mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Ideal.Pure

Goal state before:
pp: case refine_2
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : I.Pure
x : R
hx : x ∈ I
⊢ x ∈ RingHom.ker (RingHom.pi fun p ↦ algebraMap R (Localization.AtPrime (↑p).asIdeal))

Tactic:
rw [RingHom.mem_ker]
    ext p
    rw [RingHom.pi_apply, Pi.zero_apply]
    exact Ideal.le_ker_atPrime_of_forall_exists_eq_mul
      (fun x hx ↦ Ideal.exists_eq_mul_of_pure hx) p.2 hx

Goal state after:
Module: Mathlib.RingTheory.KrullDimension.LocalRing

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsLocalRing R
inst✝ : IsDomain R
x✝ : ¬IsField R ∧ ∀ (x : R), x ≠ 0 → IsLocalRing.maximalIdeal R ≤ (Ideal.span {x}).radical
hn : ¬IsField R
h : ∀ (x : R), x ≠ 0 → IsLocalRing.maximalIdeal R ≤ (Ideal.span {x}).radical
this : ¬ringKrullDim R ≤ 0
I : Ideal R
hI : I ≠ ⊥
hI_prime : I.IsPrime
⊢ ∃ x ∈ I, x ≠ 0

Tactic:
apply by_contradiction fun h ↦ (hI (le_antisymm (fun x ↦ ?_) bot_le))
      simp only [ne_eq, not_exists, not_and, not_not] at h
      exact fun hx ↦ h x hx

Goal state after:
Module: Mathlib.RingTheory.KrullDimension.PID

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
inst✝² : IsDomain R
inst✝¹ : IsPrincipalIdealRing R
m : Ideal R
inst✝ : m.IsMaximal
h : ¬IsField R
⊢ m.height = 1

Tactic:
by
  refine le_antisymm ?_ ?_
  · suffices h : (m.height : WithBot ℕ∞) ≤ 1 by norm_cast at h
    rw [← IsPrincipalIdealRing.ringKrullDim_eq_one _ h]
    exact Ideal.height_le_ringKrullDim_of_ne_top Ideal.IsPrime.ne_top'
  · apply le_of_eq_of_le _ (Ideal.height_add_one_le_of_lt_of_isPrime (Ideal.bot_lt_of_maximal m h))
    simp

Goal state after:
Module: Mathlib.RingTheory.Ideal.KrullsHeightTheorem

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsNoetherianRing R
p : Ideal R
s : Finset R
hI : p ∈ (span ↑s).minimalPrimes
⊢ p.height ≤ ↑s.card

Tactic:
by
  trans (Cardinal.toENat (Submodule.spanRank (Ideal.span (s : Set R))))
  · exact Ideal.height_le_spanRank_toENat_of_mem_minimalPrimes _ _ hI
  · simpa using Submodule.spanRank_span_le_card (s : Set R)

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
⊢ ringKrullDim R[X] ≤ 2 * ringKrullDim R + 1

Tactic:
rw [ringKrullDim, ringKrullDim]

Goal state after:
R : Type u_1
inst✝ : CommRing R
⊢ Order.krullDim (PrimeSpectrum R[X]) ≤ 2 * Order.krullDim (PrimeSpectrum R) + 1
Module: Mathlib.RingTheory.KrullDimension.Polynomial

Goal state before:
pp: case e'_3
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsNoetherianRing R
ι : Type u_2
inst✝ : Finite ι
α✝ β✝ : Type u_2
e : α✝ ≃ β✝
H : ringKrullDim (MvPolynomial α✝ R) = ringKrullDim R + ↑(Nat.card α✝)
⊢ ringKrullDim R + ↑(Nat.card α✝) = ringKrullDim R + ↑(Nat.card β✝)

Tactic:
rw [Nat.card_congr e]

Goal state after:
Module: Mathlib.RingTheory.Ideal.KrullsHeightTheorem

Goal state before:
pp: R : Type u_1
inst✝⁷ : CommRing R
inst✝⁶ : IsNoetherianRing R
S : Type u_2
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : IsNoetherianRing S
p : Ideal R
inst✝² : p.IsPrime
P : Ideal S
inst✝¹ : P.IsPrime
inst✝ : P.LiesOver p
s : Finset R
hp : p ∈ (span ↑s).minimalPrimes
heq : ↑s.card = p.height
P' : Ideal (S ⧸ map (algebraMap R S) p) := map (Quotient.mk (map (algebraMap R S) p)) P
s' : Finset (S ⧸ map (algebraMap R S) p)
hP' : P' ∈ (span ↑s').minimalPrimes
heq' : ↑s'.card = P'.height
hsP'sub : ↑s' ⊆ ↑P'
this : Set.SurjOn ⇑(Quotient.mk (map (algebraMap R S) p)) ↑P ↑s'
o : Finset S
ho : ↑o ⊆ ↑P
hinj : Set.InjOn ⇑(Quotient.mk (map (algebraMap R S) p)) ↑o
himgo : Finset.image (⇑(Quotient.mk (map (algebraMap R S) p))) o = s'
t : Finset S := Finset.image (⇑(algebraMap R S)) s ∪ o
h : P.height ≤ ↑t.card
⊢ P.height ≤ p.height + (map (Quotient.mk (map (algebraMap R S) p)) P).height

Tactic:
rw [← heq, ← heq']
    apply le_trans h
    norm_cast
    refine le_trans (Finset.card_union_le _ _) (add_le_add Finset.card_image_le ?_)
    rw [← himgo, Finset.card_image_of_injOn hinj]

Goal state after:
Module: Mathlib.RingTheory.LittleWedderburn

Goal state before:
pp: D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Finite D
hD : InductionHyp D
val✝ : Fintype D
Z : Subring D := Subring.center D
hZ : Z ≠ ⊤
this✝ : Field ↥Z := hD.field ⋯
q : ℕ := card ↥Z
card_Z : q = card ↥Z
hq : 1 < q
n : ℕ := finrank (↥Z) D
card_D : card D = q ^ n
h1qn : 1 ≤ q ^ n
Φₙ : ℤ[X] := cyclotomic n ℤ
key : ↑q - 1 + ↑(∑ x ∈ (ConjClasses.noncenter Dˣ).toFinset, x.carrier.toFinset.card) = ↑q ^ n - 1
this : eval (↑q) Φₙ ∣ ↑(∑ x ∈ (ConjClasses.noncenter Dˣ).toFinset, x.carrier.toFinset.card)
contra : eval (↑q) Φₙ ∣ eval (↑q) (X ^ n) - eval (↑q) 1
⊢ False

Tactic:
eval_X_pow,

Goal state after:
D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Finite D
hD : InductionHyp D
val✝ : Fintype D
Z : Subring D := Subring.center D
hZ : Z ≠ ⊤
this✝ : Field ↥Z := hD.field ⋯
q : ℕ := card ↥Z
card_Z : q = card ↥Z
hq : 1 < q
n : ℕ := finrank (↥Z) D
card_D : card D = q ^ n
h1qn : 1 ≤ q ^ n
Φₙ : ℤ[X] := cyclotomic n ℤ
key : ↑q - 1 + ↑(∑ x ∈ (ConjClasses.noncenter Dˣ).toFinset, x.carrier.toFinset.card) = ↑q ^ n - 1
this : eval (↑q) Φₙ ∣ ↑(∑ x ∈ (ConjClasses.noncenter Dˣ).toFinset, x.carrier.toFinset.card)
contra : eval (↑q) Φₙ ∣ ↑q ^ n - eval (↑q) 1
⊢ False
Module: Mathlib.RingTheory.LocalRing.Etale

Goal state before:
pp: case refine_2
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : IsLocalRing S
inst✝³ : IsLocalRing R
inst✝² : Module.Finite R S
inst✝¹ : FaithfulSMul R S
inst✝ : Etale R S
this : Module.Free R S
e : (S ⧸ Ideal.map (algebraMap R S) (maximalIdeal R)) ≃ₗ[R ⧸ maximalIdeal R] S ⧸ maximalIdeal S
⊢ Module.finrank R S = Module.finrank (ResidueField R) (ResidueField S)

Tactic:
← finrank_quotient_map (R := R) (S := S)

Goal state after:
case refine_2
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : IsLocalRing S
inst✝³ : IsLocalRing R
inst✝² : Module.Finite R S
inst✝¹ : FaithfulSMul R S
inst✝ : Etale R S
this : Module.Free R S
e : (S ⧸ Ideal.map (algebraMap R S) (maximalIdeal R)) ≃ₗ[R ⧸ maximalIdeal R] S ⧸ maximalIdeal S
⊢ Module.finrank (R ⧸ maximalIdeal R) (S ⧸ Ideal.map (algebraMap R S) (maximalIdeal R)) =
    Module.finrank (ResidueField R) (ResidueField S)
Module: Mathlib.RingTheory.LaurentSeries

Goal state before:
pp: K : Type u_2
inst✝ : Field K
⊢ (idealX K).intValuation X = exp (-1)

Tactic:
← Polynomial.coe_X,

Goal state after:
K : Type u_2
inst✝ : Field K
⊢ (idealX K).intValuation ↑Polynomial.X = exp (-1)
Module: Mathlib.RingTheory.Localization.Pi

Goal state before:
pp: ι : Type u_1
R : ι → Type u_2
inst✝³ : (i : ι) → CommSemiring (R i)
S' : Type u_4
inst✝² : CommSemiring S'
inst✝¹ : Algebra ((i : ι) → R i) S'
M : Submonoid ((i : ι) → R i)
inst✝ : Finite ι
n : (i : ι) → R i
hn : n ∈ Submonoid.pi Set.univ fun i ↦ Submonoid.map (Pi.evalRingHom R i) M
⊢ ∃ m ∈ M, n ∣ m

Tactic:
choose m mem eq using hn
    have := Fintype.ofFinite ι
    refine ⟨∏ i, m i ⟨⟩, prod_mem fun i _ ↦ mem i _, pi_dvd_iff.mpr fun i ↦ ?_⟩
    rw [Fintype.prod_apply]
    exact (eq i ⟨⟩).symm.dvd.trans (Finset.dvd_prod_of_mem _ <| Finset.mem_univ _)

Goal state after:
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem

Goal state before:
pp: case add_single
R : Type u_3
n m : ℕ
inst✝¹ : CommSemiring R
r : R
hr : r ≠ 0
hnm : n ≤ m
inst✝ : Nontrivial R
i : Fin n
b✝ : ℕ
f✝ : Fin n →₀ ℕ
a✝¹ : i ∉ f✝.support
a✝ : b✝ ≠ 0
ih : ⇑(ofLex (supDegree (⇑toLex) (esymmAlgHomMonomial (Fin m) f✝ r))) = (accumulate n m) ⇑f✝
this : ↑i < m
⊢ (accumulate n m) ⇑f✝ + b✝ • ⇑(ofLex (supDegree (⇑toLex) (esymm (Fin m) R (↑i + 1)))) =
    (accumulate n m) ⇑f✝ + (accumulate n m) ⇑fun₀ | i => b✝

pp: case add_single
R : Type u_3
n m : ℕ
inst✝¹ : CommSemiring R
r : R
hr : r ≠ 0
hnm : n ≤ m
inst✝ : Nontrivial R
i : Fin n
b✝ : ℕ
f✝ : Fin n →₀ ℕ
a✝¹ : i ∉ f✝.support
a✝ : b✝ ≠ 0
ih : ⇑(ofLex (supDegree (⇑toLex) (esymmAlgHomMonomial (Fin m) f✝ r))) = (accumulate n m) ⇑f✝
this : ↑i < m
⊢ Monic (⇑toLex) (esymm (Fin m) R (↑i + 1))

pp: case add_single.hq
R : Type u_3
n m : ℕ
inst✝¹ : CommSemiring R
r : R
hr : r ≠ 0
hnm : n ≤ m
inst✝ : Nontrivial R
i : Fin n
b✝ : ℕ
f✝ : Fin n →₀ ℕ
a✝¹ : i ∉ f✝.support
a✝ : b✝ ≠ 0
ih : ⇑(ofLex (supDegree (⇑toLex) (esymmAlgHomMonomial (Fin m) f✝ r))) = (accumulate n m) ⇑f✝
this : ↑i < m
⊢ Monic (⇑toLex) (esymm (Fin m) R (↑i + 1) ^ b✝)

pp: case add_single.hp
R : Type u_3
n m : ℕ
inst✝¹ : CommSemiring R
r : R
hr : r ≠ 0
hnm : n ≤ m
inst✝ : Nontrivial R
i : Fin n
b✝ : ℕ
f✝ : Fin n →₀ ℕ
a✝¹ : i ∉ f✝.support
a✝ : b✝ ≠ 0
ih : ⇑(ofLex (supDegree (⇑toLex) (esymmAlgHomMonomial (Fin m) f✝ r))) = (accumulate n m) ⇑f✝
this : ↑i < m
⊢ esymmAlgHomMonomial (Fin m) f✝ r ≠ 0

Tactic:
supDegree_esymm this,

Goal state after:
case add_single
R : Type u_3
n m : ℕ
inst✝¹ : CommSemiring R
r : R
hr : r ≠ 0
hnm : n ≤ m
inst✝ : Nontrivial R
i : Fin n
b✝ : ℕ
f✝ : Fin n →₀ ℕ
a✝¹ : i ∉ f✝.support
a✝ : b✝ ≠ 0
ih : ⇑(ofLex (supDegree (⇑toLex) (esymmAlgHomMonomial (Fin m) f✝ r))) = (accumulate n m) ⇑f✝
this : ↑i < m
⊢ ((accumulate n m) ⇑f✝ + b✝ • (accumulate n m) ⇑fun₀ | i => 1) =
    (accumulate n m) ⇑f✝ + (accumulate n m) ⇑fun₀ | i => b✝

case add_single
R : Type u_3
n m : ℕ
inst✝¹ : CommSemiring R
r : R
hr : r ≠ 0
hnm : n ≤ m
inst✝ : Nontrivial R
i : Fin n
b✝ : ℕ
f✝ : Fin n →₀ ℕ
a✝¹ : i ∉ f✝.support
a✝ : b✝ ≠ 0
ih : ⇑(ofLex (supDegree (⇑toLex) (esymmAlgHomMonomial (Fin m) f✝ r))) = (accumulate n m) ⇑f✝
this : ↑i < m
⊢ Monic (⇑toLex) (esymm (Fin m) R (↑i + 1))

case add_single.hq
R : Type u_3
n m : ℕ
inst✝¹ : CommSemiring R
r : R
hr : r ≠ 0
hnm : n ≤ m
inst✝ : Nontrivial R
i : Fin n
b✝ : ℕ
f✝ : Fin n →₀ ℕ
a✝¹ : i ∉ f✝.support
a✝ : b✝ ≠ 0
ih : ⇑(ofLex (supDegree (⇑toLex) (esymmAlgHomMonomial (Fin m) f✝ r))) = (accumulate n m) ⇑f✝
this : ↑i < m
⊢ Monic (⇑toLex) (esymm (Fin m) R (↑i + 1) ^ b✝)

case add_single.hp
R : Type u_3
n m : ℕ
inst✝¹ : CommSemiring R
r : R
hr : r ≠ 0
hnm : n ≤ m
inst✝ : Nontrivial R
i : Fin n
b✝ : ℕ
f✝ : Fin n →₀ ℕ
a✝¹ : i ∉ f✝.support
a✝ : b✝ ≠ 0
ih : ⇑(ofLex (supDegree (⇑toLex) (esymmAlgHomMonomial (Fin m) f✝ r))) = (accumulate n m) ⇑f✝
this : ↑i < m
⊢ esymmAlgHomMonomial (Fin m) f✝ r ≠ 0
Module: Mathlib.RingTheory.MvPowerSeries.Expand

Goal state before:
pp: σ : Type u_1
R : Type u_3
S : Type u_4
inst✝¹ : CommRing R
inst✝ : CommRing S
p : ℕ
hp : p ≠ 0
f : R →+* S
φ : MvPowerSeries σ R
⊢ (map f) ((expand p hp) φ) = (expand p hp) ((map f) φ)

Tactic:
by
  simp [expand, map_subst (HasSubst.X_pow hp)]

Goal state after:
Module: Mathlib.RingTheory.Morita.Matrix

Goal state before:
pp: R : Type u
ι : Type v
inst✝² : Ring R
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
M : ModuleCat (Matrix ι ι R)
j : ι
x : Matrix ι ι R
m : ↑M
i : ι
this✝ : Module R ↑M := Module.compHom (↑M) (scalar ι)
this : IsScalarTower R (Matrix ι ι R) ↑M
⊢ ↑⟨single j i 1 • x • m, ⋯⟩ = ↑(((RingHom.id (Matrix ι ι R)) x • fun i ↦ ⟨single j i 1 • m, ⋯⟩) i)

Tactic:
simp only [← mul_smul, RingHom.id_apply, Module.smul_apply,
      AddSubmonoidClass.coe_finsetSum, SetLike.val_smul, ← smul_assoc, ← Finset.sum_smul]

Goal state after:
R : Type u
ι : Type v
inst✝² : Ring R
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
M : ModuleCat (Matrix ι ι R)
j : ι
x : Matrix ι ι R
m : ↑M
i : ι
this✝ : Module R ↑M := Module.compHom (↑M) (scalar ι)
this : IsScalarTower R (Matrix ι ι R) ↑M
⊢ (single j i 1 * x) • m = (∑ i_1, x i i_1 • single j i_1 1) • m
Module: Mathlib.RingTheory.MvPowerSeries.Expand

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
p : ℕ
hp : p ≠ 0
n m : σ →₀ ℕ
h : n ≤ m
inst✝ : DecidableEq σ
f : MvPowerSeries σ R
⊢ (MvPolynomial.expand p) ((trunc' R n) f) = (trunc' R (p • n)) ((expand p hp) ↑((trunc' R m) f))

Tactic:
trunc'_expand,

Goal state after:
σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
p : ℕ
hp : p ≠ 0
n m : σ →₀ ℕ
h : n ≤ m
inst✝ : DecidableEq σ
f : MvPowerSeries σ R
⊢ (MvPolynomial.expand p) ((trunc' R n) f) = (MvPolynomial.expand p) ((trunc' R n) ↑((trunc' R m) f))
Module: Mathlib.RingTheory.MvPowerSeries.Expand

Goal state before:
pp: case h
σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
p : ℕ
hp : p ≠ 0
inst✝ : ExpChar R p
f : MvPowerSeries σ R
n : σ →₀ ℕ
⊢ (trunc' R (p • n)) ((map (frobenius R p)) ((expand p hp) f)) = (trunc' R (p • n)) (f ^ p)

Tactic:
have : (((trunc' R (p • n) f).expand p).map (frobenius R p)).toMvPowerSeries =
      MvPowerSeries.map (frobenius R p) ((trunc' R (p • n) f).expand p) := by
      simp only [MvPolynomial.map_expand, ← expand_eq_expand p hp, map_expand]
      congr

Goal state after:
case h
σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
p : ℕ
hp : p ≠ 0
inst✝ : ExpChar R p
f : MvPowerSeries σ R
n : σ →₀ ℕ
this :
  ↑((MvPolynomial.map (frobenius R p)) ((MvPolynomial.expand p) ((trunc' R (p • n)) f))) =
    (map (frobenius R p)) ↑((MvPolynomial.expand p) ((trunc' R (p • n)) f))
⊢ (trunc' R (p • n)) ((map (frobenius R p)) ((expand p hp) f)) = (trunc' R (p • n)) (f ^ p)
Module: Mathlib.RingTheory.MvPowerSeries.Expand

Goal state before:
pp: case h
σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
p : ℕ
hp : p ≠ 0
inst✝ : ExpChar R p
f : MvPowerSeries σ R
n : σ →₀ ℕ
this :
  ↑((MvPolynomial.map (frobenius R p)) ((MvPolynomial.expand p) ((trunc' R (p • n)) f))) =
    (map (frobenius R p)) ↑((MvPolynomial.expand p) ((trunc' R (p • n)) f))
⊢ (MvPolynomial.map (frobenius R p)) ((trunc' R (p • n)) ((expand p hp) f)) = (trunc' R (p • n)) (f ^ p)

Tactic:
trunc'_expand,

Goal state after:
case h
σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
p : ℕ
hp : p ≠ 0
inst✝ : ExpChar R p
f : MvPowerSeries σ R
n : σ →₀ ℕ
this :
  ↑((MvPolynomial.map (frobenius R p)) ((MvPolynomial.expand p) ((trunc' R (p • n)) f))) =
    (map (frobenius R p)) ↑((MvPolynomial.expand p) ((trunc' R (p • n)) f))
⊢ (MvPolynomial.map (frobenius R p)) ((MvPolynomial.expand p) ((trunc' R n) f)) = (trunc' R (p • n)) (f ^ p)
Module: Mathlib.RingTheory.WittVector.WittPolynomial

Goal state before:
pp: case e_a.e_a.e_a
p n k : ℕ
hk : k ∈ range (n + 1)
⊢ n + 1 - k = n - k + 1

Tactic:
mem_range

Goal state after:
case e_a.e_a.e_a
p n k : ℕ
hk : k < n + 1
⊢ n + 1 - k = n - k + 1
Module: Mathlib.RingTheory.WittVector.WittPolynomial

Goal state before:
pp: case ind
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
hp : Fact (Nat.Prime p)
inst✝ : Invertible ↑p
n : ℕ
IH : ∀ m < n, constantCoeff (xInTermsOfW p R m) = 0
m : ℕ
H : m ∈ range n
⊢ constantCoeff (C (↑p ^ m) * xInTermsOfW p R m ^ p ^ (n - m)) = 0

Tactic:
mem_range

Goal state after:
case ind
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
hp : Fact (Nat.Prime p)
inst✝ : Invertible ↑p
n : ℕ
IH : ∀ m < n, constantCoeff (xInTermsOfW p R m) = 0
m : ℕ
H : m < n
⊢ constantCoeff (C (↑p ^ m) * xInTermsOfW p R m ^ p ^ (n - m)) = 0
Module: Mathlib.RingTheory.WittVector.WittPolynomial

Goal state before:
pp: case ind
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
i j : ℕ
hj : j ∈ range n
H : i ∈ (xInTermsOfW p ℚ j ^ p ^ (n - j)).vars
⊢ i ∈ range n

Tactic:
mem_range

Goal state after:
case ind
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
i j : ℕ
hj : j < n
H : i ∈ (xInTermsOfW p ℚ j ^ p ^ (n - j)).vars
⊢ i ∈ range n
Module: Mathlib.RingTheory.WittVector.WittPolynomial

Goal state before:
pp: case ind
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
i j : ℕ
hj : j < n
H : i ∈ range (j + 1)
⊢ i ∈ range n

Tactic:
mem_range

Goal state after:
case ind
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
i j : ℕ
hj : j < n
H : i < j + 1
⊢ i ∈ range n
Module: Mathlib.RingTheory.WittVector.WittPolynomial

Goal state before:
pp: case ind.hpq
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
j : ℕ
hj : j ∈ range n
H : n ∈ (xInTermsOfW p ℚ j ^ p ^ (n - j)).vars
⊢ False

Tactic:
mem_range

Goal state after:
case ind.hpq
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
j : ℕ
hj : j < n
H : n ∈ (xInTermsOfW p ℚ j ^ p ^ (n - j)).vars
⊢ False
Module: Mathlib.RingTheory.WittVector.WittPolynomial

Goal state before:
pp: case ind.hpq
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
j : ℕ
hj : j < n
H : n ∈ range (j + 1)
⊢ False

Tactic:
mem_range

Goal state after:
case ind.hpq
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
j : ℕ
hj : j < n
H : n < j + 1
⊢ False
Module: Mathlib.RingTheory.WittVector.WittPolynomial

Goal state before:
pp: case ind
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
i j : ℕ
hj : j < n
H : i < j + 1
⊢ i ∈ range n

Tactic:
mem_range

Goal state after:
case ind
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
i j : ℕ
hj : j < n
H : i < j + 1
⊢ i < n
Module: Mathlib.RingTheory.WittVector.WittPolynomial

Goal state before:
pp: case ind.e_a
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : Invertible ↑p
n : ℕ
H : ∀ m < n, (bind₁ (W_ R)) (xInTermsOfW p R m) = X m
i : ℕ
h : i ∈ range n
⊢ C (↑p ^ i) * X i ^ p ^ (n - i) = (bind₁ (W_ R)) (C (↑p ^ i) * xInTermsOfW p R i ^ p ^ (n - i))

Tactic:
mem_range

Goal state after:
case ind.e_a
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : Invertible ↑p
n : ℕ
H : ∀ m < n, (bind₁ (W_ R)) (xInTermsOfW p R m) = X m
i : ℕ
h : i < n
⊢ C (↑p ^ i) * X i ^ p ^ (n - i) = (bind₁ (W_ R)) (C (↑p ^ i) * xInTermsOfW p R i ^ p ^ (n - i))
Module: Mathlib.RingTheory.WittVector.StructurePolynomial

Goal state before:
pp: case h
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH : ∀ m < n, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
c : idx × ℕ →₀ ℕ
sum_induction_steps :
  (map (Int.castRingHom ℚ)) (∑ i ∈ Finset.range n, C (↑p ^ i) * wittStructureInt p Φ i ^ p ^ (n - i)) =
    ∑ i ∈ Finset.range n, C (↑p ^ i) * wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) i ^ p ^ (n - i)
⊢ (↑(coeff c
            ((bind₁ fun i ↦ (rename fun i_1 ↦ (i, i_1)) (W_ ℤ n)) Φ -
              ∑ i ∈ Finset.range n, C (↑p ^ i) * wittStructureInt p Φ i ^ p ^ (n - i))) /
        ↑↑(p ^ n)).den =
    1

Tactic:
Rat.den_div_intCast_eq_one_iff

Goal state after:
case h
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH : ∀ m < n, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
c : idx × ℕ →₀ ℕ
sum_induction_steps :
  (map (Int.castRingHom ℚ)) (∑ i ∈ Finset.range n, C (↑p ^ i) * wittStructureInt p Φ i ^ p ^ (n - i)) =
    ∑ i ∈ Finset.range n, C (↑p ^ i) * wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) i ^ p ^ (n - i)
⊢ ↑(p ^ n) ∣
    coeff c
      ((bind₁ fun i ↦ (rename fun i_1 ↦ (i, i_1)) (W_ ℤ n)) Φ -
        ∑ i ∈ Finset.range n, C (↑p ^ i) * wittStructureInt p Φ i ^ p ^ (n - i))

case h.hn
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH : ∀ m < n, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
c : idx × ℕ →₀ ℕ
sum_induction_steps :
  (map (Int.castRingHom ℚ)) (∑ i ∈ Finset.range n, C (↑p ^ i) * wittStructureInt p Φ i ^ p ^ (n - i)) =
    ∑ i ∈ Finset.range n, C (↑p ^ i) * wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) i ^ p ^ (n - i)
⊢ ↑(p ^ n) ≠ 0
Module: Mathlib.RingTheory.Perfection

Goal state before:
pp: M✝ : Type u_1
inst✝³ : CommMonoid M✝
p✝ p : ℕ
M : Type u_2
inst✝² : CommMonoid M
inst✝¹ : PerfectRing M p
N : Type u_3
inst✝ : CommMonoid N
f : M →* N
r : M
n : ℕ
⊢ f ((powMulEquiv M p).symm ((powMulEquiv M p ^ n).symm r) ^ p) = (fun n ↦ f ((powMulEquiv M (p ^ n)).symm r)) n

Tactic:
powMulEquiv_symm_pow_p,

Goal state after:
M✝ : Type u_1
inst✝³ : CommMonoid M✝
p✝ p : ℕ
M : Type u_2
inst✝² : CommMonoid M
inst✝¹ : PerfectRing M p
N : Type u_3
inst✝ : CommMonoid N
f : M →* N
r : M
n : ℕ
⊢ f ((powMulEquiv M p ^ n).symm r) = (fun n ↦ f ((powMulEquiv M (p ^ n)).symm r)) n
Module: Mathlib.RingTheory.Perfection

Goal state before:
pp: M✝ : Type u_1
inst✝³ : CommMonoid M✝
p✝ p : ℕ
M : Type u_2
inst✝² : CommMonoid M
inst✝¹ : PerfectRing M p
N : Type u_3
inst✝ : CommMonoid N
f : M →* Perfection N p
m : M
n : ℕ
⊢ (coeffMonoidHom N p (0 + n)) (f ((powMulEquiv M (p ^ n)).symm m ^ p ^ n)) = (coeffMonoidHom N p n) (f m)

Tactic:
powMulEquiv_symm_pow_p,

Goal state after:
M✝ : Type u_1
inst✝³ : CommMonoid M✝
p✝ p : ℕ
M : Type u_2
inst✝² : CommMonoid M
inst✝¹ : PerfectRing M p
N : Type u_3
inst✝ : CommMonoid N
f : M →* Perfection N p
m : M
n : ℕ
⊢ (coeffMonoidHom N p (0 + n)) (f m) = (coeffMonoidHom N p n) (f m)
Module: Mathlib.RingTheory.Perfection

Goal state before:
pp: R : Type u_1
inst✝¹ : CommSemiring R
p : ℕ
hp : Fact (Nat.Prime p)
inst✝ : CharP R p
⊢ pthRoot R p = ↑(frobeniusEquiv (Perfection R p) p).symm

Tactic:
by
  ext : 1
  simpa [RingEquiv.eq_symm_apply] using ext <| coeffMonoidHom_pow_p' _

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

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
IH :
  ∀ m < n,
    X m ^ p + C ↑p * (MvPolynomial.map (Int.castRingHom ℚ)) (frobeniusPolyAux p m) =
      (bind₁ (wittPolynomial p ℚ ∘ fun n ↦ n + 1)) (xInTermsOfW p ℚ m)
h1 : ↑p ^ n * ⅟↑p ^ n = 1
⊢ X n ^ p + C ↑p * (MvPolynomial.map (Int.castRingHom ℚ)) (frobeniusPolyAux p n) =
    X n ^ p + ((∑ x ∈ range n, C (↑p ^ x) * X x ^ p ^ (n + 1 - x)) * C (⅟↑p ^ n) + C ↑p * X (n + 1)) -
      (∑ x ∈ range n, C (↑p ^ x) * (bind₁ (wittPolynomial p ℚ ∘ fun n ↦ n + 1)) (xInTermsOfW p ℚ x) ^ p ^ (n - x)) *
        C (⅟↑p ^ n)

Tactic:
← add_sub,

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
IH :
  ∀ m < n,
    X m ^ p + C ↑p * (MvPolynomial.map (Int.castRingHom ℚ)) (frobeniusPolyAux p m) =
      (bind₁ (wittPolynomial p ℚ ∘ fun n ↦ n + 1)) (xInTermsOfW p ℚ m)
h1 : ↑p ^ n * ⅟↑p ^ n = 1
⊢ X n ^ p + C ↑p * (MvPolynomial.map (Int.castRingHom ℚ)) (frobeniusPolyAux p n) =
    X n ^ p +
      ((∑ x ∈ range n, C (↑p ^ x) * X x ^ p ^ (n + 1 - x)) * C (⅟↑p ^ n) + C ↑p * X (n + 1) -
        (∑ x ∈ range n, C (↑p ^ x) * (bind₁ (wittPolynomial p ℚ ∘ fun n ↦ n + 1)) (xInTermsOfW p ℚ x) ^ p ^ (n - x)) *
          C (⅟↑p ^ n))
Module: Mathlib.RingTheory.WittVector.Frobenius

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
IH :
  ∀ m < n,
    X m ^ p + C ↑p * (MvPolynomial.map (Int.castRingHom ℚ)) (frobeniusPolyAux p m) =
      (bind₁ (wittPolynomial p ℚ ∘ fun n ↦ n + 1)) (xInTermsOfW p ℚ m)
h1 : ↑p ^ n * ⅟↑p ^ n = 1
⊢ C ↑p * X (n + 1) +
      -(C ↑p *
          (MvPolynomial.map (Int.castRingHom ℚ))
            (∑ i ∈ range n,
              ∑ j ∈ range (p ^ (n - i)),
                (X i ^ p) ^ (p ^ (n - i) - (j + 1)) * frobeniusPolyAux p i ^ (j + 1) *
                  C ↑((p ^ (n - i)).choose (j + 1) / p ^ (n - i - v p (j + 1)) * p ^ (j - v p (j + 1))))) =
    C ↑p * X (n + 1) + (∑ x ∈ range n, C (↑p ^ x) * X x ^ p ^ (n + 1 - x)) * C (⅟↑p ^ n) -
      (∑ x ∈ range n, C (↑p ^ x) * (bind₁ (wittPolynomial p ℚ ∘ fun n ↦ n + 1)) (xInTermsOfW p ℚ x) ^ p ^ (n - x)) *
        C (⅟↑p ^ n)

Tactic:
← add_sub,

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
IH :
  ∀ m < n,
    X m ^ p + C ↑p * (MvPolynomial.map (Int.castRingHom ℚ)) (frobeniusPolyAux p m) =
      (bind₁ (wittPolynomial p ℚ ∘ fun n ↦ n + 1)) (xInTermsOfW p ℚ m)
h1 : ↑p ^ n * ⅟↑p ^ n = 1
⊢ C ↑p * X (n + 1) +
      -(C ↑p *
          (MvPolynomial.map (Int.castRingHom ℚ))
            (∑ i ∈ range n,
              ∑ j ∈ range (p ^ (n - i)),
                (X i ^ p) ^ (p ^ (n - i) - (j + 1)) * frobeniusPolyAux p i ^ (j + 1) *
                  C ↑((p ^ (n - i)).choose (j + 1) / p ^ (n - i - v p (j + 1)) * p ^ (j - v p (j + 1))))) =
    C ↑p * X (n + 1) +
      ((∑ x ∈ range n, C (↑p ^ x) * X x ^ p ^ (n + 1 - x)) * C (⅟↑p ^ n) -
        (∑ x ∈ range n, C (↑p ^ x) * (bind₁ (wittPolynomial p ℚ ∘ fun n ↦ n + 1)) (xInTermsOfW p ℚ x) ^ p ^ (n - x)) *
          C (⅟↑p ^ n))