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

Goal state before:
pp: case inr.inr
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
⊢ f ↑n = ↑n ^ s

Tactic:
]

Goal state after:
case inr.inr
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
⊢ f ↑n = ↑n ^ s
Module: Mathlib.NumberTheory.NumberField.House

Goal state before:
pp: case calc_6
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)
p q : ℕ
hpq : p < q
x : β × (K →+* ℂ) → ℤ
inst✝² : Fintype β
A : ℝ
habs : ∀ (k : α) (l : β), house ((algebraMap (𝓞 K) K) (a k l)) ≤ A
inst✝¹ : DecidableEq (K →+* ℂ)
inst✝ : Fintype α
Apos : 0 ≤ A
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖asiegel K a‖) ^ (↑p / (↑q - ↑p))
h : ℕ := finrank ℚ K
l : β
H₀ : 0 ≤ supOfBasis K
H₁ : 0 < ↑q - ↑p
⊢ ↑h * c₂ K * (c₁ K * ↑q * A) ^ (↑p / (↑q - ↑p)) ≤ c₁ K * (c₁ K * ↑q * A) ^ (↑p / (↑q - ↑p))

Tactic:
]

Goal state after:
case calc_6
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)
p q : ℕ
hpq : p < q
x : β × (K →+* ℂ) → ℤ
inst✝² : Fintype β
A : ℝ
habs : ∀ (k : α) (l : β), house ((algebraMap (𝓞 K) K) (a k l)) ≤ A
inst✝¹ : DecidableEq (K →+* ℂ)
inst✝ : Fintype α
Apos : 0 ≤ A
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖asiegel K a‖) ^ (↑p / (↑q - ↑p))
h : ℕ := finrank ℚ K
l : β
H₀ : 0 ≤ supOfBasis K
H₁ : 0 < ↑q - ↑p
⊢ ↑h * c₂ K * (c₁ K * ↑q * A) ^ (↑p / (↑q - ↑p)) ≤ c₁ K * (c₁ K * ↑q * A) ^ (↑p / (↑q - ↑p))
Module: Mathlib.NumberTheory.Padics.AddChar

Goal state before:
pp: p : ℕ
inst✝⁵ : Fact (Nat.Prime p)
R : Type u_1
inst✝⁴ : NormedRing R
inst✝³ : Algebra ℤ_[p] R
inst✝² : IsBoundedSMul ℤ_[p] R
inst✝¹ : IsUltrametricDist R
inst✝ : CompleteSpace R
r : R
hr : Tendsto (fun x ↦ r ^ x) atTop (𝓝 0)
⊢ 1 = 1

Tactic:
]

Goal state after:
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
⊢ f ↑m = f ↑m

Tactic:
]

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

Goal state before:
pp: p : ℕ
inst✝⁵ : Fact (Nat.Prime p)
R : Type u_1
inst✝⁴ : NormedRing R
inst✝³ : Algebra ℤ_[p] R
inst✝² : IsBoundedSMul ℤ_[p] R
inst✝¹ : IsUltrametricDist R
inst✝ : CompleteSpace R
r : R
hr : Tendsto (fun x ↦ r ^ x) atTop (𝓝 0)
a b : ℤ_[p]
F : C(ℤ_[p], R) := mahlerSeries fun x ↦ r ^ x
n : ℕ
⊢ ∑ i ∈ Finset.range (n + 1), n.choose i • r ^ i = ∑ m ∈ Finset.range (n + 1), r ^ m * 1 ^ (n - m) * ↑(n.choose m)

Tactic:
]

Goal state after:
p : ℕ
inst✝⁵ : Fact (Nat.Prime p)
R : Type u_1
inst✝⁴ : NormedRing R
inst✝³ : Algebra ℤ_[p] R
inst✝² : IsBoundedSMul ℤ_[p] R
inst✝¹ : IsUltrametricDist R
inst✝ : CompleteSpace R
r : R
hr : Tendsto (fun x ↦ r ^ x) atTop (𝓝 0)
a b : ℤ_[p]
F : C(ℤ_[p], R) := mahlerSeries fun x ↦ r ^ x
n : ℕ
⊢ ∑ i ∈ Finset.range (n + 1), n.choose i • r ^ i = ∑ m ∈ Finset.range (n + 1), r ^ m * 1 ^ (n - m) * ↑(n.choose m)
Module: Mathlib.NumberTheory.Padics.AddChar

Goal state before:
pp: p : ℕ
inst✝⁵ : Fact (Nat.Prime p)
R : Type u_1
inst✝⁴ : NormedRing R
inst✝³ : Algebra ℤ_[p] R
inst✝² : IsBoundedSMul ℤ_[p] R
inst✝¹ : IsUltrametricDist R
inst✝ : CompleteSpace R
r : R
hr : Tendsto (fun x ↦ r ^ x) atTop (𝓝 0)
a b : ℤ_[p]
F : C(ℤ_[p], R) := mahlerSeries fun x ↦ r ^ x
n i : ℕ
hi : i ∈ Finset.range (n + 1)
⊢ r ^ i * ↑(n.choose i) = r ^ i * ↑(n.choose i)

Tactic:
]

Goal state after:
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
hphqh : p * h < q * h
⊢ 0 < p ∧ 0 < h ∨ p < 0 ∧ h < 0

Tactic:
]

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
⊢ 0 < p ∧ 0 < h ∨ p < 0 ∧ h < 0
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
⊢ f ↑n = f ↑n

Tactic:
]

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

Goal state before:
pp: p : ℕ
inst✝⁵ : Fact (Nat.Prime p)
R : Type u_1
inst✝⁴ : NormedRing R
inst✝³ : Algebra ℤ_[p] R
inst✝² : IsBoundedSMul ℤ_[p] R
inst✝¹ : IsUltrametricDist R
inst✝ : CompleteSpace R
r : R
hr : Tendsto (fun x ↦ r ^ x) atTop (𝓝 0)
⊢ 1 + r = 1 + r

Tactic:
]

Goal state after:
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
hphqh : p * h < q * h
h0ph : 0 < p * h
⊢ p * finrank ℚ K = p * h

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.Ostrowski

Goal state before:
pp: case inr.inr
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
hfn : f ↑n = ↑n ^ logb (↑n) (f ↑n)
⊢ ↑n ^ s = ↑n ^ s

Tactic:
]

Goal state after:
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
hphqh : p * h < q * h
h0ph : 0 < p * h
hfinp : Fintype.card (α × (K →+* ℂ)) = p * h
⊢ q * finrank ℚ K = q * h

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.Ostrowski

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
⊢ ¬∃ c, 0 < c ∧ (fun x ↦ real x ^ c) = ⇑(padic p)

Tactic:
]

Goal state after:
p : ℕ
inst✝ : Fact (Nat.Prime p)
⊢ ¬∃ c, 0 < c ∧ (fun x ↦ real x ^ c) = ⇑(padic p)
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
hphqh : p * h < q * h
h0ph : 0 < p * h
hfinp : Fintype.card (α × (K →+* ℂ)) = p * h
hfinq : Fintype.card (β × (K →+* ℂ)) = q * h
⊢ p * h < q * h

Tactic:
]

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
h0ph : 0 < p * h
hfinp : Fintype.card (α × (K →+* ℂ)) = p * h
hfinq : Fintype.card (β × (K →+* ℂ)) = q * h
⊢ p * h < q * h
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
hphqh : p * h < q * h
h0ph : 0 < p * h
hfinp : Fintype.card (α × (K →+* ℂ)) = p * h
hfinq : Fintype.card (β × (K →+* ℂ)) = q * h
⊢ 0 < p * h

Tactic:
]

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
h0ph : 0 < p * h
hfinp : Fintype.card (α × (K →+* ℂ)) = p * h
hfinq : Fintype.card (β × (K →+* ℂ)) = q * h
⊢ 0 < p * h
Module: Mathlib.NumberTheory.Padics.Complex

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
⊢ Valued.v ((algebraMap ℚ_[p] (PadicAlgCl p)) ↑p) = 1 / ↑p

Tactic:
]

Goal state after:
p : ℕ
inst✝ : Fact (Nat.Prime p)
⊢ Valued.v ((algebraMap ℚ_[p] (PadicAlgCl p)) ↑p) = 1 / ↑p
Module: Mathlib.NumberTheory.Padics.Complex

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

Tactic:
]

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

Goal state before:
pp: case h
p : ℕ
hp : Fact (Nat.Prime p)
x : ℚ_[p]
hx : 1 < ‖x‖
⊢ 1 < ‖x‖

Tactic:
]

Goal state after:
case h
p : ℕ
hp : Fact (Nat.Prime p)
x : ℚ_[p]
hx : 1 < ‖x‖
⊢ 1 < ‖x‖
Module: Mathlib.NumberTheory.Padics.Complex

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
⊢ (algebraMap (PadicAlgCl p) ℂ_[p]) ↑n = (algebraMap (PadicAlgCl p) ℂ_[p]) ↑n

Tactic:
]

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

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
⊢ 1 / ↑p = 1 / ↑p

Tactic:
]

