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

Goal state before:
pp: K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
α : Type u_2
β : Type u_3
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
h0p : 0 < p
hpq : p < q
inst✝² : Fintype β
cardβ : Fintype.card β = q
A : ℝ
habs : ∀ (k : α) (l : β), house ((algebraMap (𝓞 K) K) (a k l)) ≤ A
inst✝¹ : DecidableEq (K →+* ℂ)
inst✝ : Fintype α
cardα : Fintype.card α = p
h : ℕ := finrank ℚ K
⊢ ∃ ξ, ξ ≠ 0 ∧ a *ᵥ ξ = 0 ∧ ∀ (l : β), house ↑(ξ l) ≤ c₁ K * (c₁ K * ↑q * A) ^ (↑p / (↑q - ↑p))

Tactic:
have hphqh : p * h < q * h := by gcongr; exact finrank_pos

Goal state after:
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
α : Type u_2
β : Type u_3
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
h0p : 0 < p
hpq : p < q
inst✝² : Fintype β
cardβ : Fintype.card β = q
A : ℝ
habs : ∀ (k : α) (l : β), house ((algebraMap (𝓞 K) K) (a k l)) ≤ A
inst✝¹ : DecidableEq (K →+* ℂ)
inst✝ : Fintype α
cardα : Fintype.card α = p
h : ℕ := finrank ℚ K
hphqh : p * h < q * h
⊢ ∃ ξ, ξ ≠ 0 ∧ a *ᵥ ξ = 0 ∧ ∀ (l : β), house ↑(ξ l) ≤ c₁ K * (c₁ K * ↑q * A) ^ (↑p / (↑q - ↑p))
Module: Mathlib.NumberTheory.Ostrowski

Goal state before:
pp: f : AbsoluteValue ℚ ℝ
notbdd : ¬∀ (n : ℕ), f ↑n ≤ 1
m : ℕ
hm : ¬f ↑m ≤ 1
hfm1 : 1 < f ↑m
oneltm : 1 < m
s : ℝ := logb (↑m) (f ↑m)
hs : s = logb (↑m) (f ↑m)
hs0 : 0 < s
n : ℕ
h : 1 < n
hfm : f ↑m = ↑m ^ s
⊢ ↑n ≠ 0

Tactic:
exact_mod_cast ne_zero_of_lt h

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

Goal state before:
pp: M : Type u_1
G : Type u_2
inst✝⁴ : TopologicalSpace M
inst✝³ : CompactSpace M
inst✝² : AddCommMonoid M
inst✝¹ : SeminormedAddCommGroup G
inst✝ : IsUltrametricDist G
h : M
f : C(M, G)
m : M
n : ℕ
⊢ ‖Δ_[h]^[n] (⇑f) m‖ ≤ ‖f‖

Tactic:
rw [fwdDiff_iter_eq_sum_shift]
  refine norm_sum_le_of_forall_le_of_nonneg (norm_nonneg f) fun i _ ↦ ?_
  exact (norm_zsmul_le _ _).trans (f.norm_coe_le_norm _)

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

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : Module ℤ_[p] E
inst✝¹ : IsBoundedSMul ℤ_[p] E
inst✝ : IsUltrametricDist E
f : C(ℤ_[p], E)
s t : ℕ
hst : ∀ (x y : ℤ_[p]), ‖x - y‖ ≤ ↑p ^ (-↑t) → ‖f x - f y‖ ≤ ‖f‖ / ↑p ^ s
n i : ℕ
x✝ : i ∈ range (n + 1)
⊢ ‖↑i + ↑(p ^ t) - ↑i‖ ≤ ↑p ^ (-↑t)

Tactic:
Nat.cast_pow,

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : Module ℤ_[p] E
inst✝¹ : IsBoundedSMul ℤ_[p] E
inst✝ : IsUltrametricDist E
f : C(ℤ_[p], E)
s t : ℕ
hst : ∀ (x y : ℤ_[p]), ‖x - y‖ ≤ ↑p ^ (-↑t) → ‖f x - f y‖ ≤ ‖f‖ / ↑p ^ s
n i : ℕ
x✝ : i ∈ range (n + 1)
⊢ ‖↑i + ↑p ^ t - ↑i‖ ≤ ↑p ^ (-↑t)
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)
n j : ℕ
hj : j ∈ range (n + 1)
⊢ ((-1) ^ (n - j) * ↑(n.choose j)) • (mahlerSeries a) (j • 1) =
    ((-1) ^ (n - j) * ↑(n.choose j)) • ∑ x ∈ range (n + 1), (j • 1).choose x • a x

Tactic:
nsmul_one,

Goal state after:
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)
n j : ℕ
hj : j ∈ range (n + 1)
⊢ ((-1) ^ (n - j) * ↑(n.choose j)) • (mahlerSeries a) ↑j =
    ((-1) ^ (n - j) * ↑(n.choose j)) • ∑ x ∈ range (n + 1), (j • 1).choose x • a x
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)
n j : ℕ
hj : j ∈ range (n + 1)
⊢ ((-1) ^ (n - j) * ↑(n.choose j)) • (mahlerSeries a) ↑j =
    ((-1) ^ (n - j) * ↑(n.choose j)) • ∑ x ∈ range (n + 1), (j • 1).choose x • a x

Tactic:
nsmul_one,

Goal state after:
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)
n j : ℕ
hj : j ∈ range (n + 1)
⊢ ((-1) ^ (n - j) * ↑(n.choose j)) • (mahlerSeries a) ↑j =
    ((-1) ^ (n - j) * ↑(n.choose j)) • ∑ x ∈ range (n + 1), (↑j).choose x • a x
Module: Mathlib.NumberTheory.Padics.HeightOneSpectrum

Goal state before:
pp: R : Type u_2
inst✝² : CommRing R
inst✝¹ : Algebra R ℚ
inst✝ : IsIntegralClosure R ℤ ℚ
v : HeightOneSpectrum R
⊢ Ideal.span {↑(natGenerator v)} = Ideal.map (IsIntegralClosure.intEquiv R) v.asIdeal

Tactic:
simp [natGenerator]

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

Goal state before:
pp: R : Type u_2
inst✝² : CommRing R
inst✝¹ : Algebra R ℚ
inst✝ : IsIntegralClosure R ℤ ℚ
v : HeightOneSpectrum R
n : ℕ
⊢ natGenerator v ∣ n ↔ ↑n ∈ Ideal.map (IsIntegralClosure.intEquiv R) v.asIdeal

Tactic:
rw [← span_natGenerator, Ideal.mem_span_singleton]

Goal state after:
R : Type u_2
inst✝² : CommRing R
inst✝¹ : Algebra R ℚ
inst✝ : IsIntegralClosure R ℤ ℚ
v : HeightOneSpectrum R
n : ℕ
⊢ natGenerator v ∣ n ↔ ↑(natGenerator v) ∣ ↑n
Module: Mathlib.NumberTheory.Pell

Goal state before:
pp: d : ℤ
a b : Solution₁ d
hax : 0 < a.x
hay : 0 < a.y
hbx : 0 < b.x
hby : 0 < b.y
⊢ 0 < (a * b).y

Tactic:
simp only [y_mul]

Goal state after:
d : ℤ
a b : Solution₁ d
hax : 0 < a.x
hay : 0 < a.y
hbx : 0 < b.x
hby : 0 < b.y
⊢ 0 < a.x * b.y + a.y * b.x
Module: Mathlib.NumberTheory.Pell

Goal state before:
pp: d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
ξ : ℝ := √↑d
hξ : Irrational ξ
M : ℤ
hM₁ : 2 * |ξ| + 1 < ↑M
hM : {q | |q.num ^ 2 - d * ↑q.den ^ 2| < M}.Infinite
m : ℤ
hm : {q | q.num ^ 2 - d * ↑q.den ^ 2 = m}.Infinite
hm₀ : m ≠ 0
this : NeZero m.natAbs
f : ℚ → ZMod m.natAbs × ZMod m.natAbs := fun q ↦ (↑q.num, ↑q.den)
q₁ : ℚ
h₁ : q₁.num ^ 2 - d * ↑q₁.den ^ 2 = m
q₂ : ℚ
h₂ : q₂.num ^ 2 - d * ↑q₂.den ^ 2 = m
hne : q₁ ≠ q₂
hqf : f q₁ = f q₂
hq1 : ↑q₁.num = ↑q₂.num
hq2 : ↑q₁.den = ↑q₂.den
⊢ m ∣ q₂.num ^ 2 - d * ↑(q₂.den ^ 2)

Tactic:
Nat.cast_pow,

Goal state after:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
ξ : ℝ := √↑d
hξ : Irrational ξ
M : ℤ
hM₁ : 2 * |ξ| + 1 < ↑M
hM : {q | |q.num ^ 2 - d * ↑q.den ^ 2| < M}.Infinite
m : ℤ
hm : {q | q.num ^ 2 - d * ↑q.den ^ 2 = m}.Infinite
hm₀ : m ≠ 0
this : NeZero m.natAbs
f : ℚ → ZMod m.natAbs × ZMod m.natAbs := fun q ↦ (↑q.num, ↑q.den)
q₁ : ℚ
h₁ : q₁.num ^ 2 - d * ↑q₁.den ^ 2 = m
q₂ : ℚ
h₂ : q₂.num ^ 2 - d * ↑q₂.den ^ 2 = m
hne : q₁ ≠ q₂
hqf : f q₁ = f q₂
hq1 : ↑q₁.num = ↑q₂.num
hq2 : ↑q₁.den = ↑q₂.den
⊢ m ∣ q₂.num ^ 2 - d * ↑q₂.den ^ 2
Module: Mathlib.NumberTheory.Pell

Goal state before:
pp: d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
⊢ ∃ a, 1 < a.x ∧ 0 < a.y

Tactic:
obtain ⟨x, y, h, hy⟩ := exists_of_not_isSquare h₀ hd
  refine ⟨mk |x| |y| (by rwa [sq_abs, sq_abs]), ?_, abs_pos.mpr hy⟩
  rw [x_mk, ← one_lt_sq_iff_one_lt_abs, eq_add_of_sub_eq h, lt_add_iff_pos_right]
  exact mul_pos h₀ (sq_pos_of_ne_zero hy)

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

Goal state before:
pp: R : Type u
inst✝¹⁶ : CommRing R
S : Type v
inst✝¹⁵ : CommRing S
inst✝¹⁴ : Algebra R S
p : Ideal R
K : Type u_1
inst✝¹³ : Field K
inst✝¹² : Algebra R K
L : Type u_2
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra S L
inst✝⁹ : IsFractionRing S L
inst✝⁸ : IsDomain R
inst✝⁷ : IsDomain S
inst✝⁶ : Algebra K L
inst✝⁵ : Module.Finite R S
inst✝⁴ : Algebra R L
inst✝³ : IsScalarTower R S L
inst✝² : IsScalarTower R K L
inst✝¹ : Algebra.IsAlgebraic R S
inst✝ : IsTorsionFree R K
hp : p ≠ ⊤
b : Set S
hb' : Submodule.span R b ⊔ Submodule.restrictScalars R (map f p) = ⊤
hRL : Function.Injective ⇑(algebraMap R L)
M : Submodule R S := Submodule.span R b
n : ℕ
a : Fin n → S ⧸ M
ha : Submodule.span R (Set.range a) = ⊤
smul_top_eq : p • ⊤ = ⊤
x : S ⧸ M
a'' : Fin n →₀ R
ha'' : ∀ (i : Fin n), a'' i ∈ p
hx : (a''.sum fun i c ↦ c • a i) = x
⊢ ∑ i, (fun i ↦ a'' i) i • a i = a''.sum fun i c ↦ c • a i

Tactic:
Finsupp.sum_fintype

Goal state after:
R : Type u
inst✝¹⁶ : CommRing R
S : Type v
inst✝¹⁵ : CommRing S
inst✝¹⁴ : Algebra R S
p : Ideal R
K : Type u_1
inst✝¹³ : Field K
inst✝¹² : Algebra R K
L : Type u_2
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra S L
inst✝⁹ : IsFractionRing S L
inst✝⁸ : IsDomain R
inst✝⁷ : IsDomain S
inst✝⁶ : Algebra K L
inst✝⁵ : Module.Finite R S
inst✝⁴ : Algebra R L
inst✝³ : IsScalarTower R S L
inst✝² : IsScalarTower R K L
inst✝¹ : Algebra.IsAlgebraic R S
inst✝ : IsTorsionFree R K
hp : p ≠ ⊤
b : Set S
hb' : Submodule.span R b ⊔ Submodule.restrictScalars R (map f p) = ⊤
hRL : Function.Injective ⇑(algebraMap R L)
M : Submodule R S := Submodule.span R b
n : ℕ
a : Fin n → S ⧸ M
ha : Submodule.span R (Set.range a) = ⊤
smul_top_eq : p • ⊤ = ⊤
x : S ⧸ M
a'' : Fin n →₀ R
ha'' : ∀ (i : Fin n), a'' i ∈ p
hx : (a''.sum fun i c ↦ c • a i) = x
⊢ ∑ i, (fun i ↦ a'' i) i • a i = ∑ i, a'' i • a i

case h
R : Type u
inst✝¹⁶ : CommRing R
S : Type v
inst✝¹⁵ : CommRing S
inst✝¹⁴ : Algebra R S
p : Ideal R
K : Type u_1
inst✝¹³ : Field K
inst✝¹² : Algebra R K
L : Type u_2
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra S L
inst✝⁹ : IsFractionRing S L
inst✝⁸ : IsDomain R
inst✝⁷ : IsDomain S
inst✝⁶ : Algebra K L
inst✝⁵ : Module.Finite R S
inst✝⁴ : Algebra R L
inst✝³ : IsScalarTower R S L
inst✝² : IsScalarTower R K L
inst✝¹ : Algebra.IsAlgebraic R S
inst✝ : IsTorsionFree R K
hp : p ≠ ⊤
b : Set S
hb' : Submodule.span R b ⊔ Submodule.restrictScalars R (map f p) = ⊤
hRL : Function.Injective ⇑(algebraMap R L)
M : Submodule R S := Submodule.span R b
n : ℕ
a : Fin n → S ⧸ M
ha : Submodule.span R (Set.range a) = ⊤
smul_top_eq : p • ⊤ = ⊤
x : S ⧸ M
a'' : Fin n →₀ R
ha'' : ∀ (i : Fin n), a'' i ∈ p
hx : (a''.sum fun i c ↦ c • a i) = x
⊢ ∀ (i : Fin n), 0 • a i = 0
Module: Mathlib.NumberTheory.RamificationInertia.Valuation

Goal state before:
pp: A : Type u_1
B : Type u_4
inst✝⁶ : CommRing A
inst✝⁵ : IsDedekindDomain A
inst✝⁴ : CommRing B
inst✝³ : IsDedekindDomain B
inst✝² : Algebra A B
inst✝¹ : Module.IsTorsionFree A B
v : HeightOneSpectrum A
w : HeightOneSpectrum B
inst✝ : w.asIdeal.LiesOver v.asIdeal
x : A
hx : x ≠ 0
⊢ Ideal.span {x} ≠ ⊥

Tactic:
simp [hx]

Goal state after:
Module: Mathlib.NumberTheory.RatFunc.Ostrowski

Goal state before:
pp: case inr
K : Type u_1
Γ : Type u_2
inst✝³ : Field K
inst✝² : LinearOrderedCommGroupWithZero Γ
v : Valuation K⟮X⟯ Γ
inst✝¹ : DecidableEq K⟮X⟯
inst✝ : IsTrivialOn K v
hlt : 1 < v X
f : K⟮X⟯
hf : f ≠ 0
hlt' : 1 < (inftyValuation K) X
⊢ v X ^ f.intDegree < 1 ↔ (inftyValuation K) f < 1

Tactic:
valuation_eq_valuation_X_zpow_intDegree_of_one_lt_valuation_X hlt' hf

Goal state after:
case inr
K : Type u_1
Γ : Type u_2
inst✝³ : Field K
inst✝² : LinearOrderedCommGroupWithZero Γ
v : Valuation K⟮X⟯ Γ
inst✝¹ : DecidableEq K⟮X⟯
inst✝ : IsTrivialOn K v
hlt : 1 < v X
f : K⟮X⟯
hf : f ≠ 0
hlt' : 1 < (inftyValuation K) X
⊢ v X ^ f.intDegree < 1 ↔ (inftyValuation K) X ^ f.intDegree < 1
Module: Mathlib.NumberTheory.Rayleigh

Goal state before:
pp: r s : ℝ
hrs : r.HolderConjugate s
j : ℤ
⊢ j ∈ {x | ∃ k, beattySeq r k = x} → j ∉ {x | ∃ k, beattySeq' s k = x}

Tactic:
⟨k, h₁⟩

Goal state after:
r s : ℝ
hrs : r.HolderConjugate s
j k : ℤ
h₁ : beattySeq r k = j
⊢ j ∉ {x | ∃ k, beattySeq' s k = x}
Module: Mathlib.NumberTheory.Rayleigh

Goal state before:
pp: case inr.right
r : ℝ
j : ℤ
h : r > 0
h✝ : ↑j < ↑⌊(↑j + 1) / r⌋ * r
⊢ ↑⌊(↑j + 1) / r⌋ * r ≤ ↑j + 1

Tactic:
exact sub_nonneg.1 (Int.sub_floor_div_mul_nonneg (j + 1 : ℝ) h)

Goal state after:
Module: Mathlib.NumberTheory.Rayleigh

Goal state before:
pp: case a.inr
r s : ℝ
hrs : r.HolderConjugate s
j k : ℤ
hk : k > 0
hjk : beattySeq' s k = j
⊢ 1 < ⌈↑k * s⌉

Tactic:
Int.lt_ceil,

Goal state after:
case a.inr
r s : ℝ
hrs : r.HolderConjugate s
j k : ℤ
hk : k > 0
hjk : beattySeq' s k = j
⊢ ↑1 < ↑k * s
Module: Mathlib.NumberTheory.Rayleigh

Goal state before:
pp: r s : ℝ
hrs : r.HolderConjugate s
j : ℤ
hb₁ : ∀ s ≥ 0, j ∈ {x | ∃ k > 0, beattySeq s k = x} ↔ j ∈ {x | ∃ k, beattySeq s k = x}
s✝ : ℝ
hs : s✝ ≥ 0
x✝ : j ∈ {x | ∃ k, beattySeq' s✝ k = x}
k : ℤ
hj : 1 < ⌈↑k * s✝⌉
hk : beattySeq' s✝ k = j
⊢ k > 0

Tactic:
Int.lt_ceil,

Goal state after:
r s : ℝ
hrs : r.HolderConjugate s
j : ℤ
hb₁ : ∀ s ≥ 0, j ∈ {x | ∃ k > 0, beattySeq s k = x} ↔ j ∈ {x | ∃ k, beattySeq s k = x}
s✝ : ℝ
hs : s✝ ≥ 0
x✝ : j ∈ {x | ∃ k, beattySeq' s✝ k = x}
k : ℤ
hj : ↑1 < ↑k * s✝
hk : beattySeq' s✝ k = j
⊢ k > 0
Module: Mathlib.NumberTheory.SumFourSquares

Goal state before:
pp: m a b c d : ℤ
h : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 2 * m
this :
  ∀ (f : Fin 4 → ZMod 2),
    f 0 ^ 2 + f 1 ^ 2 + f 2 ^ 2 + f 3 ^ 2 = 0 →
      ∃ i, f i ^ 2 + f ((swap i 0) 1) ^ 2 = 0 ∧ f ((swap i 0) 2) ^ 2 + f ((swap i 0) 3) ^ 2 = 0
f : Fin 4 → ℤ := ![a, b, c, d]
⊢ ↑(f 0) ^ 2 + ↑(f 1) ^ 2 + ↑(f 2) ^ 2 + ↑(f 3) ^ 2 = 0

Tactic:
rw [← @zero_mul (ZMod 2) _ m, ← show ((2 : ℤ) : ZMod 2) = 0 from rfl, ← Int.cast_mul, ← h]

Goal state after:
m a b c d : ℤ
h : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 2 * m
this :
  ∀ (f : Fin 4 → ZMod 2),
    f 0 ^ 2 + f 1 ^ 2 + f 2 ^ 2 + f 3 ^ 2 = 0 →
      ∃ i, f i ^ 2 + f ((swap i 0) 1) ^ 2 = 0 ∧ f ((swap i 0) 2) ^ 2 + f ((swap i 0) 3) ^ 2 = 0
f : Fin 4 → ℤ := ![a, b, c, d]
⊢ ↑(f 0) ^ 2 + ↑(f 1) ^ 2 + ↑(f 2) ^ 2 + ↑(f 3) ^ 2 = ↑(a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2)
Module: Mathlib.NumberTheory.Zsqrtd.GaussianInt

Goal state before:
pp: x y : ℤ[i]
⊢ normSq (1 / 2 + 1 / 2 * I) < 1

Tactic:
simp [normSq]; norm_num

Goal state after:
Module: Mathlib.NumberTheory.Zsqrtd.QuadraticReciprocity

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
hpi : Prime ↑p
hp2 : p = 2
⊢ p % 4 = 3

Tactic:
have := hpi.irreducible.isUnit_or_isUnit (a := ⟨1, 1⟩) (b := ⟨1, -1⟩)
      simp [hp2, Zsqrtd.ext_iff, ← norm_eq_one_iff, norm_def] at this

Goal state after:
Module: Mathlib.NumberTheory.SumFourSquares

Goal state before:
pp: case pos
p : ℕ
hp : Prime p
this✝ : Fact (Prime p)
natAbs_iff :
  ∀ {a b c d : ℤ} {k : ℕ},
    a.natAbs ^ 2 + b.natAbs ^ 2 + c.natAbs ^ 2 + d.natAbs ^ 2 = k ↔ a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = ↑k
hm : ∃ m < p, 0 < m ∧ ∃ a b c d, a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = m * p
a b c d m : ℕ
hmin : ∀ m_1 < 2 * m, ¬(m_1 < p ∧ 0 < m_1 ∧ ∃ a b c d, a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = m_1 * p)
hmp : 2 * m < p
hm₀ : 0 < m
hm₁ : 1 < 2 * m
this : NeZero (2 * m)
hm₂ : m < 2 * m
habcd : ↑a ^ 2 + ↑b ^ 2 + ↑c ^ 2 + ↑d ^ 2 = 2 * (↑m * ↑p)
⊢ False

Tactic:
obtain ⟨_, _, _, _, h⟩ := sum_four_squares_of_two_mul_sum_four_squares habcd

Goal state after:
case pos
p : ℕ
hp : Prime p
this✝ : Fact (Prime p)
natAbs_iff :
  ∀ {a b c d : ℤ} {k : ℕ},
    a.natAbs ^ 2 + b.natAbs ^ 2 + c.natAbs ^ 2 + d.natAbs ^ 2 = k ↔ a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = ↑k
hm : ∃ m < p, 0 < m ∧ ∃ a b c d, a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = m * p
a b c d m : ℕ
hmin : ∀ m_1 < 2 * m, ¬(m_1 < p ∧ 0 < m_1 ∧ ∃ a b c d, a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = m_1 * p)
hmp : 2 * m < p
hm₀ : 0 < m
hm₁ : 1 < 2 * m
this : NeZero (2 * m)
hm₂ : m < 2 * m
habcd : ↑a ^ 2 + ↑b ^ 2 + ↑c ^ 2 + ↑d ^ 2 = 2 * (↑m * ↑p)
w✝³ w✝² w✝¹ w✝ : ℤ
h : w✝³ ^ 2 + w✝² ^ 2 + w✝¹ ^ 2 + w✝ ^ 2 = ↑m * ↑p
⊢ False
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:
mul_rat_iff (Int.cast_ne_zero.2 hm)

Goal state after:
p x : ℝ
m : ℤ
hm : m ≠ 0
⊢ LiouvilleWith p x ↔ LiouvilleWith p x
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
⊢ ∃ᶠ (b : ℕ) in atTop, ∃ a ∈ Finset.Icc 0 ↑b, |x - ↑a / ↑b| < 1 / ↑b ^ (2 + 1 / ↑(n + 1))

Tactic:
refine ((hxp.frequently_lt_rpow_neg hn).and_eventually (eventually_ge_atTop 1)).mono ?_

Goal state after:
p : ℝ
hp : p > 2
n : ℕ
hn : 2 + 1 / (↑n + 1) < p
x : ℝ
hxp : LiouvilleWith p x
hx01 : x ∈ Ico 0 1
⊢ ∀ (x_1 : ℕ),
    (∃ m, x ≠ ↑m / ↑x_1 ∧ |x - ↑m / ↑x_1| < ↑x_1 ^ (-(2 + 1 / (↑n + 1)))) ∧ 1 ≤ x_1 →
      ∃ a ∈ Finset.Icc 0 ↑x_1, |x - ↑a / ↑x_1| < 1 / ↑x_1 ^ (2 + 1 / ↑(n + 1))
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:
simp [neg_div, hne]

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

Goal state before:
pp: x : ℝ
hx : Liouville x
n N : ℕ
hN : ∀ (b : ℕ), N ≤ b → ∀ (a : ℤ), x ≠ ↑a / ↑b → 1 / ↑b ^ n ≤ |x - ↑a / ↑b|
this✝ : ∀ b > 1, ∀ᶠ (m : ℕ) in atTop, ∀ (a : ℤ), 1 / ↑b ^ m ≤ |x - ↑a / ↑b|
this : ∀ᶠ (m : ℕ) in atTop, ∀ b < N, 1 < b → ∀ (a : ℤ), 1 / ↑b ^ m ≤ |x - ↑a / ↑b|
m : ℕ
hm : ∀ b < N, 1 < b → ∀ (a : ℤ), 1 / ↑b ^ m ≤ |x - ↑a / ↑b|
hnm : n ≤ m
a : ℤ
b : ℕ
hb : 1 < ↑b
hne : x ≠ ↑a / ↑↑b
hlt : |x - ↑a / ↑↑b| < 1 / ↑↑b ^ m
⊢ False

Tactic:
norm_cast at hb

Goal state after:
x : ℝ
hx : Liouville x
n N : ℕ
hN : ∀ (b : ℕ), N ≤ b → ∀ (a : ℤ), x ≠ ↑a / ↑b → 1 / ↑b ^ n ≤ |x - ↑a / ↑b|
this✝ : ∀ b > 1, ∀ᶠ (m : ℕ) in atTop, ∀ (a : ℤ), 1 / ↑b ^ m ≤ |x - ↑a / ↑b|
this : ∀ᶠ (m : ℕ) in atTop, ∀ b < N, 1 < b → ∀ (a : ℤ), 1 / ↑b ^ m ≤ |x - ↑a / ↑b|
m : ℕ
hm : ∀ b < N, 1 < b → ∀ (a : ℤ), 1 / ↑b ^ m ≤ |x - ↑a / ↑b|
hnm : n ≤ m
a : ℤ
b : ℕ
hne : x ≠ ↑a / ↑↑b
hlt : |x - ↑a / ↑↑b| < 1 / ↑↑b ^ m
hb : 1 < b
⊢ False
Module: Mathlib.Order.Height

Goal state before:
pp: α : Type u_1
r : α → α → Prop
s : Set α
n : ℕ
h : ↑n ≤ s.chainHeight r
h' : n = 0
⊢ ∅ ⊆ s ∧ ∅.encard = ↑n ∧ IsChain r ∅

Tactic:
by simp [h']

Goal state after:
Module: Mathlib.Order.Filter.Cocardinal

Goal state before:
pp: case hmono
α : Type u
c : Cardinal.{u}
hreg✝ hreg : c.IsRegular
⊢ ∀ t ∈ {s | #↑s < c}, ∀ s ⊆ t, s ∈ {s | #↑s < c}

Tactic:
· exact fun _ hSc _ ht ↦ lt_of_le_of_lt (mk_le_mk_of_subset ht) hSc

Goal state after:
Module: Mathlib.Order.Nucleus

Goal state before:
pp: X : Type u_1
inst✝ : Frame X
n✝ m✝ : Nucleus X
x✝ y✝ : X
m n : Nucleus X
x y : X
this : Nonempty X
k : X
hxyk : k ≥ x ⊓ y
l : X
hlx : ∀ x_1 ≥ x, l ⊓ m x_1 ≤ n x_1
hly : ∀ i ≥ y, l ⊓ m i ≤ n i
hlk : l ≤ m k
⊢ n (x ⊔ k) ⊓ n (y ⊔ k) = n k

Tactic:
rw [← map_inf, ← sup_inf_right, sup_eq_right.2 hxyk]

Goal state after:
Module: Mathlib.Order.Partition.Basic

Goal state before:
pp: α : Type u_1
x : α
u : Set α
P : Partition u
hx : x ∈ ⋃₀ ↑P
⊢ ∃ t ∈ P, x ∈ t

Tactic:
mem_sUnion

Goal state after:
α : Type u_1
x : α
u : Set α
P : Partition u
hx : ∃ t ∈ ↑P, x ∈ t
⊢ ∃ t ∈ P, x ∈ t
Module: Mathlib.Order.Partition.Basic

Goal state before:
pp: α : Type u_1
x : α
u : Set α
P : Partition u
hx : x ∈ ⋃₀ ↑P
⊢ ∃! t, t ∈ P ∧ x ∈ t

Tactic:
mem_sUnion

Goal state after:
α : Type u_1
x : α
u : Set α
P : Partition u
hx : ∃ t ∈ ↑P, x ∈ t
⊢ ∃! t, t ∈ P ∧ x ∈ t
Module: Mathlib.Order.Partition.Basic

Goal state before:
pp: α : Type u_1
x : α
u t : Set α
P : Partition u
ht : t ∈ P
hxt : x ∈ t
y : α
⊢ y ∈ t ↔ y ∈ P.partOf x

Tactic:
exact ⟨(⟨t, ht, hxt, ·⟩), fun ⟨s, hsP, hxs, hys⟩ ↦ (P.eq_of_mem_of_mem ht hsP hxt hxs) ▸ hys⟩

Goal state after:
Module: Mathlib.Order.Sublocale

Goal state before:
pp: X : Type u_1
inst✝ : Order.Frame X
S : Sublocale X
x : X
hx : x ∈ S
⊢ ↑(S.restrict x) = x

Tactic:
simp [hx]

Goal state after:
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: case inr
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝¹ : LinearOrder ι
inst✝ : OrderBot ι
𝓕 : Filtration ι m
i j : ι
s : Set Ω
hs : MeasurableSet s
hij : i ≤ j
⊢ MeasurableSet (Set.Ioi i ×ˢ (s \ s) ∪ (Set.Ioi i \ Set.Ioi j) ×ˢ s)

Tactic:
← Set.prod_sdiff_prod

Goal state after:
case inr
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝¹ : LinearOrder ι
inst✝ : OrderBot ι
𝓕 : Filtration ι m
i j : ι
s : Set Ω
hs : MeasurableSet s
hij : i ≤ j
⊢ MeasurableSet (Set.Ioi i ×ˢ s \ Set.Ioi j ×ˢ s)
Module: Mathlib.Probability.Martingale.Basic

Goal state before:
pp: Ω : Type u_1
E : Type u_2
ι : Type u_3
inst✝⁶ : Preorder ι
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
ℱ : Filtration ι m0
inst✝³ : CompleteSpace E
inst✝² : PartialOrder E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : SigmaFiniteFiltration μ ℱ
f : ι → Ω → E
hadp : StronglyAdapted ℱ f
hint : ∀ (i : ι), Integrable (f i) μ
hf : ∀ (i j : ι), i ≤ j → 0 ≤ᵐ[μ] μ[f j - f i | ↑ℱ i]
i j : ι
hij : i ≤ j
⊢ f i ≤ᵐ[μ] μ[f j | ↑ℱ i]

Tactic:
rw [← condExp_of_stronglyMeasurable (ℱ.le _) (hadp _) (hint _), ← eventually_sub_nonneg]

Goal state after:
Ω : Type u_1
E : Type u_2
ι : Type u_3
inst✝⁶ : Preorder ι
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
ℱ : Filtration ι m0
inst✝³ : CompleteSpace E
inst✝² : PartialOrder E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : SigmaFiniteFiltration μ ℱ
f : ι → Ω → E
hadp : StronglyAdapted ℱ f
hint : ∀ (i : ι), Integrable (f i) μ
hf : ∀ (i j : ι), i ≤ j → 0 ≤ᵐ[μ] μ[f j - f i | ↑ℱ i]
i j : ι
hij : i ≤ j
⊢ 0 ≤ᵐ[μ] μ[f j | ↑ℱ i] - μ[f i | ↑ℱ i]
Module: Mathlib.Probability.Martingale.Centering

Goal state before:
pp: Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
ℱ : Filtration ℕ m0
inst✝¹ : CompleteSpace E
inst✝ : SigmaFiniteFiltration μ ℱ
f g : ℕ → Ω → E
hf : Martingale f ℱ μ
hg : StronglyAdapted ℱ fun n ↦ g (n + 1)
hg0 : g 0 = 0
hgint : ∀ (n : ℕ), Integrable (g n) μ
n : ℕ
h : ℕ → Ω → E := f - martingalePart (f + g) ℱ μ
hhdef : h = f - martingalePart (f + g) ℱ μ
hh : h = predictablePart (f + g) ℱ μ - g
hhpred : StronglyAdapted ℱ fun n ↦ h (n + 1)
this : IsStronglyPredictable ℱ g
hhmgle : Martingale h ℱ μ
ω : Ω
hω : (f n - martingalePart (f + g) ℱ μ n) ω = (f - martingalePart (f + g) ℱ μ) 0 ω
⊢ (f 0 - ((f + g) 0 - predictablePart (f + g) ℱ μ 0)) ω = 0 ω

Tactic:
simp [hg0]

Goal state after:
Module: Mathlib.Probability.Martingale.Basic

Goal state before:
pp: Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
𝒢 : Filtration ℕ m0
inst✝⁵ : CompleteSpace E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : ClosedIciTopology E
inst✝¹ : IsOrderedModule ℝ E
inst✝ : IsFiniteMeasure μ
f : ℕ → Ω → E
hadp : StronglyAdapted 𝒢 f
hint : ∀ (i : ℕ), Integrable (f i) μ
hf : ∀ (i : ℕ), f i ≤ᵐ[μ] μ[f (i + 1) | ↑𝒢 i]
i j k : ℕ
hik : i ≤ k
hk : f i ≤ᵐ[μ] μ[f k | ↑𝒢 i]
ω : Ω
hω1 : f i ω ≤ μ[f k | ↑𝒢 i] ω
hω2 : μ[f k | ?m.143] ω ≤ μ[μ[f (k + 1) | ↑𝒢 k] | ?m.143] ω
hω3 : μ[μ[f (k + 1) | ↑𝒢 k] | ↑𝒢 i] ω = μ[f (k + 1) | ↑𝒢 i] ω
⊢ μ[f k | ↑𝒢 i] ω ≤ μ[f (k + 1) | ↑𝒢 i] ω

Tactic:
hω2,

Goal state after:
Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
𝒢 : Filtration ℕ m0
inst✝⁵ : CompleteSpace E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : ClosedIciTopology E
inst✝¹ : IsOrderedModule ℝ E
inst✝ : IsFiniteMeasure μ
f : ℕ → Ω → E
hadp : StronglyAdapted 𝒢 f
hint : ∀ (i : ℕ), Integrable (f i) μ
hf : ∀ (i : ℕ), f i ≤ᵐ[μ] μ[f (i + 1) | ↑𝒢 i]
i j k : ℕ
hik : i ≤ k
hk : f i ≤ᵐ[μ] μ[f k | ↑𝒢 i]
ω : Ω
hω1 : f i ω ≤ μ[f k | ↑𝒢 i] ω
hω2 : μ[f k | ↑𝒢 i] ω ≤ μ[μ[f (k + 1) | ↑𝒢 k] | ↑𝒢 i] ω
hω3 : μ[μ[f (k + 1) | ↑𝒢 k] | ↑𝒢 i] ω = μ[f (k + 1) | ↑𝒢 i] ω
⊢ μ[μ[f (k + 1) | ↑𝒢 k] | ↑𝒢 i] ω ≤ μ[f (k + 1) | ↑𝒢 i] ω
Module: Mathlib.Probability.Martingale.Basic

Goal state before:
pp: Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
𝒢 : Filtration ℕ m0
inst✝¹ : CompleteSpace E
inst✝ : IsFiniteMeasure μ
f : ℕ → Ω → E
hadp : StronglyAdapted 𝒢 f
hint : ∀ (i : ℕ), Integrable (f i) μ
hf : ∀ (i : ℕ), f i =ᵐ[μ] μ[f (i + 1) | ↑𝒢 i]
i j k : ℕ
hik : i ≤ k
hk : μ[f k | ↑𝒢 i] =ᵐ[μ] f i
ω : Ω
hω1 : μ[f k | ↑𝒢 i] ω = f i ω
hω2 : μ[f k | ?m.127] ω = μ[μ[f (k + 1) | ↑𝒢 k] | ?m.127] ω
hω3 : μ[μ[f (k + 1) | ↑𝒢 k] | ↑𝒢 i] ω = μ[f (k + 1) | ↑𝒢 i] ω
⊢ μ[f (k + 1) | ↑𝒢 i] ω = μ[f k | ↑𝒢 i] ω

Tactic:
hω2,

Goal state after:
Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
𝒢 : Filtration ℕ m0
inst✝¹ : CompleteSpace E
inst✝ : IsFiniteMeasure μ
f : ℕ → Ω → E
hadp : StronglyAdapted 𝒢 f
hint : ∀ (i : ℕ), Integrable (f i) μ
hf : ∀ (i : ℕ), f i =ᵐ[μ] μ[f (i + 1) | ↑𝒢 i]
i j k : ℕ
hik : i ≤ k
hk : μ[f k | ↑𝒢 i] =ᵐ[μ] f i
ω : Ω
hω1 : μ[f k | ↑𝒢 i] ω = f i ω
hω2 : μ[f k | ↑𝒢 i] ω = μ[μ[f (k + 1) | ↑𝒢 k] | ↑𝒢 i] ω
hω3 : μ[μ[f (k + 1) | ↑𝒢 k] | ↑𝒢 i] ω = μ[f (k + 1) | ↑𝒢 i] ω
⊢ μ[f (k + 1) | ↑𝒢 i] ω = μ[μ[f (k + 1) | ↑𝒢 k] | ↑𝒢 i] ω