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

Goal state before:
pp: R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : FinitePresentation R S
P : Presentation R S (Fin (Presentation.ofFinitePresentationVars R S))
  (Fin (Presentation.ofFinitePresentationRels R S)) :=
  Presentation.ofFinitePresentation R S
⊢ FiniteType R (Presentation.ofFinitePresentation R S).Ring

Tactic:
infer_instance

Goal state after:
Module: Mathlib.RingTheory.Extension.Cotangent.Basic

Goal state before:
pp: R : Type u
S : Type v
inst✝¹² : CommRing R
inst✝¹¹ : CommRing S
inst✝¹⁰ : Algebra R S
ι : Type w
ι' : Type u_1
P✝ : Generators R S ι
S' : Type u_2
inst✝⁹ : CommRing S'
inst✝⁸ : Algebra R S'
T : Type w
inst✝⁷ : CommRing T
inst✝⁶ : Algebra R T
inst✝⁵ : Algebra S T
inst✝⁴ : IsScalarTower R S T
inst✝³ : Algebra S' T
inst✝² : IsScalarTower R S' T
inst✝¹ : FinitePresentation R S
inst✝ : Projective S Ω[S⁄R]
P : Presentation R S (Fin (Presentation.ofFinitePresentationVars R S))
  (Fin (Presentation.ofFinitePresentationRels R S)) :=
  Presentation.ofFinitePresentation R S
⊢ FiniteType R (Presentation.ofFinitePresentation R S).Ring

Tactic:
infer_instance

Goal state after:
Module: Mathlib.RingTheory.Extension.Cotangent.Basic

Goal state before:
pp: R : Type u
S : Type v
inst✝¹² : CommRing R
inst✝¹¹ : CommRing S
inst✝¹⁰ : Algebra R S
ι : Type w
ι' : Type u_1
P✝ : Generators R S ι
S' : Type u_2
inst✝⁹ : CommRing S'
inst✝⁸ : Algebra R S'
T : Type w
inst✝⁷ : CommRing T
inst✝⁶ : Algebra R T
inst✝⁵ : Algebra S T
inst✝⁴ : IsScalarTower R S T
inst✝³ : Algebra S' T
inst✝² : IsScalarTower R S' T
inst✝¹ : FinitePresentation R S
inst✝ : Projective S Ω[S⁄R]
P : Presentation R S (Fin (Presentation.ofFinitePresentationVars R S))
  (Fin (Presentation.ofFinitePresentationRels R S)) :=
  Presentation.ofFinitePresentation R S
this : FiniteType R P.toExtension.Ring
⊢ ⊤.FG

Tactic:
Submodule.fg_top,

Goal state after:
R : Type u
S : Type v
inst✝¹² : CommRing R
inst✝¹¹ : CommRing S
inst✝¹⁰ : Algebra R S
ι : Type w
ι' : Type u_1
P✝ : Generators R S ι
S' : Type u_2
inst✝⁹ : CommRing S'
inst✝⁸ : Algebra R S'
T : Type w
inst✝⁷ : CommRing T
inst✝⁶ : Algebra R T
inst✝⁵ : Algebra S T
inst✝⁴ : IsScalarTower R S T
inst✝³ : Algebra S' T
inst✝² : IsScalarTower R S' T
inst✝¹ : FinitePresentation R S
inst✝ : Projective S Ω[S⁄R]
P : Presentation R S (Fin (Presentation.ofFinitePresentationVars R S))
  (Fin (Presentation.ofFinitePresentationRels R S)) :=
  Presentation.ofFinitePresentation R S
this : FiniteType R P.toExtension.Ring
⊢ P.toExtension.cotangentComplex.ker.FG
Module: Mathlib.RingTheory.Support

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_3
P : Type u_4
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : AddCommGroup P
inst✝ : Module R P
f : M →ₗ[R] N
g : N →ₗ[R] P
h : Function.Exact ⇑f ⇑g
hf : Function.Injective ⇑f
hg : Function.Surjective ⇑g
x : PrimeSpectrum R
⊢ (∀ (m : M), ∃ r ∉ x.asIdeal, r • m = 0) →
    (∀ (m : P), ∃ r ∉ x.asIdeal, r • m = 0) → ∀ (m : N), ∃ r ∉ x.asIdeal, r • m = 0

Tactic:
intro H₁ H₂ m

Goal state after:
R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_3
P : Type u_4
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : AddCommGroup P
inst✝ : Module R P
f : M →ₗ[R] N
g : N →ₗ[R] P
h : Function.Exact ⇑f ⇑g
hf : Function.Injective ⇑f
hg : Function.Surjective ⇑g
x : PrimeSpectrum R
H₁ : ∀ (m : M), ∃ r ∉ x.asIdeal, r • m = 0
H₂ : ∀ (m : P), ∃ r ∉ x.asIdeal, r • m = 0
m : N
⊢ ∃ r ∉ x.asIdeal, r • m = 0
Module: Mathlib.Data.Fin.Parity

Goal state before:
pp: n : ℕ
R : Type u_1
inst✝¹ : Monoid R
inst✝ : HasDistribNeg R
i : Fin (n + 1)
j : Fin n
⊢ (-1) ^ (↑(i.succAbove j) + ↑(j.predAbove i)) = -(-1) ^ (↑i + ↑j)

Tactic:
← neg_one_mul (_ ^ _),

Goal state after:
n : ℕ
R : Type u_1
inst✝¹ : Monoid R
inst✝ : HasDistribNeg R
i : Fin (n + 1)
j : Fin n
⊢ (-1) ^ (↑(i.succAbove j) + ↑(j.predAbove i)) = -1 * (-1) ^ (↑i + ↑j)
Module: Mathlib.LinearAlgebra.Alternating.Uncurry.Fin

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_4
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : AddCommGroup N
inst✝¹ : Module R M
inst✝ : Module R N
n : ℕ
f : M →ₗ[R] M →ₗ[R] M [⋀^Fin n]→ₗ[R] N
v : Fin (n + 2) → M
i j : Fin (n + 1)
hj : j ∈ Finset.Ici i
⊢ (-1) ^ ↑i • (-1) ^ ↑j • ((f (v i.castSucc)) (i.castSucc.removeNth v j)) (j.removeNth (i.castSucc.removeNth v)) +
      (-1) ^ (↑j + 1) • (-1) ^ ↑i • ((f (v j.succ)) (j.succ.removeNth v i)) (i.removeNth (j.succ.removeNth v)) =
    (-1) ^ (↑i + ↑j) •
      (((f (v i.castSucc)) (v j.succ)) (j.removeNth (i.castSucc.removeNth v)) -
        ((f (v j.succ)) (v i.castSucc)) (j.removeNth (i.castSucc.removeNth v)))

Tactic:
rw [Finset.mem_Ici] at hj

Goal state after:
R : Type u_1
M : Type u_2
N : Type u_4
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : AddCommGroup N
inst✝¹ : Module R M
inst✝ : Module R N
n : ℕ
f : M →ₗ[R] M →ₗ[R] M [⋀^Fin n]→ₗ[R] N
v : Fin (n + 2) → M
i j : Fin (n + 1)
hj : i ≤ j
⊢ (-1) ^ ↑i • (-1) ^ ↑j • ((f (v i.castSucc)) (i.castSucc.removeNth v j)) (j.removeNth (i.castSucc.removeNth v)) +
      (-1) ^ (↑j + 1) • (-1) ^ ↑i • ((f (v j.succ)) (j.succ.removeNth v i)) (i.removeNth (j.succ.removeNth v)) =
    (-1) ^ (↑i + ↑j) •
      (((f (v i.castSucc)) (v j.succ)) (j.removeNth (i.castSucc.removeNth v)) -
        ((f (v j.succ)) (v i.castSucc)) (j.removeNth (i.castSucc.removeNth v)))
Module: Mathlib.LinearAlgebra.ExteriorPower.Pairing

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : LinearOrder ι
x : ι → M
f : ι → Module.Dual R M
h₀ : ∀ ⦃i j : ι⦄, i ≠ j → (f i) (x j) = 0
n : ℕ
a b : Fin n ↪o ι
h : a ≠ b
⊢ ((pairingDual R M n) ((ιMulti R n) (f ∘ ⇑a))) ((ιMulti R n) (x ∘ ⇑b)) = 0

Tactic:
by
  simp only [pairingDual_ιMulti_ιMulti, Function.comp_apply, Matrix.det_apply]
  refine Finset.sum_eq_zero (fun σ _ ↦ ?_)
  simp only [Matrix.of_apply, smul_eq_iff_eq_inv_smul, smul_zero]
  by_contra h'
  apply h
  have : a = b ∘ σ := by
    ext i
    by_contra hi
    exact h' (Finset.prod_eq_zero (i := i) (by simp) (h₀ hi))
  have hσ : Monotone σ := fun i j hij ↦ by
    have h'' := congr_fun this
    dsimp at h''
    rw [← a.map_rel_iff] at hij
    simpa only [← b.map_rel_iff, ← h'']
  have hσ' : Monotone σ.symm := fun i j hij ↦ by
    obtain ⟨i, rfl⟩ := σ.surjective i
    obtain ⟨j, rfl⟩ := σ.surjective j
    simp only [Equiv.symm_apply_apply]
    by_contra! h
    obtain rfl : i = j := σ.injective (le_antisymm hij (hσ h.le))
    simp only [lt_self_iff_false] at h
  obtain rfl : σ = 1 := by
    ext i : 1
    exact DFunLike.congr_fun (Subsingleton.elim (σ.toOrderIso hσ hσ') (OrderIso.refl _)) i
  ext
  apply congr_fun this

Goal state after:
Module: Mathlib.LinearAlgebra.ExteriorPower.Basis

Goal state before:
pp: R : Type u_1
K : Type u_2
M : Type u_3
E : Type u_4
n : ℕ
inst✝⁶ : CommRing R
inst✝⁵ : Field K
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup E
inst✝¹ : Module K E
inst✝ : Free R M
⊢ Free R ↥(⋀[R]^n M)

Tactic:
have ⟨I, b⟩ := Module.Free.exists_basis R M
  let : LinearOrder I := linearOrderOfSTO WellOrderingRel
  exact Module.Free.of_basis (b.exteriorPower n)

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

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsLocalRing R
inst✝ : FinitePresentation R M
ι : Type u_5
v : ι → M
hli : LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v)
hsp : Submodule.span k (Set.range (⇑((TensorProduct.mk R k M) 1) ∘ v)) = ⊤
H : Function.Injective ⇑(LinearMap.rTensor M (Submodule.subtype 𝔪))
bk : Basis ι k (k ⊗[R] M) := Basis.mk hli ⋯
⊢ ∃ b, ∀ (i : ι), b i = v i

Tactic:
have : Finite ι := Module.Finite.finite_basis bk

Goal state after:
R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsLocalRing R
inst✝ : FinitePresentation R M
ι : Type u_5
v : ι → M
hli : LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v)
hsp : Submodule.span k (Set.range (⇑((TensorProduct.mk R k M) 1) ∘ v)) = ⊤
H : Function.Injective ⇑(LinearMap.rTensor M (Submodule.subtype 𝔪))
bk : Basis ι k (k ⊗[R] M) := Basis.mk hli ⋯
this : Finite ι
⊢ ∃ b, ∀ (i : ι), b i = v i
Module: Mathlib.LinearAlgebra.LinearDisjoint

Goal state before:
pp: R : Type u
S : Type v
inst✝² : CommSemiring R
inst✝¹ : Semiring S
inst✝ : Algebra R S
M N : Submodule R S
ι : Type u_1
n : Basis ι R ↥N
H : Function.Injective ⇑(M.mulRightMap ⇑n)
⊢ M.LinearDisjoint N

Tactic:
simp_rw [mulRightMap_eq_mulMap_comp, ← Basis.coe_repr_symm,
    ← LinearEquiv.coe_lTensor, LinearEquiv.comp_coe, LinearMap.coe_comp,
    LinearEquiv.coe_coe, EquivLike.injective_comp] at H

Goal state after:
R : Type u
S : Type v
inst✝² : CommSemiring R
inst✝¹ : Semiring S
inst✝ : Algebra R S
M N : Submodule R S
ι : Type u_1
n : Basis ι R ↥N
H : Function.Injective ⇑(M.mulMap N)
⊢ M.LinearDisjoint N
Module: Mathlib.RingTheory.LocalRing.Module

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsLocalRing R
inst✝ : Flat R M
ι : Type u
f : ι → R
n : ι
s : Finset ι
hn : n ∉ s
ih : ∀ (v : ι → M), LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v) → ∑ i ∈ s, f i • v i = 0 → ∀ i ∈ s, f i = 0
v : ι → M
h : LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v)
hfv : ∑ i, f ↑i • v ↑i = 0
i : ι
hi : i ∈ insert n s
l : ℕ
a : ↥(insert n s) → Fin l → R
y : Fin l → M
hay : ∀ (i : ↥(insert n s)), (fun i ↦ v ↑i) i = ∑ j, a i j • y j
hfa : ∀ (j : Fin l), ∑ i, (fun i ↦ f ↑i) i * a i j = 0
⊢ v n ∉ 𝔪 • ⊤

Tactic:
simpa only [← LinearMap.ker_tensorProductMk] using! h.ne_zero n

Goal state after:
Module: Mathlib.LinearAlgebra.TensorPower.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
⊢ Semiring (⨁ (n : ℕ), ⨂[R]^n M)

Tactic:
infer_instance

Goal state after:
Module: Mathlib.LinearAlgebra.TensorPower.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
r s : R
⊢ (GradedMonoid.mk 0 ((↑algebraMap₀).toAddMonoidHom (r * s))).snd =
    (cast R M ⋯).symm
      (GradedMonoid.mk (0 + 0)
          (GradedMonoid.GMul.mul ((↑algebraMap₀).toAddMonoidHom r) ((↑algebraMap₀).toAddMonoidHom s))).snd

Tactic:
have := algebraMap₀_mul_algebraMap₀ (M := M) r s

Goal state after:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
r s : R
this : (cast R M ⋯) (GradedMonoid.GMul.mul (algebraMap₀ r) (algebraMap₀ s)) = algebraMap₀ (r * s)
⊢ (GradedMonoid.mk 0 ((↑algebraMap₀).toAddMonoidHom (r * s))).snd =
    (cast R M ⋯).symm
      (GradedMonoid.mk (0 + 0)
          (GradedMonoid.GMul.mul ((↑algebraMap₀).toAddMonoidHom r) ((↑algebraMap₀).toAddMonoidHom s))).snd
Module: Mathlib.LinearAlgebra.TensorPower.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
⊢ Algebra R (⨁ (n : ℕ), ⨂[R]^n M)

Tactic:
infer_instance

Goal state after:
Module: Mathlib.LinearAlgebra.PiTensorProduct.Basic

Goal state before:
pp: ι : Type u_1
R : Type u_4
inst✝⁵ : CommSemiring R
s : ι → Type u_7
inst✝⁴ : (i : ι) → AddCommMonoid (s i)
inst✝³ : (i : ι) → Module R (s i)
t : ι → Type u_11
inst✝² : (i : ι) → AddCommMonoid (t i)
inst✝¹ : (i : ι) → Module R (t i)
f : (i : ι) → s i →ₗ[R] t i
inst✝ : DecidableEq ι
i : ι
u v : s i →ₗ[R] t i
⊢ map (update f i (u + v)) = map (update f i u) + map (update f i v)

Tactic:
ext x
  simp only [LinearMap.compMultilinearMap_apply, map_tprod, map_add_smul_aux, LinearMap.add_apply,
    MultilinearMap.map_update_add]

Goal state after:
Module: Mathlib.RingTheory.Ideal.AssociatedPrime.Basic

Goal state before:
pp: case mp
R : Type u_1
M : Type u_2
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
N : Submodule R M
I : Ideal R
inst✝ : IsNoetherianRing R
⊢ N.IsAssociatedPrime I → I.IsPrime ∧ ∃ x, I = N.colon {x}

pp: case mpr
R : Type u_1
M : Type u_2
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
N : Submodule R M
I : Ideal R
inst✝ : IsNoetherianRing R
⊢ (I.IsPrime ∧ ∃ x, I = N.colon {x}) → N.IsAssociatedPrime I

Tactic:
· rintro ⟨hx, x, rfl⟩
    refine ⟨hx, exists_eq_colon_of_mem_minimalPrimes (x := x) ?_⟩
    rw [← Ideal.radical_minimalPrimes, Ideal.minimalPrimes_eq_subsingleton_self,
      Set.mem_singleton_iff]

Goal state after:
case mpr
R : Type u_1
M : Type u_2
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
N : Submodule R M
I : Ideal R
inst✝ : IsNoetherianRing R
⊢ (I.IsPrime ∧ ∃ x, I = N.colon {x}) → N.IsAssociatedPrime I
Module: Mathlib.RingTheory.Ideal.AssociatedPrime.Finiteness

Goal state before:
pp: case refine_1.refine_2
A : Type u
inst✝² : CommRing A
M : Type v
inst✝¹ : AddCommGroup M
inst✝ : Module A M
N₁ N₂ : Submodule A M
f : ↥N₂ ⧸ N₁.submoduleOf N₂ →ₗ[A] M ⧸ N₁ := (N₁.submoduleOf N₂).mapQ N₁ N₂.subtype ⋯
hf₁ : f.ker = ⊥
hf₂ : f.range = map N₁.mkQ N₂
x✝ : N₁.IsQuotientEquivQuotientPrime N₂
h : N₁ ≤ N₂
p : PrimeSpectrum A
e : (↥N₂ ⧸ N₁.submoduleOf N₂) ≃ₗ[A] A ⧸ p.asIdeal
x : M
hx : x ∈ N₂
hx' : (N₁.submoduleOf N₂).mkQ ⟨x, hx⟩ = e.symm 1
hx'' : N₁.mkQ x = f (e.symm 1)
⊢ N₂ ≤ N₁ ⊔ A ∙ x

Tactic:
have : (span A {x}).map N₁.mkQ = ((span A {1}).map e.symm.toLinearMap).map f := by
        simp only [map_span, Set.image_singleton, hx'', LinearEquiv.coe_coe]

Goal state after:
case refine_1.refine_2
A : Type u
inst✝² : CommRing A
M : Type v
inst✝¹ : AddCommGroup M
inst✝ : Module A M
N₁ N₂ : Submodule A M
f : ↥N₂ ⧸ N₁.submoduleOf N₂ →ₗ[A] M ⧸ N₁ := (N₁.submoduleOf N₂).mapQ N₁ N₂.subtype ⋯
hf₁ : f.ker = ⊥
hf₂ : f.range = map N₁.mkQ N₂
x✝ : N₁.IsQuotientEquivQuotientPrime N₂
h : N₁ ≤ N₂
p : PrimeSpectrum A
e : (↥N₂ ⧸ N₁.submoduleOf N₂) ≃ₗ[A] A ⧸ p.asIdeal
x : M
hx : x ∈ N₂
hx' : (N₁.submoduleOf N₂).mkQ ⟨x, hx⟩ = e.symm 1
hx'' : N₁.mkQ x = f (e.symm 1)
this : map N₁.mkQ (A ∙ x) = map f (map (↑e.symm) (A ∙ 1))
⊢ N₂ ≤ N₁ ⊔ A ∙ x
Module: Mathlib.RingTheory.ClassGroup.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
I : (FractionalIdeal R⁰ (FractionRing R))ˣ
⊢ Quot.mk (⇑(QuotientGroup.leftRel (toPrincipalIdeal R (FractionRing R)).range)) I =
    (QuotientGroup.mk' (toPrincipalIdeal R (FractionRing R)).range) I

Tactic:
QuotientGroup.mk'_apply

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
I : (FractionalIdeal R⁰ (FractionRing R))ˣ
⊢ Quot.mk (⇑(QuotientGroup.leftRel (toPrincipalIdeal R (FractionRing R)).range)) I = ↑I
Module: Mathlib.RingTheory.LocalProperties.Projective

Goal state before:
pp: R : Type u_1
M : Type uM
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
H : ∀ (I : Ideal R) (x : I.IsMaximal), Projective (Localization.AtPrime I) (LocalizedModule I.primeCompl M)
inst✝ : FinitePresentation R M
⊢ Module.Finite R M

Tactic:
infer_instance

Goal state after:
Module: Mathlib.Topology.LocallyConstant.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : X ≃ₜ Y
⊢ Function.LeftInverse (comap ↑e) (comap ↑e.symm)

Tactic:
intro

Goal state after:
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : X ≃ₜ Y
x✝ : LocallyConstant X Z
⊢ comap (↑e) (comap (↑e.symm) x✝) = x✝
Module: Mathlib.Topology.LocallyConstant.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : X ≃ₜ Y
⊢ Function.RightInverse (comap ↑e) (comap ↑e.symm)

Tactic:
intro

Goal state after:
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : X ≃ₜ Y
x✝ : LocallyConstant Y Z
⊢ comap (↑e.symm) (comap (↑e) x✝) = x✝
Module: Mathlib.Topology.LocallyConstant.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
inst✝ : TopologicalSpace X
e : Y ≃ Z
⊢ Function.LeftInverse (map ⇑e.symm) (map ⇑e)

Tactic:
intro

Goal state after:
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
inst✝ : TopologicalSpace X
e : Y ≃ Z
x✝ : LocallyConstant X Y
⊢ map (⇑e.symm) (map (⇑e) x✝) = x✝
Module: Mathlib.Topology.LocallyConstant.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
inst✝ : TopologicalSpace X
e : Y ≃ Z
⊢ Function.RightInverse (map ⇑e.symm) (map ⇑e)

Tactic:
intro

Goal state after:
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
inst✝ : TopologicalSpace X
e : Y ≃ Z
x✝ : LocallyConstant X Z
⊢ map (⇑e) (map (⇑e.symm) x✝) = x✝
Module: Mathlib.RingTheory.TensorProduct.IsBaseChangePi

Goal state before:
pp: case hf
R : Type u_1
S✝ : Type u_2
inst✝¹² : CommSemiring R
inst✝¹¹ : CommSemiring S✝
inst✝¹⁰ : Algebra R S✝
S : Submonoid R
M : Type u_3
N : Type u_4
M' : Type u_5
N' : Type u_6
inst✝⁹ : AddCommMonoid M
inst✝⁸ : AddCommMonoid N
inst✝⁷ : Module R M
inst✝⁶ : Module R N
inst✝⁵ : AddCommMonoid M'
inst✝⁴ : AddCommMonoid N'
inst✝³ : Module R M'
inst✝² : Module R N'
f : M →ₗ[R] M'
g : N →ₗ[R] N'
inst✝¹ : IsLocalizedModule S f
inst✝ : IsLocalizedModule S g
this✝ : Module (Localization S) M' := module S f
this : Module (Localization S) N' := module S g
⊢ IsLocalizedModule S f

Tactic:
infer_instance

Goal state after:
Module: Mathlib.RingTheory.TensorProduct.IsBaseChangePi

Goal state before:
pp: case hg
R : Type u_1
S✝ : Type u_2
inst✝¹² : CommSemiring R
inst✝¹¹ : CommSemiring S✝
inst✝¹⁰ : Algebra R S✝
S : Submonoid R
M : Type u_3
N : Type u_4
M' : Type u_5
N' : Type u_6
inst✝⁹ : AddCommMonoid M
inst✝⁸ : AddCommMonoid N
inst✝⁷ : Module R M
inst✝⁶ : Module R N
inst✝⁵ : AddCommMonoid M'
inst✝⁴ : AddCommMonoid N'
inst✝³ : Module R M'
inst✝² : Module R N'
f : M →ₗ[R] M'
g : N →ₗ[R] N'
inst✝¹ : IsLocalizedModule S f
inst✝ : IsLocalizedModule S g
this✝ : Module (Localization S) M' := module S f
this : Module (Localization S) N' := module S g
⊢ IsLocalizedModule S g

Tactic:
infer_instance

Goal state after:
Module: Mathlib.RingTheory.TensorProduct.IsBaseChangePi

Goal state before:
pp: R : Type u_1
S✝ : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S✝
inst✝⁶ : Algebra R S✝
S : Submonoid R
ι : Type u_3
inst✝⁵ : Finite ι
M : ι → Type u_4
M' : ι → Type u_5
inst✝⁴ : (i : ι) → AddCommMonoid (M i)
inst✝³ : (i : ι) → AddCommMonoid (M' i)
inst✝² : (i : ι) → Module R (M i)
inst✝¹ : (i : ι) → Module R (M' i)
f : (i : ι) → M i →ₗ[R] M' i
inst✝ : ∀ (i : ι), IsLocalizedModule S (f i)
this : (i : ι) → Module (Localization S) (M' i) := fun i ↦ module S (f i)
i : ι
⊢ IsLocalizedModule S (f i)

Tactic:
infer_instance

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.FreeLocus

Goal state before:
pp: case h'
R : Type uR
M : Type uM
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
S : Submonoid R
p : PrimeSpectrum (Localization S)
p' : Ideal R := Ideal.comap (algebraMap R (Localization S)) p.asIdeal
hp' : S ≤ p'.primeCompl
Rₚ : Type uR := Localization.AtPrime p'
Mₚ : Type (max uR uM) := LocalizedModule p'.primeCompl M
this✝¹ : Algebra (Localization S) Rₚ :=
  IsLocalization.localizationAlgebraOfSubmonoidLe (Localization S) Rₚ S p'.primeCompl hp'
this✝ : IsScalarTower R (Localization S) Rₚ
this : IsLocalization (Submonoid.map (algebraMap R (Localization S)) p'.primeCompl) Rₚ
x : Localization S
hx : x ∈ p.asIdeal.primeCompl
⊢ ∃ m, m * ↑⟨x, hx⟩ ∈ Submonoid.map (algebraMap R (Localization S)) p'.primeCompl

Tactic:
obtain ⟨x, s, rfl⟩ := IsLocalization.exists_mk'_eq S x

Goal state after:
case h'
R : Type uR
M : Type uM
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
S : Submonoid R
p : PrimeSpectrum (Localization S)
p' : Ideal R := Ideal.comap (algebraMap R (Localization S)) p.asIdeal
hp' : S ≤ p'.primeCompl
Rₚ : Type uR := Localization.AtPrime p'
Mₚ : Type (max uR uM) := LocalizedModule p'.primeCompl M
this✝¹ : Algebra (Localization S) Rₚ :=
  IsLocalization.localizationAlgebraOfSubmonoidLe (Localization S) Rₚ S p'.primeCompl hp'
this✝ : IsScalarTower R (Localization S) Rₚ
this : IsLocalization (Submonoid.map (algebraMap R (Localization S)) p'.primeCompl) Rₚ
x : R
s : ↥S
hx : IsLocalization.mk' (Localization S) x s ∈ p.asIdeal.primeCompl
⊢ ∃ m, m * ↑⟨IsLocalization.mk' (Localization S) x s, hx⟩ ∈ Submonoid.map (algebraMap R (Localization S)) p'.primeCompl
Module: Mathlib.GroupTheory.Solvable

Goal state before:
pp: case mk
G : Type u_3
toDivInvMonoid✝ : DivInvMonoid G
inv_mul_cancel✝ : ∀ (a : G), a⁻¹ * a = 1
h : ∀ (a b : G), a * b = b * a
hG' : CommGroup G := { toDivInvMonoid := toDivInvMonoid✝, inv_mul_cancel := inv_mul_cancel✝, mul_comm := h }
⊢ IsSolvable G

Tactic:
infer_instance

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.FreeLocus

Goal state before:
pp: case h'
R : Type uR
M : Type uM
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
S : Submonoid R
p : PrimeSpectrum (Localization S)
p' : Ideal R := Ideal.comap (algebraMap R (Localization S)) p.asIdeal
hp' : S ≤ p'.primeCompl
Rₚ : Type uR := Localization.AtPrime p'
Mₚ : Type (max uR uM) := LocalizedModule p'.primeCompl M
this✝⁶ : Algebra (Localization S) Rₚ :=
  IsLocalization.localizationAlgebraOfSubmonoidLe (Localization S) Rₚ S p'.primeCompl hp'
this✝⁵ : IsScalarTower R (Localization S) Rₚ
this✝⁴ : IsLocalization.AtPrime Rₚ p.asIdeal
this✝³ : Module (Localization S) Mₚ := compHom Mₚ (algebraMap (Localization S) Rₚ)
this✝² : IsScalarTower R (Localization S) Mₚ
this✝¹ : IsScalarTower (Localization S) Rₚ Mₚ
l : LocalizedModule S M →ₗ[Localization S] LocalizedModule p'.primeCompl M :=
  LinearMap.extendScalarsOfIsLocalization S (Localization S)
    (IsLocalizedModule.liftOfLE S p'.primeCompl hp' (LocalizedModule.mkLinearMap S M)
      (LocalizedModule.mkLinearMap p'.primeCompl M))
this✝ : IsLocalizedModule p'.primeCompl (↑R l)
this : IsLocalizedModule (Algebra.algebraMapSubmonoid (Localization S) p'.primeCompl) l
x : Localization S
hx : x ∈ p.asIdeal.primeCompl
⊢ ∃ m, m * ↑⟨x, hx⟩ ∈ Algebra.algebraMapSubmonoid (Localization S) p'.primeCompl

Tactic:
obtain ⟨x, s, rfl⟩ := IsLocalization.exists_mk'_eq S x

Goal state after:
case h'
R : Type uR
M : Type uM
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
S : Submonoid R
p : PrimeSpectrum (Localization S)
p' : Ideal R := Ideal.comap (algebraMap R (Localization S)) p.asIdeal
hp' : S ≤ p'.primeCompl
Rₚ : Type uR := Localization.AtPrime p'
Mₚ : Type (max uR uM) := LocalizedModule p'.primeCompl M
this✝⁶ : Algebra (Localization S) Rₚ :=
  IsLocalization.localizationAlgebraOfSubmonoidLe (Localization S) Rₚ S p'.primeCompl hp'
this✝⁵ : IsScalarTower R (Localization S) Rₚ
this✝⁴ : IsLocalization.AtPrime Rₚ p.asIdeal
this✝³ : Module (Localization S) Mₚ := compHom Mₚ (algebraMap (Localization S) Rₚ)
this✝² : IsScalarTower R (Localization S) Mₚ
this✝¹ : IsScalarTower (Localization S) Rₚ Mₚ
l : LocalizedModule S M →ₗ[Localization S] LocalizedModule p'.primeCompl M :=
  LinearMap.extendScalarsOfIsLocalization S (Localization S)
    (IsLocalizedModule.liftOfLE S p'.primeCompl hp' (LocalizedModule.mkLinearMap S M)
      (LocalizedModule.mkLinearMap p'.primeCompl M))
this✝ : IsLocalizedModule p'.primeCompl (↑R l)
this : IsLocalizedModule (Algebra.algebraMapSubmonoid (Localization S) p'.primeCompl) l
x : R
s : ↥S
hx : IsLocalization.mk' (Localization S) x s ∈ p.asIdeal.primeCompl
⊢ ∃ m, m * ↑⟨IsLocalization.mk' (Localization S) x s, hx⟩ ∈ Algebra.algebraMapSubmonoid (Localization S) p'.primeCompl
Module: Mathlib.GroupTheory.Solvable

Goal state before:
pp: case succ
G : Type u_1
inst✝¹ : Group G
inst✝ : Nontrivial G
hn : _root_.commutator G = ⊤
n : ℕ
h : derivedSeries G n = ⊤
⊢ derivedSeries G (n + 1) = ⊤

Tactic:
derivedSeries_succ,

Goal state after:
case succ
G : Type u_1
inst✝¹ : Group G
inst✝ : Nontrivial G
hn : _root_.commutator G = ⊤
n : ℕ
h : derivedSeries G n = ⊤
⊢ ⁅derivedSeries G n, derivedSeries G n⁆ = ⊤
Module: Mathlib.GroupTheory.Solvable

Goal state before:
pp: case ind.inl
G : Type u_1
inst✝¹ : Group G
inst✝ : WellFoundedLT (Subgroup G)
h : ∀ (H : Subgroup G), H ≠ ⊥ → ⁅H, H⁆ < H
hH : ∀ y < ⊥, IsSolvable ↥y
⊢ IsSolvable ↥⊥

Tactic:
infer_instance

Goal state after: