module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 575,
"column": 4
} | {
"line": 575,
"column": 44
} | [
{
"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
} | [
{
"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": 202,
"column": 4
} | {
"line": 202,
"column": 25
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"IntermediateField.instPartialOr... | adjoin_simple_le_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Eigenspace.Pi | {
"line": 171,
"column": 8
} | {
"line": 171,
"column": 71
} | [
{
"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
} | [
{
"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
} | [
{
"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.Extension | {
"line": 202,
"column": 2
} | {
"line": 203,
"column": 93
} | [
{
"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})... | 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.IntermediateField.Adjoin.Basic | {
"line": 555,
"column": 2
} | {
"line": 555,
"column": 47
} | [
{
"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",
"usedConstants": [
"algebraMap",
"Field.... | 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.IntermediateField.Adjoin.Basic | {
"line": 552,
"column": 74
} | {
"line": 563,
"column": 27
} | [
{
"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\n⊢ f.natDegree ∣ finrank K L",
"usedConstants": [
"Iff.mpr",
"WithBot.instPreorder",
"Eq.mpr",
"Polynomial.C",
... | by
have := hi.degree_pos.ne'
rw [← f.degree_map (algebraMap K L)] at this
obtain ⟨x, hx⟩ := hs.exists_eval_eq_zero this
rw [eval_map_algebraMap] at hx
have key := minpoly.Irreducible.eq_minpoly hi hx
replace hi := hi.ne_zero
rw [key, natDegree_C_mul (leadingCoeff_ne_zero.mpr hi)]
apply minpoly.degree_dv... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 771,
"column": 2
} | {
"line": 771,
"column": 37
} | [
{
"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⟯",
"usedConstants": [
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonU... | have := adjoin.finiteDimensional hy | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.Multiset.Fintype | {
"line": 198,
"column": 26
} | {
"line": 198,
"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",
"usedConstants": [
"Eq.mpr",
"Finset.univ",
"Multiset.map",
"congrArg",
"Multiset.fintypeCoe",
"Multiset.count",
"Multise... | Multiset.map_univ_coe | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.Cartan | {
"line": 252,
"column": 2
} | {
"line": 254,
"column": 7
} | [
{
"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",
"usedConstants": [
"LieAlgebra.zeroRootSubalgebra",
... | 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
} | [
{
"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",
"usedConstants": [
"LieAlgebra.zeroRootSubalgebra",
... | 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": 114,
"column": 4
} | {
"line": 115,
"column": 41
} | [
{
"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.Linear | {
"line": 226,
"column": 2
} | {
"line": 235,
"column": 64
} | [
{
"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": 226,
"column": 2
} | {
"line": 235,
"column": 64
} | [
{
"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.FieldTheory.SplittingField.Construction | {
"line": 59,
"column": 4
} | {
"line": 59,
"column": 23
} | [
{
"pp": "case neg\nK : Type v\ninst✝ : Field K\nf : K[X]\nH : ¬∃ g, Irreducible g ∧ g ∣ f\n⊢ Irreducible X",
"usedConstants": [
"Field.toSemifield",
"instIsDomain",
"Polynomial.irreducible_X",
"EuclideanDomain.toCommRing",
"Field.toEuclideanDomain"
]
}
] | exact irreducible_X | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.SplittingField.Construction | {
"line": 59,
"column": 4
} | {
"line": 59,
"column": 23
} | [
{
"pp": "case neg\nK : Type v\ninst✝ : Field K\nf : K[X]\nH : ¬∃ g, Irreducible g ∧ g ∣ f\n⊢ Irreducible X",
"usedConstants": [
"Field.toSemifield",
"instIsDomain",
"Polynomial.irreducible_X",
"EuclideanDomain.toCommRing",
"Field.toEuclideanDomain"
]
}
] | exact irreducible_X | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.SplittingField.Construction | {
"line": 59,
"column": 4
} | {
"line": 59,
"column": 23
} | [
{
"pp": "case neg\nK : Type v\ninst✝ : Field K\nf : K[X]\nH : ¬∃ g, Irreducible g ∧ g ∣ f\n⊢ Irreducible X",
"usedConstants": [
"Field.toSemifield",
"instIsDomain",
"Polynomial.irreducible_X",
"EuclideanDomain.toCommRing",
"Field.toEuclideanDomain"
]
}
] | exact irreducible_X | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Weights.Cartan | {
"line": 343,
"column": 2
} | {
"line": 343,
"column": 63
} | [
{
"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": 177,
"column": 6
} | {
"line": 177,
"column": 69
} | [
{
"pp": "n✝ : ℕ\nK✝ : Type u\ninst✝ : Field K✝\nn : ℕ\nih :\n (fun n ↦\n ∀ {K : Type u} [inst : Field K] (f : K[X]),\n f.natDegree = n → (map (algebraMap K (SplittingFieldAux n f)) f).Splits)\n n\nK : Type u\nx✝ : Field K\nf : K[X]\nhf : f.natDegree = n.succ\n⊢ (map (algebraMap (AdjoinRoot f.fac... | ← 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": 95,
"column": 40
} | {
"line": 95,
"column": 48
} | [
{
"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' :... | lie_lie, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Sl2 | {
"line": 122,
"column": 8
} | {
"line": 124,
"column": 14
} | [
{
"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... | · rw [← lie_skew, t.lie_h_e_nsmul, neg_mem_iff]
apply nsmul_mem <| subset_span _
simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Dynamics.Newton | {
"line": 76,
"column": 37
} | {
"line": 76,
"column": 46
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : R[X]\nx : S\nh : (aeval x) P = 0\n⊢ x - Ring.inverse ((aeval x) (derivative P)) * 0 = x",
"usedConstants": [
"Polynomial.derivative",
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocComm... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.Chain | {
"line": 145,
"column": 27
} | {
"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... | iSup_subtype' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.JordanChevalley | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 61
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nP : K[X]\nk : ℕ\nsep : P.Separable\nnil : minpoly K f ∣ P ^ k\n⊢ ∃ n ∈ K[f], ∃ s ∈ K[f], IsNilpotent n ∧ s.IsSemisimple ∧ f = n + s",
"usedConstants": [
"Subalgebra.instSetLike",
"Com... | set ff : adjoin K {f} := ⟨f, self_mem_adjoin_singleton K f⟩ | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.Algebra.Polynomial.Sequence | {
"line": 147,
"column": 4
} | {
"line": 153,
"column": 95
} | [
{
"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 ... | 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.Algebra.Lie.TraceForm | {
"line": 112,
"column": 93
} | {
"line": 126,
"column": 87
} | [
{
"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.LinearAlgebra.RootSystem.Defs | {
"line": 435,
"column": 57
} | {
"line": 438,
"column": 21
} | [
{
"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
} | [
{
"pp": "case h.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.coro... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 487,
"column": 2
} | {
"line": 487,
"column": 79
} | [
{
"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": 487,
"column": 2
} | {
"line": 487,
"column": 79
} | [
{
"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": 487,
"column": 2
} | {
"line": 487,
"column": 79
} | [
{
"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.Reflection | {
"line": 153,
"column": 2
} | {
"line": 154,
"column": 33
} | [
{
"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.PerfectPairing.Restrict | {
"line": 106,
"column": 14
} | {
"line": 106,
"column": 34
} | [
{
"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
} | [
{
"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": 605,
"column": 65
} | {
"line": 605,
"column": 76
} | [
{
"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... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Span.TensorProduct | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 16
} | [
{
"pp": "case h\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Algebra R A\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module A M\ninst✝ : IsScalarTower R A M\np : Submodule R M\nv : ↥(span A ↑p)\nf : ↑↑p →₀ A\nhf : (Finsupp.linearCombination A... | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Ideal.Quotient.PowTransition | {
"line": 171,
"column": 2
} | {
"line": 171,
"column": 52
} | [
{
"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",
"usedConstants": [
... | 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": 366,
"column": 71
} | {
"line": 366,
"column": 79
} | [
{
"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... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Reflection | {
"line": 368,
"column": 19
} | {
"line": 368,
"column": 33
} | [
{
"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... | 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
} | [
{
"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... | simp only [Φ'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Valuation.ValuationRing | {
"line": 287,
"column": 4
} | {
"line": 287,
"column": 36
} | [
{
"pp": "case neg.h.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",
"usedConstants": [
"CommSemiring.toSemiring",
"Ideal.instIsTwoSided_1",
"CommRin... | apply Ideal.mul_mem_right _ _ hb | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Valuation.ValuationRing | {
"line": 347,
"column": 96
} | {
"line": 368,
"column": 25
} | [
{
"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",
"usedConstants": [
"IsLocalization.IsInteger",
"mul_di... | 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.Polynomial.Chebyshev | {
"line": 1007,
"column": 47
} | {
"line": 1007,
"column": 56
} | [
{
"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)^... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 1014,
"column": 47
} | {
"line": 1014,
"column": 56
} | [
{
"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... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Valuation.ValuationRing | {
"line": 390,
"column": 2
} | {
"line": 390,
"column": 42
} | [
{
"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",
"usedConstants": [
"Iff.mpr",
"Dvd.dvd",
"CommRing.toNonUnitalCommRing",
... | refine iff_dvd_total.mpr ⟨fun a b => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 53
} | [
{
"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",
"usedConstants": [
"MulOne.toOne",
"Semigroup.toMul",
"HMul.hMul",
"... | exact fun hh => h2 (isUnit_of_dvd_one ⟨_, hh.symm⟩) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.DiscreteValuationRing.TFAE | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 30
} | [
{
"pp": "case neg.zero\nR : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsLocalRing R\ninst✝¹ : IsDomain 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, ma... | have := Nat.find_spec this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 428,
"column": 2
} | {
"line": 437,
"column": 21
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsDiscreteValuationRing R\na : R\n⊢ (addVal R) a = ⊤ ↔ a = 0",
"usedConstants": [
"Prime.irreducible",
"IsDiscreteValuationRing.addVal_zero",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Units.v... | 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": 428,
"column": 2
} | {
"line": 437,
"column": 21
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsDiscreteValuationRing R\na : R\n⊢ (addVal R) a = ⊤ ↔ a = 0",
"usedConstants": [
"Prime.irreducible",
"IsDiscreteValuationRing.addVal_zero",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Units.v... | 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.AdicCompletion.Basic | {
"line": 850,
"column": 2
} | {
"line": 853,
"column": 15
} | [
{
"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
} | [
{
"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.LocalProperties.IntegrallyClosed | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 81
} | [
{
"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",
"usedConstants": [
"IsIntegrallyClosed.of_localization",
"MaximalSpectrum",
... | refine of_localization (.range MaximalSpectrum.toPrimeSpectrum) (fun _ ↦ ?_) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Flat.Tensor | {
"line": 50,
"column": 66
} | {
"line": 50,
"column": 77
} | [
{
"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), ... | 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
} | [
{
"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))",
"usedConstants":... | Surjective, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Valuation.Basic | {
"line": 327,
"column": 2
} | {
"line": 327,
"column": 63
} | [
{
"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",
"usedConstants": [
"Lattice.toSemilatticeSup",
"Ring.toNonAssocRing",
"AddGroupWithOne.toAddGroup",
"Valuation.map_add_of... | 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.LinearAlgebra.RootSystem.RootPositive | {
"line": 68,
"column": 6
} | {
"line": 68,
"column": 23
} | [
{
"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.... | reflection_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.Finite.CanonicalBilinear | {
"line": 158,
"column": 2
} | {
"line": 158,
"column": 22
} | [
{
"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",
... | 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": 274,
"column": 30
} | {
"line": 274,
"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... | rootFormIn_self_smul_coroot | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.Finite.CanonicalBilinear | {
"line": 378,
"column": 2
} | {
"line": 379,
"column": 90
} | [
{
"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... | 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
} | [
{
"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... | 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.Reduced | {
"line": 276,
"column": 2
} | {
"line": 277,
"column": 68
} | [
{
"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
} | [
{
"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
} | [
{
"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
} | [
{
"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
} | [
{
"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... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 162,
"column": 36
} | {
"line": 162,
"column": 50
} | [
{
"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... | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 164,
"column": 19
} | {
"line": 164,
"column": 33
} | [
{
"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... | ← neg_add_rev, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RepresentationTheory.Basic | {
"line": 403,
"column": 4
} | {
"line": 403,
"column": 19
} | [
{
"pp": "case h.h.h\nk : 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 ... | simp [mul_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.RootSystem.Irreducible | {
"line": 131,
"column": 2
} | {
"line": 141,
"column": 70
} | [
{
"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... | · 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": 143,
"column": 4
} | {
"line": 143,
"column": 63
} | [
{
"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... | 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
} | [
{
"pp": "case indexEquiv.H\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.indexEqui... | 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
} | [
{
"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": 504,
"column": 6
} | {
"line": 504,
"column": 49
} | [
{
"pp": "case hr.h\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\... | simp [hf, RootPairing.map, RootPairing.map] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.RootSystem.Hom | {
"line": 504,
"column": 6
} | {
"line": 504,
"column": 49
} | [
{
"pp": "case hr.h\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\... | simp [hf, RootPairing.map, RootPairing.map] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.Hom | {
"line": 504,
"column": 6
} | {
"line": 504,
"column": 49
} | [
{
"pp": "case hr.h\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\... | 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
} | [
{
"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
} | [
{
"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\... | 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.Base | {
"line": 448,
"column": 52
} | {
"line": 448,
"column": 63
} | [
{
"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.... | smul_assoc, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 190,
"column": 2
} | {
"line": 194,
"column": 69
} | [
{
"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
} | [
{
"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.BaseExists | {
"line": 233,
"column": 2
} | {
"line": 243,
"column": 77
} | [
{
"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... | 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.LinearAlgebra.RootSystem.BaseExists | {
"line": 318,
"column": 30
} | {
"line": 318,
"column": 47
} | [
{
"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... | simpa using hsp i | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 318,
"column": 30
} | {
"line": 318,
"column": 47
} | [
{
"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... | simpa using hsp i | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 318,
"column": 30
} | {
"line": 318,
"column": 47
} | [
{
"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... | simpa using hsp i | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Charpoly.ToMatrix | {
"line": 79,
"column": 41
} | {
"line": 79,
"column": 91
} | [
{
"pp": "R : Type u_1\nM₁ : Type u_3\nM₂ : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M₁\ninst✝⁶ : Module R M₁\ninst✝⁵ : Module.Finite R M₁\ninst✝⁴ : Free R M₁\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : Module.Finite R M₂\ninst✝ : Free R M₂\ne : M₁ ≃ₗ[R] M₂\nφ : End R M₁\nb : Basis (Cho... | ← LinearMap.charpoly_toMatrix (e.conj φ) (b.map e) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 350,
"column": 2
} | {
"line": 350,
"column": 43
} | [
{
"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... | 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.RingTheory.MvPolynomial.WeightedHomogeneous | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 24
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nm : M\na b : MvPolynomial σ R\nha : a ∈ {x | IsWeightedHomogeneous w x m}\nhb : b ∈ {x | IsWeightedHomogeneous w x m}\nc : σ →₀ ℕ\nhc : coeff c (a + b) ≠ 0\n⊢ (weight w) c = m",
"usedConstants": [... | rw [coeff_add] at hc | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous | {
"line": 309,
"column": 54
} | {
"line": 309,
"column": 69
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nm : M\nmotive : (p : MvPolynomial σ R) → IsWeightedHomogeneous w p m → Prop\nzero : motive 0 ⋯\nadd :\n ∀ (p q : MvPolynomial σ R) (hp : IsWeightedHomogeneous w p m) (hq : IsWeightedHomogeneous w q m... | simpa using h 1 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous | {
"line": 309,
"column": 54
} | {
"line": 309,
"column": 69
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nm : M\nmotive : (p : MvPolynomial σ R) → IsWeightedHomogeneous w p m → Prop\nzero : motive 0 ⋯\nadd :\n ∀ (p q : MvPolynomial σ R) (hp : IsWeightedHomogeneous w p m) (hq : IsWeightedHomogeneous w q m... | simpa using h 1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous | {
"line": 309,
"column": 54
} | {
"line": 309,
"column": 69
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nm : M\nmotive : (p : MvPolynomial σ R) → IsWeightedHomogeneous w p m → Prop\nzero : motive 0 ⋯\nadd :\n ∀ (p q : MvPolynomial σ R) (hp : IsWeightedHomogeneous w p m) (hq : IsWeightedHomogeneous w q m... | simpa using h 1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.Homogeneous | {
"line": 101,
"column": 4
} | {
"line": 101,
"column": 24
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝ : CommSemiring R\nn : ℕ\na b : MvPolynomial σ R\nha : a ∈ {x | x.IsHomogeneous n}\nhb : b ∈ {x | x.IsHomogeneous n}\nc : σ →₀ ℕ\nhc : coeff c (a + b) ≠ 0\n⊢ (weight 1) c = n",
"usedConstants": [
"Nat.instMulZeroClass",
"congr... | rw [coeff_add] at hc | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous | {
"line": 593,
"column": 2
} | {
"line": 599,
"column": 37
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommSemiring R\nσ : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : LinearOrder M\ninst✝¹ : OrderBot M\ninst✝ : CanonicallyOrderedAdd M\nw : σ → M\nhw : NonTorsionWeight w\np : MvPolynomial σ R\n⊢ weightedTotalDegree w p = 0 ↔ ∀ m ∈ p.support, ∀ (x : σ), m x = 0",
... | rw [← isWeightedHomogeneous_zero_iff_weightedTotalDegree_eq_zero, IsWeightedHomogeneous]
apply forall_congr'
intro m
rw [mem_support_iff]
apply forall_congr'
intro _
exact weightedDegree_eq_zero_iff hw | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous | {
"line": 593,
"column": 2
} | {
"line": 599,
"column": 37
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommSemiring R\nσ : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : LinearOrder M\ninst✝¹ : OrderBot M\ninst✝ : CanonicallyOrderedAdd M\nw : σ → M\nhw : NonTorsionWeight w\np : MvPolynomial σ R\n⊢ weightedTotalDegree w p = 0 ↔ ∀ m ∈ p.support, ∀ (x : σ), m x = 0",
... | rw [← isWeightedHomogeneous_zero_iff_weightedTotalDegree_eq_zero, IsWeightedHomogeneous]
apply forall_congr'
intro m
rw [mem_support_iff]
apply forall_congr'
intro _
exact weightedDegree_eq_zero_iff hw | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finsupp.Weight | {
"line": 309,
"column": 2
} | {
"line": 309,
"column": 60
} | [
{
"pp": "case succ\nσ : Type u_5\ns : Set ℕ\nn : ℕ\nhn : n + 1 ≠ 0\n⊢ ⇑degree ⁻¹' ((n + 1) • s) = (n + 1) • ⇑degree ⁻¹' s",
"usedConstants": [
"Finsupp.instAddZeroClass",
"False",
"Nat.instMulZeroClass",
"Nat.recAux",
"instHSMul",
"Nat.instOne",
"congrArg",
"N... | induction n <;> simp_all [succ_nsmul, degree_preimage_add] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.LinearAlgebra.Eigenspace.Zero | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 31
} | [
{
"pp": "K : Type u_2\nM : Type u_3\ninst✝³ : Field K\ninst✝² : AddCommGroup M\ninst✝¹ : Module K M\ninst✝ : Module.Finite K M\nφ : End K M\n⊢ finrank K ↥(φ.maxGenEigenspace 0) = (charpoly φ).natTrailingDegree",
"usedConstants": [
"Submodule",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | set V := φ.maxGenEigenspace 0 | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.Algebra.Module.LinearMap.Polynomial | {
"line": 267,
"column": 6
} | {
"line": 268,
"column": 37
} | [
{
"pp": "case h₀\nR : Type u_1\nL : Type u_2\nM : Type u_3\nι : Type u_5\nιM : Type u_7\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup L\ninst✝⁸ : Module R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nφ : L →ₗ[R] End R M\ninst✝⁵ : Fintype ι\ninst✝⁴ : Fintype ιM\ninst✝³ : DecidableEq ι\ninst✝² : DecidableEq ιM... | · rintro kl - H
rw [this, if_neg H, map_zero] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Module.LinearMap.Polynomial | {
"line": 451,
"column": 24
} | {
"line": 451,
"column": 43
} | [
{
"pp": "case a\nR : Type u_1\nL : Type u_2\nM : Type u_3\nι : Type u_5\nι' : Type u_6\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : AddCommGroup L\ninst✝⁹ : Module R L\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\nφ : L →ₗ[R] End R M\ninst✝⁶ : Fintype ι\ninst✝⁵ : Fintype ι'\ninst✝⁴ : DecidableEq ι\ninst✝³ : DecidableEq ι'... | apply nilRankAux_le | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Module.LinearMap.Polynomial | {
"line": 451,
"column": 24
} | {
"line": 451,
"column": 43
} | [
{
"pp": "case a\nR : Type u_1\nL : Type u_2\nM : Type u_3\nι : Type u_5\nι' : Type u_6\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : AddCommGroup L\ninst✝⁹ : Module R L\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\nφ : L →ₗ[R] End R M\ninst✝⁶ : Fintype ι\ninst✝⁵ : Fintype ι'\ninst✝⁴ : DecidableEq ι\ninst✝³ : DecidableEq ι'... | apply nilRankAux_le | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
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