module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.Valuation.ValuationRing | {
"line": 243,
"column": 34
} | {
"line": 243,
"column": 74
} | {
"line": 243,
"column": 75
} | [
{
"pp": "A : Type u\ninst✝⁵ : CommRing A\nK : Type v\ninst✝⁴ : Field K\ninst✝³ : Algebra A K\ninst✝² : IsDomain A\ninst✝¹ : ValuationRing A\ninst✝ : IsFractionRing A K\nx✝¹ x✝ : A\n⊢ ⟨(algebraMap A K) (x✝¹ + x✝), ⋯⟩ = ⟨(algebraMap A K) x✝¹, ⋯⟩ + ⟨(algebraMap A K) x✝, ⋯⟩",
"ppTerm": "?m.107",
"assigned":... | [] | ext1; exact (algebraMap A K).map_add _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Valuation.ValuationRing | {
"line": 243,
"column": 34
} | {
"line": 243,
"column": 74
} | {
"line": 243,
"column": 75
} | [
{
"pp": "A : Type u\ninst✝⁵ : CommRing A\nK : Type v\ninst✝⁴ : Field K\ninst✝³ : Algebra A K\ninst✝² : IsDomain A\ninst✝¹ : ValuationRing A\ninst✝ : IsFractionRing A K\nx✝¹ x✝ : A\n⊢ ⟨(algebraMap A K) (x✝¹ + x✝), ⋯⟩ = ⟨(algebraMap A K) x✝¹, ⋯⟩ + ⟨(algebraMap A K) x✝, ⋯⟩",
"ppTerm": "?m.107",
"assigned":... | [] | ext1; exact (algebraMap A K).map_add _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Valuation.Basic | {
"line": 318,
"column": 4
} | {
"line": 318,
"column": 9
} | {
"line": 319,
"column": 2
} | [
{
"pp": "case inl\nR : Type u_3\nΓ₀ : Type u_4\ninst✝² : Ring R\ninst✝¹ : LinearOrderedCommMonoidWithZero Γ₀\nv : Valuation R Γ₀\nι : Type u_7\ninst✝ : DecidableEq ι\ns : Finset ι\nf : ι → R\nj : ι\nhj : j ∈ s\nhf : ∀ i ∈ s \\ {j}, v (f i) < v (f j)\nh0 : v (f j) = 0\n⊢ v (∑ i ∈ s, f i) = v (f j)",
"ppTerm"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Valuation.Basic | {
"line": 318,
"column": 4
} | {
"line": 318,
"column": 9
} | {
"line": 319,
"column": 2
} | [
{
"pp": "case inl\nR : Type u_3\nΓ₀ : Type u_4\ninst✝² : Ring R\ninst✝¹ : LinearOrderedCommMonoidWithZero Γ₀\nv : Valuation R Γ₀\nι : Type u_7\ninst✝ : DecidableEq ι\ns : Finset ι\nf : ι → R\nj : ι\nhj : j ∈ s\nhf : ∀ i ∈ s \\ {j}, v (f i) < v (f j)\nh0 : v (f j) = 0\n⊢ v (∑ i ∈ s, f i) = v (f j)",
"ppTerm"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Valuation.Basic | {
"line": 318,
"column": 4
} | {
"line": 318,
"column": 9
} | {
"line": 319,
"column": 2
} | [
{
"pp": "case inl\nR : Type u_3\nΓ₀ : Type u_4\ninst✝² : Ring R\ninst✝¹ : LinearOrderedCommMonoidWithZero Γ₀\nv : Valuation R Γ₀\nι : Type u_7\ninst✝ : DecidableEq ι\ns : Finset ι\nf : ι → R\nj : ι\nhj : j ∈ s\nhf : ∀ i ∈ s \\ {j}, v (f i) < v (f j)\nh0 : v (f j) = 0\n⊢ v (∑ i ∈ s, f i) = v (f j)",
"ppTerm"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 294,
"column": 8
} | {
"line": 294,
"column": 21
} | {
"line": 294,
"column": 22
} | [
{
"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... | ← φ'.map_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Valuation.Basic | {
"line": 530,
"column": 2
} | {
"line": 530,
"column": 7
} | {
"line": 532,
"column": 0
} | [
{
"pp": "R : Type u_3\nΓ₀ : Type u_4\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx y : R\n⊢ ((if h : v x = 0 then 0 else ↑⟨Units.mk0 (v x) ⋯, ⋯⟩) = if h : v y = 0 then 0 else ↑⟨Units.mk0 (v y) ⋯, ⋯⟩) ↔\n v x = v y",
"ppTerm": "?m.18",
"assigned": true,
"usedCo... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Valuation.Basic | {
"line": 538,
"column": 4
} | {
"line": 538,
"column": 9
} | {
"line": 539,
"column": 2
} | [
{
"pp": "R : Type u_3\nΓ₀ : Type u_4\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nγ : (ofClass v).ValueGroup₀ˣ\nu : ↥(ofClass v).valueGroup := WithZero.unzero ⋯\nhu_def : u = WithZero.unzero ⋯\na : R\nha : (ofClass v) a ≠ 0\nx : R\nhax : (ofClass v) a * ↑↑u = (ofClass v) x\n⊢... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 397,
"column": 4
} | {
"line": 397,
"column": 20
} | {
"line": 398,
"column": 2
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsDiscreteValuationRing R\nϖ : R\nhirr : Irreducible ϖ\nu v : Rˣ\nm : ℕ\nh : ↑u = ↑v\n⊢ u = v",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Units.val",
"CommSemiring.toSemiring",
"cast",
... | [] | exact mod_cast h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 414,
"column": 4
} | {
"line": 414,
"column": 77
} | {
"line": 415,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsDiscreteValuationRing R\nr : R\nu : Rˣ\nϖ : R\nhϖ : Irreducible ϖ\nn : ℕ\nhr : r = ↑u * ϖ ^ n\n⊢ emultiplicity ϖ (↑u * ϖ ^ n) = ↑n",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toN... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsDiscreteValuationRing R\nr : R\nu : Rˣ\nϖ : R\nhϖ : Irreducible ϖ\nn : ℕ\nhr : r = ↑u * ϖ ^ n\n⊢ emultiplicity ϖ (ϖ ^ n) = ↑n"
] | emultiplicity_eq_of_associated_right (Associated.symm ⟨u, mul_comm _ _⟩), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.KrullDimension.Zero | {
"line": 178,
"column": 2
} | {
"line": 178,
"column": 99
} | {
"line": 179,
"column": 2
} | [
{
"pp": "R : Type u_2\ninst✝² : CommRing R\ninst✝¹ : IsReduced R\ninst✝ : Nontrivial R\nH : Subsingleton (PrimeSpectrum R)\n⊢ IsField R",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Function.Injective.subsingleton",
"MaximalSpectrum",
"MaximalSpectrum.toPrimeSpectrum",... | [
"R : Type u_2\ninst✝² : CommRing R\ninst✝¹ : IsReduced R\ninst✝ : Nontrivial R\nH : Subsingleton (PrimeSpectrum R)\nthis : Subsingleton (MaximalSpectrum R)\n⊢ IsField R"
] | have : Subsingleton (MaximalSpectrum R) := MaximalSpectrum.toPrimeSpectrum_injective.subsingleton | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Valuation.Basic | {
"line": 653,
"column": 74
} | {
"line": 653,
"column": 79
} | {
"line": 653,
"column": 79
} | [
{
"pp": "K : Type u_7\ninst✝¹ : DivisionRing K\nΓ₀ : Type u_8\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation K Γ₀\nx✝ : ∃ x, 1 < v x\nx : K\nhx1 : 1 < v x\n⊢ v x ≠ 0 ∧ v x ≠ 1",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"not_lt_zero._simp_1",
"Eq.mpr",
"Gro... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Valuation.Basic | {
"line": 653,
"column": 74
} | {
"line": 653,
"column": 79
} | {
"line": 653,
"column": 79
} | [
{
"pp": "K : Type u_7\ninst✝¹ : DivisionRing K\nΓ₀ : Type u_8\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation K Γ₀\nx✝ : ∃ x, 1 < v x\nx : K\nhx1 : 1 < v x\n⊢ v x ≠ 0 ∧ v x ≠ 1",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"not_lt_zero._simp_1",
"Eq.mpr",
"Gro... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Valuation.Basic | {
"line": 653,
"column": 74
} | {
"line": 653,
"column": 79
} | {
"line": 653,
"column": 79
} | [
{
"pp": "K : Type u_7\ninst✝¹ : DivisionRing K\nΓ₀ : Type u_8\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation K Γ₀\nx✝ : ∃ x, 1 < v x\nx : K\nhx1 : 1 < v x\n⊢ v x ≠ 0 ∧ v x ≠ 1",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"not_lt_zero._simp_1",
"Eq.mpr",
"Gro... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.Dvr | {
"line": 76,
"column": 2
} | {
"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\np : Ideal Rₘ\nhp0 : p ≠ ⊥\nhpp : p.IsPrime\nP : Ideal Rₘ\nhpP : p < P\nhPm : P.IsMaximal\nhpP' : (IsLocali... | [] | exact IsLocalization.bot_lt_under_prime _ _ hM _ hp0 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Flat.Tensor | {
"line": 62,
"column": 26
} | {
"line": 62,
"column": 78
} | {
"line": 62,
"column": 78
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\n⊢ Baer R (CharacterModule (ULift.{u, v} M)) ↔ Baer R (CharacterModule M)",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ULift.addCommGroup",
"congrArg",
"C... | [
"R : Type u\nM : Type v\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\n⊢ Baer R (CharacterModule M) ↔ Baer R (CharacterModule M)"
] | Baer.congr (CharacterModule.congr ULift.moduleEquiv) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 364,
"column": 2
} | {
"line": 364,
"column": 67
} | {
"line": 366,
"column": 0
} | [
{
"pp": "S : Type u_2\ninst✝¹⁰ : CommRing S\ninst✝⁹ : IsDomain S\nR : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsJacobsonRing R\nRₘ : Type u_6\nSₘ : Type u_7\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nx : R\nhx : x ≠ 0\ninst✝³ : Algebra R Rₘ\ninst✝²... | [] | rwa [under_def, comap_comap, hcomm, ← bot_quotient_isMaximal_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.RingTheory.Valuation.Basic | {
"line": 855,
"column": 6
} | {
"line": 855,
"column": 53
} | {
"line": 856,
"column": 6
} | [
{
"pp": "case neg\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝⁴ : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝³ : LinearOrderedCommGroupWithZero Γ₀\ninst✝² : LinearOrderedCommGroupWithZero Γ'₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ''₀\ninst✝ : Ring R\nv : Valuation R Γ₀\nw : Val... | [
"case neg\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝⁴ : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝³ : LinearOrderedCommGroupWithZero Γ₀\ninst✝² : LinearOrderedCommGroupWithZero Γ'₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ''₀\ninst✝ : Ring R\nv : Valuation R Γ₀\nw : Valuation R Γ'₀... | generalize_proofs _ hx' hx20 hy' hy10 hx10 hy20 | Batteries.Tactic._aux_Batteries_Tactic_GeneralizeProofs___elabRules_Batteries_Tactic_generalizeProofsElab_1 | Batteries.Tactic.generalizeProofsElab |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 382,
"column": 4
} | {
"line": 382,
"column": 59
} | {
"line": 383,
"column": 4
} | [
{
"pp": "case neg\nR : Type u_5\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\nhR : IsJacobsonRing R\nP : Ideal R[X]\ninst✝ : P.IsPrime\nhP : ∀ (x : R), C x ∈ P → x = 0\nPb : ¬P = ⊥\nP' : Ideal R := comap C P\nthis : P'.IsPrime\nhR' : IsJacobsonRing (R ⧸ P')\np : R[X]\npP : p ∈ P\np0 : map (Ideal.Quotient.mk (comap... | [
"case neg\nR : Type u_5\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\nhR : IsJacobsonRing R\nP : Ideal R[X]\ninst✝ : P.IsPrime\nhP : ∀ (x : R), C x ∈ P → x = 0\nPb : ¬P = ⊥\nP' : Ideal R := comap C P\nthis : P'.IsPrime\nhR' : IsJacobsonRing (R ⧸ P')\np : R[X]\npP : p ∈ P\np0 : map (Ideal.Quotient.mk (comap C P)) p ≠ 0... | let hφ : Function.Injective ↑φ := quotientMap_injective | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Ideal.IsPrincipal | {
"line": 44,
"column": 4
} | {
"line": 45,
"column": 56
} | {
"line": 46,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\n⊢ ∀ {a b : Ideal R}, a ∈ {I | IsPrincipal I} → b ∈ {I | IsPrincipal I} → a * b ∈ {I | IsPrincipal I}",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Submodule",
"Semiring.toModule",
"HMul.hMul",
"CommSemiring.toSemiring",
... | [] | rintro _ _ ⟨x, rfl⟩ ⟨y, rfl⟩
exact ⟨x * y, span_singleton_mul_span_singleton x y⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.IsPrincipal | {
"line": 44,
"column": 4
} | {
"line": 45,
"column": 56
} | {
"line": 46,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\n⊢ ∀ {a b : Ideal R}, a ∈ {I | IsPrincipal I} → b ∈ {I | IsPrincipal I} → a * b ∈ {I | IsPrincipal I}",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Submodule",
"Semiring.toModule",
"HMul.hMul",
"CommSemiring.toSemiring",
... | [] | rintro _ _ ⟨x, rfl⟩ ⟨y, rfl⟩
exact ⟨x * y, span_singleton_mul_span_singleton x y⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Flat.TorsionFree | {
"line": 59,
"column": 2
} | {
"line": 61,
"column": 57
} | {
"line": 63,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nr : R\nhr : IsRegular r\ninst✝ : Flat R M\nh : Function.Injective ⇑(rTensor M (toSpanSingleton R R r))\n⊢ IsSMulRegular M r",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"No... | [
"R : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nr : R\nhr : IsRegular r\ninst✝ : Flat R M\nh : Function.Injective ⇑(rTensor M (toSpanSingleton R R r))\nh2 : (fun x ↦ r • x) = ⇑(↑(TensorProduct.lid R M) ∘ₗ rTensor M (toSpanSingleton R R r) ∘ₗ ↑(TensorProduct.lid R... | have h2 : (fun (x : M) ↦ r • x) = ((TensorProduct.lid R M) ∘ₗ
(rTensor M (toSpanSingleton R R r)) ∘ₗ
(TensorProduct.lid R M).symm) := by ext; simp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Flat.TorsionFree | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 33
} | {
"line": 66,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nr : R\nhr : IsRegular r\ninst✝ : Flat R M\nh : Function.Injective ⇑(rTensor M (toSpanSingleton R R r))\nh2 : (fun x ↦ r • x) = ⇑(↑(TensorProduct.lid R M) ∘ₗ rTensor M (toSpanSingleton R R r) ∘ₗ ↑(TensorP... | [] | simp [h, LinearEquiv.injective] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 628,
"column": 4
} | {
"line": 630,
"column": 30
} | {
"line": 631,
"column": 2
} | [
{
"pp": "case zero\nn : ℕ\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobsonRing R\nP : Ideal (MvPolynomial (Fin 0) R)\nhP : P.IsMaximal\n⊢ (algebraMap R (MvPolynomial (Fin 0) R ⧸ P)).IsIntegral",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"AddMon... | [] | apply RingHom.isIntegral_of_surjective
apply Function.Surjective.comp Quotient.mk_surjective
exact C_surjective (Fin 0) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 628,
"column": 4
} | {
"line": 630,
"column": 30
} | {
"line": 631,
"column": 2
} | [
{
"pp": "case zero\nn : ℕ\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobsonRing R\nP : Ideal (MvPolynomial (Fin 0) R)\nhP : P.IsMaximal\n⊢ (algebraMap R (MvPolynomial (Fin 0) R ⧸ P)).IsIntegral",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"AddMon... | [] | apply RingHom.isIntegral_of_surjective
apply Function.Surjective.comp Quotient.mk_surjective
exact C_surjective (Fin 0) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.SumsOfSquares | {
"line": 96,
"column": 94
} | {
"line": 96,
"column": 99
} | {
"line": 98,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : AddZeroClass R\ninst✝ : Mul R\nx : R\nhx : IsSquare x\n⊢ IsSumSq x",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"IsSquare.exists_mul_self",
"Exists",
"AddZeroClass.toAddZero",
"AddZero.toZero",
"Exists.cases... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Ring.SumsOfSquares | {
"line": 96,
"column": 94
} | {
"line": 96,
"column": 99
} | {
"line": 98,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : AddZeroClass R\ninst✝ : Mul R\nx : R\nhx : IsSquare x\n⊢ IsSumSq x",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"IsSquare.exists_mul_self",
"Exists",
"AddZeroClass.toAddZero",
"AddZero.toZero",
"Exists.cases... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Ring.SumsOfSquares | {
"line": 96,
"column": 94
} | {
"line": 96,
"column": 99
} | {
"line": 98,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : AddZeroClass R\ninst✝ : Mul R\nx : R\nhx : IsSquare x\n⊢ IsSumSq x",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"IsSquare.exists_mul_self",
"Exists",
"AddZeroClass.toAddZero",
"AddZero.toZero",
"Exists.cases... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.SumsOfSquares | {
"line": 101,
"column": 77
} | {
"line": 101,
"column": 82
} | {
"line": 103,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : AddZeroClass R\ninst✝ : MulOneClass R\n⊢ IsSumSq 1",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"MulOne.toMul",
"IsSquare.isSumSq",
"MulOneClass.toMulOne",
"IsSquare",
"of_eq_true",
"One.toOfNat1",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Ring.SumsOfSquares | {
"line": 101,
"column": 77
} | {
"line": 101,
"column": 82
} | {
"line": 103,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : AddZeroClass R\ninst✝ : MulOneClass R\n⊢ IsSumSq 1",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"MulOne.toMul",
"IsSquare.isSumSq",
"MulOneClass.toMulOne",
"IsSquare",
"of_eq_true",
"One.toOfNat1",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Ring.SumsOfSquares | {
"line": 101,
"column": 77
} | {
"line": 101,
"column": 82
} | {
"line": 103,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : AddZeroClass R\ninst✝ : MulOneClass R\n⊢ IsSumSq 1",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"MulOne.toMul",
"IsSquare.isSumSq",
"MulOneClass.toMulOne",
"IsSquare",
"of_eq_true",
"One.toOfNat1",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.SumsOfSquares | {
"line": 111,
"column": 19
} | {
"line": 111,
"column": 24
} | {
"line": 113,
"column": 0
} | [
{
"pp": "case zero\nR : Type u_1\ninst✝¹ : AddMonoid R\ninst✝ : Mul R\nx : R\n⊢ 0 ∈ closure {x | IsSquare x}",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"AddSubmonoid.instAddSubmonoidClass",
"AddMonoid.toAddZeroClass",
"setOf",
"Membership.mem",
"AddZeroCl... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Ring.SumsOfSquares | {
"line": 111,
"column": 19
} | {
"line": 111,
"column": 24
} | {
"line": 113,
"column": 0
} | [
{
"pp": "case sq_add\nR : Type u_1\ninst✝¹ : AddMonoid R\ninst✝ : Mul R\nx a✝ s✝ : R\nhs✝ : IsSumSq s✝\nhs_ih✝ : s✝ ∈ closure {x | IsSquare x}\n⊢ a✝ * a✝ + s✝ ∈ closure {x | IsSquare x}",
"ppTerm": "?sq_add",
"assigned": true,
"usedConstants": [
"AddSubmonoidClass.toAddMemClass",
"AddSub... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Ring.SumsOfSquares | {
"line": 127,
"column": 67
} | {
"line": 127,
"column": 72
} | {
"line": 129,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : AddCommMonoid R\ninst✝ : Mul R\nι : Type u_2\nI : Finset ι\nx : ι → R\nhx : ∀ i ∈ I, IsSquare (x i)\n⊢ IsSumSq (∑ i ∈ I, x i)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"IsSumSq.sum",
"Finset",
"AddMonoid.toAddZeroClass",
"Member... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Ring.SumsOfSquares | {
"line": 127,
"column": 67
} | {
"line": 127,
"column": 72
} | {
"line": 129,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : AddCommMonoid R\ninst✝ : Mul R\nι : Type u_2\nI : Finset ι\nx : ι → R\nhx : ∀ i ∈ I, IsSquare (x i)\n⊢ IsSumSq (∑ i ∈ I, x i)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"IsSumSq.sum",
"Finset",
"AddMonoid.toAddZeroClass",
"Member... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Ring.SumsOfSquares | {
"line": 127,
"column": 67
} | {
"line": 127,
"column": 72
} | {
"line": 129,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : AddCommMonoid R\ninst✝ : Mul R\nι : Type u_2\nI : Finset ι\nx : ι → R\nhx : ∀ i ∈ I, IsSquare (x i)\n⊢ IsSumSq (∑ i ∈ I, x i)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"IsSumSq.sum",
"Finset",
"AddMonoid.toAddZeroClass",
"Member... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.SumsOfSquares | {
"line": 135,
"column": 39
} | {
"line": 135,
"column": 44
} | {
"line": 137,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : AddCommMonoid R\ninst✝ : Mul R\nι : Type u_2\nI : Finset ι\na : ι → R\n⊢ IsSumSq (∑ i ∈ I, a i * a i)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"IsSumSq.sum",
"HMul.hMul",
"Finset",
"AddMonoid.toAddZeroClass",
"Membership.... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Ring.SumsOfSquares | {
"line": 135,
"column": 39
} | {
"line": 135,
"column": 44
} | {
"line": 137,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : AddCommMonoid R\ninst✝ : Mul R\nι : Type u_2\nI : Finset ι\na : ι → R\n⊢ IsSumSq (∑ i ∈ I, a i * a i)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"IsSumSq.sum",
"HMul.hMul",
"Finset",
"AddMonoid.toAddZeroClass",
"Membership.... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Ring.SumsOfSquares | {
"line": 135,
"column": 39
} | {
"line": 135,
"column": 44
} | {
"line": 137,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : AddCommMonoid R\ninst✝ : Mul R\nι : Type u_2\nI : Finset ι\na : ι → R\n⊢ IsSumSq (∑ i ∈ I, a i * a i)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"IsSumSq.sum",
"HMul.hMul",
"Finset",
"AddMonoid.toAddZeroClass",
"Membership.... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.SumsOfSquares | {
"line": 139,
"column": 37
} | {
"line": 139,
"column": 42
} | {
"line": 141,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nι : Type u_2\nI : Finset ι\na : ι → R\n⊢ IsSumSq (∑ i ∈ I, a i ^ 2)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"IsSumSq.sum",
"CommSemiring.toSemiring",
"Finset",
"A... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Ring.SumsOfSquares | {
"line": 139,
"column": 37
} | {
"line": 139,
"column": 42
} | {
"line": 141,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nι : Type u_2\nI : Finset ι\na : ι → R\n⊢ IsSumSq (∑ i ∈ I, a i ^ 2)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"IsSumSq.sum",
"CommSemiring.toSemiring",
"Finset",
"A... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Ring.SumsOfSquares | {
"line": 139,
"column": 37
} | {
"line": 139,
"column": 42
} | {
"line": 141,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nι : Type u_2\nI : Finset ι\na : ι → R\n⊢ IsSumSq (∑ i ∈ I, a i ^ 2)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"IsSumSq.sum",
"CommSemiring.toSemiring",
"Finset",
"A... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.SumsOfSquares | {
"line": 168,
"column": 18
} | {
"line": 168,
"column": 23
} | {
"line": 170,
"column": 0
} | [
{
"pp": "case zero\nR : Type u_2\ninst✝ : NonAssocSemiring R\n⊢ IsSumSq ↑0",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"congrArg",
"AddMonoid.toAddZeroClass",
"NonUnitalNonAssocSemiring.toMulZeroClass",
"IsSumSq.z... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Ring.SumsOfSquares | {
"line": 168,
"column": 18
} | {
"line": 168,
"column": 23
} | {
"line": 170,
"column": 0
} | [
{
"pp": "case succ\nR : Type u_2\ninst✝ : NonAssocSemiring R\nn✝ : ℕ\na✝ : IsSumSq ↑n✝\n⊢ IsSumSq ↑(n✝ + 1)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"AddMonoid.toAddSemigroup",
"congrA... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.RootPositive | {
"line": 66,
"column": 15
} | {
"line": 66,
"column": 34
} | {
"line": 66,
"column": 34
} | [
{
"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.root i)) (P.root j) + (B.form (P.root i)) (P.root j) = P.pairing i ... | [
"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.root j) = P.pairing i j * (B.form (P.root j)) (P.root j) - (B.form ... | ← eq_sub_iff_add_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.Reduced | {
"line": 90,
"column": 4
} | {
"line": 90,
"column": 69
} | {
"line": 91,
"column": 4
} | [
{
"pp": "case inl\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : CharZero R\ninst✝² : IsAddTorsionFree M\ninst✝¹ : P.IsReduced\nn : ℕ\ninst✝ : n.AtLeastTwo... | [
"case inl\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : CharZero R\ninst✝² : IsAddTorsionFree M\ninst✝¹ : P.IsReduced\nn : ℕ\ninst✝ : n.AtLeastTwo\nj : ι\nhj ... | rw [(smul_left_injective ℤ <| P.ne_zero j).eq_iff, eq_comm] at hj | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.RootSystem.Finite.CanonicalBilinear | {
"line": 257,
"column": 4
} | {
"line": 259,
"column": 71
} | {
"line": 260,
"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 : Algebra S R\ninst✝⁶ : FaithfulSMul S R\ninst✝⁵ : Module S M\n... | [] | exact (Fintype.sum_equiv (P.reflectionPerm i)
(fun j ↦ P.pairingIn S i (P.reflectionPerm i j) • P.coroot (P.reflectionPerm i j))
(fun j ↦ P.pairingIn S i j • P.coroot j) (congrFun rfl)).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.RootSystem.Finite.CanonicalBilinear | {
"line": 306,
"column": 85
} | {
"line": 306,
"column": 97
} | {
"line": 307,
"column": 4
} | [
{
"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\ni : ι\n⊢ 2 • 2 • P.CoPolarization (P.Polarization (P.root i)) =\n ((P.RootForm (P.root ... | [
"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\ni : ι\n⊢ 2 • P.CoPolarization (2 • P.Polarization (P.root i)) =\n ((P.RootForm (P.root i)) (P.root ... | ← map_nsmul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.Finite.CanonicalBilinear | {
"line": 333,
"column": 28
} | {
"line": 333,
"column": 33
} | {
"line": 333,
"column": 33
} | [
{
"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\ni : ι\nc : R\nhc : ∏ i, (P.RootForm (P.root i)) (P.root i) = (P.RootForm (P.root i)) (P.ro... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.Finite.CanonicalBilinear | {
"line": 333,
"column": 28
} | {
"line": 333,
"column": 33
} | {
"line": 333,
"column": 33
} | [
{
"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\ni : ι\nc : R\nhc : ∏ i, (P.RootForm (P.root i)) (P.root i) = (P.RootForm (P.root i)) (P.ro... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.Finite.CanonicalBilinear | {
"line": 333,
"column": 28
} | {
"line": 333,
"column": 33
} | {
"line": 333,
"column": 33
} | [
{
"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\ni : ι\nc : R\nhc : ∏ i, (P.RootForm (P.root i)) (P.root i) = (P.RootForm (P.root i)) (P.ro... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Submodule.Union | {
"line": 50,
"column": 4
} | {
"line": 50,
"column": 26
} | {
"line": 51,
"column": 4
} | [
{
"pp": "case insert.inr\nι : 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\n⊢ ↑(p j) ⊆ ⋃... | [
"case insert.inr\nι : 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\n⊢ x... | intro x (hx : x ∈ p j) | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Module.Submodule.Union | {
"line": 54,
"column": 27
} | {
"line": 54,
"column": 32
} | {
"line": 55,
"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 ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.Submodule.Union | {
"line": 66,
"column": 33
} | {
"line": 66,
"column": 38
} | {
"line": 66,
"column": 38
} | [
{
"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 ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.Submodule.Union | {
"line": 66,
"column": 33
} | {
"line": 66,
"column": 38
} | {
"line": 66,
"column": 38
} | [
{
"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 ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Submodule.Union | {
"line": 66,
"column": 33
} | {
"line": 66,
"column": 38
} | {
"line": 66,
"column": 38
} | [
{
"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 ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Submodule.Union | {
"line": 68,
"column": 66
} | {
"line": 68,
"column": 71
} | {
"line": 69,
"column": 6
} | [
{
"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 ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.Submodule.Union | {
"line": 72,
"column": 90
} | {
"line": 72,
"column": 95
} | {
"line": 72,
"column": 95
} | [
{
"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 ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.QuadraticForm.Prod | {
"line": 256,
"column": 56
} | {
"line": 256,
"column": 73
} | {
"line": 256,
"column": 73
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nP : Type u_7\nMᵢ : ι → Type u_8\ninst✝⁶ : CommSemiring R\ninst✝⁵ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝⁴ : AddCommMonoid P\ninst✝³ : (i : ι) → Module R (Mᵢ i)\ninst✝² : Module R P\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nQ : (i : ι) → QuadraticMap R (Mᵢ i) P\ni : ι\nm : ... | [
"ι : Type u_1\nR : Type u_2\nP : Type u_7\nMᵢ : ι → Type u_8\ninst✝⁶ : CommSemiring R\ninst✝⁵ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝⁴ : AddCommMonoid P\ninst✝³ : (i : ι) → Module R (Mᵢ i)\ninst✝² : Module R P\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nQ : (i : ι) → QuadraticMap R (Mᵢ i) P\ni : ι\nm : Mᵢ i\n⊢ (Q i... | Pi.single_eq_same | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.QuadraticForm.Prod | {
"line": 284,
"column": 61
} | {
"line": 284,
"column": 78
} | {
"line": 284,
"column": 78
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM₁ : Type u_3\nM₂ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\nP : Type u_7\nMᵢ : ι → Type u_8\nNᵢ : ι → Type u_9\ninst✝⁸ : CommSemiring R\ninst✝⁷ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝⁶ : (i : ι) → AddCommMonoid (Nᵢ i)\ninst✝⁵ : AddCommMonoid P\ninst✝⁴ : (i : ι) → Module R (M... | [
"ι : Type u_1\nR : Type u_2\nM₁ : Type u_3\nM₂ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\nP : Type u_7\nMᵢ : ι → Type u_8\nNᵢ : ι → Type u_9\ninst✝⁸ : CommSemiring R\ninst✝⁷ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝⁶ : (i : ι) → AddCommMonoid (Nᵢ i)\ninst✝⁵ : AddCommMonoid P\ninst✝⁴ : (i : ι) → Module R (Mᵢ i)\ninst✝³... | Pi.single_eq_same | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.QuadraticForm.Prod | {
"line": 313,
"column": 8
} | {
"line": 313,
"column": 25
} | {
"line": 313,
"column": 25
} | [
{
"pp": "case refine_2\nι : Type u_1\nR : Type u_2\nP : Type u_7\nMᵢ : ι → Type u_8\ninst✝⁵ : CommSemiring R\ninst✝⁴ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝³ : AddCommMonoid P\ninst✝² : (i : ι) → Module R (Mᵢ i)\ninst✝¹ : Module R P\ninst✝ : Fintype ι\nQ : (i : ι) → QuadraticMap R (Mᵢ i) P\nh : ∀ (x : (i : ι) →... | [
"case refine_2\nι : Type u_1\nR : Type u_2\nP : Type u_7\nMᵢ : ι → Type u_8\ninst✝⁵ : CommSemiring R\ninst✝⁴ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝³ : AddCommMonoid P\ninst✝² : (i : ι) → Module R (Mᵢ i)\ninst✝¹ : Module R P\ninst✝ : Fintype ι\nQ : (i : ι) → QuadraticMap R (Mᵢ i) P\nh : ∀ (x : (i : ι) → Mᵢ i), ∑ i,... | Pi.single_eq_same | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.Submodule.Union | {
"line": 100,
"column": 46
} | {
"line": 100,
"column": 51
} | {
"line": 101,
"column": 2
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup M\ninst✝² : Module K M\ninst✝¹ : Finite ι\ninst✝ : Infinite K\nf : ι → Dual K M\np : ι → Submodule K M := fun i ↦ LinearMap.ker (f i)\ni : ι\nh : ∃ x, (f i) x ≠ 0\n⊢ p i ≠ ⊤",
"ppTerm": "?m.47",
"assigned": true,
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.QuadraticForm.Prod | {
"line": 330,
"column": 77
} | {
"line": 330,
"column": 94
} | {
"line": 330,
"column": 94
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nMᵢ : ι → Type u_8\ninst✝⁷ : CommSemiring R\ninst✝⁶ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝⁵ : (i : ι) → Module R (Mᵢ i)\nP : Type u_10\ninst✝⁴ : Fintype ι\ninst✝³ : AddCommMonoid P\ninst✝² : PartialOrder P\ninst✝¹ : IsOrderedAddMonoid P\ninst✝ : Module R P\nQ : (i : ι) → Qua... | [
"ι : Type u_1\nR : Type u_2\nMᵢ : ι → Type u_8\ninst✝⁷ : CommSemiring R\ninst✝⁶ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝⁵ : (i : ι) → Module R (Mᵢ i)\nP : Type u_10\ninst✝⁴ : Fintype ι\ninst✝³ : AddCommMonoid P\ninst✝² : PartialOrder P\ninst✝¹ : IsOrderedAddMonoid P\ninst✝ : Module R P\nQ : (i : ι) → QuadraticMap R ... | Pi.single_eq_same | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 218,
"column": 44
} | {
"line": 218,
"column": 57
} | {
"line": 219,
"column": 4
} | [
{
"pp": "case mpr\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.IsNo... | [
"case mpr\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.IsNonZero\nn : ℤ... | clear_value k | Lean.Elab.Tactic.evalClearValue | Lean.Parser.Tactic.clearValue |
Mathlib.LinearAlgebra.QuadraticForm.Dual | {
"line": 180,
"column": 4
} | {
"line": 180,
"column": 41
} | {
"line": 181,
"column": 2
} | [
{
"pp": "ι : Type u_4\nR : Type u_5\nM : Type u_6\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : LinearMap.BilinForm R M\nhB : (BilinMap.toQuadraticMap B).PosDef\nf : Module.Dual R M\nv : ι → M\nhp : ∀ (i : ι), 0 < f (v i)\nhn : Pai... | [] | · grind only [Finset.disjoint_filter] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RepresentationTheory.Submodule | {
"line": 66,
"column": 27
} | {
"line": 66,
"column": 32
} | {
"line": 68,
"column": 0
} | [
{
"pp": "case hM\nk : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommSemiring k\ninst✝² : Monoid G\ninst✝¹ : AddCommMonoid V\ninst✝ : Module k V\nρ : Representation k G V\np : Submodule k V\nhp : ∀ (g : G), ∀ v ∈ p, (ρ g) v ∈ p\nv : V\nhv : v ∈ p\nx : k[G]\n⊢ ∀ (m : G), (ρ.asAlgebraHom ((MonoidAlgebra.of k ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RepresentationTheory.Submodule | {
"line": 66,
"column": 27
} | {
"line": 66,
"column": 32
} | {
"line": 68,
"column": 0
} | [
{
"pp": "case hadd\nk : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommSemiring k\ninst✝² : Monoid G\ninst✝¹ : AddCommMonoid V\ninst✝ : Module k V\nρ : Representation k G V\np : Submodule k V\nhp : ∀ (g : G), ∀ v ∈ p, (ρ g) v ∈ p\nv : V\nhv : v ∈ p\nx : k[G]\n⊢ ∀ (x y : k[G]), (ρ.asAlgebraHom x) v ∈ p → (ρ.... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RepresentationTheory.Submodule | {
"line": 66,
"column": 27
} | {
"line": 66,
"column": 32
} | {
"line": 68,
"column": 0
} | [
{
"pp": "case hsmul\nk : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommSemiring k\ninst✝² : Monoid G\ninst✝¹ : AddCommMonoid V\ninst✝ : Module k V\nρ : Representation k G V\np : Submodule k V\nhp : ∀ (g : G), ∀ v ∈ p, (ρ g) v ∈ p\nv : V\nhv : v ∈ p\nx : k[G]\n⊢ ∀ (r : k) (x : k[G]), (ρ.asAlgebraHom x) v ∈ ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RepresentationTheory.Basic | {
"line": 274,
"column": 6
} | {
"line": 274,
"column": 11
} | {
"line": 276,
"column": 0
} | [
{
"pp": "case hsmul\nk : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommSemiring k\ninst✝² : Monoid G\ninst✝¹ : AddCommMonoid V\ninst✝ : Module k V\nρ : Representation k G V\nx : k[G]\nv : ρ.asModule\ns : k\ny : k[G]\nhy : ∀ (t : k), (t • y) • v = t • y • v\nt : k\n⊢ t • s • y • v = t • (s • y) • v",
"p... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RepresentationTheory.Basic | {
"line": 267,
"column": 4
} | {
"line": 274,
"column": 11
} | {
"line": 276,
"column": 0
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommSemiring k\ninst✝² : Monoid G\ninst✝¹ : AddCommMonoid V\ninst✝ : Module k V\nρ : Representation k G V\nt : k\nx : k[G]\nv : ρ.asModule\n⊢ (t • x) • v = t • x • v",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"MonoidAlg... | [] | revert t
apply x.induction_on
· simp
· intro y z hy hz
simp [add_smul, hy, hz]
· intro s y hy t
rw [← smul_assoc, smul_eq_mul, hy (t * s), ← smul_eq_mul, smul_assoc]
aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RepresentationTheory.Basic | {
"line": 267,
"column": 4
} | {
"line": 274,
"column": 11
} | {
"line": 276,
"column": 0
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommSemiring k\ninst✝² : Monoid G\ninst✝¹ : AddCommMonoid V\ninst✝ : Module k V\nρ : Representation k G V\nt : k\nx : k[G]\nv : ρ.asModule\n⊢ (t • x) • v = t • x • v",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"MonoidAlg... | [] | revert t
apply x.induction_on
· simp
· intro y z hy hz
simp [add_smul, hy, hz]
· intro s y hy t
rw [← smul_assoc, smul_eq_mul, hy (t * s), ← smul_eq_mul, smul_assoc]
aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RepresentationTheory.Submodule | {
"line": 85,
"column": 25
} | {
"line": 85,
"column": 30
} | {
"line": 86,
"column": 2
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommSemiring k\ninst✝² : Monoid G\ninst✝¹ : AddCommMonoid V\ninst✝ : Module k V\nρ : Representation k G V\nq : Submodule k[G] ρ.asModule\nx✝ : ρ.asModule\n⊢ x✝ ∈\n (fun p ↦\n let __AddSubmonoid := AddSubmonoid.map ρ.asModuleEquiv.symm (↑p... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.Irreducible | {
"line": 70,
"column": 47
} | {
"line": 70,
"column": 52
} | {
"line": 71,
"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✝ : P.IsRootSystem\nx : M\nh : ∀ (i : ι), (P.coroot' i) x = 0\n⊢ x ∈ ⨅ i, ker (P.coroot' i)",
"ppTerm... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RepresentationTheory.Submodule | {
"line": 91,
"column": 20
} | {
"line": 91,
"column": 25
} | {
"line": 91,
"column": 25
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommSemiring k\ninst✝² : Monoid G\ninst✝¹ : AddCommMonoid V\ninst✝ : Module k V\nρ : Representation k G V\np q : ↥ρ.invtSubmodule\nh : p ≤ q\nx : ρ.asModule\nhx :\n x ∈\n {\n toFun := fun p ↦\n let __AddSubmonoid := AddSubmonoid.map... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RepresentationTheory.Submodule | {
"line": 91,
"column": 20
} | {
"line": 91,
"column": 25
} | {
"line": 91,
"column": 25
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommSemiring k\ninst✝² : Monoid G\ninst✝¹ : AddCommMonoid V\ninst✝ : Module k V\nρ : Representation k G V\np q : ↥ρ.invtSubmodule\nh : p ≤ q\nx : ρ.asModule\nhx :\n x ∈\n {\n toFun := fun p ↦\n let __AddSubmonoid := AddSubmonoid.map... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RepresentationTheory.Submodule | {
"line": 91,
"column": 20
} | {
"line": 91,
"column": 25
} | {
"line": 91,
"column": 25
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommSemiring k\ninst✝² : Monoid G\ninst✝¹ : AddCommMonoid V\ninst✝ : Module k V\nρ : Representation k G V\np q : ↥ρ.invtSubmodule\nh : p ≤ q\nx : ρ.asModule\nhx :\n x ∈\n {\n toFun := fun p ↦\n let __AddSubmonoid := AddSubmonoid.map... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.Hom | {
"line": 431,
"column": 4
} | {
"line": 431,
"column": 27
} | {
"line": 432,
"column": 4
} | [
{
"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\nι₂✝ : Type u_5\nM₂✝ : Type u_6\nN₂✝ : Type u_7\ninst✝⁷ : AddCommGroup M₂✝\ninst✝⁶ : Module R M₂✝\ninst✝⁵ : AddCommGroup N₂✝\ninst✝⁴... | [
"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\nι₂✝ : Type u_5\nM₂✝ : Type u_6\nN₂✝ : Type u_7\ninst✝⁷ : AddCommGroup M₂✝\ninst✝⁶ : Module R M₂✝\ninst✝⁵ : AddCommGroup N₂✝\ninst✝⁴ : Module R ... | rw [funext_iff] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.RootSystem.Hom | {
"line": 439,
"column": 4
} | {
"line": 439,
"column": 27
} | {
"line": 440,
"column": 4
} | [
{
"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\nι₂✝ : Type u_5\nM₂✝ : Type u_6\nN₂✝ : Type u_7\ninst✝⁷ : AddCommGroup M₂✝\ninst✝⁶ : Module R M₂✝\ninst✝⁵ : AddCommGroup N₂✝\ninst✝⁴... | [
"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\nι₂✝ : Type u_5\nM₂✝ : Type u_6\nN₂✝ : Type u_7\ninst✝⁷ : AddCommGroup M₂✝\ninst✝⁶ : Module R M₂✝\ninst✝⁵ : AddCommGroup N₂✝\ninst✝⁴ : Module R ... | rw [funext_iff] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 7
} | {
"line": 115,
"column": 0
} | [
{
"pp": "case inr.inr\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\ninst✝⁴ : Finite ι\ninst✝³ : CharZero R\ninst✝² : P.IsCrystallographic\ni j : ι\ninst✝¹ : IsDoma... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 9
} | {
"line": 132,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : CharZero R\ninst✝¹ : P.IsCrystallographic\ninst✝ : IsDomain R\nB : RootPositiveFo... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 147,
"column": 4
} | {
"line": 147,
"column": 9
} | {
"line": 148,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : CharZero R\ninst✝¹ : P.IsCrystallographic\ninst✝ : IsDomain R\nB : RootPositiveFo... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 151,
"column": 4
} | {
"line": 151,
"column": 9
} | {
"line": 152,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : CharZero R\ninst✝¹ : P.IsCrystallographic\ninst✝ : IsDomain R\nB : RootPositiveFo... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 7
} | {
"line": 168,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : CharZero R\ninst✝¹ : P.IsCrystallographic\ni j : ι\ninst✝ : IsDomain R\nB : P.Inv... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 165,
"column": 6
} | {
"line": 165,
"column": 38
} | {
"line": 165,
"column": 38
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝² : Finite ι\ninst✝¹ : IsAddTorsionFree M\ninst✝ : IsAddTorsionFree N\ni : ι\nh : i ∈ b.supp... | [] | coroot_eq_smul_coroot_iff.mpr hj | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 216,
"column": 6
} | {
"line": 216,
"column": 11
} | {
"line": 217,
"column": 4
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : CharZero R\ninst✝¹ : P.IsCrystallographic\ni j : ι\ninst✝ : IsDomain R\nh : 0 < P... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 289,
"column": 2
} | {
"line": 289,
"column": 7
} | {
"line": 291,
"column": 0
} | [
{
"pp": "ι : 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\ninst✝⁴ : Finite ι\ninst✝³ : CharZero R\ninst✝² : P.IsCrystallographic\ninst✝¹ : IsDomain R\ninst✝ : P.IsReduc... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 260,
"column": 74
} | {
"line": 260,
"column": 79
} | {
"line": 260,
"column": 79
} | [
{
"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 : ι\nhi : i ∈ b.support\nhj : j ∈ b.support\nhij : j ≠ i\nf : ι → ℤ := fu... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 260,
"column": 74
} | {
"line": 260,
"column": 79
} | {
"line": 260,
"column": 79
} | [
{
"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 : ι\nhi : i ∈ b.support\nhj : j ∈ b.support\nhij : j ≠ i\nf : ι → ℤ := fu... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 260,
"column": 74
} | {
"line": 260,
"column": 79
} | {
"line": 260,
"column": 79
} | [
{
"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 : ι\nhi : i ∈ b.support\nhj : j ∈ b.support\nhij : j ≠ i\nf : ι → ℤ := fu... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 264,
"column": 63
} | {
"line": 264,
"column": 68
} | {
"line": 264,
"column": 68
} | [
{
"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 : ι\nhi : i ∈ b.support\nhj : j ∈ b.support\nhij : j ≠ i\nf : ι → ℤ := fu... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 264,
"column": 63
} | {
"line": 264,
"column": 68
} | {
"line": 264,
"column": 68
} | [
{
"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 : ι\nhi : i ∈ b.support\nhj : j ∈ b.support\nhij : j ≠ i\nf : ι → ℤ := fu... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 264,
"column": 63
} | {
"line": 264,
"column": 68
} | {
"line": 264,
"column": 68
} | [
{
"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 : ι\nhi : i ∈ b.support\nhj : j ∈ b.support\nhij : j ≠ i\nf : ι → ℤ := fu... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 332,
"column": 2
} | {
"line": 332,
"column": 7
} | {
"line": 334,
"column": 0
} | [
{
"pp": "case inl.inl\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\ni j : ι\ninst✝² : IsDom... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 332,
"column": 2
} | {
"line": 332,
"column": 7
} | {
"line": 334,
"column": 0
} | [
{
"pp": "case inl.inr\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\ni j : ι\ninst✝² : IsDom... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 332,
"column": 2
} | {
"line": 332,
"column": 7
} | {
"line": 334,
"column": 0
} | [
{
"pp": "case inr.inl\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\ni j : ι\ninst✝² : IsDom... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 332,
"column": 2
} | {
"line": 332,
"column": 7
} | {
"line": 334,
"column": 0
} | [
{
"pp": "case inr.inr\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\ni j : ι\ninst✝² : IsDom... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 89,
"column": 81
} | {
"line": 89,
"column": 86
} | {
"line": 89,
"column": 87
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁷ : Finite ι\ninst✝¹⁶ : AddCommGroup M\ninst✝¹⁵ : AddCommGroup N\ninst✝¹⁴ : CommRing R\ninst✝¹³ : Module R M\ninst✝¹² : Module R N\nP : RootPairing ι R M N\ninst✝¹¹ : IsDomain R\nS : Type u_6\ninst✝¹⁰ : LinearOrder S\ninst✝⁹ : CommRing S\nin... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 89,
"column": 81
} | {
"line": 89,
"column": 86
} | {
"line": 89,
"column": 87
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁷ : Finite ι\ninst✝¹⁶ : AddCommGroup M\ninst✝¹⁵ : AddCommGroup N\ninst✝¹⁴ : CommRing R\ninst✝¹³ : Module R M\ninst✝¹² : Module R N\nP : RootPairing ι R M N\ninst✝¹¹ : IsDomain R\nS : Type u_6\ninst✝¹⁰ : LinearOrder S\ninst✝⁹ : CommRing S\nin... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 89,
"column": 81
} | {
"line": 89,
"column": 86
} | {
"line": 89,
"column": 87
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁷ : Finite ι\ninst✝¹⁶ : AddCommGroup M\ninst✝¹⁵ : AddCommGroup N\ninst✝¹⁴ : CommRing R\ninst✝¹³ : Module R M\ninst✝¹² : Module R N\nP : RootPairing ι R M N\ninst✝¹¹ : IsDomain R\nS : Type u_6\ninst✝¹⁰ : LinearOrder S\ninst✝⁹ : CommRing S\nin... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 129,
"column": 46
} | {
"line": 129,
"column": 51
} | {
"line": 129,
"column": 51
} | [
{
"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\nf : M →+ ℚ\nhf : ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
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