module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.TotallySplit | {
"line": 119,
"column": 17
} | {
"line": 119,
"column": 19
} | {
"line": 120,
"column": 4
} | [
{
"pp": "k : Type u_1\nR : Type u_2\nS : Type u_3\ninst✝⁵ : Field k\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra k R\ninst✝¹ : Algebra R S\ninst✝ : IsFiniteSplit k R\np : PrimeSpectrum R\n⊢ (fun f ↦ { asIdeal := RingHom.ker f, isPrime := ⋯ })\n ((fun p ↦ (↑(AlgEquiv.ofBijective (ofId k (R ⧸ ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.TotallySplit | {
"line": 131,
"column": 88
} | {
"line": 131,
"column": 90
} | {
"line": 132,
"column": 2
} | [
{
"pp": "k : Type u_1\nR : Type u_2\nS : Type u_3\ninst✝⁷ : Field k\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra k R\ninst✝³ : Algebra R S\ninst✝² : IsSepClosed k\ninst✝¹ : EssFiniteType k R\ninst✝ : FormallyEtale k R\n⊢ IsFiniteSplit k R",
"ppTerm": "?m.15",
"assigned": true,
"usedCo... | [] | by | [anonymous] | by |
Mathlib.RingTheory.TotallySplit | {
"line": 154,
"column": 32
} | {
"line": 154,
"column": 34
} | {
"line": 155,
"column": 6
} | [
{
"pp": "R S : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Etale R S\ninst✝ : Module.Finite R S\nhn : Module.rankAtStalk S = ↑0\ne : R ⊗[R] S ≃ₐ[R] S := TensorProduct.lid R S\n⊢ IsFiniteSplit R S",
"ppTerm": "?m.119",
"assigned": true,
"usedConstants": [
"N... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 401,
"column": 23
} | {
"line": 401,
"column": 25
} | {
"line": 401,
"column": 26
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nS : Set A\nhS : S ⊆ ↑I\nk : ℕ\nhk : k ≠ 0\nz : A\nhz : z ∈ span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}\nJ : Ideal A := span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}\nhSI : span {y | ∃ n,... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Etale.Weakly | {
"line": 47,
"column": 81
} | {
"line": 47,
"column": 83
} | {
"line": 48,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\n⊢ WeaklyEtale (ULift.{u₁, u_1} R) (ULift.{u₂, u_2} S) ↔ WeaklyEtale R S",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"AlgHom.comp_toRingHom",
"Eq.mpr",
"RingHom.Flat",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Etale.Weakly | {
"line": 56,
"column": 16
} | {
"line": 56,
"column": 18
} | {
"line": 57,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Etale R S\n⊢ (TensorProduct.lmul' R).Flat",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Algebra.to_smulCommClass",
"IsScalarTower.right",
"Algebra.Smooth.flat",... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Etale.Weakly | {
"line": 70,
"column": 81
} | {
"line": 70,
"column": 83
} | {
"line": 71,
"column": 6
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nT : Type u_3\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\ninst✝ : WeaklyEtale R S\ne : T ⊗[R] S ⊗[T] (T ⊗[R] S) ≃ₐ[T] T ⊗[R] (S ⊗[R] S) :=\n (TensorProduct.cancelBaseChange R T T (T ⊗[R] S) S).trans (TensorProdu... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.TotallySplit | {
"line": 163,
"column": 27
} | {
"line": 163,
"column": 29
} | {
"line": 164,
"column": 6
} | [
{
"pp": "n : ℕ\nih :\n ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] [Etale R S] [Module.Finite R S],\n Module.rankAtStalk S = ↑n →\n ∃ T x x_1, ∃ (_ : Module.FaithfullyFlat R T) (_ : Module.Finite R T) (_ : Etale R T), IsFiniteSplit T (T ⊗[R] S)\nR S : Type u\ninst✝... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Etale.Weakly | {
"line": 65,
"column": 16
} | {
"line": 65,
"column": 18
} | {
"line": 66,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nT : Type u_3\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\ninst✝ : WeaklyEtale R S\n⊢ (TensorProduct.lmul' T).Flat",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"RingH... | [] | by | [anonymous] | by |
Mathlib.RingTheory.TotallySplit | {
"line": 171,
"column": 24
} | {
"line": 171,
"column": 26
} | {
"line": 172,
"column": 6
} | [
{
"pp": "n : ℕ\nih :\n ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] [Etale R S] [Module.Finite R S],\n Module.rankAtStalk S = ↑n →\n ∃ T x x_1, ∃ (_ : Module.FaithfullyFlat R T) (_ : Module.Finite R T) (_ : Etale R T), IsFiniteSplit T (T ⊗[R] S)\nR S : Type u\ninst✝... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Etale.Weakly | {
"line": 88,
"column": 52
} | {
"line": 88,
"column": 54
} | {
"line": 89,
"column": 6
} | [
{
"pp": "R : Type u₁\nS : Type u₂\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\nT : Type u₃\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : WeaklyEtale (ULift.{max u₁ u₂ u₃, u₁} R) (ULift.{max u₁ u₂ u₃, u₂} S)\ninst✝ : WeaklyEtale (ULif... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.TotallySplit | {
"line": 174,
"column": 32
} | {
"line": 174,
"column": 34
} | {
"line": 175,
"column": 6
} | [
{
"pp": "n : ℕ\nih :\n ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] [Etale R S] [Module.Finite R S],\n Module.rankAtStalk S = ↑n →\n ∃ T x x_1, ∃ (_ : Module.FaithfullyFlat R T) (_ : Module.Finite R T) (_ : Etale R T), IsFiniteSplit T (T ⊗[R] S)\nR S : Type u\ninst✝... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.TotallySplit | {
"line": 179,
"column": 82
} | {
"line": 179,
"column": 84
} | {
"line": 180,
"column": 6
} | [
{
"pp": "n : ℕ\nih :\n ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] [Etale R S] [Module.Finite R S],\n Module.rankAtStalk S = ↑n →\n ∃ T x x_1, ∃ (_ : Module.FaithfullyFlat R T) (_ : Module.Finite R T) (_ : Etale R T), IsFiniteSplit T (T ⊗[R] S)\nR S : Type u\ninst✝... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.TotallySplit | {
"line": 184,
"column": 48
} | {
"line": 184,
"column": 50
} | {
"line": 185,
"column": 6
} | [
{
"pp": "n : ℕ\nih :\n ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] [Etale R S] [Module.Finite R S],\n Module.rankAtStalk S = ↑n →\n ∃ T x x_1, ∃ (_ : Module.FaithfullyFlat R T) (_ : Module.Finite R T) (_ : Etale R T), IsFiniteSplit T (T ⊗[R] S)\nR S : Type u\ninst✝... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Etale.Weakly | {
"line": 82,
"column": 61
} | {
"line": 82,
"column": 63
} | {
"line": 83,
"column": 2
} | [
{
"pp": "R : Type u₁\nS : Type u₂\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\nT : Type u₃\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : WeaklyEtale R S\ninst✝ : WeaklyEtale S T\n⊢ WeaklyEtale R T",
"ppTerm": "?m.21",
"assign... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Extension.ExtendScalars | {
"line": 68,
"column": 5
} | {
"line": 68,
"column": 7
} | {
"line": 68,
"column": 8
} | [
{
"pp": "R✝ : Type u\nS✝ : Type v\ninst✝⁵ : CommRing R✝\ninst✝⁴ : CommRing S✝\ninst✝³ : Algebra R✝ S✝\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Extension R S\n⊢ (IsScalarTower.toAlgHom R P.extendScalars.Ring S).comp (IsScalarTower.toAlgHom R P.Ring P.extendScala... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Etale.Finite | {
"line": 110,
"column": 87
} | {
"line": 110,
"column": 89
} | {
"line": 111,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝⁷ : CommRing R\nk : Type u\ninst✝⁶ : Field k\nΩ : Type w\ninst✝⁵ : Field Ω\ninst✝⁴ : Algebra R Ω\nS : Type w\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Algebra S Ω\ninst✝ : IsScalarTower R S Ω\nA B : FiniteEtale R\nf : A ⟶ B\n⊢ (baseChange R R).map f ≫ (isoMk (Algebra.TensorP... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Extension.ExtendScalars | {
"line": 79,
"column": 88
} | {
"line": 79,
"column": 90
} | {
"line": 80,
"column": 2
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Extension R S\n⊢ ↑P.cotangentExtendScalarsEquiv.symm = Cotangent.map P.toExtendScalars",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"LinearEquiv.symm",
"Algebra.Extension.comm... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Etale.Finite | {
"line": 169,
"column": 22
} | {
"line": 169,
"column": 24
} | {
"line": 170,
"column": 10
} | [
{
"pp": "R✝ : Type u\ninst✝⁹ : CommRing R✝\nk : Type u\ninst✝⁸ : Field k\nΩ✝ : Type w\ninst✝⁷ : Field Ω✝\ninst✝⁶ : Algebra R✝ Ω✝\nS✝ : Type w\ninst✝⁵ : CommRing S✝\ninst✝⁴ : Algebra R✝ S✝\ninst✝³ : Algebra S✝ Ω✝\ninst✝² : IsScalarTower R✝ S✝ Ω✝\nΩ : Type u\ninst✝¹ : Field Ω\ninst✝ : IsSepClosed Ω\nR S : (Finite... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Extension.ExtendScalars | {
"line": 86,
"column": 62
} | {
"line": 86,
"column": 64
} | {
"line": 87,
"column": 2
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Extension R S\n⊢ Function.Injective ⇑(map P.toExtendScalars)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"LinearEquiv.symm",
"Algebra.Exten... | [] | by | [anonymous] | by |
Mathlib.RingTheory.TotallySplit | {
"line": 201,
"column": 42
} | {
"line": 201,
"column": 44
} | {
"line": 202,
"column": 8
} | [
{
"pp": "n✝ : ℕ\nih :\n ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] [Etale R S] [Module.Finite R S],\n Module.rankAtStalk S = ↑n✝ →\n ∃ T x x_1, ∃ (_ : Module.FaithfullyFlat R T) (_ : Module.Finite R T) (_ : Etale R T), IsFiniteSplit T (T ⊗[R] S)\nR S : Type u\nins... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Extension.ExtendScalars | {
"line": 99,
"column": 35
} | {
"line": 99,
"column": 37
} | {
"line": 99,
"column": 38
} | [
{
"pp": "R✝ : Type u\nS✝ : Type v\ninst✝⁵ : CommRing R✝\ninst✝⁴ : CommRing S✝\ninst✝³ : Algebra R✝ S✝\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Extension R S\n⊢ (IsScalarTower.toAlgHom P.Ring (Generators.self P.Ring S).toExtension.Ring S).comp\n (ofId P.Rin... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Extension.ExtendScalars | {
"line": 113,
"column": 18
} | {
"line": 113,
"column": 20
} | {
"line": 114,
"column": 4
} | [
{
"pp": "R✝ : Type u\nS✝ : Type v\ninst✝⁵ : CommRing R✝\ninst✝⁴ : CommRing S✝\ninst✝³ : Algebra R✝ S✝\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Extension R S\nh : Function.Surjective ⇑(algebraMap R P.Ring)\nx : P.Cotangent\n⊢ x ∈ P.cotangentComplex.ker",
"pp... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Etale.Finite | {
"line": 176,
"column": 30
} | {
"line": 176,
"column": 32
} | {
"line": 177,
"column": 6
} | [
{
"pp": "R : Type u\ninst✝⁹ : CommRing R\nk : Type u\ninst✝⁸ : Field k\nΩ✝ : Type w\ninst✝⁷ : Field Ω✝\ninst✝⁶ : Algebra R Ω✝\nS : Type w\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra S Ω✝\ninst✝² : IsScalarTower R S Ω✝\nΩ : Type u\ninst✝¹ : Field Ω\ninst✝ : IsSepClosed Ω\nX : FintypeCat\n⊢ { obj... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Extension.ExtendScalars | {
"line": 140,
"column": 70
} | {
"line": 140,
"column": 72
} | {
"line": 141,
"column": 6
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Extension R S\nx : P.Ring\nx_in : x ∈ P.ker\nproperty✝ : Cotangent.mk ⟨x, x_in⟩ ∈ P.extendScalars.cotangentComplex.ker\n⊢ (algebraMap P.Ring (Generators.self P.Ring S).toExtension.Ring) x ∈ (Generators.self P.Rin... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Flat.LocallyFree | {
"line": 49,
"column": 29
} | {
"line": 49,
"column": 31
} | {
"line": 50,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\nM : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Module.Finite R M\ninst✝¹ : Flat R M\np : Ideal R\ninst✝ : p.IsPrime\nh :\n ∀ (m : Ideal R) [inst : m.IsMaximal],\n rankAtStalk M { asIdeal := m, isPrime := ⋯ } = rankAtStalk M { asIdeal := p, is... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Extension.ExtendScalars | {
"line": 131,
"column": 34
} | {
"line": 131,
"column": 36
} | {
"line": 132,
"column": 2
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Extension R S\n⊢ P.cotangentComplex ∘ₗ ↑P.h1CotangentEquivCotangent = H1Cotangent.δ R P.Ring S",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Derivation",
"Finsupp.instAddZeroC... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Flat.LocallyFree | {
"line": 39,
"column": 77
} | {
"line": 39,
"column": 79
} | {
"line": 40,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\nM : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Module.Finite R M\ninst✝¹ : Flat R M\np : Ideal R\ninst✝ : p.IsPrime\nh :\n ∀ (m : Ideal R) [inst : m.IsMaximal],\n rankAtStalk M { asIdeal := m, isPrime := ⋯ } = rankAtStalk M { asIdeal := p, is... | [] | by | [anonymous] | by |
Mathlib.RingTheory.TotallySplit | {
"line": 149,
"column": 36
} | {
"line": 149,
"column": 38
} | {
"line": 150,
"column": 2
} | [
{
"pp": "R S : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Etale R S\ninst✝ : Module.Finite R S\nn : ℕ\nhn : Module.rankAtStalk S = ↑n\n⊢ ∃ T x x_1, ∃ (_ : Module.FaithfullyFlat R T) (_ : Module.Finite R T) (_ : Etale R T), IsFiniteSplit T (T ⊗[R] S)",
"ppTerm": "?m.43",... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 394,
"column": 90
} | {
"line": 394,
"column": 92
} | {
"line": 395,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nS : Set A\nhS : S ⊆ ↑I\nk : ℕ\nhk : k ≠ 0\nz : A\nhz : z ∈ span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}\nJ : Ideal A := span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}\nhSI : span {y | ∃ n,... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 390,
"column": 95
} | {
"line": 390,
"column": 97
} | {
"line": 391,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nS : Set A\nhS : S ⊆ ↑I\nk : ℕ\nhk : k ≠ 0\nz : A\nhz : z ∈ span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}\n⊢ hI.dpow k z ∈ span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}",
"ppTerm": "?m.... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 427,
"column": 56
} | {
"line": 427,
"column": 58
} | {
"line": 427,
"column": 59
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nS : Set A\nhS : S ⊆ ↑I\nJ : hI.SubDPIdeal :=\n { carrier := span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}, isSubideal := ⋯, dpow_mem := ⋯ }\nx : A\nhx : x ∈ S\n⊢ x = hI.dpow 1 x",
"ppTerm": "?m.123",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Extension.ExtendScalars | {
"line": 148,
"column": 58
} | {
"line": 148,
"column": 60
} | {
"line": 149,
"column": 2
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Extension R S\n⊢ ↑P.h1CotangentEquivCotangent ∘ₗ Algebra.H1Cotangent.map R P.Ring S S =\n h1Cotangentι ∘ₗ H1Cotangent.map (defaultHom R S P)",
"ppTerm": "?m.107",
"assigned": true,
"usedConstants":... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Extension.ExtendScalars | {
"line": 159,
"column": 58
} | {
"line": 159,
"column": 60
} | {
"line": 160,
"column": 2
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Extension R S\n⊢ Function.Surjective ⇑(map (defaultHom R S P))",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"RingHomSurjective.ids",
"Linear... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 432,
"column": 20
} | {
"line": 432,
"column": 22
} | {
"line": 433,
"column": 6
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nS : Set A\nhS : S ⊆ ↑I\nJ : hI.SubDPIdeal :=\n { carrier := span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}, isSubideal := ⋯, dpow_mem := ⋯ }\nK : hI.SubDPIdeal\nhK : K ∈ insert ⊤ {J | S ⊆ ↑J.carrier}\n⊢ S ⊆ ↑K... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 418,
"column": 83
} | {
"line": 418,
"column": 85
} | {
"line": 419,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nS : Set A\nhS : S ⊆ ↑I\n⊢ (SubDPIdeal.span hI S).carrier = span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Ideal.span_le",
"Eq.m... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 451,
"column": 43
} | {
"line": 451,
"column": 45
} | {
"line": 452,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nB : Type u_2\ninst✝ : CommRing B\nJ : Ideal B\nhJ : DividedPowers J\nf : A →+* B\nhf : hI.IsDPMorphism hJ f\n⊢ hI.IsSubDPIdeal (RingHom.ker f ⊓ I)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 464,
"column": 21
} | {
"line": 464,
"column": 23
} | {
"line": 465,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nB : Type u_2\ninst✝ : CommRing B\nJ : Ideal B\nhJ : DividedPowers J\nf : hI.DPMorphism hJ\nx✝ : ℕ\nhn : x✝ ≠ 0\na : A\n⊢ a ∈ RingHom.ker f.toRingHom ⊓ I → hI.dpow x✝ a ∈ RingHom.ker f.toRingHom ⊓ I",
"ppTerm": "?m.46",
"assig... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 483,
"column": 42
} | {
"line": 483,
"column": 44
} | {
"line": 483,
"column": 45
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI hI' : DividedPowers I\na b : A\nha : a ∈ {a | a ∈ I ∧ ∀ (n : ℕ), hI.dpow n a = hI'.dpow n a}\nhb : b ∈ {a | a ∈ I ∧ ∀ (n : ℕ), hI.dpow n a = hI'.dpow n a}\nn : ℕ\nk : ℕ × ℕ\nx✝ : k ∈ Finset.antidiagonal n\n⊢ hI.dpow k.1 a * hI.dpow k.2 b = hI'.dpow ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 480,
"column": 26
} | {
"line": 480,
"column": 28
} | {
"line": 481,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI hI' : DividedPowers I\na b : A\nha : a ∈ {a | a ∈ I ∧ ∀ (n : ℕ), hI.dpow n a = hI'.dpow n a}\nhb : b ∈ {a | a ∈ I ∧ ∀ (n : ℕ), hI.dpow n a = hI'.dpow n a}\n⊢ a + b ∈ {a | a ∈ I ∧ ∀ (n : ℕ), hI.dpow n a = hI'.dpow n a}",
"ppTerm": "?m.36",
"a... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 484,
"column": 15
} | {
"line": 484,
"column": 17
} | {
"line": 485,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI hI' : DividedPowers I\n⊢ 0 ∈ {a | a ∈ I ∧ ∀ (n : ℕ), hI.dpow n a = hI'.dpow n a}",
"ppTerm": "?m.95",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submodule.addSubmonoidClas... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 491,
"column": 43
} | {
"line": 491,
"column": 45
} | {
"line": 491,
"column": 46
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI hI' : DividedPowers I\na x : A\nhx : x ∈ {a | a ∈ I ∧ ∀ (n : ℕ), hI.dpow n a = hI'.dpow n a}\nn : ℕ\n⊢ hI.dpow n (a * x) = hI'.dpow n (a * x)",
"ppTerm": "?m.162",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toMo... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 489,
"column": 22
} | {
"line": 489,
"column": 24
} | {
"line": 490,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI hI' : DividedPowers I\na x : A\nhx : x ∈ {a | a ∈ I ∧ ∀ (n : ℕ), hI.dpow n a = hI'.dpow n a}\n⊢ a • x ∈ {a | a ∈ I ∧ ∀ (n : ℕ), hI.dpow n a = hI'.dpow n a}",
"ppTerm": "?m.148",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 494,
"column": 76
} | {
"line": 494,
"column": 78
} | {
"line": 495,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI hI' : DividedPowers I\nx : A\n⊢ x ∈ hI.dpEqualizer hI' ↔ x ∈ I ∧ ∀ (n : ℕ), hI.dpow n x = hI'.dpow n x",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Submodule",
"Semiring.toModule",
"AddSubsemigroup.instSet... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 501,
"column": 12
} | {
"line": 501,
"column": 14
} | {
"line": 501,
"column": 15
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI hI' : DividedPowers I\nx✝ : ℕ\nhn : x✝ ≠ 0\nx : A\nhx : x ∈ hI.dpEqualizer hI'\nm : ℕ\n⊢ hI.dpow m (hI.dpow x✝ x) = hI'.dpow m (hI.dpow x✝ x)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiri... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 505,
"column": 84
} | {
"line": 505,
"column": 86
} | {
"line": 506,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI hI' : DividedPowers I\nx✝ : ℕ\nhn : x✝ ≠ 0\nx : A\nhx : x ∈ hI.dpEqualizer hI'\nm : ℕ\n⊢ hI.dpow m (hI'.dpow x✝ x) = hI'.dpow m (hI'.dpow x✝ x)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemi... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 516,
"column": 48
} | {
"line": 516,
"column": 50
} | {
"line": 516,
"column": 51
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nB : Type u_2\ninst✝ : CommSemiring B\nf : A →+* B\nK : Ideal B\nhI_le_K : Ideal.map f I ≤ K\nhK hK' : DividedPowers K\nhIK : hI.IsDPMorphism hK f\nhIK' : hI.IsDPMorphism hK' f\na : A\nha : a ∈ ↑I\nn : ℕ\n⊢ hK.dpow n (f a) = hK'.d... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 513,
"column": 42
} | {
"line": 513,
"column": 44
} | {
"line": 514,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nB : Type u_2\ninst✝ : CommSemiring B\nf : A →+* B\nK : Ideal B\nhI_le_K : Ideal.map f I ≤ K\nhK hK' : DividedPowers K\nhIK : hI.IsDPMorphism hK f\nhIK' : hI.IsDPMorphism hK' f\n⊢ Ideal.map f I ≤ hK.dpEqualizer hK'",
"ppTerm":... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 526,
"column": 16
} | {
"line": 526,
"column": 18
} | {
"line": 526,
"column": 19
} | [
{
"pp": "A✝ : Type u_1\ninst✝¹ : CommSemiring A✝\nI✝ : Ideal A✝\nhI✝ hI' : DividedPowers I✝\nA : Type u_2\ninst✝ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nJ : Ideal A\nhJ : DividedPowers (Ideal.map (Ideal.Quotient.mk J) I)\nφ : hI.DPMorphism hJ\nhφ : φ.toRingHom = Ideal.Quotient.mk J\n⊢ J ⊓ I ≤ I",
"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 530,
"column": 37
} | {
"line": 530,
"column": 39
} | {
"line": 531,
"column": 6
} | [
{
"pp": "A✝ : Type u_1\ninst✝¹ : CommSemiring A✝\nI✝ : Ideal A✝\nhI✝ hI' : DividedPowers I✝\nA : Type u_2\ninst✝ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nJ : Ideal A\nhJ : DividedPowers (Ideal.map (Ideal.Quotient.mk J) I)\nφ : hI.DPMorphism hJ\nhφ : φ.toRingHom = Ideal.Quotient.mk J\nx✝¹ : ℕ\nhn : x✝¹ ≠... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 527,
"column": 40
} | {
"line": 527,
"column": 42
} | {
"line": 528,
"column": 4
} | [
{
"pp": "A✝ : Type u_1\ninst✝¹ : CommSemiring A✝\nI✝ : Ideal A✝\nhI✝ hI' : DividedPowers I✝\nA : Type u_2\ninst✝ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nJ : Ideal A\nhJ : DividedPowers (Ideal.map (Ideal.Quotient.mk J) I)\nφ : hI.DPMorphism hJ\nhφ : φ.toRingHom = Ideal.Quotient.mk J\nx✝¹ : ℕ\nhn : x✝¹ ≠... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 567,
"column": 40
} | {
"line": 567,
"column": 42
} | {
"line": 567,
"column": 43
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nB : Type u_2\ninst✝ : CommRing B\nf : A →+* B\nhIf : hI.IsSubDPIdeal (RingHom.ker f ⊓ I)\nn : ℕ\na : A\nha : a ∈ I\n⊢ ∃ a_1, f ↑a_1 = f a",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"Semiring.toModule"... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 564,
"column": 43
} | {
"line": 564,
"column": 45
} | {
"line": 565,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nB : Type u_2\ninst✝ : CommRing B\nf : A →+* B\nhIf : hI.IsSubDPIdeal (RingHom.ker f ⊓ I)\nn : ℕ\na : A\nha : a ∈ I\n⊢ dpow hI f n (f a) = f (hI.dpow n a)",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Ad... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 578,
"column": 25
} | {
"line": 578,
"column": 27
} | {
"line": 579,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nB : Type u_2\ninst✝ : CommRing B\nf : A →+* B\nJ : Ideal B\nhf : Function.Surjective ⇑f\nhIJ : J = Ideal.map f I\nhIf : hI.IsSubDPIdeal (RingHom.ker f ⊓ I)\nn : ℕ\nx : B\nhx' : x ∉ J\n⊢ dpow hI f n x = 0",
"ppTerm": "?m.78",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 583,
"column": 22
} | {
"line": 583,
"column": 24
} | {
"line": 584,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nB : Type u_2\ninst✝ : CommRing B\nf : A →+* B\nJ : Ideal B\nhf : Function.Surjective ⇑f\nhIJ : J = Ideal.map f I\nhIf : hI.IsSubDPIdeal (RingHom.ker f ⊓ I)\nx : B\nhx : x ∈ J\n⊢ dpow hI f 0 x = 1",
"ppTerm": "?m.145",
"assign... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 586,
"column": 21
} | {
"line": 586,
"column": 23
} | {
"line": 587,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nB : Type u_2\ninst✝ : CommRing B\nf : A →+* B\nJ : Ideal B\nhf : Function.Surjective ⇑f\nhIJ : J = Ideal.map f I\nhIf : hI.IsSubDPIdeal (RingHom.ker f ⊓ I)\nx : B\nhx : x ∈ J\n⊢ dpow hI f 1 x = x",
"ppTerm": "?m.206",
"assign... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Frobenius | {
"line": 63,
"column": 81
} | {
"line": 63,
"column": 83
} | {
"line": 64,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\nx : S\n⊢ (Ideal.Quotient.mk Q) (φ x) = (Ideal.Quotient.mk Q) x ^ Nat.card (R ⧸ Ideal.under R Q)",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Frobenius | {
"line": 69,
"column": 24
} | {
"line": 69,
"column": 26
} | {
"line": 69,
"column": 27
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\nh : Infinite (R ⧸ Ideal.under R Q)\n⊢ Q = ⊤",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"NonUnitalCommRing.... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 590,
"column": 4
} | {
"line": 593,
"column": 46
} | {
"line": 594,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nB : Type u_2\ninst✝ : CommRing B\nf : A →+* B\nJ : Ideal B\nhf : Function.Surjective ⇑f\nhIJ : J = Ideal.map f I\nhIf : hI.IsSubDPIdeal (RingHom.ker f ⊓ I)\nn : ℕ\nx : B\nhn : n ≠ 0\nhx : x ∈ J\n⊢ dpow hI f n x ∈ J",
"ppTerm": "?... | [] | rw [hIJ] at hx ⊢
obtain ⟨a, ha, rfl⟩ := (mem_map_iff_of_surjective f hf).mp hx
rw [dpow_apply' hI hIf ha]
exact mem_map_of_mem _ (hI.dpow_mem hn ha) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 590,
"column": 4
} | {
"line": 593,
"column": 46
} | {
"line": 594,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nB : Type u_2\ninst✝ : CommRing B\nf : A →+* B\nJ : Ideal B\nhf : Function.Surjective ⇑f\nhIJ : J = Ideal.map f I\nhIf : hI.IsSubDPIdeal (RingHom.ker f ⊓ I)\nn : ℕ\nx : B\nhn : n ≠ 0\nhx : x ∈ J\n⊢ dpow hI f n x ∈ J",
"ppTerm": "?... | [] | rw [hIJ] at hx ⊢
obtain ⟨a, ha, rfl⟩ := (mem_map_iff_of_surjective f hf).mp hx
rw [dpow_apply' hI hIf ha]
exact mem_map_of_mem _ (hI.dpow_mem hn ha) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 589,
"column": 26
} | {
"line": 589,
"column": 28
} | {
"line": 590,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nB : Type u_2\ninst✝ : CommRing B\nf : A →+* B\nJ : Ideal B\nhf : Function.Surjective ⇑f\nhIJ : J = Ideal.map f I\nhIf : hI.IsSubDPIdeal (RingHom.ker f ⊓ I)\nn : ℕ\nx : B\nhn : n ≠ 0\nhx : x ∈ J\n⊢ dpow hI f n x ∈ J",
"ppTerm": "?... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Frobenius | {
"line": 67,
"column": 57
} | {
"line": 67,
"column": 59
} | {
"line": 68,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\n⊢ _root_.Finite (R ⧸ Ideal.under R Q)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"False",
"NonUnital... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Frobenius | {
"line": 77,
"column": 34
} | {
"line": 77,
"column": 36
} | {
"line": 78,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\n⊢ Q ≤ Ideal.comap φ Q",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Submodule",
"MulOne.toOne",
"Submodule.Quotient.in... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Frobenius | {
"line": 85,
"column": 17
} | {
"line": 85,
"column": 19
} | {
"line": 86,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ ψ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\nx : R ⧸ Ideal.under R Q\n⊢ (↑↑(Ideal.quotientMap Q ↑φ ⋯)).toFun ((algebraMap (R ⧸ Ideal.under R Q) (S ⧸ Q)) x) =\n (algebraMap (R ⧸ Ideal.under R Q) (S ⧸ Q)... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Frobenius | {
"line": 90,
"column": 51
} | {
"line": 90,
"column": 53
} | {
"line": 91,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\nx : S ⧸ Q\n⊢ H.restrict x = x ^ Nat.card (R ⧸ Ideal.under R Q)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Ideal.Quotient.commSe... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 599,
"column": 20
} | {
"line": 599,
"column": 22
} | {
"line": 599,
"column": 23
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nB : Type u_2\ninst✝ : CommRing B\nf : A →+* B\nJ : Ideal B\nhf : Function.Surjective ⇑f\nhIJ : J = Ideal.map f I\nhIf : hI.IsSubDPIdeal (RingHom.ker f ⊓ I)\nn✝ : ℕ\na : A\nha : a ∈ I\nhx : f a ∈ J\nb : A\nhb : b ∈ I\nhy : f b ∈ J\nk ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 594,
"column": 20
} | {
"line": 594,
"column": 22
} | {
"line": 595,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nB : Type u_2\ninst✝ : CommRing B\nf : A →+* B\nJ : Ideal B\nhf : Function.Surjective ⇑f\nhIJ : J = Ideal.map f I\nhIf : hI.IsSubDPIdeal (RingHom.ker f ⊓ I)\nn✝ : ℕ\nx✝ y✝ : B\nhx : x✝ ∈ J\nhy : y✝ ∈ J\n⊢ dpow hI f n✝ (x✝ + y✝) = ∑ k ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Frobenius | {
"line": 97,
"column": 37
} | {
"line": 97,
"column": 39
} | {
"line": 98,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nφ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\ninst✝ : Q.IsPrime\n⊢ Function.Injective ⇑H.restrict",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 600,
"column": 25
} | {
"line": 600,
"column": 27
} | {
"line": 601,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nB : Type u_2\ninst✝ : CommRing B\nf : A →+* B\nJ : Ideal B\nhf : Function.Surjective ⇑f\nhIJ : J = Ideal.map f I\nhIf : hI.IsSubDPIdeal (RingHom.ker f ⊓ I)\nn : ℕ\nx y : B\nhy : y ∈ J\n⊢ dpow hI f n (x * y) = x ^ n * dpow hI f n y",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Frobenius | {
"line": 102,
"column": 46
} | {
"line": 102,
"column": 48
} | {
"line": 103,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nφ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\ninst✝ : Q.IsPrime\n⊢ Ideal.comap φ Q = Q",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Ideal.Quotient.commSemiring",
"Eq.mp... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Frobenius | {
"line": 112,
"column": 47
} | {
"line": 112,
"column": 49
} | {
"line": 112,
"column": 50
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nφ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\ninst✝ : IsDomain S\nζ : S\nq : ℕ := Nat.card (R ⧸ Ideal.under R Q)\nhζ : ζ ^ 0 = 1\nhk' : ↑0 ∉ Q\n⊢ ↑0 ∈ Q",
"ppTerm": "?m.75",
"assigned": true,
"u... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Frobenius | {
"line": 112,
"column": 21
} | {
"line": 112,
"column": 23
} | {
"line": 112,
"column": 24
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nφ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\ninst✝ : IsDomain S\nζ : S\nm : ℕ\nhζ : ζ ^ m = 1\nhk' : ↑m ∉ Q\nq : ℕ := Nat.card (R ⧸ Ideal.under R Q)\n⊢ m ≠ 0",
"ppTerm": "?m.62",
"assigned": true,
... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 605,
"column": 17
} | {
"line": 605,
"column": 19
} | {
"line": 606,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nB : Type u_2\ninst✝ : CommRing B\nf : A →+* B\nJ : Ideal B\nhf : Function.Surjective ⇑f\nhIJ : J = Ideal.map f I\nhIf : hI.IsSubDPIdeal (RingHom.ker f ⊓ I)\nm✝ n✝ : ℕ\nx✝ : B\nhx : x✝ ∈ J\n⊢ dpow hI f m✝ x✝ * dpow hI f n✝ x✝ = ↑((m✝ ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Frobenius | {
"line": 116,
"column": 59
} | {
"line": 116,
"column": 61
} | {
"line": 116,
"column": 62
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nφ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\ninst✝ : IsDomain S\nζ : S\nm : ℕ\nhζ✝ : ζ ^ m = 1\nhk'✝ : ↑m ∉ Q\nq : ℕ := Nat.card (R ⧸ Ideal.under R Q)\nhm : m ≠ 0\nk : ℕ\nhk : k > 0\nhζ : IsPrimitiveRoot ζ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 609,
"column": 21
} | {
"line": 609,
"column": 23
} | {
"line": 610,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nB : Type u_2\ninst✝ : CommRing B\nf : A →+* B\nJ : Ideal B\nhf : Function.Surjective ⇑f\nhIJ : J = Ideal.map f I\nhIf : hI.IsSubDPIdeal (RingHom.ker f ⊓ I)\nm✝ n✝ : ℕ\nx✝ : B\nhn : n✝ ≠ 0\nhx : x✝ ∈ J\n⊢ dpow hI f m✝ (dpow hI f n✝ x✝... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Frobenius | {
"line": 119,
"column": 69
} | {
"line": 119,
"column": 71
} | {
"line": 119,
"column": 72
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nφ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\ninst✝ : IsDomain S\nζ : S\nm : ℕ\nhζ✝ : ζ ^ m = 1\nhk'✝ : ↑m ∉ Q\nq : ℕ := Nat.card (R ⧸ Ideal.under R Q)\nhm : m ≠ 0\nk : ℕ\nhk : k > 0\nhζ : IsPrimitiveRoot ζ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 617,
"column": 68
} | {
"line": 617,
"column": 70
} | {
"line": 618,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nB : Type u_2\ninst✝ : CommRing B\nf : A →+* B\nJ : Ideal B\nhf : Function.Surjective ⇑f\nhIJ : J = Ideal.map f I\nhIf : hI.IsSubDPIdeal (RingHom.ker f ⊓ I)\nn : ℕ\na : A\nha : a ∈ I\n⊢ (dividedPowers hI hf hIJ hIf).dpow n (f a) = f (... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 621,
"column": 33
} | {
"line": 621,
"column": 35
} | {
"line": 621,
"column": 36
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nB : Type u_2\ninst✝ : CommRing B\nf : A →+* B\nJ : Ideal B\nhf : Function.Surjective ⇑f\nhIJ : J = Ideal.map f I\nhIf : hI.IsSubDPIdeal (RingHom.ker f ⊓ I)\nn✝ : ℕ\na : A\nha : a ∈ I\n⊢ (dividedPowers hI hf hIJ hIf).dpow n✝ (f a) = f... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 625,
"column": 23
} | {
"line": 625,
"column": 25
} | {
"line": 626,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nB : Type u_2\ninst✝ : CommRing B\nf : A →+* B\nJ : Ideal B\nhf : Function.Surjective ⇑f\nhIJ : J = Ideal.map f I\nhIf : hI.IsSubDPIdeal (RingHom.ker f ⊓ I)\nhquot : DividedPowers J\nhm : hI.IsDPMorphism hquot f\nn : ℕ\nx : B\nhx : x ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Frobenius | {
"line": 117,
"column": 50
} | {
"line": 117,
"column": 52
} | {
"line": 118,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nφ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\ninst✝ : IsDomain S\nζ : S\nm : ℕ\nhζ✝ : ζ ^ m = 1\nhk'✝ : ↑m ∉ Q\nq : ℕ := Nat.card (R ⧸ Ideal.under R Q)\nhm : m ≠ 0\nk : ℕ\nhk : k > 0\nhζ : IsPrimitiveRoot ζ... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 641,
"column": 63
} | {
"line": 641,
"column": 65
} | {
"line": 642,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nJ : Ideal A\nhIJ : hI.IsSubDPIdeal (J ⊓ I)\n⊢ hI.IsSubDPIdeal (RingHom.ker (Ideal.Quotient.mk J) ⊓ I)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"RingHom.instRingHomClass",
"Semir... | [] | by | [anonymous] | by |
Mathlib.RingTheory.GradedAlgebra.Homogeneous.Subsemiring | {
"line": 47,
"column": 44
} | {
"line": 47,
"column": 46
} | {
"line": 47,
"column": 47
} | [
{
"pp": "ι : Type u_1\nσ : Type u_2\nA : Type u_3\ninst✝⁵ : AddMonoid ι\ninst✝⁴ : Semiring A\ninst✝³ : SetLike σ A\ninst✝² : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝¹ : DecidableEq ι\ninst✝ : GradedRing 𝒜\nx✝¹ x✝ : HomogeneousSubsemiring 𝒜\nx : Subsemiring A\nhx : IsHomogeneous 𝒜 x\ny : Subsemiring A\nhy : I... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Frobenius | {
"line": 128,
"column": 49
} | {
"line": 128,
"column": 51
} | {
"line": 128,
"column": 52
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nφ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\ninst✝ : IsDomain S\nζ : S\nm : ℕ\nhζ✝ : ζ ^ m = 1\nhk'✝ : ↑m ∉ Q\nq : ℕ := Nat.card (R ⧸ Ideal.under R Q)\nhm : m ≠ 0\nk : ℕ\nhk : k > 0\nhζ : IsPrimitiveRoot ζ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.GradedAlgebra.Homogeneous.Subsemiring | {
"line": 83,
"column": 60
} | {
"line": 83,
"column": 62
} | {
"line": 84,
"column": 2
} | [
{
"pp": "ι : Type u_1\nσ : Type u_2\nA : Type u_3\ninst✝⁵ : AddMonoid ι\ninst✝⁴ : Semiring A\ninst✝³ : SetLike σ A\ninst✝² : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝¹ : DecidableEq ι\ninst✝ : GradedRing 𝒜\ns : Set A\nh : ∀ (i : ι) ⦃x : A⦄, x ∈ s → ↑(((decompose 𝒜) x) i) ∈ s\ni : ι\nx : A\nhx : x ∈ Subsemiring... | [] | by | [anonymous] | by |
Mathlib.RingTheory.GradedAlgebra.AlgHom | {
"line": 62,
"column": 25
} | {
"line": 62,
"column": 27
} | {
"line": 63,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\nU : Type u_4\nV : Type u_5\nA : Type u_6\nB : Type u_7\nC : Type u_8\nD : Type u_9\nι : Type u_10\ninst✝¹⁴ : CommSemiring R\ninst✝¹³ : Semiring A\ninst✝¹² : Semiring B\ninst✝¹¹ : Semiring C\ninst✝¹⁰ : Semiring D\ninst✝⁹ : Algebra R A\ninst✝⁸ : Algebra R B\ninst... | [] | by | [anonymous] | by |
Mathlib.RingTheory.GradedAlgebra.AlgHom | {
"line": 101,
"column": 89
} | {
"line": 101,
"column": 91
} | {
"line": 102,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_6\nB : Type u_7\nι : Type u_10\ninst✝⁸ : CommSemiring R\ninst✝⁷ : Semiring A\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra R B\ninst✝³ : DecidableEq ι\ninst✝² : AddMonoid ι\n𝒜 : ι → Submodule R A\nℬ : ι → Submodule R B\ninst✝¹ : GradedAlgebra 𝒜\ninst✝ : GradedA... | [] | by | [anonymous] | by |
Mathlib.RingTheory.GradedAlgebra.AlgHom | {
"line": 218,
"column": 9
} | {
"line": 218,
"column": 11
} | {
"line": 218,
"column": 12
} | [
{
"pp": "R : Type u_1\nA : Type u_6\nι : Type u_10\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\ninst✝² : DecidableEq ι\ninst✝¹ : AddMonoid ι\n𝒜 : ι → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : 𝒜 →ₐᵍ[R] 𝒜\nn : ℕ\n⊢ ⇑(f ^ Nat.zero) = (⇑f)^[Nat.zero]",
"ppTerm": "?m.88",
"assig... | [] | by | [anonymous] | by |
Mathlib.RingTheory.GradedAlgebra.AlgHom | {
"line": 218,
"column": 34
} | {
"line": 218,
"column": 36
} | {
"line": 218,
"column": 37
} | [
{
"pp": "R : Type u_1\nA : Type u_6\nι : Type u_10\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\ninst✝² : DecidableEq ι\ninst✝¹ : AddMonoid ι\n𝒜 : ι → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : 𝒜 →ₐᵍ[R] 𝒜\nn x✝ : ℕ\nih : ⇑(f ^ x✝) = (⇑f)^[x✝]\n⊢ ⇑(f ^ x✝.succ) = (⇑f)^[x✝.succ]",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Frobenius | {
"line": 110,
"column": 42
} | {
"line": 110,
"column": 44
} | {
"line": 111,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nφ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\ninst✝ : IsDomain S\nζ : S\nm : ℕ\nhζ : ζ ^ m = 1\nhk' : ↑m ∉ Q\n⊢ φ ζ = ζ ^ Nat.card (R ⧸ Ideal.under R Q)",
"ppTerm": "?m.43",
"assigned": true,
"u... | [] | by | [anonymous] | by |
Mathlib.RingTheory.GradedAlgebra.AlgHom | {
"line": 240,
"column": 56
} | {
"line": 240,
"column": 58
} | {
"line": 240,
"column": 59
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\nU : Type u_4\nV : Type u_5\nA : Type u_6\nB : Type u_7\nC : Type u_8\nD : Type u_9\nι : Type u_10\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : Semiring A\ninst✝¹³ : Semiring B\ninst✝¹² : Semiring C\ninst✝¹¹ : Semiring D\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : Algebra R B\nins... | [] | by | [anonymous] | by |
Mathlib.RingTheory.GradedAlgebra.Homogeneous.Subsemiring | {
"line": 102,
"column": 47
} | {
"line": 102,
"column": 49
} | {
"line": 103,
"column": 2
} | [
{
"pp": "ι : Type u_1\nσ : Type u_2\nA : Type u_3\ninst✝⁵ : AddMonoid ι\ninst✝⁴ : Semiring A\ninst✝³ : SetLike σ A\ninst✝² : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝¹ : DecidableEq ι\ninst✝ : GradedRing 𝒜\ns : Set A\nh : ∀ x ∈ s, IsHomogeneousElem 𝒜 x\n⊢ IsHomogeneous 𝒜 (Subsemiring.closure s)",
"ppTerm"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Frobenius | {
"line": 136,
"column": 17
} | {
"line": 136,
"column": 19
} | {
"line": 137,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nφ ψ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\ninst✝ : Q.IsPrime\nx : R\n⊢ (↑↑(Localization.localRingHom Q Q ↑φ ⋯)).toFun ((algebraMap R (Localization.AtPrime Q)) x) =\n (algebraMap R (Localization.AtPr... | [] | by | [anonymous] | by |
Mathlib.RingTheory.GradedAlgebra.Homogeneous.Subsemiring | {
"line": 118,
"column": 83
} | {
"line": 118,
"column": 85
} | {
"line": 119,
"column": 4
} | [
{
"pp": "ι : Type u_1\nσ : Type u_2\nA : Type u_3\ninst✝⁵ : AddMonoid ι\ninst✝⁴ : Semiring A\ninst✝³ : SetLike σ A\ninst✝² : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝¹ : DecidableEq ι\ninst✝ : GradedRing 𝒜\nR : Subsemiring A\nx : A\n⊢ x ∈ Subtype.val '' Subtype.val ⁻¹' ↑R → IsHomogeneousElem 𝒜 x",
"ppTerm"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Frobenius | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 97
} | {
"line": 153,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nφ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\ninst✝ : Q.IsPrime\n⊢ Ideal.comap (algebraMap R (Localization.AtPrime Q)) (maximalIdeal (Localization.AtPrime Q)) = Ideal.under R Q",
"ppTerm": "?m.82",
... | [] | rw [← Ideal.under_def, ← Ideal.under_under (B := S), Localization.AtPrime.under_maximalIdeal] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Frobenius | {
"line": 150,
"column": 34
} | {
"line": 150,
"column": 36
} | {
"line": 151,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nφ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\ninst✝ : Q.IsPrime\n⊢ Nat.card (R ⧸ Ideal.comap (algebraMap R (Localization.AtPrime Q)) (maximalIdeal (Localization.AtPrime Q))) =\n Nat.card (R ⧸ Ideal.under... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Frobenius | {
"line": 148,
"column": 88
} | {
"line": 148,
"column": 90
} | {
"line": 149,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nφ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\ninst✝ : Q.IsPrime\n⊢ H.localize.IsArithFrobAt (maximalIdeal (Localization.AtPrime Q))",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Frobenius | {
"line": 167,
"column": 37
} | {
"line": 167,
"column": 39
} | {
"line": 168,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nφ ψ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\nH' : ψ.IsArithFrobAt Q\ninst✝² : Q.IsPrime\nhQ : Q.primeCompl ≤ S⁰\ninst✝¹ : Algebra.IsUnramifiedAt R Q\ninst✝ : IsNoetherianRing S\n⊢ H.localize = H'.localiz... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Frobenius | {
"line": 166,
"column": 65
} | {
"line": 166,
"column": 67
} | {
"line": 167,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nφ ψ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\nH' : ψ.IsArithFrobAt Q\ninst✝² : Q.IsPrime\nhQ : Q.primeCompl ≤ S⁰\ninst✝¹ : Algebra.IsUnramifiedAt R Q\ninst✝ : IsNoetherianRing S\n⊢ φ = ψ",
"ppTerm": "... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Frobenius | {
"line": 194,
"column": 95
} | {
"line": 194,
"column": 97
} | {
"line": 195,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nG : Type u_3\ninst✝³ : Group G\ninst✝² : MulSemiringAction G S\ninst✝¹ : SMulCommClass G R S\nQ : Ideal S\nσ : G\ninst✝ : Q.IsPrime\nh : IsArithFrobAt R σ Q\n⊢ σ ∈ MulAction.stabilizer G Q",
"ppTerm": "?m.24... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Frobenius | {
"line": 201,
"column": 30
} | {
"line": 201,
"column": 32
} | {
"line": 202,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nG : Type u_3\ninst✝² : Group G\ninst✝¹ : MulSemiringAction G S\ninst✝ : SMulCommClass G R S\nQ : Ideal S\nσ σ' : G\nH : IsArithFrobAt R σ Q\nH' : IsArithFrobAt R σ' Q\n⊢ σ * σ'⁻¹ ∈ Ideal.inertia G Q",
"ppTer... | [] | by | [anonymous] | by |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.