mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.QuasiFinite.Basic

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R S
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
inst✝ : QuasiFinite R T
P : Ideal S
hP : P.IsPrime
f : P.ResidueField ⊗[R] T →ₐ[P.ResidueField] P.Fiber T :=
  TensorProduct.lift (ofId P.ResidueField (P.Fiber T)) (AlgHom.restrictScalars R TensorProduct.includeRight) ⋯
⊢ {t | ∃ m n, m ⊗ₜ[S] n = t} ⊆ ↑(AlgHom.restrictScalars S f).toLinearMap.range

Tactic:
rintro _ ⟨a, b, rfl⟩

Goal state after:
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R S
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
inst✝ : QuasiFinite R T
P : Ideal S
hP : P.IsPrime
f : P.ResidueField ⊗[R] T →ₐ[P.ResidueField] P.Fiber T :=
  TensorProduct.lift (ofId P.ResidueField (P.Fiber T)) (AlgHom.restrictScalars R TensorProduct.includeRight) ⋯
a : P.ResidueField
b : T
⊢ a ⊗ₜ[S] b ∈ ↑(AlgHom.restrictScalars S f).toLinearMap.range
Module: Mathlib.RingTheory.QuasiFinite.Basic

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
p : Ideal S
inst✝³ : p.IsPrime
inst✝² : IsArtinianRing R
inst✝¹ : EssFiniteType R S
inst✝ : QuasiFiniteAt R p
this✝⁵ : IsLocalizedModule p.primeCompl LinearMap.id
this✝⁴ : Module.Finite R (Localization.AtPrime p)
this✝³ : Module.Finite S (Localization.AtPrime p)
this✝² : IsArtinianRing (Localization.AtPrime p)
this✝¹ : IsNoetherianRing S
this✝ : Module.FinitePresentation S (Localization.AtPrime p)
r : S
hrp : r ∈ p.primeCompl
H : IsLocalizedModule.Away r (Algebra.linearMap S (Localization.AtPrime p))
this : IsLocalization (Submonoid.powers r) (Localization.AtPrime p)
φ : Localization.Away r ≃ₐ[S] Localization.AtPrime p :=
  IsLocalization.algEquiv (Submonoid.powers r) (Localization.Away r) (Localization.AtPrime p)
⊢ ∃ f ∉ p, ↑(PrimeSpectrum.basicOpen f) = {{ asIdeal := p, isPrime := inst✝³ }}

Tactic:
refine ⟨r, hrp, subset_antisymm (fun q hrq ↦ ?_) (Set.singleton_subset_iff.mpr hrp)⟩

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
p : Ideal S
inst✝³ : p.IsPrime
inst✝² : IsArtinianRing R
inst✝¹ : EssFiniteType R S
inst✝ : QuasiFiniteAt R p
this✝⁵ : IsLocalizedModule p.primeCompl LinearMap.id
this✝⁴ : Module.Finite R (Localization.AtPrime p)
this✝³ : Module.Finite S (Localization.AtPrime p)
this✝² : IsArtinianRing (Localization.AtPrime p)
this✝¹ : IsNoetherianRing S
this✝ : Module.FinitePresentation S (Localization.AtPrime p)
r : S
hrp : r ∈ p.primeCompl
H : IsLocalizedModule.Away r (Algebra.linearMap S (Localization.AtPrime p))
this : IsLocalization (Submonoid.powers r) (Localization.AtPrime p)
φ : Localization.Away r ≃ₐ[S] Localization.AtPrime p :=
  IsLocalization.algEquiv (Submonoid.powers r) (Localization.Away r) (Localization.AtPrime p)
q : PrimeSpectrum S
hrq : q ∈ ↑(PrimeSpectrum.basicOpen r)
⊢ q ∈ {{ asIdeal := p, isPrime := inst✝³ }}
Module: Mathlib.RingTheory.Smooth.Basic

Goal state before:
pp: R : Type u
A : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
B : Type u_1
inst✝¹ : CommRing B
inst✝ : FormallySmooth R A
I : Ideal B
hI : IsNilpotent I
⊢ ∀ [inst : Algebra R B], Function.Surjective (Ideal.Quotient.mkₐ R I).comp

Tactic:
apply Ideal.IsNilpotent.induction_on (S := B) I hI

Goal state after:
case h₁
R : Type u
A : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
B : Type u_1
inst✝¹ : CommRing B
inst✝ : FormallySmooth R A
I : Ideal B
hI : IsNilpotent I
⊢ ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I : Ideal S),
    I ^ 2 = ⊥ → ∀ [inst_1 : Algebra R S], Function.Surjective (Ideal.Quotient.mkₐ R I).comp

case h₂
R : Type u
A : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
B : Type u_1
inst✝¹ : CommRing B
inst✝ : FormallySmooth R A
I : Ideal B
hI : IsNilpotent I
⊢ ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I J : Ideal S),
    I ≤ J →
      (∀ [inst_1 : Algebra R S], Function.Surjective (Ideal.Quotient.mkₐ R I).comp) →
        (∀ [inst_1 : Algebra R (S ⧸ I)],
            Function.Surjective (Ideal.Quotient.mkₐ R (Ideal.map (Ideal.Quotient.mk I) J)).comp) →
          ∀ [inst_1 : Algebra R S], Function.Surjective (Ideal.Quotient.mkₐ R J).comp
Module: Mathlib.RingTheory.RamificationInertia.Ramification

Goal state before:
pp: S : Type u_1
inst✝⁴ : CommRing S
q : Ideal S
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra R S
inst✝¹ : q.IsPrime
inst✝ : Module.Finite R S
p : Ideal R := under R q
Sq : Type u_1 := Localization.AtPrime q
⊢ 0 < q.ramificationIdx R

Tactic:
ramificationIdx_def

Goal state after:
S : Type u_1
inst✝⁴ : CommRing S
q : Ideal S
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra R S
inst✝¹ : q.IsPrime
inst✝ : Module.Finite R S
p : Ideal R := under R q
Sq : Type u_1 := Localization.AtPrime q
⊢ 0 <
    (Module.length (Localization.AtPrime q)
        (Localization.AtPrime q ⧸ map (algebraMap R (Localization.AtPrime q)) (under R q))).toNat
Module: Mathlib.RingTheory.RamificationInertia.Ramification

Goal state before:
pp: S : Type u_1
inst✝⁴ : CommRing S
q : Ideal S
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra R S
inst✝¹ : q.IsPrime
inst✝ : Module.Finite R S
p : Ideal R := under R q
Sq : Type u_1 := Localization.AtPrime q
r : PrimeSpectrum (p.Fiber S) := (PrimeSpectrum.primesOverOrderIsoFiber R S p) (primesOver.mk p q)
this✝ : q = comap Algebra.TensorProduct.includeRight r.asIdeal
this : Algebra (Localization.AtPrime p) (Localization.AtPrime (comap Algebra.TensorProduct.includeRight r.asIdeal)) :=
  Localization.AtPrime.algebraOfLiesOver p (comap Algebra.TensorProduct.includeRight r.asIdeal)
⊢ IsArtinianRing (Sq ⧸ map (algebraMap R Sq) p)

Tactic:
convert (Fiber.localizationAlgEquivQuotient p r.1).toRingEquiv.isArtinianRing

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

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
inst✝⁵ : EssFiniteType R S
p : Ideal R
inst✝⁴ : p.IsPrime
q : Ideal S
inst✝³ : q.IsPrime
inst✝² : q.LiesOver p
inst✝¹ : Algebra (Localization.AtPrime p) (Localization.AtPrime q)
inst✝ : Localization.AtPrime.IsLiesOverAlgebra p q
⊢ IsUnramifiedAt R q ↔
    Algebra.IsSeparable p.ResidueField q.ResidueField ∧
      Ideal.map (algebraMap R (Localization.AtPrime q)) p = maximalIdeal (Localization.AtPrime q)

Tactic:
by
  have : EssFiniteType (Localization.AtPrime p) (Localization.AtPrime q) := .of_comp R _ _
  trans Algebra.FormallyUnramified (Localization.AtPrime p) (Localization.AtPrime q)
  · exact ⟨fun _ ↦ .of_restrictScalars R _ _,
      fun _ ↦ Algebra.FormallyUnramified.comp _ (Localization.AtPrime p) _⟩
  rw [FormallyUnramified.iff_map_maximalIdeal_eq]
  congr!
  rw [Localization.AtPrime.IsLiesOverAlgebra.algebraMap_eq,
    ← Localization.AtPrime.map_eq_maximalIdeal, Ideal.map_map, Localization.localRingHom,
    IsLocalization.map_comp, ← IsScalarTower.algebraMap_eq]

Goal state after:
Module: Mathlib.RingTheory.Etale.Kaehler

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : CommRing T
inst✝³ : Algebra R S
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
P : Extension R S
Q : Extension R T
f : P.Hom Q
alg : Algebra P.Ring Q.Ring
halg : algebraMap P.Ring Q.Ring = f.toRingHom
H : Function.Bijective ⇑(LinearMap.liftBaseChange Q.Ring (f.mapKer halg))
x : Q.Cotangent
⊢ (LinearMap.liftBaseChange T (Cotangent.map f)).toFun ((tensorCotangentInvFun f halg H) x) = x

Tactic:
obtain ⟨x, rfl⟩ := Cotangent.mk_surjective x
      obtain ⟨x, rfl⟩ := H.surjective x
      simp only [AddHom.toFun_eq_coe, LinearMap.coe_toAddHom]
      induction x with
      | zero => simp only [map_zero]
      | add x y _ _ => simp only [map_add, *]
      | tmul a b =>
        simp only [LinearMap.liftBaseChange_tmul, map_smul]
        simp [Hom.mapKer, tensorCotangentInvFun_smul_mk]

Goal state after:
Module: Mathlib.RingTheory.Etale.Kaehler

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R S
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
P : Extension R S
Q : Extension R T
f : P.Hom Q
alg : Algebra P.Ring Q.Ring
halg : algebraMap P.Ring Q.Ring = f.toRingHom
inst✝ : Module.Flat S T
H₁ : f.toRingHom.FormallyEtale
H₂ : Function.Bijective ⇑(LinearMap.liftBaseChange Q.Ring (f.mapKer halg))
⊢ T ⊗[S] P.H1Cotangent ≃ₗ[T] Q.H1Cotangent

Tactic:
by
  refine .ofBijective ((H1Cotangent.map f).liftBaseChange T) ?_
  constructor
  · rw [injective_iff_map_eq_zero]
    intro x hx
    apply Module.Flat.lTensor_preserves_injective_linearMap _ h1Cotangentι_injective
    apply (Extension.tensorCotangent f halg H₂).injective
    simp only [map_zero]
    rw [← h1Cotangentι.map_zero, ← hx]
    change ((Cotangent.map f).liftBaseChange T ∘ₗ h1Cotangentι.baseChange T) x =
      (h1Cotangentι ∘ₗ _) x
    congr 1
    ext x
    simp
  · intro x
    have : Function.Exact (h1Cotangentι.baseChange T) (P.cotangentComplex.baseChange T) :=
      Module.Flat.lTensor_exact T (LinearMap.exact_subtype_ker_map _)
    obtain ⟨a, ha⟩ := (this ((Extension.tensorCotangent f halg H₂).symm x.1)).mp (by
      apply (Extension.tensorCotangentSpaceOfFormallyEtale f H₁).injective
      rw [LinearEquiv.map_zero, ← x.2]
      have : (CotangentSpace.map f).liftBaseChange T ∘ₗ P.cotangentComplex.baseChange T =
          Q.cotangentComplex ∘ₗ (Cotangent.map f).liftBaseChange T := by
        ext x; obtain ⟨x, rfl⟩ := Cotangent.mk_surjective x; dsimp
        simp only [CotangentSpace.map_tmul,
          map_one, Hom.toAlgHom_apply, one_smul, cotangentComplex_mk]
      exact (DFunLike.congr_fun this _).trans (DFunLike.congr_arg Q.cotangentComplex
        ((tensorCotangent f halg H₂).apply_symm_apply x.1)))
    refine ⟨a, Subtype.ext (.trans ?_ ((LinearEquiv.eq_symm_apply _).mp ha))⟩
    change (h1Cotangentι ∘ₗ (H1Cotangent.map f).liftBaseChange T) _ =
      ((Cotangent.map f).liftBaseChange T ∘ₗ h1Cotangentι.baseChange T) _
    congr 1
    ext; dsimp

Goal state after:
Module: Mathlib.RingTheory.RamificationInertia.Ramification

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
p : Ideal R
q : Ideal S
inst✝³ : IsDomain R
inst✝² : IsDedekindDomain S
inst✝¹ : Module.IsTorsionFree R S
inst✝ : q.LiesOver p
hq : q.IsPrime
hp : p ≠ ⊥
⊢ p.ramificationIdx' q = q.ramificationIdx R

Tactic:
have hpS : p.map (algebraMap R S) ≠ ⊥ := map_ne_bot_of_ne_bot hp

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
p : Ideal R
q : Ideal S
inst✝³ : IsDomain R
inst✝² : IsDedekindDomain S
inst✝¹ : Module.IsTorsionFree R S
inst✝ : q.LiesOver p
hq : q.IsPrime
hp : p ≠ ⊥
hpS : map (algebraMap R S) p ≠ ⊥
⊢ p.ramificationIdx' q = q.ramificationIdx R
Module: Mathlib.RingTheory.RamificationInertia.Basic

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
p : Ideal R
inst✝⁴ : p.IsPrime
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Algebra.QuasiFinite R S
inst✝ : Fintype ↑(p.primesOver S)
this✝ : Fintype (PrimeSpectrum (p.Fiber S)) := Fintype.ofFinite (PrimeSpectrum (p.Fiber S))
q : PrimeSpectrum (p.Fiber S)
a✝ : q ∈ Finset.univ
r : Ideal S := comap Algebra.TensorProduct.includeRight q.asIdeal
this : Algebra (Localization.AtPrime p) (Localization.AtPrime r) := Localization.AtPrime.algebraOfLiesOver p r
Rp : Type u_1 := Localization.AtPrime p
⊢ (length (Localization.AtPrime r) (Localization.AtPrime r ⧸ map (algebraMap R (Localization.AtPrime r)) p)).toNat *
      finrank p.ResidueField r.ResidueField =
    finrank p.ResidueField (Localization.AtPrime q.asIdeal)

Tactic:
let Sq := Localization.AtPrime q.1

Goal state after:
R : Type u_1
inst✝⁵ : CommRing R
p : Ideal R
inst✝⁴ : p.IsPrime
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Algebra.QuasiFinite R S
inst✝ : Fintype ↑(p.primesOver S)
this✝ : Fintype (PrimeSpectrum (p.Fiber S)) := Fintype.ofFinite (PrimeSpectrum (p.Fiber S))
q : PrimeSpectrum (p.Fiber S)
a✝ : q ∈ Finset.univ
r : Ideal S := comap Algebra.TensorProduct.includeRight q.asIdeal
this : Algebra (Localization.AtPrime p) (Localization.AtPrime r) := Localization.AtPrime.algebraOfLiesOver p r
Rp : Type u_1 := Localization.AtPrime p
Sq : Type (max u_1 u_2) := Localization.AtPrime q.asIdeal
⊢ (length (Localization.AtPrime r) (Localization.AtPrime r ⧸ map (algebraMap R (Localization.AtPrime r)) p)).toNat *
      finrank p.ResidueField r.ResidueField =
    finrank p.ResidueField (Localization.AtPrime q.asIdeal)
Module: Mathlib.RingTheory.Unramified.LocalRing

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
p : Ideal R
inst✝³ : p.IsPrime
q : Ideal S
inst✝² : q.IsPrime
hq : p.primesOver S = {q}
inst✝¹ : Module.Finite R S
inst✝ : q.LiesOver p
hRS : (RingHom.ker (algebraMap R S)).FG
H : Function.Bijective ⇑(localRingHom p q (algebraMap R S) ⋯)
s : Finset S
hs : Algebra.adjoin R ↑s = ⊤
a b : S → R
hbp : ∀ (x : S), b x ∉ p
e : ∀ (x : S), (algebraMap R S) (a x) = x * (algebraMap R S) (b x)
r : R
hrp : r ∉ p
hr : ∀ (r' : R), r ∣ r' → Function.Injective ⇑(awayMap (algebraMap R S) r')
r' : R
x : S
hxs : x ∈ ↑s
r'' : R
hr' : r' = (r * ∏ i ∈ s, b i) * r''
⊢ (IsLocalization.map (Away ((algebraMap R S) r')) (algebraMap R S) ⋯)
      (IsLocalization.mk' (Away r') ((r'' * r * ∏ i ∈ s.erase x, b i) * a x) ⟨r' ^ 1, ⋯⟩) =
    (algebraMap S (Away ((algebraMap R S) r'))) x

Tactic:
simp only [pow_one, IsLocalization.map_mk', IsLocalization.mk'_eq_iff_eq_mul,
      ← map_mul (algebraMap S _), map_mul (algebraMap R _), e]

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
p : Ideal R
inst✝³ : p.IsPrime
q : Ideal S
inst✝² : q.IsPrime
hq : p.primesOver S = {q}
inst✝¹ : Module.Finite R S
inst✝ : q.LiesOver p
hRS : (RingHom.ker (algebraMap R S)).FG
H : Function.Bijective ⇑(localRingHom p q (algebraMap R S) ⋯)
s : Finset S
hs : Algebra.adjoin R ↑s = ⊤
a b : S → R
hbp : ∀ (x : S), b x ∉ p
e : ∀ (x : S), (algebraMap R S) (a x) = x * (algebraMap R S) (b x)
r : R
hrp : r ∉ p
hr : ∀ (r' : R), r ∣ r' → Function.Injective ⇑(awayMap (algebraMap R S) r')
r' : R
x : S
hxs : x ∈ ↑s
r'' : R
hr' : r' = (r * ∏ i ∈ s, b i) * r''
⊢ (algebraMap S (Away ((algebraMap R S) r')))
      ((algebraMap R S) r'' * (algebraMap R S) r * (algebraMap R S) (∏ i ∈ s.erase x, b i) *
        (x * (algebraMap R S) (b x))) =
    (algebraMap S (Away ((algebraMap R S) r'))) (x * (algebraMap R S) r')
Module: Mathlib.Algebra.Order.Archimedean.Real.Basic

Goal state before:
pp: s : Set ℝ
L : ℝ
hL : L ∈ s
U : ℝ
hU : U ∈ upperBounds s
k : ℤ
hk : U < ↑k
d : ℕ
z : ℤ
y : ℝ
yS : y ∈ s
hy : ↑z ≤ y * ↑d
⊢ y * ↑d ≤ ↑k * ↑d

Tactic:
gcongr

Goal state after:
case hbc
s : Set ℝ
L : ℝ
hL : L ∈ s
U : ℝ
hU : U ∈ upperBounds s
k : ℤ
hk : U < ↑k
d : ℕ
z : ℤ
y : ℝ
yS : y ∈ s
hy : ↑z ≤ y * ↑d
⊢ y ≤ ↑k
Module: Mathlib.Data.NNReal.Defs

Goal state before:
pp: r : ℝ
hr : 0 ≤ r
⊢ r.toNNReal = NNReal.mk r hr

Tactic:
simp_rw [Real.toNNReal, max_eq_left hr]

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

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
I : Ideal R
hI : I ≠ 0
⊢ HasFiniteMulSupport fun v ↦ ↑v.asIdeal ^ (-↑((Associates.mk v.asIdeal).count (Associates.mk I).factors))

Tactic:
by
  rw [HasFiniteMulSupport, mulSupport]
  simp_rw [zpow_neg, Ne, inv_eq_one]
  exact hasFiniteMulSupport_coe hI

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

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : Ideal R
hI : I ≠ 0
h_ne_zero :
  Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.asIdeal ^ (Associates.mk v.asIdeal).count (Associates.mk I).factors) ≠
    0
hv : Irreducible (Associates.mk v.asIdeal)
h_dvd :
  Associates.mk v.asIdeal ^ (Associates.mk v.asIdeal).count (Associates.mk I).factors ≤
    Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.asIdeal ^ (Associates.mk v.asIdeal).count (Associates.mk I).factors)
h_not_dvd :
  ¬Associates.mk v.asIdeal ^ ((Associates.mk v.asIdeal).count (Associates.mk I).factors + 1) ≤
      Associates.mk
        (∏ᶠ (v : HeightOneSpectrum R), v.asIdeal ^ (Associates.mk v.asIdeal).count (Associates.mk I).factors)
⊢ (Associates.mk v.asIdeal).count (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)).factors =
    (Associates.mk v.asIdeal).count (Associates.mk I).factors

Tactic:
Associates.prime_pow_dvd_iff_le h_ne_zero hv

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : Ideal R
hI : I ≠ 0
h_ne_zero :
  Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.asIdeal ^ (Associates.mk v.asIdeal).count (Associates.mk I).factors) ≠
    0
hv : Irreducible (Associates.mk v.asIdeal)
h_dvd :
  (Associates.mk v.asIdeal).count (Associates.mk I).factors ≤
    (Associates.mk v.asIdeal).count
      (Associates.mk
          (∏ᶠ (v : HeightOneSpectrum R), v.asIdeal ^ (Associates.mk v.asIdeal).count (Associates.mk I).factors)).factors
h_not_dvd :
  ¬(Associates.mk v.asIdeal).count (Associates.mk I).factors + 1 ≤
      (Associates.mk v.asIdeal).count
        (Associates.mk
            (∏ᶠ (v : HeightOneSpectrum R),
              v.asIdeal ^ (Associates.mk v.asIdeal).count (Associates.mk I).factors)).factors
⊢ (Associates.mk v.asIdeal).count (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)).factors =
    (Associates.mk v.asIdeal).count (Associates.mk I).factors
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
I : Ideal R
hI : I ≠ 0
⊢ ∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I = I

Tactic:
← associated_iff_eq,

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
I : Ideal R
hI : I ≠ 0
⊢ Associated (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I) I
Module: Mathlib.Algebra.Order.Archimedean.Real.Basic

Goal state before:
pp: a : ℝ
ha : 1 < a
k : ℕ
posk : 0 < k
hk : 1 / ↑k + 1 < a
⊢ (1 / ↑k * ↑k + 1) ^ (2 * k) ≤ ((1 / ↑k + 1) ^ k) ^ (2 * k)

Tactic:
gcongr

Goal state after:
case hab
a : ℝ
ha : 1 < a
k : ℕ
posk : 0 < k
hk : 1 / ↑k + 1 < a
⊢ 1 / ↑k * ↑k + 1 ≤ (1 / ↑k + 1) ^ k
Module: Mathlib.Algebra.Order.Archimedean.Real.Basic

Goal state before:
pp: a : ℝ
ha : 1 < a
k : ℕ
posk : 0 < k
hk : 1 / ↑k + 1 < a
⊢ (1 / ↑k + 1) ^ (2 * k ^ 2) < a ^ (2 * k ^ 2)

Tactic:
gcongr

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Seminorm

Goal state before:
pp: case inr
R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
inst✝² : Group E
inst✝¹ : Group F
inst✝ : Group G
p✝ q : GroupSeminorm E
f : F →* E
s : Set (GroupSeminorm E)
h : BddAbove s
x y : E
hs : s.Nonempty
this : Nonempty ↑s
p : ↑s
⊢ ↑p x + ↑p y ≤ (⨆ p, ↑p x) + ⨆ p, ↑p y

Tactic:
gcongr

Goal state after:
case h₁
R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
inst✝² : Group E
inst✝¹ : Group F
inst✝ : Group G
p✝ q : GroupSeminorm E
f : F →* E
s : Set (GroupSeminorm E)
h : BddAbove s
x y : E
hs : s.Nonempty
this : Nonempty ↑s
p : ↑s
⊢ ↑p x ≤ ⨆ p, ↑p x

case h₂
R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
inst✝² : Group E
inst✝¹ : Group F
inst✝ : Group G
p✝ q : GroupSeminorm E
f : F →* E
s : Set (GroupSeminorm E)
h : BddAbove s
x y : E
hs : s.Nonempty
this : Nonempty ↑s
p : ↑s
⊢ ↑p y ≤ ⨆ p, ↑p y
Module: Mathlib.Data.ENNReal.Basic

Goal state before:
pp: x : ℝ≥0∞
⊢ |x.toReal| = x.toReal

Tactic:
cases x <;> simp

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Seminorm

Goal state before:
pp: R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
inst✝³ : AddGroup E
inst✝² : SMul R ℝ
inst✝¹ : SMul R ℝ≥0
inst✝ : IsScalarTower R ℝ≥0 ℝ
r : R
p : AddGroupSeminorm E
x✝¹ x✝ : E
⊢ ↑(r • 1) * p (x✝¹ + x✝) ≤ ↑(r • 1) * (p x✝¹ + p x✝)

Tactic:
gcongr

Goal state after:
case hbc
R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
inst✝³ : AddGroup E
inst✝² : SMul R ℝ
inst✝¹ : SMul R ℝ≥0
inst✝ : IsScalarTower R ℝ≥0 ℝ
r : R
p : AddGroupSeminorm E
x✝¹ x✝ : E
⊢ p (x✝¹ + x✝) ≤ p x✝¹ + p x✝
Module: Mathlib.Analysis.Normed.Group.Seminorm

Goal state before:
pp: case inr
R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
inst✝ : AddGroup E
p✝ q : NonarchAddGroupSeminorm E
s : Set (NonarchAddGroupSeminorm E)
h : BddAbove s
x y : E
hs : s.Nonempty
this : Nonempty ↑s
p : ↑s
⊢ max (↑p x) (↑p y) ≤ max (⨆ p, ↑p x) (⨆ p, ↑p y)

Tactic:
gcongr

Goal state after:
case h₁
R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
inst✝ : AddGroup E
p✝ q : NonarchAddGroupSeminorm E
s : Set (NonarchAddGroupSeminorm E)
h : BddAbove s
x y : E
hs : s.Nonempty
this : Nonempty ↑s
p : ↑s
⊢ ↑p x ≤ ⨆ p, ↑p x

case h₂
R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
inst✝ : AddGroup E
p✝ q : NonarchAddGroupSeminorm E
s : Set (NonarchAddGroupSeminorm E)
h : BddAbove s
x y : E
hs : s.Nonempty
this : Nonempty ↑s
p : ↑s
⊢ ↑p y ≤ ⨆ p, ↑p y
Module: Mathlib.Analysis.Normed.Group.Seminorm

Goal state before:
pp: R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
inst✝³ : Group E
inst✝² : SMul R ℝ
inst✝¹ : SMul R ℝ≥0
inst✝ : IsScalarTower R ℝ≥0 ℝ
r : R
p : GroupSeminorm E
x✝¹ x✝ : E
⊢ ↑(r • 1) * p (x✝¹ * x✝) ≤ ↑(r • 1) * (p x✝¹ + p x✝)

Tactic:
gcongr

Goal state after:
case hbc
R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
inst✝³ : Group E
inst✝² : SMul R ℝ
inst✝¹ : SMul R ℝ≥0
inst✝ : IsScalarTower R ℝ≥0 ℝ
r : R
p : GroupSeminorm E
x✝¹ x✝ : E
⊢ p (x✝¹ * x✝) ≤ p x✝¹ + p x✝
Module: Mathlib.Analysis.Normed.Group.Seminorm

Goal state before:
pp: R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
inst✝³ : AddGroup E
inst✝² : SMul R ℝ
inst✝¹ : SMul R ℝ≥0
inst✝ : IsScalarTower R ℝ≥0 ℝ
r : R
p : NonarchAddGroupSeminorm E
x y : E
⊢ ↑(r • 1) * p (x + y) ≤ ↑(r • 1) * max (p x) (p y)

Tactic:
gcongr

Goal state after:
case hbc
R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
inst✝³ : AddGroup E
inst✝² : SMul R ℝ
inst✝¹ : SMul R ℝ≥0
inst✝ : IsScalarTower R ℝ≥0 ℝ
r : R
p : NonarchAddGroupSeminorm E
x y : E
⊢ p (x + y) ≤ max (p x) (p y)
Module: Mathlib.Analysis.Normed.Group.Seminorm

Goal state before:
pp: R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
inst✝ : Group E
p q f g : GroupNorm E
h : (fun f ↦ f.toFun) f = (fun f ↦ f.toFun) g
⊢ f = g

Tactic:
obtain ⟨⟨_, _, _, _⟩, _⟩ := f; cases g; congr

Goal state after:
Module: Mathlib.Data.ENNReal.Operations

Goal state before:
pp: case inl
p q : ℝ
hq : 0 ≤ q
h : p ≤ q
⊢ ENNReal.ofReal (p - q) = ENNReal.ofReal p - ENNReal.ofReal q

Tactic:
rw [ofReal_of_nonpos (sub_nonpos_of_le h), tsub_eq_zero_of_le (ofReal_le_ofReal h)]

Goal state after:
Module: Mathlib.Data.ENNReal.Operations

Goal state before:
pp: ι : Type u_2
inst✝¹ : Preorder ι
inst✝ : IsCodirectedOrder ι
f g : ι → ℝ≥0∞
hf : Monotone f
hg : Monotone g
i j _k : ι
x✝ : _k ≤ i ∧ _k ≤ j
hi : _k ≤ i
hj : _k ≤ j
⊢ f _k + g _k ≤ f i + g j

Tactic:
gcongr

Goal state after:
case h₁
ι : Type u_2
inst✝¹ : Preorder ι
inst✝ : IsCodirectedOrder ι
f g : ι → ℝ≥0∞
hf : Monotone f
hg : Monotone g
i j _k : ι
x✝ : _k ≤ i ∧ _k ≤ j
hi : _k ≤ i
hj : _k ≤ j
⊢ f _k ≤ f i

case h₂
ι : Type u_2
inst✝¹ : Preorder ι
inst✝ : IsCodirectedOrder ι
f g : ι → ℝ≥0∞
hf : Monotone f
hg : Monotone g
i j _k : ι
x✝ : _k ≤ i ∧ _k ≤ j
hi : _k ≤ i
hj : _k ≤ j
⊢ g _k ≤ g j
Module: Mathlib.Data.ENNReal.Operations

Goal state before:
pp: ι : Sort u_1
a : ℝ≥0∞
κ : ι → Sort u_2
f : (i : ι) → κ i → ℝ≥0∞
⊢ (⨅ i, ⨅ j, f i j) + a = ⨅ i, ⨅ j, f i j + a

Tactic:
simp only [add_comm, add_iInf₂]

Goal state after:
Module: Mathlib.Data.ENNReal.Operations

Goal state before:
pp: ι : Sort u_1
a : ℝ≥0∞
p : ι → Prop
h : ∃ i, p i
f : ι → ℝ≥0∞
⊢ (⨆ i, ⨆ (_ : p i), f i) + a = ⨆ i, ⨆ (_ : p i), f i + a

Tactic:
by simp only [add_comm, add_biSup' h]

Goal state after: