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

Goal state before:
pp: d : ℤ
h_nonsquare : ∀ (n : ℤ), d ≠ n * n
a : ℤ√d
ha : 0 * 0 = d * a.im * a.im
h : d < 0
this : a.re * a.re = 0
⊢ 0 = re 0 ∧ a.im = im 0

Tactic:
]

Goal state after:
d : ℤ
h_nonsquare : ∀ (n : ℤ), d ≠ n * n
a : ℤ√d
ha : 0 * 0 = d * a.im * a.im
h : d < 0
this : a.re * a.re = 0
⊢ 0 = re 0 ∧ a.im = im 0
Module: Mathlib.NumberTheory.PellMatiyasevic

Goal state before:
pp: a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
⊢ n - 1 = n - 1

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.Cyclotomic.Discriminant

Goal state before:
pp: p : ℕ
K : Type u
L : Type v
ζ : L
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension {p} K L
hp : Fact (Nat.Prime p)
hζ : IsPrimitiveRoot ζ p
hirr : Irreducible (cyclotomic p K)
hodd : p ≠ 2
⊢ IsCyclotomicExtension {p} K L

Tactic:
]

Goal state after:
p : ℕ
K : Type u
L : Type v
ζ : L
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension {p} K L
hp : Fact (Nat.Prime p)
hζ : IsPrimitiveRoot ζ p
hirr : Irreducible (cyclotomic p K)
hodd : p ≠ 2
⊢ IsCyclotomicExtension {p} K L
Module: Mathlib.NumberTheory.PellMatiyasevic

Goal state before:
pp: case refl
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
⊢ ↑(xn a1 i % xn a1 n) < ↑(xn a1 n) - ↑(xn a1 (n - 1))

Tactic:
]

Goal state after:
case refl
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
⊢ ↑(xn a1 i % xn a1 n) < ↑(xn a1 n) - ↑(xn a1 (n - 1))
Module: Mathlib.NumberTheory.Zsqrtd.Basic

Goal state before:
pp: d : ℤ
h_nonsquare : ∀ (n : ℤ), d ≠ n * n
a : ℤ√d
ha : a.re * a.re = d * a.im * a.im
h : d < 0
⊢ d * (a.im * a.im) ≤ 0

Tactic:
]

Goal state after:
d : ℤ
h_nonsquare : ∀ (n : ℤ), d ≠ n * n
a : ℤ√d
ha : a.re * a.re = d * a.im * a.im
h : d < 0
⊢ d * (a.im * a.im) ≤ 0
Module: Mathlib.NumberTheory.PellMatiyasevic

Goal state before:
pp: case refl.inl
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
lin : i < n
⊢ xn a1 (n - 1) + xn a1 i < xn a1 n

Tactic:
]

Goal state after:
case refl.inl
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
lin : i < n
⊢ xn a1 (n - 1) + xn a1 i < xn a1 n
Module: Mathlib.NumberTheory.Cyclotomic.Discriminant

Goal state before:
pp: case e'_3.e'_5
p : ℕ
K : Type u
L : Type v
ζ : L
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension {p} K L
hp : Fact (Nat.Prime p)
hζ : IsPrimitiveRoot ζ p
hirr : Irreducible (cyclotomic p K)
hodd : p ≠ 2
this : IsCyclotomicExtension {p ^ (0 + 1)} K L
hζ' : IsPrimitiveRoot ζ (p ^ (0 + 1))
⊢ (-1) ^ ((p - 1) / 2) = (-1) ^ ((p - 1) / 2)

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.PellMatiyasevic

Goal state before:
pp: a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
lin : i < n
⊢ xn a1 n = xn a1 n

Tactic:
]

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
d : ℤ
f : ℤ√d →+* R
⊢ ↑d = ↑d

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.PellMatiyasevic

Goal state before:
pp: a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
lin : i < n
⊢ xn a1 (n - 1) * 2 ≤ xn a1 (n - 1) * a + d a1 * yn a1 (n - 1)

Tactic:
]

Goal state after:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
lin : i < n
⊢ xn a1 (n - 1) * 2 ≤ xn a1 (n - 1) * a + d a1 * yn a1 (n - 1)
Module: Mathlib.NumberTheory.Cyclotomic.Discriminant

Goal state before:
pp: case e'_3.e'_6
p : ℕ
K : Type u
L : Type v
ζ : L
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension {p} K L
hp : Fact (Nat.Prime p)
hζ : IsPrimitiveRoot ζ p
hirr : Irreducible (cyclotomic p K)
hodd : p ≠ 2
this : IsCyclotomicExtension {p ^ (0 + 1)} K L
hζ' : IsPrimitiveRoot ζ (p ^ (0 + 1))
⊢ ↑p ^ (p - 2) = ↑p ^ (p - (1 + 1))

Tactic:
]

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

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
d : ℤ
r : { r // r * r = ↑d }
hd : ∀ (n : ℤ), d ≠ n * n
a : ℤ√d
ha : (lift r) a = 0
h_inj : Function.Injective Int.cast
this : a = 0
⊢ a = 0

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
d : ℤ
r : { r // r * r = ↑d }
hd : ∀ (n : ℤ), d ≠ n * n
a : ℤ√d
ha : (lift r) a = 0
h_inj : Function.Injective Int.cast
this : a = 0
⊢ a = 0
Module: Mathlib.NumberTheory.Zsqrtd.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
d : ℤ
r : { r // r * r = ↑d }
hd : ∀ (n : ℤ), d ≠ n * n
a : ℤ√d
ha : (lift r) a = 0
h_inj : Function.Injective Int.cast
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: d : ℤ
a : ℤ√d
⊢ a.norm = 1 ↔ ↑a.norm = 1

Tactic:
]

Goal state after:
d : ℤ
a : ℤ√d
⊢ a.norm = 1 ↔ ↑a.norm = 1
Module: Mathlib.NumberTheory.PellMatiyasevic

Goal state before:
pp: a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
lin : i < n
ll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n
npm : (n - 1).succ = n
⊢ i.succ ≤ n

Tactic:
]

Goal state after:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
lin : i < n
ll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n
npm : (n - 1).succ = n
⊢ i.succ ≤ n
Module: Mathlib.NumberTheory.DiophantineApproximation.ContinuedFractions

Goal state before:
pp: case succ
n : ℕ
ih : ∀ (ξ : ℝ), (GenContFract.of ξ).convs n = ↑(ξ.convergent n)
ξ : ℝ
⊢ ↑⌊ξ⌋ + (↑((fract ξ)⁻¹.convergent n))⁻¹ = ↑(↑⌊ξ⌋ + ((fract ξ)⁻¹.convergent n)⁻¹)

Tactic:
]

Goal state after:
case succ
n : ℕ
ih : ∀ (ξ : ℝ), (GenContFract.of ξ).convs n = ↑(ξ.convergent n)
ξ : ℝ
⊢ ↑⌊ξ⌋ + (↑((fract ξ)⁻¹.convergent n))⁻¹ = ↑(↑⌊ξ⌋ + ((fract ξ)⁻¹.convergent n)⁻¹)
Module: Mathlib.NumberTheory.DirichletCharacter.Bounds

Goal state before:
pp: F : Type u_1
inst✝ : NormedField F
n : ℕ
χ : DirichletCharacter F n
a : (ZMod n)ˣ
⊢ 1 = 1

Tactic:
]

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

Goal state before:
pp: case neg
F : Type u_1
inst✝ : NormedField F
n : ℕ
χ : DirichletCharacter F n
a : ZMod n
h : ¬IsUnit a
⊢ 0 ≤ 1

Tactic:
]

Goal state after:
case neg
F : Type u_1
inst✝ : NormedField F
n : ℕ
χ : DirichletCharacter F n
a : ZMod n
h : ¬IsUnit a
⊢ 0 ≤ 1
Module: Mathlib.NumberTheory.MulChar.Duality

Goal state before:
pp: case refine_1
M : Type u_1
R : Type u_2
inst✝¹ : CommMonoid M
inst✝ : CommRing R
a : Mˣ
x✝ : ∃ χ, χ ↑a ≠ 1
χ : MulChar M R
hχ : χ ↑a = ↑1
⊢ χ ↑a = 1

Tactic:
]

Goal state after:
case refine_1
M : Type u_1
R : Type u_2
inst✝¹ : CommMonoid M
inst✝ : CommRing R
a : Mˣ
x✝ : ∃ χ, χ ↑a ≠ 1
χ : MulChar M R
hχ : χ ↑a = ↑1
⊢ χ ↑a = 1
Module: Mathlib.NumberTheory.MulChar.Duality

Goal state before:
pp: case pos
M : Type u_1
R : Type u_2
inst✝⁴ : CommMonoid M
inst✝³ : CommRing R
inst✝² : Finite M
inst✝¹ : HasEnoughRootsOfUnity R (Monoid.exponent Mˣ)
inst✝ : Nontrivial R
a : M
hu : IsUnit a
ha : hu.unit = 1
⊢ 1 = 1

Tactic:
]

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

Goal state before:
pp: case pos
ξ : ℝ
n : ℕ
n_pos : 0 < n
f : ℤ → ℤ := fun m ↦ ⌊fract (ξ * ↑m) * (↑n + 1)⌋
hn : 0 < ↑n + 1
hfu : ∀ (m : ℤ), fract (ξ * ↑m) * (↑n + 1) < ↑n + 1
D : Finset ℤ := Icc 0 ↑n
m : ℤ
hm : m ∈ D
hf : f m = ↑n
hf' : ↑↑n ≤ fract (ξ * ↑m) * (↑n + 1)
hm₀ : 0 < m
⊢ -1 ≤ (ξ * ↑m - ↑⌊ξ * ↑m⌋) * (↑n + 1) - (↑n + 1) ∧ (ξ * ↑m - ↑⌊ξ * ↑m⌋) * (↑n + 1) - (↑n + 1) ≤ 1

Tactic:
]

Goal state after:
case pos
ξ : ℝ
n : ℕ
n_pos : 0 < n
f : ℤ → ℤ := fun m ↦ ⌊fract (ξ * ↑m) * (↑n + 1)⌋
hn : 0 < ↑n + 1
hfu : ∀ (m : ℤ), fract (ξ * ↑m) * (↑n + 1) < ↑n + 1
D : Finset ℤ := Icc 0 ↑n
m : ℤ
hm : m ∈ D
hf : f m = ↑n
hf' : ↑↑n ≤ fract (ξ * ↑m) * (↑n + 1)
hm₀ : 0 < m
⊢ -1 ≤ (ξ * ↑m - ↑⌊ξ * ↑m⌋) * (↑n + 1) - (↑n + 1) ∧ (ξ * ↑m - ↑⌊ξ * ↑m⌋) * (↑n + 1) - (↑n + 1) ≤ 1
Module: Mathlib.NumberTheory.Dioph

Goal state before:
pp: α β : Type u
S : Set (α ⊕ β → ℕ)
γ : Type u
p : Poly ((α ⊕ β) ⊕ γ)
pe : ∀ (v : α ⊕ β → ℕ), v ∈ S ↔ ∃ t, p (v ⊗ t) = 0
v : α → ℕ
x✝ : v ∈ {v | ∃ x, v ⊗ x ∈ S}
x : β → ℕ
hx : v ⊗ x ∈ S
t : γ → ℕ
ht : p ((v ⊗ x) ⊗ t) = 0
⊢ p ((v ⊗ x) ⊗ t) = 0

Tactic:
]

Goal state after:
α β : Type u
S : Set (α ⊕ β → ℕ)
γ : Type u
p : Poly ((α ⊕ β) ⊕ γ)
pe : ∀ (v : α ⊕ β → ℕ), v ∈ S ↔ ∃ t, p (v ⊗ t) = 0
v : α → ℕ
x✝ : v ∈ {v | ∃ x, v ⊗ x ∈ S}
x : β → ℕ
hx : v ⊗ x ∈ S
t : γ → ℕ
ht : p ((v ⊗ x) ⊗ t) = 0
⊢ p ((v ⊗ x) ⊗ t) = 0
Module: Mathlib.NumberTheory.PellMatiyasevic

Goal state before:
pp: case refl.inl.inr
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
lin : i < n
ll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n
npm : (n - 1).succ = n
il : i ≤ n - 1
ile : i = n - 1
⊢ xn a1 (n - 1) + xn a1 (n - 1) < xn a1 n

Tactic:
]

Goal state after:
case refl.inl.inr
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
lin : i < n
ll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n
npm : (n - 1).succ = n
il : i ≤ n - 1
ile : i = n - 1
⊢ xn a1 (n - 1) + xn a1 (n - 1) < xn a1 n
Module: Mathlib.NumberTheory.DiophantineApproximation.Basic

Goal state before:
pp: ξ : ℝ
n : ℕ
n_pos : 0 < n
f : ℤ → ℤ := fun m ↦ ⌊fract (ξ * ↑m) * (↑n + 1)⌋
hn : 0 < ↑n + 1
hfu : ∀ (m : ℤ), fract (ξ * ↑m) * (↑n + 1) < ↑n + 1
D : Finset ℤ := Icc 0 ↑n
H : ∀ m ∈ D, f m ≠ ↑n
⊢ (↑n - 0).toNat < (↑n + 1 - 0).toNat

Tactic:
]

Goal state after:
ξ : ℝ
n : ℕ
n_pos : 0 < n
f : ℤ → ℤ := fun m ↦ ⌊fract (ξ * ↑m) * (↑n + 1)⌋
hn : 0 < ↑n + 1
hfu : ∀ (m : ℤ), fract (ξ * ↑m) * (↑n + 1) < ↑n + 1
D : Finset ℤ := Icc 0 ↑n
H : ∀ m ∈ D, f m ≠ ↑n
⊢ (↑n - 0).toNat < (↑n + 1 - 0).toNat
Module: Mathlib.NumberTheory.MulChar.Duality

Goal state before:
pp: M : Type u_1
R : Type u_2
inst✝³ : CommMonoid M
inst✝² : CommRing R
inst✝¹ : Finite M
inst✝ : HasEnoughRootsOfUnity R (Monoid.exponent Mˣ)
N : Submonoid M
χ : MulChar (↥N) R
ψ : Mˣ →* Rˣ
hψ : ψ.domRestrict N.units = χ.toUnitHom.comp ↑N.unitsEquivUnitsType
x✝ : (↥N)ˣ
⊢ ((domRestrictHom N R) (ofUnitHom ψ)) ↑x✝ = χ ↑x✝

Tactic:
]

Goal state after:
M : Type u_1
R : Type u_2
inst✝³ : CommMonoid M
inst✝² : CommRing R
inst✝¹ : Finite M
inst✝ : HasEnoughRootsOfUnity R (Monoid.exponent Mˣ)
N : Submonoid M
χ : MulChar (↥N) R
ψ : Mˣ →* Rˣ
hψ : ψ.domRestrict N.units = χ.toUnitHom.comp ↑N.unitsEquivUnitsType
x✝ : (↥N)ˣ
⊢ ((domRestrictHom N R) (ofUnitHom ψ)) ↑x✝ = χ ↑x✝
Module: Mathlib.NumberTheory.DiophantineApproximation.Basic

Goal state before:
pp: case neg
ξ : ℝ
n : ℕ
n_pos : 0 < n
f : ℤ → ℤ := fun m ↦ ⌊fract (ξ * ↑m) * (↑n + 1)⌋
hn : 0 < ↑n + 1
hfu : ∀ (m : ℤ), fract (ξ * ↑m) * (↑n + 1) < ↑n + 1
D : Finset ℤ := Icc 0 ↑n
H : ∀ m ∈ D, f m ≠ ↑n
hD : #(Ico 0 ↑n) < #D
hfu' : ∀ (m : ℤ), f m ≤ ↑n
⊢ ∃ j k, 0 < k ∧ k ≤ ↑n ∧ |(ξ * ↑k - ↑j) * (↑n + 1)| ≤ 1

Tactic:
have hwd : ∀ m : ℤ, m ∈ D → f m ∈ Ico (0 : ℤ) n := fun x hx =>
      mem_Ico.mpr
        ⟨floor_nonneg.mpr (mul_nonneg (fract_nonneg (ξ * x)) hn.le), Ne.lt_of_le (H x hx) (hfu' x)⟩

Goal state after:
case neg
ξ : ℝ
n : ℕ
n_pos : 0 < n
f : ℤ → ℤ := fun m ↦ ⌊fract (ξ * ↑m) * (↑n + 1)⌋
hn : 0 < ↑n + 1
hfu : ∀ (m : ℤ), fract (ξ * ↑m) * (↑n + 1) < ↑n + 1
D : Finset ℤ := Icc 0 ↑n
H : ∀ m ∈ D, f m ≠ ↑n
hD : #(Ico 0 ↑n) < #D
hfu' : ∀ (m : ℤ), f m ≤ ↑n
hwd : ∀ m ∈ D, f m ∈ Ico 0 ↑n
⊢ ∃ j k, 0 < k ∧ k ≤ ↑n ∧ |(ξ * ↑k - ↑j) * (↑n + 1)| ≤ 1
Module: Mathlib.NumberTheory.MulChar.Duality

Goal state before:
pp: M : Type u_1
R : Type u_2
inst✝³ : CommMonoid M
inst✝² : CommRing R
inst✝¹ : Finite M
inst✝ : HasEnoughRootsOfUnity R (Monoid.exponent Mˣ)
N : Submonoid M
χ : MulChar (↥N) R
ψ : Mˣ →* Rˣ
hψ : ψ.domRestrict N.units = χ.toUnitHom.comp ↑N.unitsEquivUnitsType
x✝ : (↥N)ˣ
⊢ (ofUnitHom ((ψ.domRestrict N.units).comp ↑N.unitsEquivUnitsType.symm)) ↑x✝ = χ ↑x✝

Tactic:
]

Goal state after:
M : Type u_1
R : Type u_2
inst✝³ : CommMonoid M
inst✝² : CommRing R
inst✝¹ : Finite M
inst✝ : HasEnoughRootsOfUnity R (Monoid.exponent Mˣ)
N : Submonoid M
χ : MulChar (↥N) R
ψ : Mˣ →* Rˣ
hψ : ψ.domRestrict N.units = χ.toUnitHom.comp ↑N.unitsEquivUnitsType
x✝ : (↥N)ˣ
⊢ (ofUnitHom ((ψ.domRestrict N.units).comp ↑N.unitsEquivUnitsType.symm)) ↑x✝ = χ ↑x✝
Module: Mathlib.NumberTheory.PellMatiyasevic

Goal state before:
pp: case refl.inl.inr
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
lin : i < n
ll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n
npm : (n - 1).succ = n
il : i ≤ n - 1
ile : i = n - 1
⊢ 2 * xn a1 (n - 1) ≠ xn a1 n

Tactic:
]

Goal state after:
case refl.inl.inr
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
lin : i < n
ll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n
npm : (n - 1).succ = n
il : i ≤ n - 1
ile : i = n - 1
⊢ 2 * xn a1 (n - 1) ≠ xn a1 n
Module: Mathlib.NumberTheory.PellMatiyasevic

Goal state before:
pp: a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
lin : i < n
ll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n
npm : (n - 1).succ = n
il : i ≤ n - 1
ile : i = n - 1
e : 2 * xn a1 (n - 1) = xn a1 n
⊢ 2 * xn a1 (n - 1) = xn a1 n

Tactic:
]

Goal state after:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
lin : i < n
ll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n
npm : (n - 1).succ = n
il : i ≤ n - 1
ile : i = n - 1
e : 2 * xn a1 (n - 1) = xn a1 n
⊢ 2 * xn a1 (n - 1) = xn a1 n
Module: Mathlib.NumberTheory.MulChar.Duality

Goal state before:
pp: M : Type u_1
R : Type u_2
inst✝³ : CommMonoid M
inst✝² : CommRing R
inst✝¹ : Finite M
inst✝ : HasEnoughRootsOfUnity R (Monoid.exponent Mˣ)
m : Mˣ
χ : MulChar M R
⊢ χ ↑m = χ ↑m

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.Dioph

Goal state before:
pp: α β : Type u
S : Set (α ⊕ β → ℕ)
γ : Type u
p : Poly ((α ⊕ β) ⊕ γ)
pe : ∀ (v : α ⊕ β → ℕ), v ∈ S ↔ ∃ t, p (v ⊗ t) = 0
v : α → ℕ
x✝ : ∃ t, (Poly.map ((inl ⊗ inr ∘ inl) ⊗ inr ∘ inr) p) (v ⊗ t) = 0
t : β ⊕ γ → ℕ
ht : p ((v ⊗ t ∘ inl) ⊗ t ∘ inr) = 0
⊢ p ((v ⊗ t ∘ inl) ⊗ t ∘ inr) = 0

Tactic:
]

Goal state after:
α β : Type u
S : Set (α ⊕ β → ℕ)
γ : Type u
p : Poly ((α ⊕ β) ⊕ γ)
pe : ∀ (v : α ⊕ β → ℕ), v ∈ S ↔ ∃ t, p (v ⊗ t) = 0
v : α → ℕ
x✝ : ∃ t, (Poly.map ((inl ⊗ inr ∘ inl) ⊗ inr ∘ inr) p) (v ⊗ t) = 0
t : β ⊕ γ → ℕ
ht : p ((v ⊗ t ∘ inl) ⊗ t ∘ inr) = 0
⊢ p ((v ⊗ t ∘ inl) ⊗ t ∘ inr) = 0
Module: Mathlib.NumberTheory.MulChar.Duality

Goal state before:
pp: M : Type u_1
R : Type u_2
inst✝³ : CommMonoid M
inst✝² : CommRing R
inst✝¹ : Finite M
inst✝ : HasEnoughRootsOfUnity R (Monoid.exponent Mˣ)
χ : MulChar M R
η : MulChar (MulChar M R) R
⊢ η χ = η χ

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.PellMatiyasevic

Goal state before:
pp: a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
lin : i < n
ll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n
npm : (n - 1).succ = n
il : i ≤ n - 1
ile : i = n - 1
e : 2 * xn a1 (n - 1) = xn a1 n
a2 : a = 2
s1 : n - 1 = 0
n1 : n = 1
⊢ 2 = 2 ∧ 1 = 1 ∧ 1 - 1 = 0 ∧ 1 + 1 = 2

Tactic:
]

Goal state after:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
lin : i < n
ll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 n
npm : (n - 1).succ = n
il : i ≤ n - 1
ile : i = n - 1
e : 2 * xn a1 (n - 1) = xn a1 n
a2 : a = 2
s1 : n - 1 = 0
n1 : n = 1
⊢ 2 = 2 ∧ 1 = 1 ∧ 1 - 1 = 0 ∧ 1 + 1 = 2
Module: Mathlib.NumberTheory.MulChar.Duality

Goal state before:
pp: M : Type u_1
R : Type u_2
inst✝³ : CommMonoid M
inst✝² : CommRing R
inst✝¹ : Finite M
inst✝ : HasEnoughRootsOfUnity R (Monoid.exponent Mˣ)
H : Subgroup Mˣ
χ : MulChar M R
⊢ mulEquivToUnitHom.symm.symm χ ∈ OrderDual.ofDual ((CommGroup.subgroupOrderIsoSubgroupMonoidHom Mˣ R) H) ↔
    ∀ m ∈ H, χ ↑m = 1

Tactic:
]

Goal state after:
M : Type u_1
R : Type u_2
inst✝³ : CommMonoid M
inst✝² : CommRing R
inst✝¹ : Finite M
inst✝ : HasEnoughRootsOfUnity R (Monoid.exponent Mˣ)
H : Subgroup Mˣ
χ : MulChar M R
⊢ mulEquivToUnitHom.symm.symm χ ∈ OrderDual.ofDual ((CommGroup.subgroupOrderIsoSubgroupMonoidHom Mˣ R) H) ↔
    ∀ m ∈ H, χ ↑m = 1
Module: Mathlib.NumberTheory.MulChar.Duality

Goal state before:
pp: M : Type u_1
R : Type u_2
inst✝³ : CommMonoid M
inst✝² : CommRing R
inst✝¹ : Finite M
inst✝ : HasEnoughRootsOfUnity R (Monoid.exponent Mˣ)
H : Subgroup Mˣ
⊢ Nat.card (Mˣ ⧸ H) = Nat.card (Mˣ ⧸ H)

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.PellMatiyasevic

Goal state before:
pp: case refl.inr
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
ein : i = n
⊢ xn a1 (n - 1) < xn a1 n

Tactic:
]

Goal state after:
case refl.inr
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
lem2 : ∀ k > n, k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))
ij : i < n + 1
j2n : n + 1 ≤ 2 * n
jnn : n + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)
o : n = n ∨ n < n
ein : i = n
⊢ xn a1 (n - 1) < xn a1 n
Module: Mathlib.NumberTheory.DiophantineApproximation.Basic

Goal state before:
pp: ξ : ℝ
n : ℕ
n_pos : 0 < n
j k : ℤ
hk₀ : 0 < ↑k.toNat
hk₁ : ↑k.toNat ≤ ↑n
h : |↑↑k.toNat * ξ - ↑j| ≤ 1 / (↑n + 1)
hk : ↑k.toNat = k
⊢ ∃ k, 0 < k ∧ k ≤ n ∧ |↑k * ξ - ↑(round (↑k * ξ))| ≤ 1 / (↑n + 1)

Tactic:
]

Goal state after:
ξ : ℝ
n : ℕ
n_pos : 0 < n
j k : ℤ
hk₀ : 0 < ↑k.toNat
hk₁ : ↑k.toNat ≤ ↑n
h : |↑↑k.toNat * ξ - ↑j| ≤ 1 / (↑n + 1)
hk : ↑k.toNat = k
⊢ ∃ k, 0 < k ∧ k ≤ n ∧ |↑k * ξ - ↑(round (↑k * ξ))| ≤ 1 / (↑n + 1)
Module: Mathlib.NumberTheory.Dioph

Goal state before:
pp: α : Type u
S : Set (Option α → ℕ)
β : Type u
p : Poly (Option α ⊕ β)
pe : ∀ (v : Option α → ℕ), v ∈ S ↔ ∃ t, p (v ⊗ t) = 0
v : α → ℕ
x✝ : v ∈ {v | ∃ x, x ::ₒ v ∈ S}
x : ℕ
hx : x ::ₒ v ∈ S
t : β → ℕ
ht : p (x ::ₒ v ⊗ t) = 0
⊢ p (x ::ₒ v ⊗ t) = 0

Tactic:
]

Goal state after:
α : Type u
S : Set (Option α → ℕ)
β : Type u
p : Poly (Option α ⊕ β)
pe : ∀ (v : Option α → ℕ), v ∈ S ↔ ∃ t, p (v ⊗ t) = 0
v : α → ℕ
x✝ : v ∈ {v | ∃ x, x ::ₒ v ∈ S}
x : ℕ
hx : x ::ₒ v ∈ S
t : β → ℕ
ht : p (x ::ₒ v ⊗ t) = 0
⊢ p (x ::ₒ v ⊗ t) = 0
Module: Mathlib.NumberTheory.PellMatiyasevic

Goal state before:
pp: a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
j : ℕ
ij : i < j + 1
j2n : 2 + 1 ≤ 2 * 1
jnn : j + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)
lem2 : ∀ k > n, k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))
o : j = n ∨ n < j
jn✝ : j > n
jn : j ≠ n
s : xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n
h : i < j
x✝ : a = 2 ∧ n = 1 ∧ i = 0 ∧ j = 2
left✝¹ : a = 2
n1 : n = 1
left✝ : i = 0
j2 : j = 2
⊢ False

Tactic:
]

Goal state after:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
j : ℕ
ij : i < j + 1
j2n : 2 + 1 ≤ 2 * 1
jnn : j + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)
lem2 : ∀ k > n, k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))
o : j = n ∨ n < j
jn✝ : j > n
jn : j ≠ n
s : xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n
h : i < j
x✝ : a = 2 ∧ n = 1 ∧ i = 0 ∧ j = 2
left✝¹ : a = 2
n1 : n = 1
left✝ : i = 0
j2 : j = 2
⊢ False
Module: Mathlib.NumberTheory.DiophantineApproximation.Basic

Goal state before:
pp: ξ : ℝ
n : ℕ
n_pos : 0 < n
j k : ℤ
hk₀ : 0 < k
hk₁ : k ≤ ↑n
h : |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)
hk₀' : 0 < ↑k
hden : ↑(↑j / ↑k).den ≤ k
⊢ |ξ - ↑(↑j / ↑k)| * ↑(↑j / ↑k).den ≤ 1 / (↑n + 1)

Tactic:
]

Goal state after:
ξ : ℝ
n : ℕ
n_pos : 0 < n
j k : ℤ
hk₀ : 0 < k
hk₁ : k ≤ ↑n
h : |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)
hk₀' : 0 < ↑k
hden : ↑(↑j / ↑k).den ≤ k
⊢ |ξ - ↑(↑j / ↑k)| * ↑(↑j / ↑k).den ≤ 1 / (↑n + 1)
Module: Mathlib.NumberTheory.PellMatiyasevic

Goal state before:
pp: a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
j : ℕ
ij : i < j + 1
j2n : j + 1 ≤ 2 * n
jnn : j + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)
lem2 : ∀ k > n, k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))
o : j = n ∨ n < j
jn✝ : j > n
jn : j ≠ n
s : xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n
h : i = j
⊢ xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n

Tactic:
]

Goal state after:
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
j : ℕ
ij : i < j + 1
j2n : j + 1 ≤ 2 * n
jnn : j + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)
lem2 : ∀ k > n, k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))
o : j = n ∨ n < j
jn✝ : j > n
jn : j ≠ n
s : xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n
h : i = j
⊢ xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n
Module: Mathlib.NumberTheory.DiophantineApproximation.Basic

Goal state before:
pp: ξ : ℝ
n : ℕ
n_pos : 0 < n
j k : ℤ
hk₀ : 0 < k
hk₁ : k ≤ ↑n
h : |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)
hk₀' : 0 < ↑k
hden : ↑(↑j / ↑k).den ≤ k
⊢ |ξ - ↑j / ↑k| * |↑k| ≤ 1 / (↑n + 1)

Tactic:
← abs_mul,

Goal state after:
ξ : ℝ
n : ℕ
n_pos : 0 < n
j k : ℤ
hk₀ : 0 < k
hk₁ : k ≤ ↑n
h : |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)
hk₀' : 0 < ↑k
hden : ↑(↑j / ↑k).den ≤ k
⊢ |(ξ - ↑j / ↑k) * ↑k| ≤ 1 / (↑n + 1)
Module: Mathlib.NumberTheory.DiophantineApproximation.Basic

Goal state before:
pp: ξ : ℝ
n : ℕ
n_pos : 0 < n
j k : ℤ
hk₀ : 0 < k
hk₁ : k ≤ ↑n
h : |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)
hk₀' : 0 < ↑k
hden : ↑(↑j / ↑k).den ≤ k
⊢ |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)

Tactic:
]

Goal state after:
ξ : ℝ
n : ℕ
n_pos : 0 < n
j k : ℤ
hk₀ : 0 < k
hk₁ : k ≤ ↑n
h : |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)
hk₀' : 0 < ↑k
hden : ↑(↑j / ↑k).den ≤ k
⊢ |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)