mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Adjoin.PowerBasis

Goal state before:
pp: case neg
S : Type u_2
inst✝⁶ : CommRing S
R : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : Algebra R S
A : Type u_4
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : Algebra S A
inst✝ : IsScalarTower R S A
B : PowerBasis S A
hB : IsIntegral R B.gen
x : A
hmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)
n : ℕ
a✝¹ : Nontrivial A
a✝ : ∀ (i : Fin B.dim), IsIntegral R ((B.basis.repr x) i)
i : Fin B.dim
h✝ : ¬⟨0, ⋯⟩ = i
⊢ IsIntegral R 0

Tactic:
·

Goal state after:
case neg
S : Type u_2
inst✝⁶ : CommRing S
R : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : Algebra R S
A : Type u_4
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : Algebra S A
inst✝ : IsScalarTower R S A
B : PowerBasis S A
hB : IsIntegral R B.gen
x : A
hmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)
n : ℕ
a✝¹ : Nontrivial A
a✝ : ∀ (i : Fin B.dim), IsIntegral R ((B.basis.repr x) i)
i : Fin B.dim
h✝ : ¬⟨0, ⋯⟩ = i
⊢ IsIntegral R 0
Module: Mathlib.RingTheory.Adjoin.PowerBasis

Goal state before:
pp: case refine_2
S : Type u_2
inst✝⁶ : CommRing S
R : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : Algebra R S
A : Type u_4
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : Algebra S A
inst✝ : IsScalarTower R S A
B : PowerBasis S A
hB : IsIntegral R B.gen
x : A
hmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)
n✝ : ℕ
a✝ : Nontrivial A
n : ℕ
hn :
  ∀ m ≤ n,
    (∀ (i : Fin B.dim), IsIntegral R ((B.basis.repr x) i)) → ∀ (i : Fin B.dim), IsIntegral R ((B.basis.repr (x ^ m)) i)
⊢ (∀ (i : Fin B.dim), IsIntegral R ((B.basis.repr x) i)) →
    ∀ (i : Fin B.dim), IsIntegral R ((B.basis.repr (x ^ (n + 1))) i)

Tactic:
·

Goal state after:
case refine_2
S : Type u_2
inst✝⁶ : CommRing S
R : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : Algebra R S
A : Type u_4
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : Algebra S A
inst✝ : IsScalarTower R S A
B : PowerBasis S A
hB : IsIntegral R B.gen
x : A
hmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)
n✝ : ℕ
a✝ : Nontrivial A
n : ℕ
hn :
  ∀ m ≤ n,
    (∀ (i : Fin B.dim), IsIntegral R ((B.basis.repr x) i)) → ∀ (i : Fin B.dim), IsIntegral R ((B.basis.repr (x ^ m)) i)
⊢ (∀ (i : Fin B.dim), IsIntegral R ((B.basis.repr x) i)) →
    ∀ (i : Fin B.dim), IsIntegral R ((B.basis.repr (x ^ (n + 1))) i)
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Expand

Goal state before:
pp: case inl
p : ℕ
hp : Nat.Prime p
R : Type u_1
inst✝ : CommRing R
hdiv : ¬p ∣ 0
⊢ (expand R p) (cyclotomic 0 R) = cyclotomic (0 * p) R * cyclotomic 0 R

pp: case inr
p n : ℕ
hp : Nat.Prime p
hdiv : ¬p ∣ n
R : Type u_1
inst✝ : CommRing R
hnpos : n > 0
⊢ (expand R p) (cyclotomic n R) = cyclotomic (n * p) R * cyclotomic n R

Tactic:
·

Goal state after:
case inl
p : ℕ
hp : Nat.Prime p
R : Type u_1
inst✝ : CommRing R
hdiv : ¬p ∣ 0
⊢ (expand R p) (cyclotomic 0 R) = cyclotomic (0 * p) R * cyclotomic 0 R
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Expand

Goal state before:
pp: case inr.refine_1
p n : ℕ
hp : Nat.Prime p
hdiv : ¬p ∣ n
R : Type u_1
inst✝ : CommRing R
hnpos : n > 0
this : NeZero n
⊢ cyclotomic (n * p) ℤ * cyclotomic n ℤ ∣ (expand ℤ p) (cyclotomic n ℤ)

pp: case inr.refine_2
p n : ℕ
hp : Nat.Prime p
hdiv : ¬p ∣ n
R : Type u_1
inst✝ : CommRing R
hnpos : n > 0
this : NeZero n
⊢ ((expand ℤ p) (cyclotomic n ℤ)).natDegree ≤ (cyclotomic (n * p) ℤ * cyclotomic n ℤ).natDegree

Tactic:
·

Goal state after:
case inr.refine_1
p n : ℕ
hp : Nat.Prime p
hdiv : ¬p ∣ n
R : Type u_1
inst✝ : CommRing R
hnpos : n > 0
this : NeZero n
⊢ cyclotomic (n * p) ℤ * cyclotomic n ℤ ∣ (expand ℤ p) (cyclotomic n ℤ)
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Expand

Goal state before:
pp: case inr.refine_1.refine_1
p n : ℕ
hp : Nat.Prime p
hdiv : ¬p ∣ n
R : Type u_1
inst✝ : CommRing R
hnpos : n > 0
this : NeZero n
h : n * p = n
⊢ False

pp: case inr.refine_1.refine_2
p n : ℕ
hp : Nat.Prime p
hdiv : ¬p ∣ n
R : Type u_1
inst✝ : CommRing R
hnpos : n > 0
this : NeZero n
⊢ cyclotomic (n * p) ℚ ∣ (expand ℚ p) (cyclotomic n ℚ)

pp: case inr.refine_1.refine_3
p n : ℕ
hp : Nat.Prime p
hdiv : ¬p ∣ n
R : Type u_1
inst✝ : CommRing R
hnpos : n > 0
this : NeZero n
⊢ cyclotomic n ℚ ∣ (expand ℚ p) (cyclotomic n ℚ)

Tactic:
·

Goal state after:
case inr.refine_1.refine_1
p n : ℕ
hp : Nat.Prime p
hdiv : ¬p ∣ n
R : Type u_1
inst✝ : CommRing R
hnpos : n > 0
this : NeZero n
h : n * p = n
⊢ False
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Expand

Goal state before:
pp: case inr.refine_1.refine_2
p n : ℕ
hp : Nat.Prime p
hdiv : ¬p ∣ n
R : Type u_1
inst✝ : CommRing R
hnpos : n > 0
this : NeZero n
⊢ cyclotomic (n * p) ℚ ∣ (expand ℚ p) (cyclotomic n ℚ)

pp: case inr.refine_1.refine_3
p n : ℕ
hp : Nat.Prime p
hdiv : ¬p ∣ n
R : Type u_1
inst✝ : CommRing R
hnpos : n > 0
this : NeZero n
⊢ cyclotomic n ℚ ∣ (expand ℚ p) (cyclotomic n ℚ)

Tactic:
·

Goal state after:
case inr.refine_1.refine_2
p n : ℕ
hp : Nat.Prime p
hdiv : ¬p ∣ n
R : Type u_1
inst✝ : CommRing R
hnpos : n > 0
this : NeZero n
⊢ cyclotomic (n * p) ℚ ∣ (expand ℚ p) (cyclotomic n ℚ)
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Eval

Goal state before:
pp: case h.inl
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
h : 0 < ∑ i ∈ range n, x ^ i
⊢ 0 < eval x (cyclotomic n R)

pp: case h.inr.inl
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
h : 0 = ∑ i ∈ range n, x ^ i
⊢ 0 < eval x (cyclotomic n R)

pp: case h.inr.inr
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
h : ∑ i ∈ range n, x ^ i < 0
⊢ 0 < eval x (cyclotomic n R)

Tactic:
·

Goal state after:
case h.inl
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
h : 0 < ∑ i ∈ range n, x ^ i
⊢ 0 < eval x (cyclotomic n R)
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Eval

Goal state before:
pp: case h
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
h : 0 < ∑ i ∈ range n, x ^ i
⊢ 0 < eval x (cyclotomic n R) * ?h.inl.b

pp: case h.inl.hb
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
h : 0 < ∑ i ∈ range n, x ^ i
⊢ 0 ≤ ?h.inl.b

pp: case h.inl.b
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
h : 0 < ∑ i ∈ range n, x ^ i
⊢ R

Tactic:
·

Goal state after:
case h
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
h : 0 < ∑ i ∈ range n, x ^ i
⊢ 0 < eval x (cyclotomic n R) * ?h.inl.b
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Expand

Goal state before:
pp: case inr.refine_1.refine_3
p n : ℕ
hp : Nat.Prime p
hdiv : ¬p ∣ n
R : Type u_1
inst✝ : CommRing R
hnpos : n > 0
this : NeZero n
⊢ cyclotomic n ℚ ∣ (expand ℚ p) (cyclotomic n ℚ)

Tactic:
·

Goal state after:
case inr.refine_1.refine_3
p n : ℕ
hp : Nat.Prime p
hdiv : ¬p ∣ n
R : Type u_1
inst✝ : CommRing R
hnpos : n > 0
this : NeZero n
⊢ cyclotomic n ℚ ∣ (expand ℚ p) (cyclotomic n ℚ)
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Expand

Goal state before:
pp: case inr.refine_2
p n : ℕ
hp : Nat.Prime p
hdiv : ¬p ∣ n
R : Type u_1
inst✝ : CommRing R
hnpos : n > 0
this : NeZero n
⊢ ((expand ℤ p) (cyclotomic n ℤ)).natDegree ≤ (cyclotomic (n * p) ℤ * cyclotomic n ℤ).natDegree

Tactic:
·

Goal state after:
case inr.refine_2
p n : ℕ
hp : Nat.Prime p
hdiv : ¬p ∣ n
R : Type u_1
inst✝ : CommRing R
hnpos : n > 0
this : NeZero n
⊢ ((expand ℤ p) (cyclotomic n ℤ)).natDegree ≤ (cyclotomic (n * p) ℤ * cyclotomic n ℤ).natDegree
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Eval

Goal state before:
pp: case h.inl.hb.inl
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
i : ℕ
hi : i ≠ 1 ∧ i ∣ n ∧ i < n
hk : Odd n
⊢ 0 ≤ eval x (cyclotomic i R)

pp: case h.inl.hb.inr
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
i : ℕ
hi : i ≠ 1 ∧ i ∣ n ∧ i < n
hx : 0 < x + 1
⊢ 0 ≤ eval x (cyclotomic i R)

Tactic:
·

Goal state after:
case h.inl.hb.inl
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
i : ℕ
hi : i ≠ 1 ∧ i ∣ n ∧ i < n
hk : Odd n
⊢ 0 ≤ eval x (cyclotomic i R)
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Eval

Goal state before:
pp: case h.inl.hb.inl.refine_1
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
hk : Odd n
hi : 0 ≠ 1 ∧ 0 ∣ n ∧ 0 < n
⊢ False

pp: case h.inl.hb.inl.refine_2
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
hk : Odd n
hi : 2 ≠ 1 ∧ 2 ∣ n ∧ 2 < n
⊢ False

Tactic:
·

Goal state after:
case h.inl.hb.inl.refine_1
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
hk : Odd n
hi : 0 ≠ 1 ∧ 0 ∣ n ∧ 0 < n
⊢ False
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Eval

Goal state before:
pp: case h.inl.hb.inl.refine_2
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
hk : Odd n
hi : 2 ≠ 1 ∧ 2 ∣ n ∧ 2 < n
⊢ False

Tactic:
·

Goal state after:
case h.inl.hb.inl.refine_2
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
hk : Odd n
hi : 2 ≠ 1 ∧ 2 ∣ n ∧ 2 < n
⊢ False
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Eval

Goal state before:
pp: case h.inl.hb.inr
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
i : ℕ
hi : i ≠ 1 ∧ i ∣ n ∧ i < n
hx : 0 < x + 1
⊢ 0 ≤ eval x (cyclotomic i R)

Tactic:
·

Goal state after:
case h.inl.hb.inr
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
i : ℕ
hi : i ≠ 1 ∧ i ∣ n ∧ i < n
hx : 0 < x + 1
⊢ 0 ≤ eval x (cyclotomic i R)
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Expand

Goal state before:
pp: case inl
p : ℕ
hp : Nat.Prime p
R : Type u_1
inst✝ : CommRing R
hdiv : p ∣ 0
⊢ (expand R p) (cyclotomic 0 R) = cyclotomic (0 * p) R

pp: case inr
p n : ℕ
hp : Nat.Prime p
hdiv : p ∣ n
R : Type u_1
inst✝ : CommRing R
hzero : n > 0
⊢ (expand R p) (cyclotomic n R) = cyclotomic (n * p) R

Tactic:
·

Goal state after:
case inl
p : ℕ
hp : Nat.Prime p
R : Type u_1
inst✝ : CommRing R
hdiv : p ∣ 0
⊢ (expand R p) (cyclotomic 0 R) = cyclotomic (0 * p) R
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Eval

Goal state before:
pp: case h.inl.hb.inr.inl
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
hx : 0 < x + 1
hi : 2 ≠ 1 ∧ 2 ∣ n ∧ 2 < n
⊢ 0 ≤ eval x (cyclotomic 2 R)

pp: case h.inl.hb.inr.inr
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
i : ℕ
hi : i ≠ 1 ∧ i ∣ n ∧ i < n
hx : 0 < x + 1
hk : i ≠ 2
⊢ 0 ≤ eval x (cyclotomic i R)

Tactic:
·

Goal state after:
case h.inl.hb.inr.inl
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
hx : 0 < x + 1
hi : 2 ≠ 1 ∧ 2 ∣ n ∧ 2 < n
⊢ 0 ≤ eval x (cyclotomic 2 R)
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Expand

Goal state before:
pp: case inr.refine_1
p n : ℕ
hp : Nat.Prime p
hdiv : p ∣ n
R : Type u_1
inst✝ : CommRing R
hzero : n > 0
this : NeZero n
⊢ cyclotomic (n * p) ℤ ∣ (expand ℤ p) (cyclotomic n ℤ)

pp: case inr.refine_2
p n : ℕ
hp : Nat.Prime p
hdiv : p ∣ n
R : Type u_1
inst✝ : CommRing R
hzero : n > 0
this : NeZero n
⊢ ((expand ℤ p) (cyclotomic n ℤ)).natDegree ≤ (cyclotomic (n * p) ℤ).natDegree

Tactic:
·

Goal state after:
case inr.refine_1
p n : ℕ
hp : Nat.Prime p
hdiv : p ∣ n
R : Type u_1
inst✝ : CommRing R
hzero : n > 0
this : NeZero n
⊢ cyclotomic (n * p) ℤ ∣ (expand ℤ p) (cyclotomic n ℤ)
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Eval

Goal state before:
pp: case h.inr.inl
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
h : 0 = ∑ i ∈ range n, x ^ i
⊢ 0 < eval x (cyclotomic n R)

pp: case h.inr.inr
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
h : ∑ i ∈ range n, x ^ i < 0
⊢ 0 < eval x (cyclotomic n R)

Tactic:
·

Goal state after:
case h.inr.inl
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
h : 0 = ∑ i ∈ range n, x ^ i
⊢ 0 < eval x (cyclotomic n R)
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Expand

Goal state before:
pp: case inr.refine_2
p n : ℕ
hp : Nat.Prime p
hdiv : p ∣ n
R : Type u_1
inst✝ : CommRing R
hzero : n > 0
this : NeZero n
⊢ ((expand ℤ p) (cyclotomic n ℤ)).natDegree ≤ (cyclotomic (n * p) ℤ).natDegree

Tactic:
·

Goal state after:
case inr.refine_2
p n : ℕ
hp : Nat.Prime p
hdiv : p ∣ n
R : Type u_1
inst✝ : CommRing R
hzero : n > 0
this : NeZero n
⊢ ((expand ℤ p) (cyclotomic n ℤ)).natDegree ≤ (cyclotomic (n * p) ℤ).natDegree
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Eval

Goal state before:
pp: case h.inr.inr
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
h : ∑ i ∈ range n, x ^ i < 0
⊢ 0 < eval x (cyclotomic n R)

Tactic:
·

Goal state after:
case h.inr.inr
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
h : ∑ i ∈ range n, x ^ i < 0
⊢ 0 < eval x (cyclotomic n R)
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Eval

Goal state before:
pp: case h
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
h : ∑ i ∈ range n, x ^ i < 0
⊢ eval x (cyclotomic n R) * ?h.inr.inr.b < 0

pp: case hb
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
h : ∑ i ∈ range n, x ^ i < 0
⊢ ?h.inr.inr.b ≤ 0

pp: case h.inr.inr.b
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
h : ∑ i ∈ range n, x ^ i < 0
⊢ R

Tactic:
·

Goal state after:
case h
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
h : ∑ i ∈ range n, x ^ i < 0
⊢ eval x (cyclotomic n R) * ?h.inr.inr.b < 0
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Expand

Goal state before:
pp: case inl
p : ℕ
hp : Nat.Prime p
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
n : ℕ
h : Irreducible (cyclotomic (p ^ n) R)
hmn : 0 ≤ n
⊢ Irreducible (cyclotomic (p ^ 0) R)

pp: case inr
p : ℕ
hp : Nat.Prime p
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
n m : ℕ
hmn : m ≤ n
h : Irreducible (cyclotomic (p ^ n) R)
hm : m > 0
⊢ Irreducible (cyclotomic (p ^ m) R)

Tactic:
·

Goal state after:
case inl
p : ℕ
hp : Nat.Prime p
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
n : ℕ
h : Irreducible (cyclotomic (p ^ n) R)
hmn : 0 ≤ n
⊢ Irreducible (cyclotomic (p ^ 0) R)
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Expand

Goal state before:
pp: p : ℕ
hp : Nat.Prime p
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
m : ℕ
hm : m > 0
k : ℕ
hk : m ≤ m + k → Irreducible (cyclotomic (p ^ (m + k)) R) → Irreducible (cyclotomic (p ^ m) R)
hmn : m ≤ m + (k + 1)
h : Irreducible ((expand R p) (cyclotomic (p ^ (m + k)) R))
this : m + k ≠ 0
⊢ m ≤ m + k

Tactic:
by lia

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Eval

Goal state before:
pp: case hb.ha
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
h : Even n ∧ x + 1 < 0
h2 : 2 ∈ n.properDivisors.erase 1
⊢ 0 ≤ ∏ x_1 ∈ (n.properDivisors.erase 1).erase 2, eval x (cyclotomic x_1 R)

pp: case hb
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
h : Even n ∧ x + 1 < 0
h2 : 2 ∈ n.properDivisors.erase 1
⊢ eval x (cyclotomic 2 R) ≤ 0

Tactic:
·

Goal state after:
case hb.ha
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
h : Even n ∧ x + 1 < 0
h2 : 2 ∈ n.properDivisors.erase 1
⊢ 0 ≤ ∏ x_1 ∈ (n.properDivisors.erase 1).erase 2, eval x (cyclotomic x_1 R)
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Eval

Goal state before:
pp: case hb
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
h : Even n ∧ x + 1 < 0
h2 : 2 ∈ n.properDivisors.erase 1
⊢ eval x (cyclotomic 2 R) ≤ 0

Tactic:
·

Goal state after:
case hb
R : Type u_1
inst✝² : CommRing R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
h : Even n ∧ x + 1 < 0
h2 : 2 ∈ n.properDivisors.erase 1
⊢ eval x (cyclotomic 2 R) ≤ 0
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Expand

Goal state before:
pp: R : Type u_1
p m : ℕ
inst✝² : Fact (Nat.Prime p)
inst✝¹ : Ring R
inst✝ : CharP R p
hm : ¬p ∣ m
a : ℕ
x✝ : 0 < a + 2
hdiv : p ∣ p ^ a.succ * m
⊢ cyclotomic m R ^ ((p ^ a.succ - p ^ (a.succ - 1)) * p) = cyclotomic m R ^ (p * p ^ (a + 1) - p ^ (a + 2 - 1))

pp: R : Type u_1
p m : ℕ
inst✝² : Fact (Nat.Prime p)
inst✝¹ : Ring R
inst✝ : CharP R p
hm : ¬p ∣ m
a : ℕ
x✝ : 0 < a + 2
hdiv : p ∣ p ^ a.succ * m
⊢ ¬p ∣ m

Tactic:
·

Goal state after:
R : Type u_1
p m : ℕ
inst✝² : Fact (Nat.Prime p)
inst✝¹ : Ring R
inst✝ : CharP R p
hm : ¬p ∣ m
a : ℕ
x✝ : 0 < a + 2
hdiv : p ∣ p ^ a.succ * m
⊢ cyclotomic m R ^ ((p ^ a.succ - p ^ (a.succ - 1)) * p) = cyclotomic m R ^ (p * p ^ (a + 1) - p ^ (a + 2 - 1))