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 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.Ring.Divisibility.Lemmas | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 50
} | {
"line": 53,
"column": 2
} | [
{
"pp": "case inl\nR : Type u_1\nx y : R\nn m p : ℕ\ninst✝ : Semiring R\nhp : n + m ≤ p + 1\nh_comm : Commute x y\nhy : y ^ n = 0\nx✝ : ℕ × ℕ\ni j : ℕ\nhij : i + j = p\nhi : m ≤ i\n⊢ x ^ m ∣ x ^ (i, j).1 * y ^ (i, j).2",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"dvd_mul_of_dvd_le... | [
"case inr\nR : Type u_1\nx y : R\nn m p : ℕ\ninst✝ : Semiring R\nhp : n + m ≤ p + 1\nh_comm : Commute x y\nhy : y ^ n = 0\nx✝ : ℕ × ℕ\ni j : ℕ\nhij : i + j = p\nhi : i + 1 ≤ m\n⊢ x ^ m ∣ x ^ (i, j).1 * y ^ (i, j).2"
] | · exact dvd_mul_of_dvd_left (pow_dvd_pow x hi) _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Lie.Engel | {
"line": 223,
"column": 2
} | {
"line": 259,
"column": 11
} | {
"line": 261,
"column": 0
} | [
{
"pp": "R : Type u₁\nL : Type u₂\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsNoetherian R L\n⊢ IsEngelian R L",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"LieHom",
"LieAlgebra.toModule",
"Nontrivial",
"Module.End.instRing",
... | [] | intro M _i1 _i2 _i3 _i4 h
rw [← isNilpotent_range_toEnd_iff R]
let L' := (toEnd R L M).range
replace h : ∀ y : L', IsNilpotent (y : Module.End R M) := by
rintro ⟨-, ⟨y, rfl⟩⟩
simp [h]
change LieModule.IsNilpotent L' M
let s := {K : LieSubalgebra R L' | LieAlgebra.IsEngelian R K}
have hs : s.Nonempty... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Engel | {
"line": 223,
"column": 2
} | {
"line": 259,
"column": 11
} | {
"line": 261,
"column": 0
} | [
{
"pp": "R : Type u₁\nL : Type u₂\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsNoetherian R L\n⊢ IsEngelian R L",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"LieHom",
"LieAlgebra.toModule",
"Nontrivial",
"Module.End.instRing",
... | [] | intro M _i1 _i2 _i3 _i4 h
rw [← isNilpotent_range_toEnd_iff R]
let L' := (toEnd R L M).range
replace h : ∀ y : L', IsNilpotent (y : Module.End R M) := by
rintro ⟨-, ⟨y, rfl⟩⟩
simp [h]
change LieModule.IsNilpotent L' M
let s := {K : LieSubalgebra R L' | LieAlgebra.IsEngelian R K}
have hs : s.Nonempty... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.IntermediateField.Adjoin.Defs | {
"line": 173,
"column": 2
} | {
"line": 176,
"column": 38
} | {
"line": 177,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set (IntermediateField F E)\nhS : S.Nonempty\n⊢ (sSup S).toSubfield = sSup (toSubfield '' S)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Set.image_image",
"Eq.mpr",
"con... | [
"F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set (IntermediateField F E)\nhS : S.Nonempty\nh : toSubfield '' S = Subfield.closure '' SetLike.coe '' S\n⊢ (sSup S).toSubfield = sSup (toSubfield '' S)"
] | have h : toSubfield '' S = Subfield.closure '' SetLike.coe '' S := by
rw [Set.image_image]
congr! with x
exact x.toSubfield.closure_eq.symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 202,
"column": 4
} | {
"line": 202,
"column": 25
} | {
"line": 202,
"column": 26
} | [
{
"pp": "F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nx : E\n⊢ ∀ (s : Set (IntermediateField F E)),\n s.Nonempty → DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s → F⟮x⟯ ≤ sSup s → ∃ x_1 ∈ s, F⟮x⟯ ≤ x_1",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq... | [
"F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nx : E\n⊢ ∀ (s : Set (IntermediateField F E)), s.Nonempty → DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s → x ∈ sSup s → ∃ x_1 ∈ s, x ∈ x_1"
] | adjoin_simple_le_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 65
} | {
"line": 124,
"column": 2
} | [
{
"pp": "k : Type u\ninst✝¹ : Field k\ninst✝ : IsAlgClosed k\np : k[X]\n⊢ p.roots.card = p.natDegree",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"Polynomial.roots",
"HMul.hMul",
"Multiset.map",
"CommSemiring.toSemiring",
"Multiset.pro... | [
"k : Type u\ninst✝¹ : Field k\ninst✝ : IsAlgClosed k\np : k[X]\nw✝ : k[X]\nleft✝ : (Multiset.map (fun a ↦ X - C a) p.roots).prod * w✝ = p\nhdeg : p.roots.card + w✝.natDegree = p.natDegree\nhroots : w✝.roots = 0\n⊢ p.roots.card = p.natDegree"
] | have ⟨_, _, hdeg, hroots⟩ := exists_prod_multiset_X_sub_C_mul p | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 569,
"column": 4
} | {
"line": 569,
"column": 48
} | {
"line": 570,
"column": 4
} | [
{
"pp": "case pos\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : IsAlgClosed K\ninst✝ : CharZero K\nf g : K[X]\nhf0 : f ≠ 0\nhg0 : ¬g = 0\nhdf0 : ¬derivative f = 0\nhdg : derivative f * g ≠ 0\na : K\nhaf : eval a f = 0\nh0 : 0 < rootMultiplicity a f\n⊢ (f.IsRoot a → g.IsRoot a) → rootMultiplicity a f ≤ rootMultiplic... | [
"case pos\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : IsAlgClosed K\ninst✝ : CharZero K\nf g : K[X]\nhf0 : f ≠ 0\nhg0 : ¬g = 0\nhdf0 : ¬derivative f = 0\nhdg : derivative f * g ≠ 0\na : K\nhaf : eval a f = 0\nh0 : 0 < rootMultiplicity a f\n⊢ (f.IsRoot a → g.IsRoot a) → rootMultiplicity a f ≤ rootMultiplicity a f - 1 ... | rw [derivative_rootMultiplicity_of_root haf] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 575,
"column": 4
} | {
"line": 575,
"column": 44
} | {
"line": 576,
"column": 0
} | [
{
"pp": "case neg\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : IsAlgClosed K\ninst✝ : CharZero K\nf g : K[X]\nhf0 : f ≠ 0\nhg0 : ¬g = 0\nhdf0 : ¬derivative f = 0\nhdg : derivative f * g ≠ 0\na : K\nhaf : ¬eval a f = 0\n⊢ (f.IsRoot a → g.IsRoot a) → rootMultiplicity a f ≤ rootMultiplicity a (derivative f) + rootMul... | [] | simp [haf, rootMultiplicity_eq_zero haf] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 575,
"column": 4
} | {
"line": 575,
"column": 44
} | {
"line": 576,
"column": 0
} | [
{
"pp": "case neg\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : IsAlgClosed K\ninst✝ : CharZero K\nf g : K[X]\nhf0 : f ≠ 0\nhg0 : ¬g = 0\nhdf0 : ¬derivative f = 0\nhdg : derivative f * g ≠ 0\na : K\nhaf : ¬eval a f = 0\n⊢ (f.IsRoot a → g.IsRoot a) → rootMultiplicity a f ≤ rootMultiplicity a (derivative f) + rootMul... | [] | simp [haf, rootMultiplicity_eq_zero haf] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 575,
"column": 4
} | {
"line": 575,
"column": 44
} | {
"line": 576,
"column": 0
} | [
{
"pp": "case neg\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : IsAlgClosed K\ninst✝ : CharZero K\nf g : K[X]\nhf0 : f ≠ 0\nhg0 : ¬g = 0\nhdf0 : ¬derivative f = 0\nhdg : derivative f * g ≠ 0\na : K\nhaf : ¬eval a f = 0\n⊢ (f.IsRoot a → g.IsRoot a) → rootMultiplicity a f ≤ rootMultiplicity a (derivative f) + rootMul... | [] | simp [haf, rootMultiplicity_eq_zero haf] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 547,
"column": 2
} | {
"line": 547,
"column": 47
} | {
"line": 548,
"column": 2
} | [
{
"pp": "K : Type u\ninst✝² : Field K\nL : Type u_3\ninst✝¹ : Field L\ninst✝ : Algebra K L\nf : K[X]\nhi : Irreducible f\nhs : (Polynomial.map (algebraMap K L) f).Splits\nthis : (Polynomial.map (algebraMap K L) f).degree ≠ 0\n⊢ f.natDegree ∣ finrank K L",
"ppTerm": "?m.49",
"assigned": true,
"usedCo... | [
"K : Type u\ninst✝² : Field K\nL : Type u_3\ninst✝¹ : Field L\ninst✝ : Algebra K L\nf : K[X]\nhi : Irreducible f\nhs : (Polynomial.map (algebraMap K L) f).Splits\nthis : (Polynomial.map (algebraMap K L) f).degree ≠ 0\nx : L\nhx : eval x (Polynomial.map (algebraMap K L) f) = 0\n⊢ f.natDegree ∣ finrank K L"
] | obtain ⟨x, hx⟩ := hs.exists_eval_eq_zero this | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.FieldTheory.Extension | {
"line": 202,
"column": 2
} | {
"line": 203,
"column": 93
} | {
"line": 204,
"column": 2
} | [
{
"pp": "F : Type u_1\nE : Type u_2\nK : Type u_3\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nalg : Algebra.IsAlgebraic F E\nh : ∀ (S : Finset E), ∃ σ, ↑S ⊆ ↑σ.carrier\nthis✝¹ : ⊥.IsExtendible\nϕ : Lifts F E K\nhϕ : Maximal (fun x ↦ x ∈ {ϕ | ϕ.IsExtendible})... | [
"F : Type u_1\nE : Type u_2\nK : Type u_3\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nalg : Algebra.IsAlgebraic F E\nh : ∀ (S : Finset E), ∃ σ, ↑S ⊆ ↑σ.carrier\nthis✝² : ⊥.IsExtendible\nϕ : Lifts F E K\nhϕ : Maximal (fun x ↦ x ∈ {ϕ | ϕ.IsExtendible}) ϕ\nthis✝¹ :... | have : ϕ.carrier⟮α⟯.restrictScalars F ≤ θ.carrier := by
rw [restrictScalars_adjoin_eq_sup, sup_le_iff, adjoin_simple_le_iff]; exact ⟨hθϕ.1, hθ.1⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.Extension | {
"line": 254,
"column": 56
} | {
"line": 274,
"column": 51
} | {
"line": 276,
"column": 0
} | [
{
"pp": "F : Type u_1\nE : Type u_2\nK : Type u_3\ninst✝⁸ : Field F\ninst✝⁷ : Field E\ninst✝⁶ : Field K\ninst✝⁵ : Algebra F E\ninst✝⁴ : Algebra F K\nS : Set E\nL : Type u_4\ninst✝³ : Field L\ninst✝² : Algebra F L\ninst✝¹ : Algebra L E\ninst✝ : IsScalarTower F L E\nf : L →ₐ[F] K\nhK : ∀ s ∈ S, IsIntegral L s ∧ (... | [] | by
let L' := (IsScalarTower.toAlgHom F L E).fieldRange
let f' : L' →ₐ[F] K := f.comp (AlgEquiv.ofInjectiveField _).symm.toAlgHom
have := exists_algHom_adjoin_of_splits'' f' (S := S) fun s hs ↦ ?_
· obtain ⟨φ, hφ⟩ := this; refine ⟨φ.comp <|
inclusion (?_ : (adjoin L S).restrictScalars F ≤ (adjoin L' S).res... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Eigenspace.Pi | {
"line": 171,
"column": 8
} | {
"line": 171,
"column": 71
} | {
"line": 172,
"column": 6
} | [
{
"pp": "case h'\nι : Type u_1\nK : Type u_3\ninst✝³ : Field K\nn : ℕ\nih :\n ∀ m < n,\n ∀ {M : Type u_4} [inst : AddCommGroup M] [inst_1 : Module K M] [FiniteDimensional K M] (f : ι → End K M),\n (∀ (i j : ι) (φ : K), MapsTo ⇑(f i) ↑((f j).maxGenEigenspace φ) ↑((f j).maxGenEigenspace φ)) →\n (∀... | [] | exact fun j ↦ Module.End.genEigenspace_restrict_eq_top _ (h' j) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.Eigenspace.Pi | {
"line": 171,
"column": 8
} | {
"line": 171,
"column": 71
} | {
"line": 172,
"column": 6
} | [
{
"pp": "case h'\nι : Type u_1\nK : Type u_3\ninst✝³ : Field K\nn : ℕ\nih :\n ∀ m < n,\n ∀ {M : Type u_4} [inst : AddCommGroup M] [inst_1 : Module K M] [FiniteDimensional K M] (f : ι → End K M),\n (∀ (i j : ι) (φ : K), MapsTo ⇑(f i) ↑((f j).maxGenEigenspace φ) ↑((f j).maxGenEigenspace φ)) →\n (∀... | [] | exact fun j ↦ Module.End.genEigenspace_restrict_eq_top _ (h' j) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Eigenspace.Pi | {
"line": 171,
"column": 8
} | {
"line": 171,
"column": 71
} | {
"line": 172,
"column": 6
} | [
{
"pp": "case h'\nι : Type u_1\nK : Type u_3\ninst✝³ : Field K\nn : ℕ\nih :\n ∀ m < n,\n ∀ {M : Type u_4} [inst : AddCommGroup M] [inst_1 : Module K M] [FiniteDimensional K M] (f : ι → End K M),\n (∀ (i j : ι) (φ : K), MapsTo ⇑(f i) ↑((f j).maxGenEigenspace φ) ↑((f j).maxGenEigenspace φ)) →\n (∀... | [] | exact fun j ↦ Module.End.genEigenspace_restrict_eq_top _ (h' j) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 763,
"column": 2
} | {
"line": 763,
"column": 37
} | {
"line": 764,
"column": 2
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx y : L\nhx : IsIntegral K x\nhy : IsIntegral K y\nthis : FiniteDimensional K ↥K⟮x⟯\n⊢ FiniteDimensional K ↥K⟮x, y⟯",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"instSMulOfMul",
"In... | [
"K : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx y : L\nhx : IsIntegral K x\nhy : IsIntegral K y\nthis✝ : FiniteDimensional K ↥K⟮x⟯\nthis : FiniteDimensional K ↥K⟮y⟯\n⊢ FiniteDimensional K ↥K⟮x, y⟯"
] | have := adjoin.finiteDimensional hy | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Lie.Weights.Linear | {
"line": 228,
"column": 2
} | {
"line": 237,
"column": 64
} | {
"line": 239,
"column": 0
} | [
{
"pp": "R : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : LieRing L\ninst✝⁷ : LieAlgebra R L\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : LieRingModule L M\ninst✝³ : LieModule R L M\ninst✝² : LieRing.IsNilpotent L\ninst✝¹ : LinearWeights R L M\ninst✝ : IsNoetherian R M\nχ : Wei... | [] | replace hχ : Nontrivial (shiftedGenWeightSpace R L M χ) :=
(LieSubmodule.nontrivial_iff_ne_bot R L M).mpr χ.genWeightSpace_ne_bot
obtain ⟨⟨⟨m, _⟩, hm₁⟩, hm₂⟩ :=
@exists_ne _ (nontrivial_max_triv_of_isNilpotent R L (shiftedGenWeightSpace R L M χ)) 0
simp_rw [mem_maxTrivSubmodule, Subtype.ext_iff,
ZeroMem... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Weights.Linear | {
"line": 228,
"column": 2
} | {
"line": 237,
"column": 64
} | {
"line": 239,
"column": 0
} | [
{
"pp": "R : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : LieRing L\ninst✝⁷ : LieAlgebra R L\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : LieRingModule L M\ninst✝³ : LieModule R L M\ninst✝² : LieRing.IsNilpotent L\ninst✝¹ : LinearWeights R L M\ninst✝ : IsNoetherian R M\nχ : Wei... | [] | replace hχ : Nontrivial (shiftedGenWeightSpace R L M χ) :=
(LieSubmodule.nontrivial_iff_ne_bot R L M).mpr χ.genWeightSpace_ne_bot
obtain ⟨⟨⟨m, _⟩, hm₁⟩, hm₂⟩ :=
@exists_ne _ (nontrivial_max_triv_of_isNilpotent R L (shiftedGenWeightSpace R L M χ)) 0
simp_rw [mem_maxTrivSubmodule, Subtype.ext_iff,
ZeroMem... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Multiset.Fintype | {
"line": 195,
"column": 26
} | {
"line": 195,
"column": 47
} | {
"line": 195,
"column": 47
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nβ : Type u_3\nm : Multiset α\nf : α → β\n⊢ map f (map (fun x ↦ x.fst) Finset.univ.val) = map f m",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.univ",
"Multiset.map",
"congrArg",
"Multiset.fintyp... | [
"α : Type u_1\ninst✝ : DecidableEq α\nβ : Type u_3\nm : Multiset α\nf : α → β\n⊢ map f m = map f m"
] | Multiset.map_univ_coe | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.Cartan | {
"line": 252,
"column": 2
} | {
"line": 254,
"column": 7
} | {
"line": 256,
"column": 0
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\nH : LieSubalgebra R L\ninst✝¹ : LieRing.IsNilpotent ↥H\ninst✝ : IsNoetherian R L\n⊢ H.toLieSubmodule = rootSpace H 0 ↔ H.IsCartanSubalgebra",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [... | [] | rw [← zeroRootSubalgebra_eq_iff_is_cartan, ← LieSubalgebra.toSubmodule_inj,
← LieSubmodule.toSubmodule_inj]
aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Weights.Cartan | {
"line": 252,
"column": 2
} | {
"line": 254,
"column": 7
} | {
"line": 256,
"column": 0
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\nH : LieSubalgebra R L\ninst✝¹ : LieRing.IsNilpotent ↥H\ninst✝ : IsNoetherian R L\n⊢ H.toLieSubmodule = rootSpace H 0 ↔ H.IsCartanSubalgebra",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [... | [] | rw [← zeroRootSubalgebra_eq_iff_is_cartan, ← LieSubalgebra.toSubmodule_inj,
← LieSubmodule.toSubmodule_inj]
aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Weights.Basic | {
"line": 113,
"column": 4
} | {
"line": 114,
"column": 41
} | {
"line": 116,
"column": 2
} | [
{
"pp": "case h\nR : Type u_2\nL : Type u_3\ninst✝¹⁴ : CommRing R\ninst✝¹³ : LieRing L\ninst✝¹² : LieAlgebra R L\nM₁ : Type u_5\nM₂ : Type u_6\nM₃ : Type u_7\ninst✝¹¹ : AddCommGroup M₁\ninst✝¹⁰ : Module R M₁\ninst✝⁹ : LieRingModule L M₁\ninst✝⁸ : LieModule R L M₁\ninst✝⁷ : AddCommGroup M₂\ninst✝⁶ : Module R M₂\... | [] | rw [← LinearMap.comp_apply, Module.End.commute_pow_left_of_commute h_comm_square,
LinearMap.comp_apply, hk, map_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.Weights.Cartan | {
"line": 343,
"column": 2
} | {
"line": 343,
"column": 63
} | {
"line": 345,
"column": 0
} | [
{
"pp": "case neg\nL : Type u_2\ninst✝⁵ : LieRing L\nK : Type u_4\ninst✝⁴ : Field K\ninst✝³ : LieAlgebra K L\ninst✝² : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nα : Weight K (↥H) L\nhα : ¬α.IsZero\n⊢ genWeightSpace L ⇑α ≤ H.toLieSubmodule ⊔... | [] | · exact le_sup_of_le_right <| le_iSup₂_of_le α hα (le_refl _) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.FieldTheory.SplittingField.Construction | {
"line": 59,
"column": 4
} | {
"line": 59,
"column": 23
} | {
"line": 61,
"column": 0
} | [
{
"pp": "case neg\nK : Type v\ninst✝ : Field K\nf : K[X]\nH : ¬∃ g, Irreducible g ∧ g ∣ f\n⊢ Irreducible X",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Field.toSemifield",
"Field.toCommRing",
"instIsDomain",
"Polynomial.irreducible_X"
],
"usedFVars": [
... | [] | exact irreducible_X | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.SplittingField.Construction | {
"line": 59,
"column": 4
} | {
"line": 59,
"column": 23
} | {
"line": 61,
"column": 0
} | [
{
"pp": "case neg\nK : Type v\ninst✝ : Field K\nf : K[X]\nH : ¬∃ g, Irreducible g ∧ g ∣ f\n⊢ Irreducible X",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Field.toSemifield",
"Field.toCommRing",
"instIsDomain",
"Polynomial.irreducible_X"
],
"usedFVars": [
... | [] | exact irreducible_X | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.SplittingField.Construction | {
"line": 59,
"column": 4
} | {
"line": 59,
"column": 23
} | {
"line": 61,
"column": 0
} | [
{
"pp": "case neg\nK : Type v\ninst✝ : Field K\nf : K[X]\nH : ¬∃ g, Irreducible g ∧ g ∣ f\n⊢ Irreducible X",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Field.toSemifield",
"Field.toCommRing",
"instIsDomain",
"Polynomial.irreducible_X"
],
"usedFVars": [
... | [] | exact irreducible_X | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.SplittingField.Construction | {
"line": 177,
"column": 6
} | {
"line": 177,
"column": 69
} | {
"line": 177,
"column": 69
} | [
{
"pp": "n✝ : ℕ\nK✝ : Type u\ninst✝ : Field K✝\nn : ℕ\nih : ∀ {K : Type u} [inst : Field K] (f : K[X]), f.natDegree = n → (map (algebraMap K (SplittingFieldAux n f)) f).Splits\nK : Type u\nx✝ : Field K\nf : K[X]\nhf : f.natDegree = n.succ\n⊢ (map (algebraMap (AdjoinRoot f.factor) (SplittingFieldAux n f.removeFa... | [
"n✝ : ℕ\nK✝ : Type u\ninst✝ : Field K✝\nn : ℕ\nih : ∀ {K : Type u} [inst : Field K] (f : K[X]), f.natDegree = n → (map (algebraMap K (SplittingFieldAux n f)) f).Splits\nK : Type u\nx✝ : Field K\nf : K[X]\nhf : f.natDegree = n.succ\n⊢ (map (algebraMap (AdjoinRoot f.factor) (SplittingFieldAux n f.removeFactor))\n ... | ← X_sub_C_mul_removeFactor f fun h => by rw [h] at hf; cases hf | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Sl2 | {
"line": 96,
"column": 40
} | {
"line": 96,
"column": 48
} | {
"line": 96,
"column": 49
} | [
{
"pp": "R : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\nh e f : L\ninst✝ : IsAddTorsionFree M\nt : IsSl2Triple h e f\nm : M\nμ ρ : R\nhm : m ≠ 0\nhm' :... | [
"R : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\nh e f : L\ninst✝ : IsAddTorsionFree M\nt : IsSl2Triple h e f\nm : M\nμ ρ : R\nhm : m ≠ 0\nhm' : ⁅h, m⁆ = μ ... | lie_lie, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Sl2 | {
"line": 123,
"column": 8
} | {
"line": 125,
"column": 14
} | {
"line": 126,
"column": 8
} | [
{
"pp": "case mem.mem.inl.inr.inr\nR : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nf x y u v : L\nt : IsSl2Triple v u f\n__spread✝⁻⁰ : Submodule R L := s... | [
"case mem.mem.inr.inl.inl\nR : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nh x y u v : L\nt : IsSl2Triple h v u\n__spread✝⁻⁰ : Submodule R L := span R {v, u,... | · rw [← lie_skew, t.lie_h_e_nsmul, neg_mem_iff]
apply nsmul_mem <| subset_span _
simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Lie.Weights.Chain | {
"line": 145,
"column": 27
} | {
"line": 145,
"column": 40
} | {
"line": 145,
"column": 40
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\nM : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\nH : LieSubalgebra R L\nα χ : ↥H → R\np q : ℤ\ninst✝ : LieRing.IsNilpotent ↥H\nhq : genWeightSpa... | [
"R : Type u_1\nL : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\nM : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\nH : LieSubalgebra R L\nα χ : ↥H → R\np q : ℤ\ninst✝ : LieRing.IsNilpotent ↥H\nhq : genWeightSpace M (q • α ... | iSup_subtype' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.TraceForm | {
"line": 141,
"column": 93
} | {
"line": 155,
"column": 87
} | {
"line": 157,
"column": 0
} | [
{
"pp": "R : Type u_1\nL : Type u_3\nM : Type u_4\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : LieRingModule L M\ninst✝⁶ : LieModule R L M\ninst✝⁵ : Free R M\ninst✝⁴ : IsDomain R\ninst✝³ : IsPrincipalIdealRing R\ninst✝² : LieRing.Is... | [] | by
set d := finrank R (genWeightSpace M χ)
have h₁ : χ y • d • χ x - χ y • χ x • (d : R) = 0 := by simp [mul_comm (χ x)]
have h₂ : χ x • d • χ y = d • (χ x * χ y) := by
simpa [nsmul_eq_mul, smul_eq_mul] using mul_left_comm (χ x) d (χ y)
have := traceForm_eq_zero_of_isNilpotent R L (shiftedGenWeightSpace R L... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Sequence | {
"line": 147,
"column": 4
} | {
"line": 153,
"column": 95
} | {
"line": 156,
"column": 4
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : Ring R\nS : Sequence R\nm : ℕ\nhCoeff : ∀ i < m, IsUnit (↑S i).leadingCoeff\na✝ : Nontrivial R\nn : ℕ\nih : ∀ m_1 < n, ∀ ⦃P : R[X]⦄, P ∈ degreeLT R m → P.natDegree = m_1 → P ∈ span R (↑S '' Set.Iio m)\nP : R[X]\nhP : P ∈ degreeLT R m\nhp : P.natDegree = n\np_ne_zero : P ... | [
"case neg\nR : Type u_1\ninst✝ : Ring R\nS : Sequence R\nm : ℕ\nhCoeff : ∀ i < m, IsUnit (↑S i).leadingCoeff\na✝ : Nontrivial R\nn : ℕ\nih : ∀ m_1 < n, ∀ ⦃P : R[X]⦄, P ∈ degreeLT R m → P.natDegree = m_1 → P ∈ span R (↑S '' Set.Iio m)\nP : R[X]\nhP : P ∈ degreeLT R m\nhp : P.natDegree = n\np_ne_zero : P ≠ 0\nhn : n ... | have hPhead : P.leadingCoeff = head.leadingCoeff := by
rw [degree_eq_natDegree p_ne_zero, head_degree_eq_natDegree] at head_degree_eq
nth_rw 2 [← coeff_natDegree]
rw_mod_cast [← head_degree_eq, hp]
dsimp [head]
nth_rw 2 [← S.natDegree_eq n]
rw [coeff_smul, coeff_smul, coeff_natDegree... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.Reflection | {
"line": 153,
"column": 2
} | {
"line": 154,
"column": 33
} | {
"line": 156,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u_1\nM : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nx : M\nf : Dual R M\ninst✝² : IsDomain R\ninst✝¹ : NeZero 2\ninst✝ : IsTorsionFree R M\nh : f x = 2\np : Submodule R M\nhp : Disjoint p (R ∙ x)\nh' : p ≤ LinearMap.ker f\ny : M\nhy : y ∈ p\n⊢ y... | [] | · have hy' : f y = 0 := by simpa using h' hy
simpa [reflection_apply, hy'] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.RootSystem.Defs | {
"line": 435,
"column": 57
} | {
"line": 438,
"column": 21
} | {
"line": 439,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\nx : M\nthis : ∀ (x : M), -x ∈ range ⇑P.root → x ∈ range ⇑P.root\n⊢ -x ∈ range ⇑P.root ↔ x ∈ range ⇑P.root",
... | [] | by
refine ⟨this x, fun h ↦ ?_⟩
rw [← neg_neg x] at h
exact this (-x) h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.RootSystem.Defs | {
"line": 516,
"column": 18
} | {
"line": 516,
"column": 29
} | {
"line": 516,
"column": 30
} | [
{
"pp": "case calc_5\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\ni j : ι\nhij : P.reflectionPerm i = P.reflectionPerm j\nx : M\n⊢ (2 • (P.toLinearMap x) (P.coroot... | [
"case calc_5\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\ni j : ι\nhij : P.reflectionPerm i = P.reflectionPerm j\nx : M\n⊢ 2 • (P.toLinearMap x) (P.coroot j) • P.root ... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 488,
"column": 2
} | {
"line": 488,
"column": 79
} | {
"line": 490,
"column": 0
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : CharZero K\nα : Weight K (↥H) L\n⊢ corootSpace ⇑α = ⊥ ↔ ... | [] | simp [← LieSubmodule.toSubmodule_eq_bot, coe_corootSpace_eq_span_singleton α] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 488,
"column": 2
} | {
"line": 488,
"column": 79
} | {
"line": 490,
"column": 0
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : CharZero K\nα : Weight K (↥H) L\n⊢ corootSpace ⇑α = ⊥ ↔ ... | [] | simp [← LieSubmodule.toSubmodule_eq_bot, coe_corootSpace_eq_span_singleton α] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 488,
"column": 2
} | {
"line": 488,
"column": 79
} | {
"line": 490,
"column": 0
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : CharZero K\nα : Weight K (↥H) L\n⊢ corootSpace ⇑α = ⊥ ↔ ... | [] | simp [← LieSubmodule.toSubmodule_eq_bot, coe_corootSpace_eq_span_singleton α] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.PerfectPairing.Restrict | {
"line": 106,
"column": 14
} | {
"line": 106,
"column": 34
} | {
"line": 107,
"column": 2
} | [
{
"pp": "case smul\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝¹⁸ : CommRing R\ninst✝¹⁷ : AddCommGroup M\ninst✝¹⁶ : Module R M\ninst✝¹⁵ : AddCommGroup N\ninst✝¹⁴ : Module R N\np : M →ₗ[R] N →ₗ[R] R\ninst✝¹³ : p.IsPerfPair\nS : Type u_4\nM' : Type u_5\nN' : Type u_6\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain ... | [] | rw [map_smul]; aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.PerfectPairing.Restrict | {
"line": 106,
"column": 14
} | {
"line": 106,
"column": 34
} | {
"line": 107,
"column": 2
} | [
{
"pp": "case smul\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝¹⁸ : CommRing R\ninst✝¹⁷ : AddCommGroup M\ninst✝¹⁶ : Module R M\ninst✝¹⁵ : AddCommGroup N\ninst✝¹⁴ : Module R N\np : M →ₗ[R] N →ₗ[R] R\ninst✝¹³ : p.IsPerfPair\nS : Type u_4\nM' : Type u_5\nN' : Type u_6\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain ... | [] | rw [map_smul]; aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 606,
"column": 65
} | {
"line": 606,
"column": 76
} | {
"line": 606,
"column": 77
} | [
{
"pp": "case h\nK : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : CharZero K\nα : Weight K (↥H) L\nhα : α.IsNonZer... | [
"case h\nK : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : CharZero K\nα : Weight K (↥H) L\nhα : α.IsNonZero\ne f : L\n... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Reflection | {
"line": 366,
"column": 71
} | {
"line": 366,
"column": 79
} | {
"line": 366,
"column": 80
} | [
{
"pp": "case succ\nR : Type u_1\nM : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nx : M\nΦ : Set M\nhΦ₁ : Φ.Finite\nhΦ₂ : span R Φ = ⊤\nf g : Dual R M\nhf₁ : f x = 2\nhf₂ : MapsTo (⇑(preReflection x f)) Φ Φ\nhg... | [
"case succ\nR : Type u_1\nM : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nx : M\nΦ : Set M\nhΦ₁ : Φ.Finite\nhΦ₂ : span R Φ = ⊤\nf g : Dual R M\nhf₁ : f x = 2\nhf₂ : MapsTo (⇑(preReflection x f)) Φ Φ\nhg₁ : g x = 2\... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Quotient.PowTransition | {
"line": 171,
"column": 2
} | {
"line": 171,
"column": 52
} | {
"line": 172,
"column": 2
} | [
{
"pp": "R : Type u_3\ninst✝ : CommRing R\nI : Ideal R\nn : ℕ\nnpos : n > 0\na : R ⧸ I ^ (n + 1)\nh : IsUnit ((factorPow I ⋯) a)\nb : R ⧸ I ^ n\nright✝ : b * (factorPow I ⋯) a = 1\nb' : R ⧸ I ^ n.succ\nhb' : (factor ⋯) b' = b\nhb : a * b' - 1 ∈ RingHom.ker (factorPow I ⋯)\n⊢ IsUnit a",
"ppTerm": "?m.140",
... | [
"R : Type u_3\ninst✝ : CommRing R\nI : Ideal R\nn : ℕ\nnpos : n > 0\na : R ⧸ I ^ (n + 1)\nh : IsUnit ((factorPow I ⋯) a)\nb : R ⧸ I ^ n\nright✝ : b * (factorPow I ⋯) a = 1\nb' : R ⧸ I ^ n.succ\nhb' : (factor ⋯) b' = b\nhb : a * b' - 1 ∈ map (mk (I ^ n.succ)) (I ^ n)\n⊢ IsUnit a"
] | rw [factor_ker (pow_le_pow_right n.le_succ)] at hb | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Reflection | {
"line": 368,
"column": 19
} | {
"line": 368,
"column": 33
} | {
"line": 368,
"column": 34
} | [
{
"pp": "case succ\nR : Type u_1\nM : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nx : M\nΦ : Set M\nhΦ₁ : Φ.Finite\nhΦ₂ : span R Φ = ⊤\nf g : Dual R M\nhf₁ : f x = 2\nhf₂ : MapsTo (⇑(preReflection x f)) Φ Φ\nhg... | [
"case succ\nR : Type u_1\nM : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nx : M\nΦ : Set M\nhΦ₁ : Φ.Finite\nhΦ₂ : span R Φ = ⊤\nf g : Dual R M\nhf₁ : f x = 2\nhf₂ : MapsTo (⇑(preReflection x f)) Φ Φ\nhg₁ : g x = 2\... | Nat.cast_succ, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Reflection | {
"line": 387,
"column": 4
} | {
"line": 387,
"column": 18
} | {
"line": 388,
"column": 4
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nx : M\nΦ : Set M\nhΦ₁ : Finite ↑Φ\nhx : x ∈ span R Φ\nf g : Dual R M\nhf₁ : f x = 2\nhf₂ : MapsTo (⇑(preReflection x f)) Φ Φ\nhg₁ : g x = 2... | [
"R : Type u_1\nM : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nx : M\nΦ : Set M\nhΦ₁ : Finite ↑Φ\nhx : x ∈ span R Φ\nf g : Dual R M\nhf₁ : f x = 2\nhf₂ : MapsTo (⇑(preReflection x f)) Φ Φ\nhg₁ : g x = 2\nhg₂ : Maps... | simp only [Φ'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Length | {
"line": 189,
"column": 6
} | {
"line": 189,
"column": 85
} | {
"line": 190,
"column": 4
} | [
{
"pp": "case pos\nR : Type u_1\nM : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nN : Type u_3\nP : Type u_4\ninst✝³ : AddCommGroup N\ninst✝² : AddCommGroup P\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : N →ₗ[R] M\ng : M →ₗ[R] P\nhf : Function.Injective ⇑f\nhg : Function.Surjective ... | [] | simp_rw [r, RelSeries.smash_length, Nat.cast_add, s', t', RelSeries.map_length] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.RingTheory.Length | {
"line": 226,
"column": 4
} | {
"line": 226,
"column": 11
} | {
"line": 226,
"column": 12
} | [
{
"pp": "case h_option\nR : Type u_1\ninst✝ : Ring R\n⊢ ∀ (α : Type u_5) [inst : Fintype α],\n (∀ (M : α → Type u_6) [inst_1 : (i : α) → AddCommGroup (M i)] [inst_2 : (i : α) → Module R (M i)],\n length R ((i : α) → M i) = ∑ i, length R (M i)) →\n ∀ (M : Option α → Type u_6) [inst_1 : (i : Option... | [
"case h_option\nR : Type u_1\ninst✝ : Ring R\nι : Type u_5\n⊢ ∀ [inst : Fintype ι],\n (∀ (M : ι → Type u_6) [inst_1 : (i : ι) → AddCommGroup (M i)] [inst_2 : (i : ι) → Module R (M i)],\n length R ((i : ι) → M i) = ∑ i, length R (M i)) →\n ∀ (M : Option ι → Type u_6) [inst_1 : (i : Option ι) → AddComm... | intro ι | Lean.Elab.Tactic.evalIntro | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 1007,
"column": 47
} | {
"line": 1007,
"column": 56
} | {
"line": 1007,
"column": 57
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℤ\nk : ℕ\nh :\n eval 0 ((⇑derivative)^[k + 2] (T R n)) =\n (2 * ↑k + 1) * 0 * eval 0 ((⇑derivative)^[k + 1] (T R n)) - (↑n ^ 2 - ↑k ^ 2) * eval 0 ((⇑derivative)^[k] (T R n))\n⊢ eval 0 ((⇑derivative)^[k + 2] (T R n)) = -(↑n ^ 2 - ↑k ^ 2) * eval 0 ((⇑derivative)^... | [
"R : Type u_1\ninst✝ : CommRing R\nn : ℤ\nk : ℕ\nh :\n eval 0 ((⇑derivative)^[k + 2] (T R n)) =\n 0 * eval 0 ((⇑derivative)^[k + 1] (T R n)) - (↑n ^ 2 - ↑k ^ 2) * eval 0 ((⇑derivative)^[k] (T R n))\n⊢ eval 0 ((⇑derivative)^[k + 2] (T R n)) = -(↑n ^ 2 - ↑k ^ 2) * eval 0 ((⇑derivative)^[k] (T R n))"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 1014,
"column": 47
} | {
"line": 1014,
"column": 56
} | {
"line": 1014,
"column": 57
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℤ\nk : ℕ\nh :\n eval 0 ((⇑derivative)^[k + 2] (U R n)) =\n (2 * ↑k + 3) * 0 * eval 0 ((⇑derivative)^[k + 1] (U R n)) -\n ((↑n + 1) ^ 2 - (↑k + 1) ^ 2) * eval 0 ((⇑derivative)^[k] (U R n))\n⊢ eval 0 ((⇑derivative)^[k + 2] (U R n)) = -((↑n + 1) ^ 2 - (↑k + 1... | [
"R : Type u_1\ninst✝ : CommRing R\nn : ℤ\nk : ℕ\nh :\n eval 0 ((⇑derivative)^[k + 2] (U R n)) =\n 0 * eval 0 ((⇑derivative)^[k + 1] (U R n)) - ((↑n + 1) ^ 2 - (↑k + 1) ^ 2) * eval 0 ((⇑derivative)^[k] (U R n))\n⊢ eval 0 ((⇑derivative)^[k + 2] (U R n)) = -((↑n + 1) ^ 2 - (↑k + 1) ^ 2) * eval 0 ((⇑derivative)^[k]... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 53
} | {
"line": 91,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : IsLocalRing R\ninst✝ : IsDomain R\nϖ : R\nhϖ : ϖ ≠ 0\nh : maximalIdeal R = span {ϖ}\nh2 : ¬IsUnit ϖ\na b : R\nhab : ϖ = ϖ * (ϖ * (a * b))\n⊢ ϖ * (a * b) ≠ 1",
"ppTerm": "?m.177",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
... | [] | exact fun hh => h2 (isUnit_of_dvd_one ⟨_, hh.symm⟩) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 850,
"column": 2
} | {
"line": 853,
"column": 15
} | {
"line": 855,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\nI : Ideal R\nM : Type u_4\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type u_5\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\na : ℕ → ℕ\nha : StrictMono a\nf : (n : ℕ) → M →ₗ[R] N ⧸ I ^ a n • ⊤\nhf : ∀ {m : ℕ}, Submodule.factorPow I N ⋯ ∘ₗ f (m + 1) = f m\ninst... | [] | apply DFunLike.coe_injective
apply IsHausdorff.StrictMono.funext I ha
intro n m
simp [← hF n] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 850,
"column": 2
} | {
"line": 853,
"column": 15
} | {
"line": 855,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\nI : Ideal R\nM : Type u_4\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type u_5\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\na : ℕ → ℕ\nha : StrictMono a\nf : (n : ℕ) → M →ₗ[R] N ⧸ I ^ a n • ⊤\nhf : ∀ {m : ℕ}, Submodule.factorPow I N ⋯ ∘ₗ f (m + 1) = f m\ninst... | [] | apply DFunLike.coe_injective
apply IsHausdorff.StrictMono.funext I ha
intro n m
simp [← hF n] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Valuation.ValuationRing | {
"line": 287,
"column": 4
} | {
"line": 287,
"column": 36
} | {
"line": 289,
"column": 0
} | [
{
"pp": "case neg.inr\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : Nontrivial A\ninst✝ : PreValuationRing A\nα β : Ideal A\na : A\nh₁ : a ∈ α\nh₂ : a ∉ β\nb : A\nhb : b ∈ β\nc : A\nh : b * c = a\n⊢ b * c ∈ β",
"ppTerm": "?neg.inr✝",
"assigned": true,
"usedConstants": [
"CommSemiring.toSemiring",... | [] | apply Ideal.mul_mem_right _ _ hb | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 29
} | {
"line": 80,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nh : ∀ (P : Ideal R), P.IsPrime → P.jacobson = P\nI : Ideal R\nhI : I.IsRadical\nx : R\nhx : x ∈ I.jacobson\nP : Ideal R\nhP : P ∈ {J | I ≤ J ∧ J.IsPrime}\n⊢ x ∈ P",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"co... | [
"R : Type u_1\ninst✝ : CommRing R\nh : ∀ (P : Ideal R), P.IsPrime → P.jacobson = P\nI : Ideal R\nhI : I.IsRadical\nx : R\nhx : x ∈ I.jacobson\nP : Ideal R\nhP : I ≤ P ∧ P.IsPrime\n⊢ x ∈ P"
] | rw [Set.mem_setOf_eq] at hP | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 446,
"column": 2
} | {
"line": 455,
"column": 21
} | {
"line": 457,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsDiscreteValuationRing R\na : R\n⊢ (addVal R) a = ⊤ ↔ a = 0",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Prime.irreducible",
"IsDiscreteValuationRing.addVal_zero",
"NonUnitalNonAssocCommRing.toN... | [] | have hi := (Classical.choose_spec (exists_prime R)).irreducible
constructor
· contrapose
intro h
obtain ⟨n, ha⟩ := associated_pow_irreducible h hi
obtain ⟨u, rfl⟩ := ha.symm
rw [mul_comm, addVal_def' u hi n]
nofun
· rintro rfl
exact addVal_zero | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 446,
"column": 2
} | {
"line": 455,
"column": 21
} | {
"line": 457,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsDiscreteValuationRing R\na : R\n⊢ (addVal R) a = ⊤ ↔ a = 0",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Prime.irreducible",
"IsDiscreteValuationRing.addVal_zero",
"NonUnitalNonAssocCommRing.toN... | [] | have hi := (Classical.choose_spec (exists_prime R)).irreducible
constructor
· contrapose
intro h
obtain ⟨n, ha⟩ := associated_pow_irreducible h hi
obtain ⟨u, rfl⟩ := ha.symm
rw [mul_comm, addVal_def' u hi n]
nofun
· rintro rfl
exact addVal_zero | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Valuation.Basic | {
"line": 327,
"column": 2
} | {
"line": 327,
"column": 63
} | {
"line": 328,
"column": 2
} | [
{
"pp": "R : Type u_3\nΓ₀ : Type u_4\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommMonoidWithZero Γ₀\nv : Valuation R Γ₀\nx y : R\nh : v (y - x) < v x\n⊢ v y = v x",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"AddGroupWithOne.toAddGroup",
"Valu... | [
"R : Type u_3\nΓ₀ : Type u_4\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommMonoidWithZero Γ₀\nv : Valuation R Γ₀\nx y : R\nh : v (y - x) < v x\nthis : v (y - x + x) = max (v (y - x)) (v x)\n⊢ v y = v x"
] | have := Valuation.map_add_of_distinct_val v (ne_of_gt h).symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Valuation.ValuationRing | {
"line": 347,
"column": 96
} | {
"line": 368,
"column": 25
} | {
"line": 370,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\n⊢ ValuationRing R ↔ ∀ (x : K), IsLocalization.IsInteger R x ∨ IsLocalization.IsInteger R x⁻¹",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
... | [] | by
constructor
· intro H x
obtain ⟨x : R, y, hy, rfl⟩ := IsFractionRing.div_surjective R x
have := (map_ne_zero_iff _ (IsFractionRing.injective R K)).mpr (nonZeroDivisors.ne_zero hy)
obtain ⟨s, rfl | rfl⟩ := ValuationRing.cond x y
· exact Or.inr
⟨s, eq_inv_of_mul_eq_one_left <| by rwa [mul_d... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 532,
"column": 4
} | {
"line": 538,
"column": 50
} | {
"line": 539,
"column": 2
} | [
{
"pp": "R✝ : Type u\ninst✝⁵ : CommRing R✝\ninst✝⁴ : IsDomain R✝\ninst✝³ : IsDiscreteValuationRing R✝\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsDiscreteValuationRing R\nk : ℕ∞\n⊢ OrderDual.toDual ((addVal R) (generator (ENat.recTopCoe ⊥ (fun n ↦ maximalIdeal R ^ n) k))) = OrderDual.toDu... | [] | induction k with
| top => simp
| coe k =>
obtain ⟨ϖ, hϖ⟩ := exists_irreducible R
rw [OrderDual.toDual_inj, ENat.recTopCoe_coe, hϖ.maximalIdeal_eq,
span_singleton_pow, ← hϖ.addVal_pow k, addVal_eq_iff_associated]
exact associated_generator_span_self (ϖ ^ k) | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.RingTheory.Valuation.ValuationRing | {
"line": 390,
"column": 2
} | {
"line": 390,
"column": 42
} | {
"line": 391,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDomain R\nK : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Algebra R K\ninst✝² : IsFractionRing R K\ninst✝¹ : IsLocalRing R\ninst✝ : IsBezout R\n⊢ ValuationRing R",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Dvd.dvd"... | [
"R : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDomain R\nK : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Algebra R K\ninst✝² : IsFractionRing R K\ninst✝¹ : IsLocalRing R\ninst✝ : IsBezout R\na b : R\n⊢ a ∣ b ∨ b ∣ a"
] | refine iff_dvd_total.mpr ⟨fun a b => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 293,
"column": 4
} | {
"line": 293,
"column": 85
} | {
"line": 294,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nP : Ideal R[X]\npX : R[X]\nhpX : pX ∈ P\ninst✝³ : Algebra (R ⧸ comap C P) Rₘ\ninst✝² : IsLocalization.Away (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff Rₘ\ninst✝¹ : Algebra (R[X] ⧸ P) S... | [
"R : Type u_1\ninst✝⁶ : CommRing R\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nP : Ideal R[X]\npX : R[X]\nhpX : pX ∈ P\ninst✝³ : Algebra (R ⧸ comap C P) Rₘ\ninst✝² : IsLocalization.Away (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff Rₘ\ninst✝¹ : Algebra (R[X] ⧸ P) Sₘ\ninst✝ :\n... | refine (hp.symm ▸ this).of_mul_unit φ' p (algebraMap (R[X] ⧸ P) Sₘ (φ q')) q'' ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.LocalProperties.IntegrallyClosed | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 81
} | {
"line": 75,
"column": 2
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nh : ∀ (p : Ideal R), p ≠ ⊥ → ∀ [inst : p.IsMaximal], IsIntegrallyClosed (Localization.AtPrime p)\nhf : ¬IsField R\n⊢ IsIntegrallyClosed R",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"IsIntegrallyClosed.of_l... | [
"case neg.refine_1\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nh : ∀ (p : Ideal R), p ≠ ⊥ → ∀ [inst : p.IsMaximal], IsIntegrallyClosed (Localization.AtPrime p)\nhf : ¬IsField R\nx✝ : PrimeSpectrum R\n⊢ x✝ ∈ Set.range MaximalSpectrum.toPrimeSpectrum → IsIntegrallyClosed (Localization.AtPrime x✝.asIdeal)"... | refine of_localization (.range MaximalSpectrum.toPrimeSpectrum) (fun _ ↦ ?_) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.DedekindDomain.Dvr | {
"line": 70,
"column": 61
} | {
"line": 76,
"column": 54
} | {
"line": 76,
"column": 54
} | [
{
"pp": "R : Type u_2\nRₘ : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\ninst✝² : CommRing Rₘ\ninst✝¹ : Algebra R Rₘ\nM : Submonoid R\ninst✝ : IsLocalization M Rₘ\nhM : M ≤ R⁰\nh : DimensionLEOne R\n⊢ ∀ {p : Ideal Rₘ}, p ≠ ⊥ → p.IsPrime → p.IsMaximal",
"ppTerm": "?m.17",
"assigned": true,
"us... | [] | by
intro p hp0 hpp
refine Ideal.isMaximal_def.mpr ⟨hpp.ne_top, Ideal.maximal_of_no_maximal fun P hpP hPm => ?_⟩
have hpP' : (⟨p, hpp⟩ : { p : Ideal Rₘ // p.IsPrime }) < ⟨P, hPm.isPrime⟩ := hpP
rw [← (IsLocalization.orderIsoOfPrime M Rₘ).lt_iff_lt] at hpP'
refine h.not_lt_lt ⊥ (p.under R) (P.under R) ⟨?_, hpP'... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.DiscreteValuationRing.TFAE | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 30
} | {
"line": 115,
"column": 4
} | [
{
"pp": "case neg.zero\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsLocalRing R\ninst✝ : IsDedekindDomain R\nne_bot : ¬maximalIdeal R = ⊥\na : R\nha₁ : a ∈ maximalIdeal R\nha₂ : a ≠ 0\nhle : Ideal.span {a} ≤ maximalIdeal R\nthis✝ : (Ideal.span {a}).radical = maximalIdeal R\nthis : ∃ n, maximalIdeal R ^ n ≤ Id... | [
"case neg.zero\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsLocalRing R\ninst✝ : IsDedekindDomain R\nne_bot : ¬maximalIdeal R = ⊥\na : R\nha₁ : a ∈ maximalIdeal R\nha₂ : a ≠ 0\nhle : Ideal.span {a} ≤ maximalIdeal R\nthis✝¹ : (Ideal.span {a}).radical = maximalIdeal R\nthis✝ : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {... | have := Nat.find_spec this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Flat.Tensor | {
"line": 50,
"column": 66
} | {
"line": 50,
"column": 77
} | {
"line": 50,
"column": 78
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\n⊢ (∀ ⦃X Y : Type v⦄ [inst : AddCommGroup X] [inst_1 : AddCommGroup Y] [inst_2 : Module R X] [inst_3 : Module R Y]\n (f : X →ₗ[R] Y), Function.Injective ⇑f → ∀ (g : X →ₗ[R] CharacterModule M), ∃ h, ∀ (x : X), ... | [
"R : Type u\nM : Type v\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\n⊢ (∀ ⦃X Y : Type v⦄ [inst : AddCommGroup X] [inst_1 : AddCommGroup Y] [inst_2 : Module R X] [inst_3 : Module R Y]\n (f : X →ₗ[R] Y), Function.Injective ⇑f → ∀ (g : X →ₗ[R] CharacterModule M), ∃ h, ∀ (x : X), h (f x) = g ... | Surjective, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Flat.Tensor | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 15
} | {
"line": 70,
"column": 16
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\n⊢ (∀ (I : Ideal R) (g : ↥I →ₗ[R] CharacterModule M), ∃ g', ∀ (x : R) (mem : x ∈ I), g' x = g ⟨x, mem⟩) ↔\n ∀ (I : Ideal R), Surjective ⇑(lcomp R (CharacterModule M) (Submodule.subtype I))",
"ppTerm": "?m.28... | [
"R : Type u\nM : Type v\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\n⊢ (∀ (I : Ideal R) (g : ↥I →ₗ[R] CharacterModule M), ∃ g', ∀ (x : R) (mem : x ∈ I), g' x = g ⟨x, mem⟩) ↔\n ∀ (I : Ideal R) (b : ↥I →ₗ[R] CharacterModule M), ∃ a, (lcomp R (CharacterModule M) (Submodule.subtype I)) a = b"
] | Surjective, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 500,
"column": 6
} | {
"line": 501,
"column": 69
} | {
"line": 502,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nP : Ideal R[X]\nhP : P.IsMaximal\ninst✝ : Nontrivial R\nhR : IsJacobsonRing R\nhP' : ∀ (x : R), C x ∈ P → x = 0\nP' : Ideal R := comap C P\npX : R[X]\nhpX : pX ∈ P\nhp0 : map (Ideal.Quotient.mk (comap C P)) pX ≠ 0\na : R ⧸ P' := (map (Ideal.Quotient.mk P') pX).leading... | [
"case refine_1\nR : Type u_1\ninst✝¹ : CommRing R\nP : Ideal R[X]\nhP : P.IsMaximal\ninst✝ : Nontrivial R\nhR : IsJacobsonRing R\nhP' : ∀ (x : R), C x ∈ P → x = 0\nP' : Ideal R := comap C P\npX : R[X]\nhpX : pX ∈ P\nhp0 : map (Ideal.Quotient.mk (comap C P)) pX ≠ 0\na : R ⧸ P' := (map (Ideal.Quotient.mk P') pX).lead... | refine RingHom.IsIntegral.trans (algebraMap (R ⧸ P') (Localization M))
(IsLocalization.map (Localization M') φ M.le_comap_map) ?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.RootSystem.RootPositive | {
"line": 68,
"column": 6
} | {
"line": 68,
"column": 23
} | {
"line": 68,
"column": 24
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_4\nN : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\nB : P.InvariantForm\ni j : ι\n⊢ (B.form ((P.reflection j) (P.root i))) ((P.reflection j) (P.root j)) =\n P.... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_4\nN : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\nB : P.InvariantForm\ni j : ι\n⊢ (B.form (P.root i - (P.coroot' j) (P.root i) • P.root j)) ((P.reflection j) (P.root j)) =\... | reflection_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.Finite.CanonicalBilinear | {
"line": 158,
"column": 2
} | {
"line": 158,
"column": 22
} | {
"line": 159,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\ninst✝ : Fintype ι\ny : N\nhy : y ∈ (P.Polarization.domRestrict (P.rootSpan R)).range\n⊢ y ∈ P.corootSpan R",
... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\ninst✝ : Fintype ι\ny : N\nx : ↥(P.rootSpan R)\nhx : (P.Polarization.domRestrict (P.rootSpan R)) x = y\n⊢ y ∈ P.corootSpan... | obtain ⟨x, hx⟩ := hy | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.LinearAlgebra.RootSystem.Finite.CanonicalBilinear | {
"line": 279,
"column": 30
} | {
"line": 279,
"column": 57
} | {
"line": 279,
"column": 57
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹³ : CommRing R\ninst✝¹² : AddCommGroup M\ninst✝¹¹ : Module R M\ninst✝¹⁰ : AddCommGroup N\ninst✝⁹ : Module R N\nP : RootPairing ι R M N\nS : Type u_5\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra S R\ninst✝⁶ : FaithfulSMul S R\ninst✝⁵ : Module S M\n... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹³ : CommRing R\ninst✝¹² : AddCommGroup M\ninst✝¹¹ : Module R M\ninst✝¹⁰ : AddCommGroup N\ninst✝⁹ : Module R N\nP : RootPairing ι R M N\nS : Type u_5\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra S R\ninst✝⁶ : FaithfulSMul S R\ninst✝⁵ : Module S M\ninst✝⁴ : IsS... | rootFormIn_self_smul_coroot | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.Finite.CanonicalBilinear | {
"line": 383,
"column": 2
} | {
"line": 384,
"column": 90
} | {
"line": 385,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹³ : CommRing R\ninst✝¹² : AddCommGroup M\ninst✝¹¹ : Module R M\ninst✝¹⁰ : AddCommGroup N\ninst✝⁹ : Module R N\nP : RootPairing ι R M N\nS : Type u_5\ninst✝⁸ : CommRing S\ninst✝⁷ : LinearOrder S\ninst✝⁶ : IsStrictOrderedRing S\ninst✝⁵ : Algeb... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹³ : CommRing R\ninst✝¹² : AddCommGroup M\ninst✝¹¹ : Module R M\ninst✝¹⁰ : AddCommGroup N\ninst✝⁹ : Module R N\nP : RootPairing ι R M N\nS : Type u_5\ninst✝⁸ : CommRing S\ninst✝⁷ : LinearOrder S\ninst✝⁶ : IsStrictOrderedRing S\ninst✝⁵ : Algebra S R\ninst... | have : s = (P.posRootForm S).posForm x x :=
FaithfulSMul.algebraMap_injective S R <| (P.algebraMap_posRootForm_posForm S x x) ▸ hs | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.RootSystem.Reduced | {
"line": 238,
"column": 4
} | {
"line": 239,
"column": 89
} | {
"line": 240,
"column": 4
} | [
{
"pp": "case refine_1\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\ni j : ι\ninst✝⁴ : Finite ι\ninst✝³ : CharZero R\ninst✝² : IsDomain R\ninst✝¹ : IsTorsionFree R... | [
"case refine_1\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\ni j : ι\ninst✝⁴ : Finite ι\ninst✝³ : CharZero R\ninst✝² : IsDomain R\ninst✝¹ : IsTorsionFree R M\ninst✝ : ... | have : ¬ LinearIndependent R ![P.root i, P.root j] := by
rw [← coxeterWeight_eq_four_iff_not_linearIndependent, coxeterWeight, h₁, h₂]; simp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.RootSystem.Finite.Nondegenerate | {
"line": 284,
"column": 2
} | {
"line": 284,
"column": 78
} | {
"line": 285,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁶ : Fintype ι\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Field R\ninst✝² : Module R M\ninst✝¹ : Module R N\nP : RootPairing ι R M N\ninst✝ : P.IsAnisotropic\nthis✝ : IsReflexive R M\nthis : IsReflexive R N\naux : finrank R M ... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁶ : Fintype ι\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Field R\ninst✝² : Module R M\ninst✝¹ : Module R N\nP : RootPairing ι R M N\ninst✝ : P.IsAnisotropic\nthis✝ : IsReflexive R M\nthis : IsReflexive R N\naux : finrank R M = finrank R ... | convert! Submodule.finrank_mono P.corootSpan_dualAnnihilator_le_ker_rootForm | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.LinearAlgebra.RootSystem.Reduced | {
"line": 276,
"column": 2
} | {
"line": 277,
"column": 68
} | {
"line": 279,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\ninst✝⁹ : AddCommGroup N\ninst✝⁸ : Module R N\nP : RootPairing ι R M N\nS : Type u_5\ni j : ι\ninst✝⁷ : Finite ι\ninst✝⁶ : CharZero R\ninst✝⁵ : IsDomain R\ninst✝⁴ : IsTorsionFree... | [] | rw [linearIndependent_iff_coxeterWeight_ne_four, ← P.algebraMap_coxeterWeightIn S,
← map_ofNat (algebraMap S R), (algebraMap_injective S R).ne_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.RootSystem.Reduced | {
"line": 276,
"column": 2
} | {
"line": 277,
"column": 68
} | {
"line": 279,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\ninst✝⁹ : AddCommGroup N\ninst✝⁸ : Module R N\nP : RootPairing ι R M N\nS : Type u_5\ni j : ι\ninst✝⁷ : Finite ι\ninst✝⁶ : CharZero R\ninst✝⁵ : IsDomain R\ninst✝⁴ : IsTorsionFree... | [] | rw [linearIndependent_iff_coxeterWeight_ne_four, ← P.algebraMap_coxeterWeightIn S,
← map_ofNat (algebraMap S R), (algebraMap_injective S R).ne_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.Reduced | {
"line": 276,
"column": 2
} | {
"line": 277,
"column": 68
} | {
"line": 279,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\ninst✝⁹ : AddCommGroup N\ninst✝⁸ : Module R N\nP : RootPairing ι R M N\nS : Type u_5\ni j : ι\ninst✝⁷ : Finite ι\ninst✝⁶ : CharZero R\ninst✝⁵ : IsDomain R\ninst✝⁴ : IsTorsionFree... | [] | rw [linearIndependent_iff_coxeterWeight_ne_four, ← P.algebraMap_coxeterWeightIn S,
← map_ofNat (algebraMap S R), (algebraMap_injective S R).ne_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Submodule.Union | {
"line": 84,
"column": 6
} | {
"line": 84,
"column": 23
} | {
"line": 85,
"column": 4
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\ninst✝² : Field K\ninst✝¹ : AddCommGroup M\ninst✝ : Module K M\np : ι → Submodule K M\nh₁ : ∀ (i : ι), p i ≠ ⊤\nj : ι\ns : Finset ι\nhj : j ∉ s\nhs : s.Nonempty\nh₂ : ↑s.card + 1 < ENat.card K\nhj' : ↑(p j) ∪ ⋃ i ∈ s, ↑(p i) = univ\nx : M\nhx : x ∈ p j\nhx₀ : x ... | [] | rwa [sub_ne_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 97,
"column": 26
} | {
"line": 97,
"column": 35
} | {
"line": 97,
"column": 36
} | [
{
"pp": "case pos\nK : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : IsKilling K L\ninst✝² : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsZe... | [
"case pos\nK : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : IsKilling K L\ninst✝² : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsZero\n⊢ 0 = ↑(... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 162,
"column": 36
} | {
"line": 162,
"column": 50
} | {
"line": 162,
"column": 51
} | [
{
"pp": "case neg.a\nK : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : IsKilling K L\ninst✝² : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : ¬α.I... | [
"case neg.a\nK : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : IsKilling K L\ninst✝² : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : ¬α.IsZero\ne : (... | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 164,
"column": 19
} | {
"line": 164,
"column": 33
} | {
"line": 164,
"column": 34
} | [
{
"pp": "case neg.a\nK : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : IsKilling K L\ninst✝² : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : ¬α.I... | [
"case neg.a\nK : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : IsKilling K L\ninst✝² : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : ¬α.IsZero\ne : (... | ← neg_add_rev, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RepresentationTheory.Basic | {
"line": 402,
"column": 4
} | {
"line": 402,
"column": 19
} | {
"line": 404,
"column": 0
} | [
{
"pp": "k : Type u_1\ninst✝² : Semiring k\nG : Type u_2\ninst✝¹ : Monoid G\nH : Type u_3\ninst✝ : MulAction G H\nx y : G\nz w : H\n⊢ (((Finsupp.lmapDomain k k fun x_1 ↦ (x * y) • x_1) ∘ₗ Finsupp.lsingle z) 1) w =\n ((((Finsupp.lmapDomain k k fun x_1 ↦ x • x_1) * Finsupp.lmapDomain k k fun x ↦ y • x) ∘ₗ Fins... | [] | simp [mul_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.RootSystem.Irreducible | {
"line": 130,
"column": 2
} | {
"line": 140,
"column": 70
} | {
"line": 141,
"column": 2
} | [
{
"pp": "case refine_1\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\ninst✝ : Nontrivial M\nh : IsSimpleOrder ↥P.weylGroupRootRep.invtSubmodule\nq : Submodule R M\n... | [
"case refine_2\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\ninst✝ : Nontrivial M\nh : ∀ (q : Submodule R M), (∀ (i : ι), q ∈ invtSubmodule ↑(P.reflection i)) → q ≠ ⊥ ... | · suffices ∀ g : P.weylGroup, q ∈ invtSubmodule (P.weylGroupRootRep g) by
let q' : P.weylGroupRootRep.invtSubmodule :=
⟨q, (Representation.mem_invtSubmodule P.weylGroupRootRep).mpr this⟩
suffices q' = ⊤ by simpa [q']
apply (IsSimpleOrder.eq_bot_or_eq_top _).resolve_left
simpa [q']
ri... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.RootSystem.Irreducible | {
"line": 142,
"column": 4
} | {
"line": 142,
"column": 63
} | {
"line": 143,
"column": 4
} | [
{
"pp": "case refine_2.inr\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\ninst✝ : Nontrivial M\nh : ∀ (q : Submodule R M), (∀ (i : ι), q ∈ invtSubmodule ↑(P.reflect... | [
"case refine_2.inr\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\ninst✝ : Nontrivial M\nh : ∀ (q : Submodule R M), (∀ (i : ι), q ∈ invtSubmodule ↑(P.reflection i)) → q ... | suffices (q : Submodule R M) = ⊤ by right; simpa using this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.LinearAlgebra.RootSystem.Hom | {
"line": 208,
"column": 4
} | {
"line": 208,
"column": 40
} | {
"line": 209,
"column": 4
} | [
{
"pp": "case indexEquiv\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\nf g : P.End\nhfg : (weightHom P) f = (weightHom P) g\nx : ι\n⊢ f.indexEquiv x = g.indexEquiv ... | [
"case indexEquiv\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\nf g : P.End\nhfg : (weightHom P) f = (weightHom P) g\nx : ι\n⊢ P.root (f.indexEquiv x) = P.root (g.indexE... | refine Embedding.injective P.root ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 396,
"column": 19
} | {
"line": 396,
"column": 72
} | {
"line": 396,
"column": 72
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : IsKilling K L\ninst✝² : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nα✝ β : Weight K (↥H) L\nx✝ : ↥LieSubalgebra.... | [] | by simpa using root_apply_coroot <| by simpa using hα | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.RootSystem.Hom | {
"line": 503,
"column": 6
} | {
"line": 503,
"column": 49
} | {
"line": 504,
"column": 2
} | [
{
"pp": "case hr\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁴ : CommRing R\ninst✝¹³ : AddCommGroup M\ninst✝¹² : Module R M\ninst✝¹¹ : AddCommGroup N\ninst✝¹⁰ : Module R N\nι₂ : Type u_5\nM₂ : Type u_6\nN₂ : Type u_7\ninst✝⁹ : AddCommGroup M₂\ninst✝⁸ : Module R M₂\ninst✝⁷ : AddCommGroup N₂\ni... | [] | simp [hf, RootPairing.map, RootPairing.map] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.RootSystem.Hom | {
"line": 503,
"column": 6
} | {
"line": 503,
"column": 49
} | {
"line": 504,
"column": 2
} | [
{
"pp": "case hr\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁴ : CommRing R\ninst✝¹³ : AddCommGroup M\ninst✝¹² : Module R M\ninst✝¹¹ : AddCommGroup N\ninst✝¹⁰ : Module R N\nι₂ : Type u_5\nM₂ : Type u_6\nN₂ : Type u_7\ninst✝⁹ : AddCommGroup M₂\ninst✝⁸ : Module R M₂\ninst✝⁷ : AddCommGroup N₂\ni... | [] | simp [hf, RootPairing.map, RootPairing.map] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.Hom | {
"line": 503,
"column": 6
} | {
"line": 503,
"column": 49
} | {
"line": 504,
"column": 2
} | [
{
"pp": "case hr\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁴ : CommRing R\ninst✝¹³ : AddCommGroup M\ninst✝¹² : Module R M\ninst✝¹¹ : AddCommGroup N\ninst✝¹⁰ : Module R N\nι₂ : Type u_5\nM₂ : Type u_6\nN₂ : Type u_7\ninst✝⁹ : AddCommGroup M₂\ninst✝⁸ : Module R M₂\ninst✝⁷ : AddCommGroup N₂\ni... | [] | simp [hf, RootPairing.map, RootPairing.map] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 83
} | {
"line": 116,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ni j : ↥b.support\nhij : i ≠ j\nthis : {↑j, ↑i} ⊆ ↑b.support\n⊢ LinearIndepOn R id (range ![P.root ... | [] | simpa [image_pair] using LinearIndepOn.id_image <| b.linearIndepOn_root.mono this | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.RootSystem.Chain | {
"line": 310,
"column": 2
} | {
"line": 310,
"column": 70
} | {
"line": 311,
"column": 2
} | [
{
"pp": "case pos\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Finite ι\ninst✝⁷ : CommRing R\ninst✝⁶ : CharZero R\ninst✝⁵ : IsDomain R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\ninst✝ : P.IsCrystallographic\ni j : ι\... | [
"case pos\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Finite ι\ninst✝⁷ : CommRing R\ninst✝⁶ : CharZero R\ninst✝⁵ : IsDomain R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\ninst✝ : P.IsCrystallographic\ni j : ι\nh : LinearI... | rw [← Iic_chainBotCoeff_eq h, mem_Iic, not_le, Nat.lt_one_iff] at h' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 190,
"column": 2
} | {
"line": 194,
"column": 69
} | {
"line": 196,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Finite ι\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Field R\ninst✝⁴ : CharZero R\ninst✝³ : Module R M\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : P.IsRootSystem\ninst✝ : P.IsCrystallographic\ns : Set ι\nf : M ... | [] | letI := P.indexNeg
refine ⟨?_, fun ⟨hli, sp⟩ ↦ P.eq_baseOf_of_linearIndepOn_of_mem_or_neg_mem_closure s hli sp f hf⟩
rintro rfl
exact ⟨P.linearIndepOn_root_baseOf f hf', fun i ↦
mem_or_neg_mem_closure_baseOf P.root f i (by simp_all) (by simp)⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 190,
"column": 2
} | {
"line": 194,
"column": 69
} | {
"line": 196,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Finite ι\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Field R\ninst✝⁴ : CharZero R\ninst✝³ : Module R M\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : P.IsRootSystem\ninst✝ : P.IsCrystallographic\ns : Set ι\nf : M ... | [] | letI := P.indexNeg
refine ⟨?_, fun ⟨hli, sp⟩ ↦ P.eq_baseOf_of_linearIndepOn_of_mem_or_neg_mem_closure s hli sp f hf⟩
rintro rfl
exact ⟨P.linearIndepOn_root_baseOf f hf', fun i ↦
mem_or_neg_mem_closure_baseOf P.root f i (by simp_all) (by simp)⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 470,
"column": 52
} | {
"line": 470,
"column": 63
} | {
"line": 470,
"column": 64
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝ : CharZero R\ni j k : ι\nz : ℤ\nhk : P.root k = P.root i + z • P.root j\nf : ι → ℤ\nhf : P.... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝ : CharZero R\ni j k : ι\nz : ℤ\nhk : P.root k = P.root i + z • P.root j\nf : ι → ℤ\nhf : P.root i = ∑ j... | smul_assoc, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Int.Star | {
"line": 29,
"column": 2
} | {
"line": 29,
"column": 69
} | {
"line": 30,
"column": 2
} | [
{
"pp": "n : ℕ\nhn : Even n\nx : ℤ\nhx : x ∈ nonneg ℤ\n⊢ x ∈ closure (range fun x ↦ x ^ n)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"one_pow",
"Eq.mpr",
"MulOne.toOne",
"instHSMul",
"HMul.hMul",
"Monoid.toMulOneClass"... | [
"n : ℕ\nhn : Even n\nx : ℤ\nhx : x ∈ nonneg ℤ\nthis : x = x.natAbs • 1 ^ n\n⊢ x ∈ closure (range fun x ↦ x ^ n)"
] | have : x = x.natAbs • 1 ^ n := by simpa [eq_comm (a := x)] using hx | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 350,
"column": 2
} | {
"line": 350,
"column": 43
} | {
"line": 351,
"column": 2
} | [
{
"pp": "case neg\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Module R N\nP : RootPairing ι R M N\ninst✝⁵ : Finite ι\ninst✝⁴ : CharZero R\ninst✝³ : P.IsCrystallographic\ninst✝² : IsDomain R\ninst✝¹... | [
"case neg\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Module R N\nP : RootPairing ι R M N\ninst✝⁵ : Finite ι\ninst✝⁴ : CharZero R\ninst✝³ : P.IsCrystallographic\ninst✝² : IsDomain R\ninst✝¹ : P.IsReduc... | have key := B.apply_eq_or_of_apply_ne hij | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 417,
"column": 25
} | {
"line": 417,
"column": 84
} | {
"line": 419,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf1 f2 g1 g2 : CauSeq β abv\nhf : f1 ≈ f2\nhg : g1 ≈ g2\n⊢ f1 + g1 ≈ f2 + g2",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants"... | [] | by simpa only [← add_sub_add_comm] using! add_limZero hf hg | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 233,
"column": 2
} | {
"line": 243,
"column": 77
} | {
"line": 244,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Field R\ninst✝⁵ : CharZero R\ninst✝⁴ : Module R M\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsRootSystem\ninst✝¹ : P.IsCrystallographic\ninst✝ : P.IsRedu... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Field R\ninst✝⁵ : CharZero R\ninst✝⁴ : Module R M\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsRootSystem\ninst✝¹ : P.IsCrystallographic\ninst✝ : P.IsReduced\nf : M →... | replace huv : u • (Nat.cast ∘ a) + v • (Nat.cast (R := ℚ) ∘ b) = Pi.single ri 1 := by
simp_rw [← ha, ← hb, Finset.smul_sum, ← Finset.sum_add_distrib, ← Nat.cast_smul_eq_nsmul ℚ,
smul_smul, ← add_smul] at huv
have aux : P.root i = ∑ x, Pi.single (M := fun x : P.root '' s ↦ ℚ) ri 1 x • (x : M) := by
s... | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.Data.Real.Basic | {
"line": 206,
"column": 18
} | {
"line": 206,
"column": 57
} | {
"line": 207,
"column": 2
} | [
{
"pp": "x : ℝ\n⊢ { cauchy := ↑0 } = 0",
"ppTerm": "?m.167",
"assigned": true,
"usedConstants": [
"Real",
"Real.cauchy",
"CauSeq.Completion.instNatCastCauchy",
"Real.ext_cauchy",
"Real.instZero",
"abs",
"congrArg",
"IsAbsoluteValue.abs_isAbsoluteValue"... | [] | by apply ext_cauchy; simp [cauchy_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Real.Basic | {
"line": 289,
"column": 33
} | {
"line": 289,
"column": 58
} | {
"line": 291,
"column": 0
} | [
{
"pp": "⊢ mk 0 = 0",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"abs",
"Real.ofCauchy_zero",
"congrArg",
"IsAbsoluteValue.abs_isAbsoluteValue",
"Rat",
"CauSeq.Completion.Cauchy",
"Rat.linearOr... | [] | rw [← ofCauchy_zero]; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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