Goal state after:
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
hphqh : p * h < q * h
h0ph : 0 < p * h
hfinp : Fintype.card (α × (K →+* ℂ)) = p * h
hfinq : Fintype.card (β × (K →+* ℂ)) = q * h
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
hmulvec0 : asiegel K a *ᵥ x = 0
hxbound : ‖x‖ ≤ (↑q * ↑h * ‖asiegel K a‖) ^ (↑p / (↑q - ↑p))
⊢ ∃ ξ, ξ ≠ 0 ∧ a *ᵥ ξ = 0 ∧ ∀ (l : β), house ↑(ξ l) ≤ c₁ K * (c₁ K * ↑q * A) ^ (↑p / (↑q - ↑p))

Tactic:
]

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
h0ph : 0 < p * h
hfinp : Fintype.card (α × (K →+* ℂ)) = p * h
hfinq : Fintype.card (β × (K →+* ℂ)) = q * h
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
hmulvec0 : asiegel K a *ᵥ x = 0
hxbound : ‖x‖ ≤ (↑q * ↑h * ‖asiegel K a‖) ^ (↑p / (↑q - ↑p))
⊢ ∃ ξ, ξ ≠ 0 ∧ a *ᵥ ξ = 0 ∧ ∀ (l : β), house ↑(ξ l) ≤ c₁ K * (c₁ K * ↑q * A) ^ (↑p / (↑q - ↑p))
Module: Mathlib.NumberTheory.Padics.MahlerBasis

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
k : ℕ
⊢ ‖↑(Polynomial.eval x (ascPochhammer ℤ_[p] k))‖ ≤ ‖↑k.factorial‖

Tactic:
]

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
k : ℕ
⊢ ‖↑(Polynomial.eval x (ascPochhammer ℤ_[p] k))‖ ≤ ‖↑k.factorial‖
Module: Mathlib.NumberTheory.Padics.Complex

Goal state before:
pp: case h
p : ℕ
hp : Fact (Nat.Prime p)
x : PadicAlgCl p
⊢ ↑(Valued.v x) = ↑(Valued.v x)

Tactic:
]

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

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
k : ℕ
⊢ (ascPochhammer ℕ k).smeval x = (ascPochhammer ℕ k).smeval x

Tactic:
]

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:
fwdDiff_iter_eq_sum_shift

Goal state after:
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
⊢ ‖∑ k ∈ range (n + 1), ((-1) ^ (n - k) * ↑(n.choose k)) • f (m + k • h)‖ ≤ ‖f‖
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
⊢ ‖∑ k ∈ range (n + 1), ((-1) ^ (n - k) * ↑(n.choose k)) • f (m + k • h)‖ ≤ ‖f‖

Tactic:
]

Goal state after:
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
⊢ ‖∑ k ∈ range (n + 1), ((-1) ^ (n - k) * ↑(n.choose k)) • f (m + k • h)‖ ≤ ‖f‖
Module: Mathlib.NumberTheory.Padics.MahlerBasis

Goal state before:
pp: M : Type u_1
G : Type u_2
inst✝¹ : AddCommMonoidWithOne M
inst✝ : AddCommGroup G
f : M → G
n R : ℕ
hR : 1 ≤ R
⊢ Δ_[1]^[n + R] f 0 = Δ_[1]^[n + R] f 0

Tactic:
]

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

Goal state before:
pp: M : Type u_1
G : Type u_2
inst✝¹ : AddCommMonoidWithOne M
inst✝ : AddCommGroup G
f : M → G
n R : ℕ
hR : 1 ≤ R
⊢ Δ_[1]^[n + R] f 0 = R.choose (R - 1 + 1) • Δ_[1]^[n + R] f 0

Tactic:
rw [Nat.sub_add_cancel hR, Nat.choose_self, one_smul]

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

Goal state before:
pp: M : Type u_1
G : Type u_2
inst✝¹ : AddCommMonoidWithOne M
inst✝ : AddCommGroup G
f : M → G
n R : ℕ
hR : 1 ≤ R
aux : Δ_[1]^[n + R] f 0 = R.choose (R - 1 + 1) • Δ_[1]^[n + R] f 0
⊢ ∑ k ∈ range (R + 1), R.choose k • Δ_[1]^[n + k] f 0 - Δ_[1]^[n] f 0 =
    ∑ k ∈ range (n + 1), ((-1) ^ (n - k) * ↑(n.choose k)) • (f (↑k + ↑R) - f ↑k)

Tactic:
]

Goal state after:
M : Type u_1
G : Type u_2
inst✝¹ : AddCommMonoidWithOne M
inst✝ : AddCommGroup G
f : M → G
n R : ℕ
hR : 1 ≤ R
aux : Δ_[1]^[n + R] f 0 = R.choose (R - 1 + 1) • Δ_[1]^[n + R] f 0
⊢ ∑ k ∈ range (R + 1), R.choose k • Δ_[1]^[n + k] f 0 - Δ_[1]^[n] f 0 =
    ∑ k ∈ range (n + 1), ((-1) ^ (n - k) * ↑(n.choose k)) • (f (↑k + ↑R) - f ↑k)
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 : ℕ
⊢ ‖-∑ j ∈ range (p ^ t - 1), (p ^ t).choose (j + 1) • Δ_[1]^[n + (j + 1)] (⇑f) 0 +
        ∑ k ∈ range (n + 1), ((-1) ^ (n - k) * ↑(n.choose k)) • (f (↑k + ↑(p ^ t)) - f ↑k)‖ ≤
    max (↑((range (p ^ t - 1)).sup fun j ↦ ‖Δ_[1]^[n + (j + 1)] (⇑f) 0‖₊ / ↑p)) (‖f‖ / ↑p ^ s)

Tactic:
]

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 : ℕ
⊢ ‖-∑ j ∈ range (p ^ t - 1), (p ^ t).choose (j + 1) • Δ_[1]^[n + (j + 1)] (⇑f) 0 +
        ∑ k ∈ range (n + 1), ((-1) ^ (n - k) * ↑(n.choose k)) • (f (↑k + ↑(p ^ t)) - f ↑k)‖ ≤
    max (↑((range (p ^ t - 1)).sup fun j ↦ ‖Δ_[1]^[n + (j + 1)] (⇑f) 0‖₊ / ↑p)) (‖f‖ / ↑p ^ s)
Module: Mathlib.NumberTheory.Padics.MahlerBasis

Goal state before:
pp: case refine_1
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 : ℕ
⊢ ‖∑ j ∈ range (p ^ t - 1), (p ^ t).choose (j + 1) • Δ_[1]^[n + (j + 1)] (⇑f) 0‖₊ ≤
    (range (p ^ t - 1)).sup fun j ↦ ‖Δ_[1]^[n + (j + 1)] (⇑f) 0‖₊ / ↑p

Tactic:
]

Goal state after:
case refine_1
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 : ℕ
⊢ ‖∑ j ∈ range (p ^ t - 1), (p ^ t).choose (j + 1) • Δ_[1]^[n + (j + 1)] (⇑f) 0‖₊ ≤
    (range (p ^ t - 1)).sup fun j ↦ ‖Δ_[1]^[n + (j + 1)] (⇑f) 0‖₊ / ↑p
Module: Mathlib.NumberTheory.Padics.MahlerBasis

Goal state before:
pp: case refine_1
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
hi : i ∈ range (p ^ t - 1)
⊢ ‖↑((p ^ t).choose (i + 1)) • Δ_[1]^[n + (i + 1)] (⇑f) 0‖₊ ≤ (↑p)⁻¹ * ‖Δ_[1]^[n + (i + 1)] (⇑f) 0‖₊

Tactic:
]

Goal state after:
case refine_1
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
hi : i ∈ range (p ^ t - 1)
⊢ ‖↑((p ^ t).choose (i + 1)) • Δ_[1]^[n + (i + 1)] (⇑f) 0‖₊ ≤ (↑p)⁻¹ * ‖Δ_[1]^[n + (i + 1)] (⇑f) 0‖₊
Module: Mathlib.NumberTheory.Padics.MahlerBasis

Goal state before:
pp: case refine_1
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
hi : i < p ^ t - 1
⊢ ‖↑((p ^ t).choose (i + 1))‖₊ ≤ (↑p)⁻¹

Tactic:
]

Goal state after:
case refine_1
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
hi : i < p ^ t - 1
⊢ ‖↑((p ^ t).choose (i + 1))‖₊ ≤ (↑p)⁻¹
Module: Mathlib.NumberTheory.Padics.MahlerBasis

Goal state before:
pp: case refine_1
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
hi : i < p ^ t - 1
this : 0 < (p ^ t).choose (i + 1)
⊢ 1 ≤ padicValNat p ((p ^ t).choose (i + 1))

Tactic:
]

Goal state after:
case refine_1
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
hi : i < p ^ t - 1
this : 0 < (p ^ t).choose (i + 1)
⊢ 1 ≤ padicValNat p ((p ^ t).choose (i + 1))
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)
⊢ ‖↑((-1) ^ (n - i) * ↑(n.choose i)) • (f (↑i + ↑(p ^ t)) - f ↑i)‖ ≤
    ‖↑(-1) ^ (n - i) * ↑(n.choose i)‖ * ‖f (↑i + ↑(p ^ t)) - f ↑i‖

Tactic:
]

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)
⊢ ‖↑((-1) ^ (n - i) * ↑(n.choose i)) • (f (↑i + ↑(p ^ t)) - f ↑i)‖ ≤
    ‖↑(-1) ^ (n - i) * ↑(n.choose i)‖ * ‖f (↑i + ↑(p ^ t)) - f ↑i‖
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)
⊢ (↑p ^ t)⁻¹ ≤ (↑p ^ t)⁻¹

Tactic:
]

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

Goal state before:
pp: case succ
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
k : ℕ
IH : ∀ (n : ℕ), k ≤ s → ‖Δ_[1]^[n + k * p ^ t] (⇑f) 0‖ ≤ ‖f‖ / ↑p ^ k
n : ℕ
hk : k + 1 ≤ s
⊢ ‖Δ_[1]^[n + k * p ^ t + p ^ t] (⇑f) 0‖ ≤ ‖f‖ / ↑p ^ (k + 1)

Tactic:
]

Goal state after:
case succ
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
k : ℕ
IH : ∀ (n : ℕ), k ≤ s → ‖Δ_[1]^[n + k * p ^ t] (⇑f) 0‖ ≤ ‖f‖ / ↑p ^ k
n : ℕ
hk : k + 1 ≤ s
⊢ ‖Δ_[1]^[n + k * p ^ t + p ^ t] (⇑f) 0‖ ≤ ‖f‖ / ↑p ^ (k + 1)
Module: Mathlib.NumberTheory.Padics.WithVal

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
hp0' : 0 < ↑p
hp0 : 0 < (↑p)⁻¹
hp1' : 1 < ↑p
hp1 : (↑p)⁻¹ < 1
⊢ (∀ (i : ℕ),
      True →
        ∃ j,
          True ∧
            ∀ (x y : WithVal (Rat.padicValuation p)),
              (x, y) ∈ {p_1 | Valued.v.restrict (p_1.2 - p_1.1) < ↑j} →
                (((Rat.castHom ℚ_[p]).comp (WithVal.equiv (Rat.padicValuation p)).toRingHom) x,
                    ((Rat.castHom ℚ_[p]).comp (WithVal.equiv (Rat.padicValuation p)).toRingHom) y) ∈
                  {p_1 | dist p_1.1 p_1.2 ≤ (↑p)⁻¹ ^ i}) ∧
    ∀ (j : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ˣ),
      True →
        ∃ i,
          True ∧
            ∀ (x y : WithVal (Rat.padicValuation p)),
              (((Rat.castHom ℚ_[p]).comp (WithVal.equiv (Rat.padicValuation p)).toRingHom) x,
                    ((Rat.castHom ℚ_[p]).comp (WithVal.equiv (Rat.padicValuation p)).toRingHom) y) ∈
                  {p_1 | dist p_1.1 p_1.2 ≤ (↑p)⁻¹ ^ i} →
                (x, y) ∈ {p_1 | Valued.v.restrict (p_1.2 - p_1.1) < ↑j}

Tactic:
]

Goal state after:
p : ℕ
inst✝ : Fact (Nat.Prime p)
hp0' : 0 < ↑p
hp0 : 0 < (↑p)⁻¹
hp1' : 1 < ↑p
hp1 : (↑p)⁻¹ < 1
⊢ (∀ (i : ℕ),
      True →
        ∃ j,
          True ∧
            ∀ (x y : WithVal (Rat.padicValuation p)),
              (x, y) ∈ {p_1 | Valued.v.restrict (p_1.2 - p_1.1) < ↑j} →
                (((Rat.castHom ℚ_[p]).comp (WithVal.equiv (Rat.padicValuation p)).toRingHom) x,
                    ((Rat.castHom ℚ_[p]).comp (WithVal.equiv (Rat.padicValuation p)).toRingHom) y) ∈
                  {p_1 | dist p_1.1 p_1.2 ≤ (↑p)⁻¹ ^ i}) ∧
    ∀ (j : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ˣ),
      True →
        ∃ i,
          True ∧
            ∀ (x y : WithVal (Rat.padicValuation p)),
              (((Rat.castHom ℚ_[p]).comp (WithVal.equiv (Rat.padicValuation p)).toRingHom) x,
                    ((Rat.castHom ℚ_[p]).comp (WithVal.equiv (Rat.padicValuation p)).toRingHom) y) ∈
                  {p_1 | dist p_1.1 p_1.2 ≤ (↑p)⁻¹ ^ i} →
                (x, y) ∈ {p_1 | Valued.v.restrict (p_1.2 - p_1.1) < ↑j}