mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.WittVector.Complete

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝² : CommRing k
inst✝¹ : CharP k p
inst✝ : PerfectRing k p
x : 𝕎 k
n : ℕ
h : ∀ m < n, x.coeff m = 0
⊢ (⇑verschiebung)^[n] (x.shift n) =
    (⇑verschiebung)^[n] ((⇑frobenius)^[n] ((⇑(frobeniusEquiv p k).symm)^[n] (x.shift n)))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.Complete

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝² : CommRing k
inst✝¹ : CharP k p
inst✝ : PerfectRing k p
x : 𝕎 k
n : ℕ
⊢ x ∈ Ideal.span {↑p ^ n} ↔ ∀ m < n, x.coeff m = 0

Tactic:
simp_rw [Ideal.mem_span_singleton, dvd_def, mul_comm]
  refine ⟨fun ⟨u, hu⟩ m hm ↦ ?_, fun h ↦ ?_⟩
  · rw [hu, mul_pow_charP_coeff_zero _ hm]
  · use (frobeniusEquiv p k).symm^[n] (x.shift n)
    rw [← iterate_verschiebung_iterate_frobenius]
    calc
    _ = verschiebung^[n] (x.shift n) := by
      simpa using eq_iterate_verschiebung (x := x) (n := n) h
    _ = _ := by
      congr
      rw [← Function.comp_apply (f := frobenius^[n]), ← Function.Commute.comp_iterate]
      · rw [← WittVector.frobeniusEquiv_apply, ← RingEquiv.coe_trans]
        simp
      · rw [Function.Commute, Function.Semiconj, ← WittVector.frobeniusEquiv_apply]
        simp only [RingEquiv.apply_symm_apply, RingEquiv.symm_apply_apply, implies_true]

Goal state after:
Module: Mathlib.RingTheory.WittVector.Complete

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝² : CommRing k
inst✝¹ : CharP k p
inst✝ : PerfectRing k p
x : 𝕎 k
n : ℕ
⊢ x ∈ Ideal.span {↑p ^ n} ↔ ∀ m < n, x.coeff m = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.Complete

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝² : CommRing k
inst✝¹ : CharP k p
inst✝ : PerfectRing k p
⊢ RingHom.ker constantCoeff = Ideal.span {↑p}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.Complete

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝² : CommRing k
inst✝¹ : CharP k p
inst✝ : PerfectRing k p
⊢ ∀ (x : 𝕎 k), (∀ (n : ℕ), x ≡ 0 [SMOD Ideal.span {↑p} ^ n • ⊤]) → x = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.Complete

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝² : CommRing k
inst✝¹ : CharP k p
inst✝ : PerfectRing k p
⊢ ∀ (f : ℕ → 𝕎 k),
    (∀ {m n : ℕ}, m ≤ n → f m ≡ f n [SMOD Ideal.span {↑p} ^ m • ⊤]) →
      ∃ L, ∀ (n : ℕ), f n ≡ L [SMOD Ideal.span {↑p} ^ n • ⊤]

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.TeichmullerSeries

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
R : Type u_1
inst✝ : CommRing R
α : Type u_2
x : α → 𝕎 R
a : α
S' : Finset α
ha : a ∉ S'
hind :
  (∀ (n : ℕ), Subsingleton ↑{r | r ∈ S' ∧ (x r).coeff n ≠ 0}) →
    ∀ (n : ℕ), (∑ s ∈ S', x s).coeff n = ∑ s ∈ S', (x s).coeff n
h : ∀ (n : ℕ), Subsingleton ↑{r | r ∈ insert a S' ∧ (x r).coeff n ≠ 0}
n : ℕ
⊢ ∀ (n : ℕ), Subsingleton ↑{r | r ∈ S' ∧ (x r).coeff n ≠ 0}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.TeichmullerSeries

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
R : Type u_1
inst✝ : CommRing R
α : Type u_2
x : α → 𝕎 R
a : α
S' : Finset α
ha : a ∉ S'
h : ∀ (n : ℕ), Subsingleton ↑{r | r ∈ insert a S' ∧ (x r).coeff n ≠ 0}
n : ℕ
this : ∀ (n : ℕ), Subsingleton ↑{r | r ∈ S' ∧ (x r).coeff n ≠ 0}
hind : ∀ (n : ℕ), (∑ s ∈ S', x s).coeff n = ∑ s ∈ S', (x s).coeff n
⊢ ∀ (n : ℕ), (x a).coeff n = 0 ∨ (∑ s ∈ S', x s).coeff n = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Perfectoid.FontaineTheta

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
p : ℕ
inst✝ : Fact (Nat.Prime p)
n : ℕ
⊢ RingHom.ker (map (Ideal.Quotient.mk 𝔭)) ≤ RingHom.ker ((Ideal.Quotient.mk (𝔭 ^ (n + 1))).comp (ghostComponent n))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Perfectoid.FontaineTheta

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
p : ℕ
inst✝ : Fact (Nat.Prime p)
x : 𝕎 (R ⧸ 𝔭)
h : 𝔭 ^ (0 + 1) = 𝔭
⊢ (quotEquivOfEq h) ((ghostComponentModPPow 0) x) = (ghostComponent 0) x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Perfectoid.FontaineTheta

Goal state before:
pp: R : Type u
inst✝³ : CommRing R
p : ℕ
inst✝² : Fact (Nat.Prime p)
inst✝¹ : Fact ¬IsUnit ↑p
inst✝ : IsAdicComplete 𝔭 R
n : ℕ
x : R♭
⊢ (ghostComponentModPPow n) ((teichmuller p) ((PreTilt.coeff n) x)) = (Ideal.Quotient.mk (𝔭 ^ (n + 1))) (untilt x)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Perfectoid.FontaineTheta

Goal state before:
pp: R : Type u
inst✝³ : CommRing R
p : ℕ
inst✝² : Fact (Nat.Prime p)
inst✝¹ : Fact ¬IsUnit ↑p
inst✝ : IsAdicComplete 𝔭 R
n : ℕ
x : R♭
⊢ (fontaineThetaModPPow R p n) ((teichmuller p) x) = (Ideal.Quotient.mk (𝔭 ^ (n + 1))) (untilt x)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Perfectoid.FontaineTheta

Goal state before:
pp: R : Type u
inst✝³ : CommRing R
p : ℕ
inst✝² : Fact (Nat.Prime p)
inst✝¹ : Fact ¬IsUnit ↑p
inst✝ : IsAdicComplete 𝔭 R
n : ℕ
⊢ ↑p = (Ideal.Quotient.mk (𝔭 ^ (n + 1))) ↑p

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Perfectoid.FontaineTheta

Goal state before:
pp: case h
R : Type u
inst✝³ : CommRing R
p : ℕ
inst✝² : Fact (Nat.Prime p)
inst✝¹ : Fact ¬IsUnit ↑p
inst✝ : IsAdicComplete 𝔭 R
n : ℕ
⊢ ↑p ^ (n + 1) = 0

Tactic:
have : p = Ideal.Quotient.mk (𝔭 ^ (n + 1)) p := by
      simp [map_natCast]

Goal state after:
case h
R : Type u
inst✝³ : CommRing R
p : ℕ
inst✝² : Fact (Nat.Prime p)
inst✝¹ : Fact ¬IsUnit ↑p
inst✝ : IsAdicComplete 𝔭 R
n : ℕ
this : ↑p = (Ideal.Quotient.mk (𝔭 ^ (n + 1))) ↑p
⊢ ↑p ^ (n + 1) = 0
Module: Mathlib.RingTheory.WittVector.TeichmullerSeries

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
R : Type u_1
inst✝ : CommRing R
α : Type u_2
S : Finset α
x : α → 𝕎 R
h : ∀ (n : ℕ), Subsingleton ↑{r | r ∈ S ∧ (x r).coeff n ≠ 0}
n : ℕ
⊢ (∑ s ∈ S, x s).coeff n = ∑ s ∈ S, (x s).coeff n

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Perfectoid.FontaineTheta

Goal state before:
pp: R : Type u
inst✝³ : CommRing R
p : ℕ
inst✝² : Fact (Nat.Prime p)
inst✝¹ : Fact ¬IsUnit ↑p
inst✝ : IsAdicComplete 𝔭 R
n : ℕ
⊢ (factorPowSucc 𝔭 (n + 1)).comp (fontaineThetaModPPow R p (n + 1)) = fontaineThetaModPPow R p n

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.TeichmullerSeries

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharP R p
n : ℕ
x : R
⊢ ((teichmuller p) x * ↑p ^ n).coeff n = x ^ p ^ n

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Perfectoid.FontaineTheta

Goal state before:
pp: R : Type u
inst✝³ : CommRing R
p : ℕ
inst✝² : Fact (Nat.Prime p)
inst✝¹ : Fact ¬IsUnit ↑p
inst✝ : IsAdicComplete 𝔭 R
n : ℕ
x : 𝕎 R♭
⊢ (factorPowSucc 𝔭 (n + 1)) ((fontaineThetaModPPow R p (n + 1)) x) = (fontaineThetaModPPow R p n) x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.TeichmullerSeries

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharP R p
x : R
m n : ℕ
h : m ≠ n
⊢ ((teichmuller p) x * ↑p ^ n).coeff m = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Perfectoid.FontaineTheta

Goal state before:
pp: R : Type u
inst✝³ : CommRing R
p : ℕ
inst✝² : Fact (Nat.Prime p)
inst✝¹ : Fact ¬IsUnit ↑p
inst✝ : IsAdicComplete 𝔭 R
x : 𝕎 R♭
⊢ (Ideal.Quotient.mk 𝔭) ((fontaineTheta R p) x) = (PreTilt.coeff 0) (x.coeff 0)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Perfectoid.FontaineTheta

Goal state before:
pp: R : Type u
inst✝³ : CommRing R
p : ℕ
inst✝² : Fact (Nat.Prime p)
inst✝¹ : Fact ¬IsUnit ↑p
inst✝ : IsAdicComplete 𝔭 R
x : R♭
⊢ (fontaineTheta R p) ((teichmuller p) x) = untilt x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Perfectoid.FontaineTheta

Goal state before:
pp: R : Type u
inst✝³ : CommRing R
p : ℕ
inst✝² : Fact (Nat.Prime p)
inst✝¹ : Fact ¬IsUnit ↑p
inst✝ : IsAdicComplete 𝔭 R
hF : Function.Surjective ⇑(_root_.frobenius (ModP R p) p)
⊢ Ideal.map (fontaineTheta R p) (span {↑p}) = 𝔭

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Perfectoid.FontaineTheta

Goal state before:
pp: R : Type u
inst✝³ : CommRing R
p : ℕ
inst✝² : Fact (Nat.Prime p)
inst✝¹ : Fact ¬IsUnit ↑p
inst✝ : IsAdicComplete 𝔭 R
hF : Function.Surjective ⇑(_root_.frobenius (ModP R p) p)
this : Ideal.map (fontaineTheta R p) (span {↑p}) = 𝔭
⊢ IsHausdorff (Ideal.map (fontaineTheta R p) (span {↑p})) R

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Perfectoid.FontaineTheta

Goal state before:
pp: R : Type u
inst✝³ : CommRing R
p : ℕ
inst✝² : Fact (Nat.Prime p)
inst✝¹ : Fact ¬IsUnit ↑p
inst✝ : IsAdicComplete 𝔭 R
hF : Function.Surjective ⇑(_root_.frobenius (ModP R p) p)
this : Ideal.map (fontaineTheta R p) (span {↑p}) = 𝔭
x✝ : IsHausdorff (Ideal.map (fontaineTheta R p) (span {↑p})) R
h : Function.Surjective (⇑(Ideal.Quotient.mk 𝔭) ∘ ⇑(fontaineTheta R p))
⊢ Function.Surjective (⇑(Ideal.Quotient.mk (Ideal.map (fontaineTheta R p) (span {↑p}))) ∘ ⇑(fontaineTheta R p))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Perfectoid.FontaineTheta

Goal state before:
pp: R : Type u
inst✝³ : CommRing R
p : ℕ
inst✝² : Fact (Nat.Prime p)
inst✝¹ : Fact ¬IsUnit ↑p
inst✝ : IsAdicComplete 𝔭 R
hF : Function.Surjective ⇑(_root_.frobenius (ModP R p) p)
this : Ideal.map (fontaineTheta R p) (span {↑p}) = 𝔭
x✝ : IsHausdorff (Ideal.map (fontaineTheta R p) (span {↑p})) R
⊢ ⇑(Ideal.Quotient.mk 𝔭) ∘ ⇑(fontaineTheta R p) = (fun x ↦ (PreTilt.coeff 0) x) ∘ fun x ↦ x.coeff 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Perfectoid.BDeRham

Goal state before:
pp: R : Type u
inst✝³ : CommRing R
p : ℕ
inst✝² : Fact (Nat.Prime p)
inst✝¹ : Fact ¬IsUnit ↑p
inst✝ : IsAdicComplete (span {↑p}) R
⊢ IsUnit (((algebraMap R (Localization.Away ↑p)).comp (fontaineTheta R p)) ↑p)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Perfectoid.FontaineTheta

Goal state before:
pp: R : Type u
inst✝³ : CommRing R
p : ℕ
inst✝² : Fact (Nat.Prime p)
inst✝¹ : Fact ¬IsUnit ↑p
inst✝ : IsAdicComplete 𝔭 R
hF : Function.Surjective ⇑(_root_.frobenius (ModP R p) p)
⊢ Function.Surjective ⇑(fontaineTheta R p)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Dickson

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
k : ℕ
a : R
⊢ dickson k a 2 = X ^ 2 - C a * (3 - ↑k)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Dickson

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
k : ℕ
a : R
n : ℕ
⊢ dickson k a (n + 2) = X * dickson k a (n + 1) - C a * dickson k a n

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Dickson

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
k : ℕ
a : R
n : ℕ
h : 2 ≤ n
⊢ dickson k a n = X * dickson k a (n - 1) - C a * dickson k a (n - 2)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Dickson

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
k : ℕ
a : R
f : R →+* S
⊢ map f (dickson k a 0) = dickson k (f a) 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Dickson

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
k : ℕ
a : R
f : R →+* S
⊢ map f (dickson k a 1) = dickson k (f a) 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.TeichmullerSeries

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
R : Type u_1
inst✝² : CommRing R
inst✝¹ : CharP R p
inst✝ : PerfectRing R p
x : 𝕎 R
n : ℕ
⊢ ↑p ^ (n + 1) ∣ x - ∑ i ≤ n, (teichmuller p) (((_root_.frobeniusEquiv R p).symm ^ i) (x.coeff i)) * ↑p ^ i

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Dickson

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
k : ℕ
a : R
f : R →+* S
n : ℕ
⊢ map f (dickson k a (n + 2)) = dickson k (f a) (n + 2)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Dickson

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
⊢ dickson 2 0 0 = X ^ 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Dickson

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
⊢ dickson 2 0 1 = X ^ 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.TeichmullerSeries

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
R : Type u_1
inst✝³ : CommRing R
inst✝² : CharP R p
inst✝¹ : PerfectRing R p
S : Type u_2
inst✝ : CommRing S
f g : 𝕎 R →+* S
h : ∀ (x : R), f ((teichmuller p) x) = g ((teichmuller p) x)
n : ℕ
hn : ↑p ^ n = 0
x c : 𝕎 R
hc : x - ∑ i ≤ n, (teichmuller p) (((_root_.frobeniusEquiv R p).symm ^ i) (x.coeff i)) * ↑p ^ i = ↑p ^ (n + 1) * c
⊢ f x =
    f (x - ∑ i ≤ n, (teichmuller p) (((_root_.frobeniusEquiv R p).symm ^ i) (x.coeff i)) * ↑p ^ i) +
      f (∑ i ≤ n, (teichmuller p) (((_root_.frobeniusEquiv R p).symm ^ i) (x.coeff i)) * ↑p ^ i)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Dickson

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℕ
⊢ dickson 2 0 (n + 2) = X ^ (n + 2)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Eisenstein.Distinguished

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R[X]
I : Ideal R
distinguish : f.IsDistinguishedAt I
⊢ map (Ideal.Quotient.mk I) f = X ^ f.natDegree

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Dickson

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
x y : R
h : x * y = 1
⊢ eval (x + y) (dickson 1 1 0) = x ^ 0 + y ^ 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Dickson

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
x y : R
h : x * y = 1
⊢ eval (x + y) (dickson 1 1 1) = x ^ 1 + y ^ 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Eisenstein.Distinguished

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
I : Ideal R
f h : R⟦X⟧
g : R[X]
distinguish : g.IsDistinguishedAt I
notMem : PowerSeries.constantCoeff h ∉ I
eq : f = ↑g * h
H : (PowerSeries.map (Ideal.Quotient.mk I)) f = 0
⊢ (PowerSeries.map (Ideal.Quotient.mk I)) f = ↑(X ^ g.natDegree) * (PowerSeries.map (Ideal.Quotient.mk I)) h

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Eisenstein.Distinguished

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
I : Ideal R
f h : R⟦X⟧
g : R[X]
distinguish : g.IsDistinguishedAt I
notMem : PowerSeries.constantCoeff h ∉ I
eq : f = ↑g * h
H : (PowerSeries.map (Ideal.Quotient.mk I)) f = 0
⊢ False

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Eisenstein.Distinguished

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
I : Ideal R
f h : R⟦X⟧
g : R[X]
distinguish : g.IsDistinguishedAt I
notMem : PowerSeries.constantCoeff h ∉ I
eq : f = ↑g * h
this : Nontrivial R
⊢ (PowerSeries.map (Ideal.Quotient.mk I)) f = ↑(X ^ g.natDegree) * (PowerSeries.map (Ideal.Quotient.mk I)) h

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.TeichmullerSeries

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
R : Type u_1
inst✝³ : CommRing R
inst✝² : CharP R p
inst✝¹ : PerfectRing R p
S : Type u_2
inst✝ : CommRing S
f g : 𝕎 R →+* S
h : ∀ (x : R), f ((teichmuller p) x) = g ((teichmuller p) x)
n : ℕ
hn : ↑p ^ n = 0
x c : 𝕎 R
hc : x - ∑ i ≤ n, (teichmuller p) (((_root_.frobeniusEquiv R p).symm ^ i) (x.coeff i)) * ↑p ^ i = ↑p ^ (n + 1) * c
⊢ f (x - ∑ i ≤ n, (teichmuller p) (((_root_.frobeniusEquiv R p).symm ^ i) (x.coeff i)) * ↑p ^ i) +
      f (∑ i ≤ n, (teichmuller p) (((_root_.frobeniusEquiv R p).symm ^ i) (x.coeff i)) * ↑p ^ i) =
    ∑ i ≤ n, f ((teichmuller p) (((_root_.frobeniusEquiv R p).symm ^ i) (x.coeff i))) * ↑p ^ i

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Dickson

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
x y : R
h : x * y = 1
n : ℕ
⊢ eval (x + y) (dickson 1 1 (n + 2)) = x ^ (n + 2) + y ^ (n + 2)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Dickson

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : Invertible 2
⊢ 2 * C ⅟2 = 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Dickson

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : Invertible 2
⊢ C ⅟2 * 2 = 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.TeichmullerSeries

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
R : Type u_1
inst✝³ : CommRing R
inst✝² : CharP R p
inst✝¹ : PerfectRing R p
S : Type u_2
inst✝ : CommRing S
f g : 𝕎 R →+* S
h : ∀ (x : R), f ((teichmuller p) x) = g ((teichmuller p) x)
n : ℕ
hn : ↑p ^ n = 0
x c : 𝕎 R
hc : x - ∑ i ≤ n, (teichmuller p) (((_root_.frobeniusEquiv R p).symm ^ i) (x.coeff i)) * ↑p ^ i = ↑p ^ (n + 1) * c
⊢ ∑ i ≤ n, f ((teichmuller p) (((_root_.frobeniusEquiv R p).symm ^ i) (x.coeff i))) * ↑p ^ i =
    ∑ i ≤ n, g ((teichmuller p) (((_root_.frobeniusEquiv R p).symm ^ i) (x.coeff i))) * ↑p ^ i

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Dickson

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
⊢ dickson 1 1 0 = Chebyshev.C R ↑0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Eisenstein.Distinguished

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
I : Ideal R
f h : R⟦X⟧
g : R[X]
distinguish : g.IsDistinguishedAt I
notMem : PowerSeries.constantCoeff h ∉ I
eq : f = ↑g * h
⊢ g.degree = ↑(((PowerSeries.map (Ideal.Quotient.mk I)) f).order.lift ⋯)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Dickson

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
⊢ dickson 1 1 1 = Chebyshev.C R ↑1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Eisenstein.Distinguished

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
I : Ideal R
f h : R⟦X⟧
g : R[X]
distinguish : g.IsDistinguishedAt I
notMem : PowerSeries.constantCoeff h ∉ I
eq : f = ↑g * h
⊢ ↑g.natDegree = ((PowerSeries.map (Ideal.Quotient.mk I)) f).order

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Dickson

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℕ
⊢ dickson 1 1 (n + 2) = Chebyshev.C R ↑(n + 2)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Dickson

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
⊢ dickson 2 1 0 = Chebyshev.S R ↑0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Dickson

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
⊢ dickson 2 1 1 = Chebyshev.S R ↑1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Dickson

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℕ
⊢ dickson 2 1 (n + 2) = Chebyshev.S R ↑(n + 2)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Hermite.Basic

Goal state before:
pp: n : ℕ
⊢ hermite (n + 1) = X * hermite n - derivative (hermite n)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Dickson

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
m n : ℕ
⊢ 1 = (Int.castRingHom R) 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Hermite.Basic

Goal state before:
pp: n : ℕ
⊢ hermite n = (fun p ↦ X * p - derivative p)^[n] 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Hermite.Basic

Goal state before:
pp: ⊢ hermite 1 = X

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Hermite.Basic

Goal state before:
pp: n : ℕ
⊢ (hermite (n + 1)).coeff 0 = -(hermite n).coeff 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Hermite.Basic

Goal state before:
pp: n k : ℕ
⊢ (hermite (n + 1)).coeff (k + 1) = (hermite n).coeff k - (↑k + 2) * (hermite n).coeff (k + 2)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Hermite.Basic

Goal state before:
pp: n : ℕ
ih : ∀ (k : ℕ), (hermite n).coeff (n + k + 1) = 0
k : ℕ
⊢ n + k + 1 + 2 = n + (k + 2) + 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Hermite.Basic

Goal state before:
pp: n k : ℕ
hnk : n < k
⊢ (hermite n).coeff k = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Hermite.Basic

Goal state before:
pp: n : ℕ
⊢ (hermite n).coeff n = 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.TeichmullerSeries

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
R : Type u_1
inst✝³ : CommRing R
inst✝² : CharP R p
inst✝¹ : PerfectRing R p
S : Type u_2
inst✝ : CommRing S
f g : 𝕎 R →+* S
h : ∀ (x : R), f ((teichmuller p) x) = g ((teichmuller p) x)
n : ℕ
hn : ↑p ^ n = 0
x c : 𝕎 R
hc : x - ∑ i ≤ n, (teichmuller p) (((_root_.frobeniusEquiv R p).symm ^ i) (x.coeff i)) * ↑p ^ i = ↑p ^ (n + 1) * c
⊢ ∑ i ≤ n, g ((teichmuller p) (((_root_.frobeniusEquiv R p).symm ^ i) (x.coeff i))) * ↑p ^ i =
    g (x - ∑ i ≤ n, (teichmuller p) (((_root_.frobeniusEquiv R p).symm ^ i) (x.coeff i)) * ↑p ^ i) +
      g (∑ i ≤ n, (teichmuller p) (((_root_.frobeniusEquiv R p).symm ^ i) (x.coeff i)) * ↑p ^ i)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Hermite.Basic

Goal state before:
pp: n : ℕ
⊢ (hermite n).degree = ↑n

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Hermite.Basic

Goal state before:
pp: n : ℕ
⊢ (hermite n).leadingCoeff = 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Hermite.Gaussian

Goal state before:
pp: n : ℕ
x : ℝ
ih : (deriv^[n] fun y ↦ Real.exp (-(y ^ 2 / 2))) = fun x ↦ (-1) ^ n * ((aeval x) (hermite n) * Real.exp (-(x ^ 2 / 2)))
⊢ DifferentiableAt ℝ (fun y ↦ -(y ^ 2 / 2)) x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.TeichmullerSeries

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
R : Type u_1
inst✝³ : CommRing R
inst✝² : CharP R p
inst✝¹ : PerfectRing R p
S : Type u_2
inst✝ : CommRing S
f g : 𝕎 R →+* S
h : ∀ (x : R), f ((teichmuller p) x) = g ((teichmuller p) x)
n : ℕ
hn : ↑p ^ n = 0
x c : 𝕎 R
hc : x - ∑ i ≤ n, (teichmuller p) (((_root_.frobeniusEquiv R p).symm ^ i) (x.coeff i)) * ↑p ^ i = ↑p ^ (n + 1) * c
⊢ g (x - ∑ i ≤ n, (teichmuller p) (((_root_.frobeniusEquiv R p).symm ^ i) (x.coeff i)) * ↑p ^ i) +
      g (∑ i ≤ n, (teichmuller p) (((_root_.frobeniusEquiv R p).symm ^ i) (x.coeff i)) * ↑p ^ i) =
    g x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Hermite.Basic

Goal state before:
pp: n : ℕ
ih : ∀ {k : ℕ}, Odd (n + k) → (hermite n).coeff k = 0
k : ℕ
hnk : Odd (n + 1 + (k + 1))
⊢ n.succ + k.succ = n + k + 2

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Hermite.Gaussian

Goal state before:
pp: n : ℕ
x : ℝ
ih : (deriv^[n] fun y ↦ Real.exp (-(y ^ 2 / 2))) = fun x ↦ (-1) ^ n * ((aeval x) (hermite n) * Real.exp (-(x ^ 2 / 2)))
⊢ deriv (fun y ↦ Real.exp (-(y ^ 2 / 2))) x = -x * Real.exp (-(x ^ 2 / 2))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Hermite.Basic

Goal state before:
pp: n k : ℕ
hnk : Odd (n + k)
⊢ (hermite n).coeff k = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.TeichmullerSeries

Goal state before:
pp: p : ℕ
hp✝ : Fact (Nat.Prime p)
R : Type u_1
inst✝³ : CommRing R
inst✝² : CharP R p
inst✝¹ : PerfectRing R p
S : Type u_2
inst✝ : CommRing S
f g : 𝕎 R →+* S
hp : IsNilpotent ↑p
h : ∀ (x : R), f ((teichmuller p) x) = g ((teichmuller p) x)
⊢ f = g

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Hermite.Basic

Goal state before:
pp: x✝ : ℕ
⊢ (hermite (2 * 0 + x✝)).coeff x✝ = (-1) ^ 0 * ↑(2 * 0 - 1)‼ * ↑((2 * 0 + x✝).choose x✝)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Hermite.Basic

Goal state before:
pp: n : ℕ
⊢ 2 * (n + 1) - 1 = 2 * n + 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Dickson

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
m n : ℕ
⊢ dickson 1 1 (m * n) = (dickson 1 1 m).comp (dickson 1 1 n)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Dickson

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
m n : ℕ
⊢ (dickson 1 1 m).comp (dickson 1 1 n) = (dickson 1 1 n).comp (dickson 1 1 m)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Hermite.Basic

Goal state before:
pp: n : ℕ
⊢ (hermite (2 * (n + 1) + 0)).coeff 0 = (-1) ^ (n + 1) * ↑(2 * (n + 1) - 1)‼ * ↑((2 * (n + 1) + 0).choose 0)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Dickson

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
K : Type := FractionRing (ZMod p)[X]
f : ZMod p →+* K := (algebraMap (ZMod p)[X] (FractionRing (ZMod p)[X])).comp C
⊢ CharP K p

Tactic:
by

Goal state after: