (2 + 1 / ↑(n + 1))
⊢ x ∈
    ⋃ m,
      (fun x ↦ x + ↑m) ⁻¹'
        ⋃ n, ⋃ (_ : n > 0), {x | ∃ᶠ (b : ℕ) in atTop, ∃ a ∈ Finset.Icc 0 ↑b, |x - ↑a / ↑b| < 1 / ↑b ^ (2 + 1 / ↑n)}

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith

Goal state before:
pp: p x : ℝ
n : ℕ
hn : n ≠ 0
⊢ LiouvilleWith p (↑n * x) ↔ LiouvilleWith p x

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith

Goal state before:
pp: p x : ℝ
n : ℕ
h : LiouvilleWith p x
hn : n ≠ 0
⊢ LiouvilleWith p (↑n * x)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Wilson

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
⊢ ↑(p - 1)! = ∏ x ∈ Ico 1 (p - 1).succ, ↑x

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Wilson

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
⊢ ∏ x, ↑x = -1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Wilson

Goal state before:
pp: case refine_2
p : ℕ
inst✝ : Fact (Nat.Prime p)
hp : 0 < p
a₁✝ : (ZMod p)ˣ
ha₁✝ : a₁✝ ∈ univ
a₂✝ : (ZMod p)ˣ
ha₂✝ : a₂✝ ∈ univ
h : (↑a₁✝).val = (↑a₂✝).val
⊢ a₁✝ = a₂✝

Tactic:
rw [Units.ext_iff]

Goal state after:
case refine_2
p : ℕ
inst✝ : Fact (Nat.Prime p)
hp : 0 < p
a₁✝ : (ZMod p)ˣ
ha₁✝ : a₁✝ ∈ univ
a₂✝ : (ZMod p)ˣ
ha₂✝ : a₂✝ ∈ univ
h : (↑a₁✝).val = (↑a₂✝).val
⊢ ↑a₁✝ = ↑a₂✝
Module: Mathlib.NumberTheory.Transcendental.Liouville.Measure

Goal state before:
pp: p : ℝ
hp : p > 2
n : ℕ
hn : 2 + 1 / (↑n + 1) < p
x : ℝ
hxp : LiouvilleWith p x
hx01 : x ∈ Ico 0 1
b : ℕ
a : ℤ
hlt : |x - ↑a / ↑b| < 1 / ↑b ^ (2 + 1 / ↑n.succ)
hb : 1 ≤ ↑b
hb0 : 0 < ↑b
⊢ |x - ↑a / ↑b| < 1 / ↑b

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Wilson

Goal state before:
pp: case refine_4
p : ℕ
inst✝ : Fact (Nat.Prime p)
hp : 0 < p
⊢ ∀ a ∈ univ, ↑a = ↑(↑a).val

Tactic:
· rintro a -; simp only [cast_id, natCast_val]

Goal state after:
Module: Mathlib.NumberTheory.Wilson

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
⊢ ↑(p - 1)! = -1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith

Goal state before:
pp: p x : ℝ
h : LiouvilleWith p x
r : ℚ
C : ℝ
_hC₀ : 0 < C
hC : ∃ᶠ (n : ℕ) in atTop, 1 ≤ n ∧ ∃ m, x ≠ ↑m / ↑n ∧ |x - ↑m / ↑n| < C / ↑n ^ p
n : ℕ
hn : 1 ≤ n
m : ℤ
hne : x ≠ ↑m / ↑n
hlt : |x - ↑m / ↑n| < C / ↑n ^ p
⊢ ↑r.den ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Wilson

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
⊢ ∏ x ∈ Ico 1 p, ↑x = -1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Wilson

Goal state before:
pp: n : ℕ
h : ↑(n - 1)! = -1
h1 : n ≠ 1
⊢ Prime n

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith

Goal state before:
pp: p x : ℝ
h : LiouvilleWith p x
r : ℚ
C : ℝ
_hC₀ : 0 < C
hC : ∃ᶠ (n : ℕ) in atTop, 1 ≤ n ∧ ∃ m, x ≠ ↑m / ↑n ∧ |x - ↑m / ↑n| < C / ↑n ^ p
n : ℕ
hn : 1 ≤ n
m : ℤ
hne : x ≠ ↑m / ↑n
hlt : |x - ↑m / ↑n| < C / ↑n ^ p
⊢ ↑n ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Wilson

Goal state before:
pp: n : ℕ
h : n ≠ 1
⊢ Prime n ↔ ↑(n - 1)! = -1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith

Goal state before:
pp: p x : ℝ
h : LiouvilleWith p x
r : ℚ
C : ℝ
_hC₀ : 0 < C
hC : ∃ᶠ (n : ℕ) in atTop, 1 ≤ n ∧ ∃ m, x ≠ ↑m / ↑n ∧ |x - ↑m / ↑n| < C / ↑n ^ p
n : ℕ
hn : 1 ≤ n
m : ℤ
hne : x ≠ ↑m / ↑n
hlt : |x - ↑m / ↑n| < C / ↑n ^ p
⊢ ↑(↑r.den * m + r.num * ↑n) / ↑(r.den • id n) = ↑m / ↑n + ↑r

Tactic:
by

Goal state after:
Module: Mathlib.Order.Bounded

Goal state before:
pp: α : Type u_1
s : Set α
inst✝ : LinearOrder α
⊢ Unbounded (fun x1 x2 ↦ x1 ≤ x2) s ↔ ∀ (a : α), ∃ b, b ∈ s ∧ a < b

Tactic:
by

Goal state after:
Module: Mathlib.Order.Bounded

Goal state before:
pp: α : Type u_1
s : Set α
inst✝ : LinearOrder α
⊢ Unbounded (fun x1 x2 ↦ x1 < x2) s ↔ ∀ (a : α), ∃ b, b ∈ s ∧ a ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Order.Bounded

Goal state before:
pp: α : Type u_1
s : Set α
inst✝¹ : Preorder α
inst✝ : NoMaxOrder α
⊢ Bounded (fun x1 x2 ↦ x1 ≤ x2) s ↔ Bounded (fun x1 x2 ↦ x1 < x2) s

Tactic:
by

Goal state after:
Module: Mathlib.Order.Bounded

Goal state before:
pp: α : Type u_1
s : Set α
inst✝¹ : Preorder α
inst✝ : NoMaxOrder α
⊢ Unbounded (fun x1 x2 ↦ x1 < x2) s ↔ Unbounded (fun x1 x2 ↦ x1 ≤ x2) s

Tactic:
by

Goal state after:
Module: Mathlib.Order.Bounded

Goal state before:
pp: α : Type u_1
inst✝¹ : Preorder α
inst✝ : NoMaxOrder α
a : α
⊢ Bounded (fun x1 x2 ↦ x1 < x2) (Iic a)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Bounded

Goal state before:
pp: α : Type u_1
inst✝¹ : Preorder α
inst✝ : NoMinOrder α
a : α
⊢ Bounded (fun x1 x2 ↦ x1 > x2) (Ici a)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Bounded

Goal state before:
pp: α : Type u_1
r : α → α → Prop
s : Set α
H : ∀ (a b : α), ∃ m, ∀ (c : α), r c a ∨ r c b → r c m
a : α
⊢ Bounded r (s ∩ {b | ¬r b a}) ↔ Bounded r s

Tactic:
by

Goal state after:
Module: Mathlib.Order.Bounded

Goal state before:
pp: α : Type u_1
r : α → α → Prop
s : Set α
H : ∀ (a b : α), ∃ m, ∀ (c : α), r c a ∨ r c b → r c m
a : α
⊢ Unbounded r (s ∩ {b | ¬r b a}) ↔ Unbounded r s

Tactic:
by

Goal state after:
Module: Mathlib.Order.Bounded

Goal state before:
pp: α : Type u_1
s : Set α
inst✝ : SemilatticeSup α
a : α
⊢ Unbounded (fun x1 x2 ↦ x1 ≤ x2) (s ∩ {b | ¬b ≤ a}) ↔ Unbounded (fun x1 x2 ↦ x1 ≤ x2) s

Tactic:
by

Goal state after:
Module: Mathlib.Order.Bounded

Goal state before:
pp: α : Type u_1
s : Set α
inst✝ : LinearOrder α
a : α
⊢ Bounded (fun x1 x2 ↦ x1 ≤ x2) (s ∩ {b | a < b}) ↔ Bounded (fun x1 x2 ↦ x1 ≤ x2) s

Tactic:
by

Goal state after:
Module: Mathlib.Order.Bounded

Goal state before:
pp: α : Type u_1
s : Set α
inst✝ : LinearOrder α
a : α
⊢ Unbounded (fun x1 x2 ↦ x1 ≤ x2) (s ∩ {b | a < b}) ↔ Unbounded (fun x1 x2 ↦ x1 ≤ x2) s

Tactic:
by

Goal state after:
Module: Mathlib.Order.Bounded

Goal state before:
pp: α : Type u_1
s : Set α
inst✝ : LinearOrder α
a : α
⊢ Bounded (fun x1 x2 ↦ x1 ≤ x2) (s ∩ {b | a ≤ b}) ↔ Bounded (fun x1 x2 ↦ x1 ≤ x2) s

Tactic:
by

Goal state after:
Module: Mathlib.Order.Bounded

Goal state before:
pp: α : Type u_1
s : Set α
inst✝ : LinearOrder α
a : α
⊢ Unbounded (fun x1 x2 ↦ x1 ≤ x2) (s ∩ {b | a ≤ b}) ↔ Unbounded (fun x1 x2 ↦ x1 ≤ x2) s

Tactic:
by

Goal state after:
Module: Mathlib.Order.Bounded

Goal state before:
pp: α : Type u_1
s : Set α
inst✝ : SemilatticeSup α
a : α
⊢ Unbounded (fun x1 x2 ↦ x1 < x2) (s ∩ {b | ¬b < a}) ↔ Unbounded (fun x1 x2 ↦ x1 < x2) s

Tactic:
by

Goal state after:
Module: Mathlib.Order.Bounded

Goal state before:
pp: α : Type u_1
s : Set α
inst✝ : LinearOrder α
a : α
⊢ Bounded (fun x1 x2 ↦ x1 < x2) (s ∩ {b | a ≤ b}) ↔ Bounded (fun x1 x2 ↦ x1 < x2) s

Tactic:
by

Goal state after:
Module: Mathlib.Order.Bounded

Goal state before:
pp: α : Type u_1
s : Set α
inst✝ : LinearOrder α
a : α
⊢ Unbounded (fun x1 x2 ↦ x1 < x2) (s ∩ {b | a ≤ b}) ↔ Unbounded (fun x1 x2 ↦ x1 < x2) s

Tactic:
by

Goal state after:
Module: Mathlib.Order.Bounded

Goal state before:
pp: α : Type u_1
s : Set α
inst✝¹ : LinearOrder α
inst✝ : NoMaxOrder α
a : α
⊢ Bounded (fun x1 x2 ↦ x1 < x2) (s ∩ {b | a < b}) ↔ Bounded (fun x1 x2 ↦ x1 < x2) s

Tactic:
by

Goal state after:
Module: Mathlib.Order.Bounded

Goal state before:
pp: α : Type u_1
s : Set α
inst✝¹ : LinearOrder α
inst✝ : NoMaxOrder α
a : α
⊢ Unbounded (fun x1 x2 ↦ x1 < x2) (s ∩ {b | a < b}) ↔ Unbounded (fun x1 x2 ↦ x1 < x2) s

Tactic:
by

Goal state after:
Module: Mathlib.Order.Bounds.Lattice

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
⊢ GaloisConnection (⇑OrderDual.toDual ∘ upperBounds) (lowerBounds ∘ ⇑OrderDual.ofDual)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Bounds.Lattice

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
ι : Sort u_2
s : ι → Set α
u : ι → α
hs : ∀ (i : ι), IsLUB (s i) (u i)
c : α
⊢ IsLUB (range u) c ↔ IsLUB (⋃ i, s i) c

Tactic:
by

Goal state after:
Module: Mathlib.Order.Bounds.Lattice

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
ι : Sort u_2
s : ι → Set α
u : ι → α
hs : ∀ (i : ι), IsGLB (s i) (u i)
c : α
⊢ IsGLB (range u) c ↔ IsGLB (⋃ i, s i) c

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.Measure

Goal state before:
pp: ⊢ {x | ∃ p > 2, LiouvilleWith p x} ⊆
    ⋃ m,
      (fun x ↦ x + ↑m) ⁻¹'
        ⋃ n, ⋃ (_ : n > 0), {x | ∃ᶠ (b : ℕ) in atTop, ∃ a ∈ Finset.Icc 0 ↑b, |x - ↑a / ↑b| < 1 / ↑b ^ (2 + 1 / ↑n)}

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith

Goal state before:
pp: p x : ℝ
h : LiouvilleWith p x
r : ℚ
C : ℝ
_hC₀ : 0 < C
hC : ∃ᶠ (n : ℕ) in atTop, 1 ≤ n ∧ ∃ m, x ≠ ↑m / ↑n ∧ |x - ↑m / ↑n| < C / ↑n ^ p
n : ℕ
hn : 1 ≤ n
m : ℤ
hne : x ≠ ↑m / ↑n
hlt : |x - ↑m / ↑n| < C / ↑n ^ p
this : ↑(↑r.den * m + r.num * ↑n) / ↑(r.den • id n) = ↑m / ↑n + ↑r
⊢ x + ↑r ≠ ↑m / ↑n + ↑r

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith

Goal state before:
pp: p x : ℝ
h : LiouvilleWith p x
r : ℚ
C : ℝ
_hC₀ : 0 < C
hC : ∃ᶠ (n : ℕ) in atTop, 1 ≤ n ∧ ∃ m, x ≠ ↑m / ↑n ∧ |x - ↑m / ↑n| < C / ↑n ^ p
n : ℕ
hn : 1 ≤ n
m : ℤ
hne : x ≠ ↑m / ↑n
hlt : |x - ↑m / ↑n| < C / ↑n ^ p
this : ↑(↑r.den * m + r.num * ↑n) / ↑(r.den • id n) = ↑m / ↑n + ↑r
⊢ ↑r.den ^ p ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith

Goal state before:
pp: p x : ℝ
h : LiouvilleWith p x
r : ℚ
⊢ LiouvilleWith p (x + ↑r)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith

Goal state before:
pp: p x : ℝ
r : ℚ
h : LiouvilleWith p (x + ↑r)
⊢ LiouvilleWith p x

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith

Goal state before:
pp: p x : ℝ
r : ℚ
⊢ LiouvilleWith p (↑r + x) ↔ LiouvilleWith p x

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith

Goal state before:
pp: p x : ℝ
m : ℤ
⊢ LiouvilleWith p (x + ↑m) ↔ LiouvilleWith p x

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith

Goal state before:
pp: p x : ℝ
m : ℤ
⊢ LiouvilleWith p (↑m + x) ↔ LiouvilleWith p x

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith

Goal state before:
pp: p x : ℝ
n : ℕ
⊢ LiouvilleWith p (x + ↑n) ↔ LiouvilleWith p x

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith

Goal state before:
pp: p x : ℝ
n : ℕ
⊢ LiouvilleWith p (↑n + x) ↔ LiouvilleWith p x

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.Measure

Goal state before:
pp: n : ℕ
hn : 1 ≤ n
r : ℝ
hr : 2 + 1 / ↑n = r
⊢ 2 < r

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith

Goal state before:
pp: p x C : ℝ
hC : ∃ᶠ (n : ℕ) in atTop, ∃ m, x ≠ ↑m / ↑n ∧ |x - ↑m / ↑n| < C / ↑n ^ p
n : ℕ
m : ℤ
hne : x ≠ ↑m / ↑n
hlt : |x - ↑m / ↑n| < C / ↑n ^ p
⊢ -x ≠ ↑(-m) / ↑n

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.Measure

Goal state before:
pp: r : ℝ
hr : 2 < r
B : ℤ → ℕ → Set ℝ := fun a b ↦ ball (↑a / ↑b) (1 / ↑b ^ r)
a : ℤ
b : ℕ
⊢ volume (B a b) = ↑(2 / ↑b ^ r)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.Measure

Goal state before:
pp: r : ℝ
hr : 2 < r
B : ℤ → ℕ → Set ℝ := fun a b ↦ ball (↑a / ↑b) (1 / ↑b ^ r)
hB : ∀ (a : ℤ) (b : ℕ), volume (B a b) = ↑(2 / ↑b ^ r)
b : ℕ
⊢ ∑ a ∈ Finset.Icc 0 ↑b, volume (B a b) = ↑((↑b + 1) * (2 / ↑b ^ r))

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Transcendental.Liouville.Measure

Goal state before:
pp: r : ℝ
hr : 2 < r
B : ℤ → ℕ → Set ℝ := fun a b ↦ ball (↑a / ↑b) (1 / ↑b ^ r)
hB : ∀ (a : ℤ) (b : ℕ), volume (B a b) = ↑(2 / ↑b ^ r)
b : ℕ
⊢ 1 - r ≠ 0

Tactic:
by

Goal state after: