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

Goal state before:
pp: s : Set ℕ
h0 : ¬setGcd s = 0
t : Finset ℕ
hts : ↑t ⊆ s
a : ↥t → ℤ
eq : ∑ i, a i • ↑↑i = ↑(setGcd s)
x : ℕ
hxs : x ∈ s
hx : x ≠ 0
n : ℕ := x / setGcd s * ∑ i, (-a i).toNat * ↑i
c : ℕ
ge : n + c ≥ n
dvd : setGcd s ∣ n + c
q : ℕ := c / x
r : ℕ := c % x
rx : ℕ
hrx : r = setGcd s * rx
i : ↥t
a✝ : i ∈ Finset.univ
hai : 0 ≤ a i
⊢ -a i ≤ 0

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S' : Solution' hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
⊢ λ ^ 2 ∣ S'.a + S'.b ∨ λ ^ 2 ∣ S'.a + ↑η * S'.b ∨ λ ^ 2 ∣ S'.a + ↑η ^ 2 * S'.b

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FermatPsp

Goal state before:
pp: b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (b ^ 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
hi_A : 1 < A
hi_B : 1 < B
hi_b : 0 < b
hi_bsquared : 0 < b ^ 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)
hd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
ha₃ : p ∣ b ^ (p - 1) - 1
k : ℕ
hk : p = 2 * k + 1
c : ℕ
hc : p - 1 = 2 * c
this : b ^ 2 - 1 ∣ (b ^ 2) ^ c - 1
⊢ b ^ 2 - 1 ∣ b ^ (2 * c) - 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FrobeniusNumber

Goal state before:
pp: s : Set ℕ
h0 : ¬setGcd s = 0
t : Finset ℕ
hts : ↑t ⊆ s
a : ↥t → ℤ
eq : ∑ i, a i • ↑↑i = ↑(setGcd s)
x : ℕ
hxs : x ∈ s
hx : x ≠ 0
n : ℕ := x / setGcd s * ∑ i, (-a i).toNat * ↑i
c : ℕ
ge : n + c ≥ n
dvd : setGcd s ∣ n + c
q : ℕ := c / x
r : ℕ := c % x
rx : ℕ
hrx : r = setGcd s * rx
i : ↥t
a✝ : i ∈ Finset.univ
hai : ¬0 ≤ a i
⊢ a i ≤ 0

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FermatPsp

Goal state before:
pp: b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (b ^ 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
hi_A : 1 < A
hi_B : 1 < B
hi_b : 0 < b
hi_bsquared : 0 < b ^ 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
p_odd : Odd p
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)
hd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
ha₃ : p ∣ b ^ (p - 1) - 1
⊢ b ^ 2 - 1 ∣ b ^ (p - 1) - 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FrobeniusNumber

Goal state before:
pp: s : Set ℕ
h0 : ¬setGcd s = 0
t : Finset ℕ
hts : ↑t ⊆ s
a : ↥t → ℤ
eq : ∑ i, a i • ↑↑i = ↑(setGcd s)
x : ℕ
hxs : x ∈ s
hx : x ≠ 0
n : ℕ := x / setGcd s * ∑ i, (-a i).toNat * ↑i
c : ℕ
ge : n + c ≥ n
dvd : setGcd s ∣ n + c
q : ℕ := c / x
r : ℕ := c % x
rx : ℕ
hrx : r = setGcd s * rx
i : ↥t
a✝ : i ∈ Finset.univ
hai : ¬0 ≤ a i
⊢ 0 ≤ -a i

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Different

Goal state before:
pp: A : Type u_1
B : Type u_3
inst✝¹¹ : CommRing A
inst✝¹⁰ : CommRing B
inst✝⁹ : Algebra A B
inst✝⁸ : IsDomain A
inst✝⁷ : IsDedekindDomain A
inst✝⁶ : IsDedekindDomain B
inst✝⁵ : IsTorsionFree A B
inst✝⁴ : Module.Finite A B
inst✝³ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
p : Ideal A
inst✝² : p.IsMaximal
hp : p ≠ ⊥
P : Ideal B
inst✝¹ : P.IsMaximal
inst✝ : P.LiesOver p
H : ¬Algebra.IsSeparable (A ⧸ p) (B ⧸ P)
⊢ P ∣ differentIdeal A B

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FermatPsp

Goal state before:
pp: b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (b ^ 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
hi_A : 1 < A
hi_B : 1 < B
hi_b : 0 < b
hi_bsquared : 0 < b ^ 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
p_odd : Odd p
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)
hd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
ha₃ : p ∣ b ^ (p - 1) - 1
ha₄ : b ^ 2 - 1 ∣ b ^ (p - 1) - 1
q : 2 * p * (b ^ 2 - 1) ∣ b * (b ^ (p - 1) - 1) * (b ^ p + b)
⊢ 2 * p * (b ^ 2 - 1) ∣ (b ^ 2 - 1) * (A * B - 1)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Different

Goal state before:
pp: A : Type u_1
B : Type u_3
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : IsDomain A
inst✝⁵ : IsDedekindDomain A
inst✝⁴ : IsDedekindDomain B
inst✝³ : IsTorsionFree A B
inst✝² : Module.Finite A B
inst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
inst✝ : ⊥.IsPrime
K : Type u_1 := FractionRing A
L : Type u_3 := FractionRing B
⊢ IsLocalization B⁰ (Localization.AtPrime ⊥)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Different

Goal state before:
pp: A : Type u_1
B : Type u_3
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : IsDomain A
inst✝⁵ : IsDedekindDomain A
inst✝⁴ : IsDedekindDomain B
inst✝³ : IsTorsionFree A B
inst✝² : Module.Finite A B
inst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
inst✝ : ⊥.IsPrime
K : Type u_1 := FractionRing A
L : Type u_3 := FractionRing B
this : IsLocalization B⁰ (Localization.AtPrime ⊥)
⊢ FormallyUnramified K L

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FermatPsp

Goal state before:
pp: b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (b ^ 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
hi_A : 1 < A
hi_B : 1 < B
hi_b : 0 < b
hi_bsquared : 0 < b ^ 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
p_odd : Odd p
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)
hd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
ha₃ : p ∣ b ^ (p - 1) - 1
ha₄ : b ^ 2 - 1 ∣ b ^ (p - 1) - 1
⊢ ¬p ∣ b ^ 2 - 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FrobeniusNumber

Goal state before:
pp: s : Set ℕ
⊢ ∃ t n, ↑t ⊆ s ∧ ∀ m ≥ n, setGcd s ∣ m → m ∈ span ↑t

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FermatPsp

Goal state before:
pp: b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (b ^ 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
hi_A : 1 < A
hi_B : 1 < B
hi_b : 0 < b
hi_bsquared : 0 < b ^ 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
p_odd : Odd p
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)
hd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
ha₃ : p ∣ b ^ (p - 1) - 1
ha₄ : b ^ 2 - 1 ∣ b ^ (p - 1) - 1
⊢ 2 * p * (b ^ 2 - 1) ∣ (b ^ 2 - 1) * (A * B - 1)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FrobeniusNumber

Goal state before:
pp: m✝ n✝¹ : ℕ
s✝ t✝ : Set ℕ
n✝ : ℕ
s : Submodule ℕ ℕ
t : Finset ℕ
n : ℕ
hts : ↑t ⊆ ↑s
hn : ∀ m ≥ n, setGcd ↑s ∣ m → m ∈ Ideal.span ↑t
m : ℕ
hm : m ∈ s
le : n ≤ m
⊢ ↑t ⊆ ↑(t ∪ {m ∈ Finset.range n | m ∈ s})

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FermatPsp

Goal state before:
pp: b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (b ^ 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
hi_A : 1 < A
hi_B : 1 < B
hi_b : 0 < b
hi_bsquared : 0 < b ^ 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
p_odd : Odd p
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)
hd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
ha₃ : p ∣ b ^ (p - 1) - 1
ha₄ : b ^ 2 - 1 ∣ b ^ (p - 1) - 1
ha₅ : 2 * p * (b ^ 2 - 1) ∣ (b ^ 2 - 1) * (A * B - 1)
⊢ 2 * p ∣ A * B - 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FrobeniusNumber

Goal state before:
pp: m✝ n✝¹ : ℕ
s✝ t✝ : Set ℕ
n✝ : ℕ
s : Submodule ℕ ℕ
t : Finset ℕ
n : ℕ
hts : ↑t ⊆ ↑s
hn : ∀ m ≥ n, setGcd ↑s ∣ m → m ∈ Ideal.span ↑t
m : ℕ
hm : m ∈ s
gt : m < n
⊢ m ∈ ↑(t ∪ {m ∈ Finset.range n | m ∈ s})

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S' : Solution' hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
⊢ ∃ a' b', a' ^ 3 + b' ^ 3 = ↑S'.u * S'.c ^ 3 ∧ IsCoprime a' b' ∧ ¬λ ∣ a' ∧ ¬λ ∣ b' ∧ λ ^ 2 ∣ a' + b'

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FrobeniusNumber

Goal state before:
pp: m n✝ : ℕ
s✝ t : Set ℕ
n : ℕ
s : Submodule ℕ ℕ
⊢ s.FG

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S' : Solution' hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
⊢ ∃ S₁, S₁.multiplicity = S'.multiplicity

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FrobeniusNumber

Goal state before:
pp: s : AddSubmonoid ℕ
⊢ s.FG

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FermatPsp

Goal state before:
pp: b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (b ^ 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
hi_A : 1 < A
hi_B : 1 < B
hi_b : 0 < b
hi_bsquared : 0 < b ^ 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
p_odd : Odd p
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)
hd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
ha₃ : p ∣ b ^ (p - 1) - 1
ha₄ : b ^ 2 - 1 ∣ b ^ (p - 1) - 1
ha₅ : 2 * p * (b ^ 2 - 1) ∣ (b ^ 2 - 1) * (A * B - 1)
ha₆ : 2 * p ∣ A * B - 1
⊢ A * B ∣ b ^ (2 * p) - 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝ : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
⊢ S.a + ↑η * S.b = S.a + S.b + λ * S.b

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝ : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
⊢ 2 ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FrobeniusNumber

Goal state before:
pp: s : Set ℕ
x✝ : ∃ n, FrobeniusNumber n s
n : ℕ
hn : n ∉ AddSubmonoid.closure s ∧ ∀ k > n, k ∈ AddSubmonoid.closure s
⊢ n + 1 > n

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝ : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
⊢ λ * S.b = λ * S.b

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FrobeniusNumber

Goal state before:
pp: s : Set ℕ
x✝ : ∃ n, FrobeniusNumber n s
n : ℕ
hn : n ∉ AddSubmonoid.closure s ∧ ∀ k > n, k ∈ AddSubmonoid.closure s
⊢ n + 2 > n

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FrobeniusNumber

Goal state before:
pp: s : Set ℕ
x✝ : ∃ n, FrobeniusNumber n s
n : ℕ
hn : FrobeniusNumber n s
⊢ setGcd s = 1 ∧ 1 ∉ s

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Different

Goal state before:
pp: A : Type u_1
B : Type u_3
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : IsDomain A
inst✝⁵ : IsDedekindDomain A
inst✝⁴ : IsDedekindDomain B
inst✝³ : IsTorsionFree A B
inst✝² : Module.Finite A B
inst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
P : Ideal B
inst✝ : P.IsPrime
hPbot : P ≠ ⊥
hp : Ideal.under A P ≠ ⊥
hp' : ¬Ideal.map (algebraMap A B) (Ideal.under A P) = ⊥
this✝¹ : P.IsMaximal
this✝ : Algebra (Localization.AtPrime (Ideal.under A P)) (Localization.AtPrime P) :=
  Localization.AtPrime.algebraOfLiesOver (Ideal.under A P) P
H : ¬P ∣ differentIdeal A B
this : Algebra.IsSeparable (A ⧸ Ideal.under A P) (B ⧸ P)
⊢ Algebra.IsSeparable (Ideal.under A P).ResidueField P.ResidueField

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝ : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
⊢ λ ^ 2 * S.b = λ * (λ * S.b)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝ : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
⊢ 2 * λ * S.b = λ * (2 * S.b)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝ : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
⊢ λ ∣ S.a + ↑η ^ 2 * S.b

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FermatPsp

Goal state before:
pp: b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (b ^ 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
hi_A : 1 < A
hi_B : 1 < B
hi_b : 0 < b
hi_bsquared : 0 < b ^ 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
p_odd : Odd p
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)
hd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
ha₃ : p ∣ b ^ (p - 1) - 1
ha₄ : b ^ 2 - 1 ∣ b ^ (p - 1) - 1
ha₅ : 2 * p * (b ^ 2 - 1) ∣ (b ^ 2 - 1) * (A * B - 1)
ha₇ : A * B ∣ b ^ (2 * p) - 1
q : ℕ
hq : A * B - 1 = 2 * p * q
⊢ b ^ (2 * p) - 1 ∣ b ^ (A * B - 1) - 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FrobeniusNumber

Goal state before:
pp: s : Set ℕ
h : setGcd s = 1 ∧ 1 ∉ s
⊢ ∃ n, FrobeniusNumber n s

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
⊢ ¬λ ^ 2 ∣ S.a + ↑η * S.b

Tactic:
by

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

Goal state before:
pp: n : ℕ
⊢ harmonic n = ∑ i ∈ Finset.Icc 1 n, (↑i)⁻¹

Tactic:
by

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

Goal state before:
pp: n : ℕ
⊢ 0 ∉ Set.uIcc 1 ↑(n + 1)

Tactic:
by

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

Goal state before:
pp: n : ℕ
⊢ 0 < ↑1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FermatPsp

Goal state before:
pp: b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (b ^ 2 - 1)
⊢ (b.psp_from_prime p).FermatPsp b

Tactic:
by

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

Goal state before:
pp: n : ℕ
⊢ Real.log ↑(n + 1) ≤ ↑(harmonic n)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FermatPsp

Goal state before:
pp: b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_gt_two : 2 < p
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
⊢ 0 < B

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FermatPsp

Goal state before:
pp: b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_gt_two : 2 < p
⊢ p < b.psp_from_prime p

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FermatPsp

Goal state before:
pp: b : ℕ
h : 1 ≤ b
m : ℕ
b_ge_two : 2 ≤ b
p : ℕ
hp₁ : b * (b ^ 2 - 1) + 1 + m ≤ p
hp₂ : Prime p
h₁ : 0 < b
h₂ : 4 ≤ b ^ 2
⊢ 1 < 4

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FermatPsp

Goal state before:
pp: b : ℕ
h : 1 ≤ b
m : ℕ
b_ge_two : 2 ≤ b
p : ℕ
hp₁ : b * (b ^ 2 - 1) + 1 + m ≤ p
hp₂ : Prime p
h₁ : 0 < b
h₂ : 4 ≤ b ^ 2
h₃ : 0 < b ^ 2 - 1
h₄ : 0 < b * (b ^ 2 - 1)
⊢ b * (b ^ 2 - 1) < p

Tactic:
by

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

Goal state before:
pp: n : ℕ
hn0 : ¬n = 0
hn : 1 ≤ n
⊢ 2 ≤ n + 1

Tactic:
by

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

Goal state before:
pp: n : ℕ
hn0 : ¬n = 0
hn : 1 ≤ n
⊢ 1 < ↑2

Tactic:
by

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

Goal state before:
pp: n : ℕ
⊢ ↑(harmonic n) ≤ 1 + Real.log ↑n

Tactic:
by

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

Goal state before:
pp: y : ℝ
hy : 0 ≤ y
⊢ Real.log y ≤ ↑(harmonic ⌊y⌋₊)

Tactic:
by

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

Goal state before:
pp: y : ℝ
hy : 1 ≤ y
⊢ ↑(harmonic ⌊y⌋₊) ≤ 1 + Real.log y

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FermatPsp

Goal state before:
pp: b : ℕ
h : 1 ≤ b
m : ℕ
b_ge_two : ¬2 ≤ b
⊢ b = 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FermatPsp

Goal state before:
pp: b : ℕ
h : 1 ≤ b
m : ℕ
b_ge_two : ¬2 ≤ b
h₁ : b = 1
⊢ 2 ≠ 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FermatPsp

Goal state before:
pp: b : ℕ
h : 1 ≤ b
m : ℕ
b_ge_two : ¬2 ≤ b
h₁ : b = 1
⊢ m + 2 ≠ 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FermatPsp

Goal state before:
pp: b : ℕ
h : 1 ≤ b
m : ℕ
b_ge_two : ¬2 ≤ b
h₁ : b = 1
this : ¬Prime (2 * (m + 2))
⊢ 1 < 2 * (m + 2)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FermatPsp

Goal state before:
pp: b : ℕ
h : 1 ≤ b
m : ℕ
b_ge_two : ¬2 ≤ b
h₁ : b = 1
this : ¬Prime (2 * (m + 2))
⊢ m ≤ 2 * (m + 2)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Harmonic.Int

Goal state before:
pp: n : ℕ
Hn : n ≠ 0
⊢ 0 < harmonic n

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Harmonic.Int

Goal state before:
pp: n : ℕ
ih : padicValRat 2 (harmonic n) = -↑(Nat.log 2 n)
hn : n ≠ 0
⊢ padicValRat 2 (harmonic n) ≠ padicValRat 2 (↑(n + 1))⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FermatPsp

Goal state before:
pp: b : ℕ
h : 1 ≤ b
m : ℕ
⊢ ∃ n, n.FermatPsp b ∧ m ≤ n

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FermatPsp

Goal state before:
pp: b : ℕ
h : 1 ≤ b
⊢ ∃ᶠ (n : ℕ) in Filter.atTop, n.FermatPsp b

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Harmonic.Int

Goal state before:
pp: n : ℕ
⊢ padicValRat 2 (harmonic n) = -↑(Nat.log 2 n)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Harmonic.Int

Goal state before:
pp: n : ℕ
hn : n ≠ 0
⊢ ‖↑(harmonic n)‖ = 2 ^ Nat.log 2 n

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Harmonic.Int

Goal state before:
pp: n : ℕ
h : 2 ≤ n
⊢ ¬(harmonic n).isInt = true

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.HeckeRing.Defs

Goal state before:
pp: G : Type u_1
inst✝ : Group G
Δ : Submonoid G
H₁ H₂ : Subgroup G
h : IsHeckeTriple Δ H₁ H₂
⊢ Δ ≤ (commensurator H₁).toSubmonoid

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.HeckeRing.Defs

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
Δ : Submonoid G
H₁ H₂ : Subgroup G
inst✝ : IsHeckeTriple Δ H₁ H₂
g : ↥Δ
⊢ (ConjAct.toConjAct ↑g • H₂).Commensurable H₁

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
⊢ ¬λ ^ 2 ∣ S.a + ↑η ^ 2 * S.b

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
p : 𝓞 K
hp : Prime p
hpab : p ∣ S.a + S.b
hpaηb : p ∣ S.a + ↑η * S.b
⊢ Associated p λ

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
p : 𝓞 K
hp : Prime p
hpab : p ∣ S.a + S.b
hpaηsqb : p ∣ S.a + ↑η ^ 2 * S.b
⊢ Associated p λ

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
p : 𝓞 K
hp : Prime p
hpaηb : p ∣ 1 * S.a + ↑η * S.b
hpaηsqb : p ∣ 1 * S.a + ↑η ^ 2 * S.b
this : p ∣ ↑η ^ 2 - ↑η
⊢ (↑η ^ 2 - ↑η) * ↑η * ↑η = (-↑η - 1 - ↑η) * (-↑η - 1)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
p : 𝓞 K
hp : Prime p
hpaηb : p ∣ 1 * S.a + ↑η * S.b
hpaηsqb : p ∣ 1 * S.a + ↑η ^ 2 * S.b
this : p ∣ ↑η ^ 2 - ↑η
⊢ (-↑η - 1 - ↑η) * (-↑η - 1) = 2 * ↑η ^ 2 + 3 * ↑η + 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
p : 𝓞 K
hp : Prime p
hpaηb : p ∣ 1 * S.a + ↑η * S.b
hpaηsqb : p ∣ 1 * S.a + ↑η ^ 2 * S.b
this : p ∣ ↑η ^ 2 - ↑η
⊢ 2 * ↑η ^ 2 + 3 * ↑η + 1 = λ

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
p : 𝓞 K
hp : Prime p
hpaηb : p ∣ S.a + ↑η * S.b
hpaηsqb : p ∣ S.a + ↑η ^ 2 * S.b
⊢ Associated p λ

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
h : λ ∣ S.y
⊢ False

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
h : λ ∣ S.z
⊢ False

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
h : λ ^ S.multiplicity ∣ S.c
⊢ (λ ^ S.multiplicity) ^ 3 ∣ ↑S.u * S.c ^ 3

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
⊢ λ ^ (3 * S.multiplicity - 2) ∣ S.a + S.b

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
h : λ ∣ S.w
⊢ False

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
h : λ ∣ S.x
⊢ False

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
r s t w : 𝓞 K
hr : ¬λ ∣ r
hs : ¬λ ∣ s
Hp : ∀ {p : 𝓞 K}, Prime p → p ∣ t → p ∣ w → Associated p λ
H₁ : ∀ {q : 𝓞 K}, q ∣ r → q ∣ t
H₂ : ∀ {q : 𝓞 K}, q ∣ s → q ∣ w
x✝ : r = 0
hz : s = 0
⊢ λ ∣ s

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
r s t w : 𝓞 K
hr : ¬λ ∣ r
hs : ¬λ ∣ s
Hp : ∀ {p : 𝓞 K}, Prime p → p ∣ t → p ∣ w → Associated p λ
H₁ : ∀ {q : 𝓞 K}, q ∣ r → q ∣ t
H₂ : ∀ {q : 𝓞 K}, q ∣ s → q ∣ w
⊢ IsCoprime r s

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
hh : λ ^ (3 * S.multiplicity - 2) * S.x * λ * S.y * λ * S.z = ↑S.u * λ ^ (3 * S.multiplicity) * S.w ^ 3
⊢ S.x * S.y * S.z = ↑S.u * S.w ^ 3

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
⊢ S.x * S.y * S.z = ↑S.u * S.w ^ 3

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.FLT.Three

Goal state before:
pp: K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
h₁ : IsCoprime (S.x * S.y) S.z
⊢ S.w ^ 3 * ↑S.u = S.x * S.y * S.z

Tactic:
by

Goal state after: