module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Probability.Process.LocalProperty | {
"line": 294,
"column": 2
} | {
"line": 294,
"column": 83
} | {
"line": 295,
"column": 2
} | [
{
"pp": "ι : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁵ : ConditionallyCompleteLinearOrderBot ι\ninst✝⁴ : TopologicalSpace ι\ninst✝³ : OrderTopology ι\n𝓕 : Filtration ι mΩ\ninst✝² : SecondCountableTopology ι\ninst✝¹ : IsFiniteMeasure P\ninst✝ : NoMaxOrder ι\nτ : ℕ → Ω → WithTop ι\nσ ... | [
"ι : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁵ : ConditionallyCompleteLinearOrderBot ι\ninst✝⁴ : TopologicalSpace ι\ninst✝³ : OrderTopology ι\n𝓕 : Filtration ι mΩ\ninst✝² : SecondCountableTopology ι\ninst✝¹ : IsFiniteMeasure P\ninst✝ : NoMaxOrder ι\nτ : ℕ → Ω → WithTop ι\nσ : ℕ → ℕ → Ω ... | refine ⟨nk, hnk, fun n ↦ (hτ.isStoppingTime n).min ((hσ _).isStoppingTime _), ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RepresentationTheory.Intertwining | {
"line": 486,
"column": 6
} | {
"line": 488,
"column": 40
} | {
"line": 488,
"column": 40
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : CommSemiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G ... | [] | induction n with
| zero => rfl
| succ n ih => simp [ih, pow_succ] | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.RepresentationTheory.Character | {
"line": 112,
"column": 48
} | {
"line": 113,
"column": 56
} | {
"line": 113,
"column": 57
} | [
{
"pp": "k : Type u\ninst✝³ : Field k\nG : Type v\ninst✝² : Group G\ninst✝¹ : Fintype G\ninst✝ : Invertible ↑(Fintype.card G)\nV W : FDRep k G\n⊢ ↑(finrank k ↥(invariants (of (linHom V.ρ W.ρ)).ρ)) = ↑(finrank k (V ⟶ W))",
"ppTerm": "?m.88",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"k : Type u\ninst✝³ : Field k\nG : Type v\ninst✝² : Group G\ninst✝¹ : Fintype G\ninst✝ : Invertible ↑(Fintype.card G)\nV W : FDRep k G\n⊢ ↑(finrank k ↥(invariants (of (linHom V.ρ W.ρ)).ρ)) = ↑(finrank k ↥(linHom V.ρ W.ρ).invariants)"
] | ← LinearEquiv.finrank_eq
(Representation.linHom.invariantsEquivFDRepHom V W), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.StrongLaw | {
"line": 730,
"column": 15
} | {
"line": 730,
"column": 16
} | {
"line": 730,
"column": 16
} | [
{
"pp": "case hindep\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁵ : IsProbabilityMeasure μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nX : ℕ → Ω → E\nhint : Integrable (X 0) μ\nh' : StronglyMe... | [
"case hindep\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁵ : IsProbabilityMeasure μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nX : ℕ → Ω → E\nhint : Integrable (X 0) μ\nh' : StronglyMeasurable (X ... | I | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.StrongLaw | {
"line": 734,
"column": 15
} | {
"line": 734,
"column": 16
} | {
"line": 734,
"column": 16
} | [
{
"pp": "case hident\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁵ : IsProbabilityMeasure μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nX : ℕ → Ω → E\nhint : Integrable (X 0) μ\nh' : StronglyMe... | [
"case hident\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁵ : IsProbabilityMeasure μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nX : ℕ → Ω → E\nhint : Integrable (X 0) μ\nh' : StronglyMeasurable (X ... | I | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.StrongLaw | {
"line": 742,
"column": 2
} | {
"line": 742,
"column": 88
} | {
"line": 745,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁵ : IsProbabilityMeasure μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nX : ℕ → Ω → E\nhint : Integrable (X 0) μ\nh' : StronglyMeasurable (X 0... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁵ : IsProbabilityMeasure μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nX : ℕ → Ω → E\nhint : Integrable (X 0) μ\nh' : StronglyMeasurable (X 0)\nhindep : ... | obtain ⟨δ, δpos, hδ⟩ : ∃ δ, 0 < δ ∧ δ + δ + δ < ε := ⟨ε/4, by positivity, by linarith⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RepresentationTheory.Coinvariants | {
"line": 129,
"column": 34
} | {
"line": 129,
"column": 84
} | {
"line": 131,
"column": 0
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nX : Type u_5\ninst✝⁷ : CommRing k\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AddCommGroup W\ninst✝² : Module k W\ninst✝¹ : AddCommGroup X\ninst✝ : Module k X\nρ : Representation k G V\nτ : Representation k G W\nυ : ... | [] | simpa using congr($((f.isIntertwining' g).symm) x) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RepresentationTheory.Coinvariants | {
"line": 129,
"column": 34
} | {
"line": 129,
"column": 84
} | {
"line": 131,
"column": 0
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nX : Type u_5\ninst✝⁷ : CommRing k\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AddCommGroup W\ninst✝² : Module k W\ninst✝¹ : AddCommGroup X\ninst✝ : Module k X\nρ : Representation k G V\nτ : Representation k G W\nυ : ... | [] | simpa using congr($((f.isIntertwining' g).symm) x) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RepresentationTheory.Coinvariants | {
"line": 129,
"column": 34
} | {
"line": 129,
"column": 84
} | {
"line": 131,
"column": 0
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nX : Type u_5\ninst✝⁷ : CommRing k\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AddCommGroup W\ninst✝² : Module k W\ninst✝¹ : AddCommGroup X\ninst✝ : Module k X\nρ : Representation k G V\nτ : Representation k G W\nυ : ... | [] | simpa using congr($((f.isIntertwining' g).symm) x) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.StrongLaw | {
"line": 819,
"column": 2
} | {
"line": 819,
"column": 19
} | {
"line": 821,
"column": 0
} | [
{
"pp": "case e_a.e_f\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nX : ℕ → Ω → E\nhint : Integrable (X 0) μ\nhindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on... | [] | exact (h₁ i).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 57,
"column": 2
} | {
"line": 61,
"column": 50
} | {
"line": 62,
"column": 2
} | [
{
"pp": "case refine_1\nk : Type u_1\nG : Type u_2\ninst✝⁴ : CommRing k\ninst✝³ : Group G\nV : Type u_4\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nρ : Representation k G V\ng : G\ninst✝ : Finite G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\n⊢ (Set.range fun gv ↦ (ρ gv.1) gv.2 - gv.2) ⊆ ↑(ρ g - LinearMap.id).ra... | [
"case refine_2\nk : Type u_1\nG : Type u_2\ninst✝⁴ : CommRing k\ninst✝³ : Group G\nV : Type u_4\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nρ : Representation k G V\ng : G\ninst✝ : Finite G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\n⊢ (ρ g - LinearMap.id).range ≤ Coinvariants.ker ρ"
] | · rintro a ⟨⟨γ, α⟩, rfl⟩
rcases mem_powers_iff_mem_zpowers.2 (hg γ) with ⟨i, rfl⟩
induction i with | zero => exact ⟨0, by simp⟩ | succ n _ =>
use (Fin.partialSum (fun (j : Fin (n + 1)) => ρ (g ^ (j : ℕ)) α) (Fin.last _))
simpa using ρ.apply_sub_id_partialSum_eq _ _ _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality | {
"line": 99,
"column": 35
} | {
"line": 102,
"column": 64
} | {
"line": 104,
"column": 0
} | [
{
"pp": "k G H : Type u\ninst✝³ : CommRing k\ninst✝² : Group G\ninst✝¹ : Group H\nA : Rep k H\nB : Rep k G\nf : G →* H\nφ : res f A ⟶ B\nhf : Function.Injective ⇑f\ninst✝ : Epi φ\ni : ℕ\n⊢ Epi ((cochainsMap f φ).f i)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi... | [] | by
simpa [ModuleCat.epi_iff_surjective] using!
((Rep.epi_iff_surjective φ).1 inferInstance).comp_left.comp <|
LinearMap.funLeft_surjective_of_injective k A _ hf.comp_left | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree | {
"line": 888,
"column": 41
} | {
"line": 891,
"column": 17
} | {
"line": 891,
"column": 17
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nx : ↥(cocycles₂ A)\n⊢ (ConcreteCategory.hom (inhomogeneousCochains.d A 2)) ((ConcreteCategory.hom (cochainsIso₂ A).inv) ⇑x) = 0",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.modul... | [] | by
rw [← LinearMap.comp_apply, ← ModuleCat.hom_comp, ← inhomogeneousCochains.d_def,
eq_d₂₃_comp_inv, ModuleCat.hom_comp, LinearMap.comp_apply, cocycles₂.d₂₃_apply x,
map_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RepresentationTheory.Homological.GroupHomology.LongExactSequence | {
"line": 174,
"column": 4
} | {
"line": 174,
"column": 82
} | {
"line": 175,
"column": 4
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nX : ShortComplex (Rep k G)\nhX : X.ShortExact\nz : ↥(cycles₂ X.X₃)\ny : G × G →₀ ↑X.X₂\nhy : (mapRange.linearMap (Rep.Hom.hom X.g).toLinearMap) y = ↑z\nx : G →₀ ↑X.X₁\nhx : (mapRange.linearMap (Rep.Hom.hom X.f).toLinearMap) x = (ConcreteCategory.hom (... | [
"k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nX : ShortComplex (Rep k G)\nhX : X.ShortExact\nz : ↥(cycles₂ X.X₃)\ny : G × G →₀ ↑X.X₂\nhy : (mapRange.linearMap (Rep.Hom.hom X.g).toLinearMap) y = ↑z\nx : G →₀ ↑X.X₁\nhx : (mapRange.linearMap (Rep.Hom.hom X.f).toLinearMap) x = (ConcreteCategory.hom (d₂₁ X.X₂)) y... | conv_rhs => rw [← LinearMap.comp_apply, ← ModuleCat.hom_comp, eq_d₂₁_comp_inv] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convRHS_1 | Mathlib.Tactic.Conv.convRHS |
Mathlib.RepresentationTheory.Tannaka | {
"line": 116,
"column": 2
} | {
"line": 119,
"column": 25
} | {
"line": 121,
"column": 0
} | [
{
"pp": "k G : Type u\ninst✝³ : CommRing k\ninst✝² : Group G\ninst✝¹ : Finite G\ninst✝ : Nontrivial k\n⊢ Function.Injective ⇑(equivHom k G)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"CategoryTheory.InducedCategory.homMk_hom",
"Pi.Function.module",
"CategoryTheory.Fun... | [] | intro s t h
classical
apply_fun (fun x ↦ (x.hom.hom.app rightFDRep).hom (single t 1) 1) at h
simp_all [single_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RepresentationTheory.Tannaka | {
"line": 116,
"column": 2
} | {
"line": 119,
"column": 25
} | {
"line": 121,
"column": 0
} | [
{
"pp": "k G : Type u\ninst✝³ : CommRing k\ninst✝² : Group G\ninst✝¹ : Finite G\ninst✝ : Nontrivial k\n⊢ Function.Injective ⇑(equivHom k G)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"CategoryTheory.InducedCategory.homMk_hom",
"Pi.Function.module",
"CategoryTheory.Fun... | [] | intro s t h
classical
apply_fun (fun x ↦ (x.hom.hom.app rightFDRep).hom (single t 1) 1) at h
simp_all [single_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Equiv | {
"line": 61,
"column": 4
} | {
"line": 65,
"column": 34
} | {
"line": 66,
"column": 2
} | [
{
"pp": "σ✝ : Type u_1\nR✝ : Type u_2\nn✝ : ℕ\ninst✝³ : CommRing R✝\ninst✝² : Finite σ✝\nσ : Type u_3\nR : Type u_4\ninst✝¹ : Finite σ\ninst✝ : CommRing R\nn : ℕ\n⊢ (Ideal.Quotient.mk (MvPolynomial.idealOfVars σ R ^ n)) ((truncTotal n) 1) = 1",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [... | [] | by_cases! h : n = 0
· have := Ideal.Quotient.subsingleton_iff.mpr
(show MvPolynomial.idealOfVars σ R ^ n = ⊤ by simp [h])
exact Subsingleton.allEq ..
rw [truncTotal_one h, map_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.Equiv | {
"line": 61,
"column": 4
} | {
"line": 65,
"column": 34
} | {
"line": 66,
"column": 2
} | [
{
"pp": "σ✝ : Type u_1\nR✝ : Type u_2\nn✝ : ℕ\ninst✝³ : CommRing R✝\ninst✝² : Finite σ✝\nσ : Type u_3\nR : Type u_4\ninst✝¹ : Finite σ\ninst✝ : CommRing R\nn : ℕ\n⊢ (Ideal.Quotient.mk (MvPolynomial.idealOfVars σ R ^ n)) ((truncTotal n) 1) = 1",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [... | [] | by_cases! h : n = 0
· have := Ideal.Quotient.subsingleton_iff.mpr
(show MvPolynomial.idealOfVars σ R ^ n = ⊤ by simp [h])
exact Subsingleton.allEq ..
rw [truncTotal_one h, map_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.ClassGroup.ExtendedHom | {
"line": 54,
"column": 9
} | {
"line": 54,
"column": 28
} | {
"line": 54,
"column": 28
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra A B\ninst✝² : Module.IsTorsionFree A B\ninst✝¹ : IsDomain A\ninst✝ : IsDomain B\nα : (FractionRing A)ˣ\n⊢ (IsFractionRing.map ⋯) ↑α ≠ 0",
"ppTerm": "?m.91",
"assigned": true,
"usedConstants": [
"Uni... | [] | by simp [α.ne_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.ClassGroup.ExtendedHom | {
"line": 111,
"column": 6
} | {
"line": 111,
"column": 28
} | {
"line": 111,
"column": 29
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹² : CommRing A\ninst✝¹¹ : CommRing B\ninst✝¹⁰ : Algebra A B\ninst✝⁹ : Module.IsTorsionFree A B\ninst✝⁸ : IsDedekindDomain A\nC : Type u_3\ninst✝⁷ : CommRing C\ninst✝⁶ : Algebra B C\ninst✝⁵ : Algebra A C\ninst✝⁴ : IsScalarTower A B C\ninst✝³ : Module.IsTorsionFree B C\n... | [
"A : Type u_1\nB : Type u_2\ninst✝¹² : CommRing A\ninst✝¹¹ : CommRing B\ninst✝¹⁰ : Algebra A B\ninst✝⁹ : Module.IsTorsionFree A B\ninst✝⁸ : IsDedekindDomain A\nC : Type u_3\ninst✝⁷ : CommRing C\ninst✝⁶ : Algebra B C\ninst✝⁵ : Algebra A C\ninst✝⁴ : IsScalarTower A B C\ninst✝³ : Module.IsTorsionFree B C\ninst✝² : Mod... | extendedHom_mk0 A B I, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 221,
"column": 6
} | {
"line": 221,
"column": 23
} | {
"line": 221,
"column": 24
} | [
{
"pp": "case succ\nA : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\na : A\nhI : DividedPowers I\nha : a ∈ I\nn : ℕ\nih : ↑n ! * hI.dpow n a = a ^ n\n⊢ ↑n ! * (↑((n + 1).choose n) * hI.dpow (n + 1) a) = a ^ (n + 1)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"No... | [
"case succ\nA : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\na : A\nhI : DividedPowers I\nha : a ∈ I\nn : ℕ\nih : ↑n ! * hI.dpow n a = a ^ n\n⊢ ↑n ! * (hI.dpow n a * hI.dpow 1 a) = a ^ (n + 1)"
] | ← hI.mul_dpow ha, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.SelmerGroup | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 45
} | {
"line": 114,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type v\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nx✝¹ x✝ : Kˣ\n⊢ v.valuationOfNeZeroToFun (x✝¹ * x✝) = v.valuationOfNeZeroToFun x✝¹ * v.valuationOfNeZeroToFun x✝",
"ppTerm": "?m.50",... | [
"R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type v\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nx✝¹ x✝ : Kˣ\n⊢ ↑(v.valuationOfNeZeroToFun (x✝¹ * x✝)) = ↑(v.valuationOfNeZeroToFun x✝¹) * ↑(v.valuationOfNeZeroToFun x✝)"
] | rw [← WithZero.coe_inj, WithZero.coe_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 298,
"column": 6
} | {
"line": 298,
"column": 70
} | {
"line": 299,
"column": 6
} | [
{
"pp": "case insert\nA : Type u_1\ninst✝² : CommSemiring A\nM : Type u_2\ninst✝¹ : AddCommMonoid M\nI : AddSubmonoid M\ndpow : ℕ → M → A\ndpow_zero : ∀ {x : M}, x ∈ I → dpow 0 x = 1\ndpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0\nι : Type u_3\ninst✝ : DecidableEq ι\nx : ι → M\ndpow_add : ∀ {n : ℕ} {x y : M}... | [
"case insert\nA : Type u_1\ninst✝² : CommSemiring A\nM : Type u_2\ninst✝¹ : AddCommMonoid M\nI : AddSubmonoid M\ndpow : ℕ → M → A\ndpow_zero : ∀ {x : M}, x ∈ I → dpow 0 x = 1\ndpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0\nι : Type u_3\ninst✝ : DecidableEq ι\nx : ι → M\ndpow_add : ∀ {n : ℕ} {x y : M}, x ∈ I → y ... | dpow_add (hx a (mem_insert_self a s)) (I.sum_mem fun i ↦ hx' i), | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 663,
"column": 4
} | {
"line": 663,
"column": 41
} | {
"line": 664,
"column": 4
} | [
{
"pp": "case h\nk G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\nS : Subgroup G\ninst✝ : S.Normal\nx : ↥(cycles₁ A)\nhx :\n (ConcreteCategory.hom (H1π (A.quotientToCoinvariants S)))\n ((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.toCoinvariantsMkQ S))) x) =\n 0\ny : ↑(Mo... | [
"case h\nk G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\nS : Subgroup G\ninst✝ : S.Normal\nx : ↥(cycles₁ A)\nhx :\n (ConcreteCategory.hom (H1π (A.quotientToCoinvariants S)))\n ((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.toCoinvariantsMkQ S))) x) =\n 0\ny : ↑(ModuleCat.of k... | refine (H1π_eq_iff _ _).2 ⟨W + δ, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.DividedPowers.DPMorphism | {
"line": 142,
"column": 4
} | {
"line": 148,
"column": 28
} | {
"line": 149,
"column": 2
} | [
{
"pp": "A✝ : Type u_1\nB✝ : Type u_2\ninst✝³ : CommSemiring A✝\ninst✝² : CommSemiring B✝\nI✝ : Ideal A✝\nJ✝ : Ideal B✝\nhI✝ : DividedPowers I✝\nhJ✝ : DividedPowers J✝\nA : Type u_3\nB : Type u_4\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nhI : DividedPowers I\nhJ : DividedPowers... | [] | simp only [mem_setOf_eq, map_add] at hx hy ⊢
refine ⟨I.add_mem hx.1 hy.1, fun n ↦ ?_⟩
rw [hI.dpow_add hx.1 hy.1, map_sum,
hJ.dpow_add (hf (mem_map_of_mem f hx.1)) (hf (mem_map_of_mem f hy.1))]
apply congr_arg
ext k
rw [map_mul, hx.2, hy.2] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DividedPowers.DPMorphism | {
"line": 142,
"column": 4
} | {
"line": 148,
"column": 28
} | {
"line": 149,
"column": 2
} | [
{
"pp": "A✝ : Type u_1\nB✝ : Type u_2\ninst✝³ : CommSemiring A✝\ninst✝² : CommSemiring B✝\nI✝ : Ideal A✝\nJ✝ : Ideal B✝\nhI✝ : DividedPowers I✝\nhJ✝ : DividedPowers J✝\nA : Type u_3\nB : Type u_4\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nhI : DividedPowers I\nhJ : DividedPowers... | [] | simp only [mem_setOf_eq, map_add] at hx hy ⊢
refine ⟨I.add_mem hx.1 hy.1, fun n ↦ ?_⟩
rw [hI.dpow_add hx.1 hy.1, map_sum,
hJ.dpow_add (hf (mem_map_of_mem f hx.1)) (hf (mem_map_of_mem f hy.1))]
apply congr_arg
ext k
rw [map_mul, hx.2, hy.2] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 59
} | {
"line": 114,
"column": 4
} | [
{
"pp": "case neg\nA : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nn : ℕ\nhn_fac : IsUnit ↑(n - 1)!\nhnI : I ^ n = 0\nm : ℕ\nx : A\nhx : x ∈ I\ny : A\nhy : y ∈ I\nhmn : n ≤ m\nh_sub : I ^ m ≤ I ^ n\nhxy : (x + y) ^ m = 0\nk : ℕ × ℕ\nhk : k ∈ Finset.antidiagonal m\n⊢ y ^ ... | [
"case neg\nA : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nn : ℕ\nhn_fac : IsUnit ↑(n - 1)!\nhnI : I ^ n = 0\nm : ℕ\nx : A\nhx : x ∈ I\ny : A\nhy : y ∈ I\nhmn : n ≤ m\nh_sub : I ^ m ≤ I ^ n\nhxy : (x + y) ^ m = 0\nk : ℕ × ℕ\nhk : k ∈ Finset.antidiagonal m\n⊢ y ^ k.2 * x ^ k.... | rw [← Finset.mem_antidiagonal.mp hk, add_comm, pow_add] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 117,
"column": 2
} | {
"line": 118,
"column": 56
} | {
"line": 120,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nm : ℕ\na x : A\nhx : x ∈ I\n⊢ dpow I m (a * x) = a ^ m * dpow I m x",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semi... | [] | rw [dpow_eq_of_mem (Ideal.mul_mem_left I _ hx), dpow_eq_of_mem hx,
mul_pow, ← mul_assoc, mul_comm _ (a ^ m), mul_assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 117,
"column": 2
} | {
"line": 118,
"column": 56
} | {
"line": 120,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nm : ℕ\na x : A\nhx : x ∈ I\n⊢ dpow I m (a * x) = a ^ m * dpow I m x",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semi... | [] | rw [dpow_eq_of_mem (Ideal.mul_mem_left I _ hx), dpow_eq_of_mem hx,
mul_pow, ← mul_assoc, mul_comm _ (a ^ m), mul_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 117,
"column": 2
} | {
"line": 118,
"column": 56
} | {
"line": 120,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nm : ℕ\na x : A\nhx : x ∈ I\n⊢ dpow I m (a * x) = a ^ m * dpow I m x",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semi... | [] | rw [dpow_eq_of_mem (Ideal.mul_mem_left I _ hx), dpow_eq_of_mem hx,
mul_pow, ← mul_assoc, mul_comm _ (a ^ m), mul_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DividedPowers.Padic | {
"line": 108,
"column": 6
} | {
"line": 113,
"column": 67
} | {
"line": 115,
"column": 0
} | [
{
"pp": "case hmn\np : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nhn : n ≠ 0\nx : ℤ_[p]\nhx : x ∈ Ideal.span {↑p}\nhx0 : ¬x = 0\nhlt : ↑(padicValNat p n !) < ↑n\nhnorm : 0 < ‖↑n !‖\n⊢ -↑x.valuation * ↑n < -(↑n !).valuation",
"ppTerm": "?hmn",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup"... | [] | simp only [neg_mul, Padic.valuation_natCast, neg_lt_neg_iff]
apply lt_of_lt_of_le hlt
conv_lhs => rw [← one_mul (n : ℤ)]
gcongr
norm_cast
rwa [← PadicInt.mem_span_pow_iff_le_valuation x hx0, pow_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DividedPowers.Padic | {
"line": 108,
"column": 6
} | {
"line": 113,
"column": 67
} | {
"line": 115,
"column": 0
} | [
{
"pp": "case hmn\np : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nhn : n ≠ 0\nx : ℤ_[p]\nhx : x ∈ Ideal.span {↑p}\nhx0 : ¬x = 0\nhlt : ↑(padicValNat p n !) < ↑n\nhnorm : 0 < ‖↑n !‖\n⊢ -↑x.valuation * ↑n < -(↑n !).valuation",
"ppTerm": "?hmn",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup"... | [] | simp only [neg_mul, Padic.valuation_natCast, neg_lt_neg_iff]
apply lt_of_lt_of_le hlt
conv_lhs => rw [← one_mul (n : ℤ)]
gcongr
norm_cast
rwa [← PadicInt.mem_span_pow_iff_le_valuation x hx0, pow_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 139,
"column": 8
} | {
"line": 139,
"column": 29
} | {
"line": 139,
"column": 30
} | [
{
"pp": "case refine_1\nA : Type u_1\ninst✝ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nJ : Ideal A\nhIJ : hI.IsSubDPIdeal (J ⊓ I)\nn : ℕ\na b : A\nha : a ∈ I\nhb : b ∈ I\nhab : a - b ∈ J\nhab' : a - b ∈ I\n⊢ hI.dpow n a - hI.dpow n b ∈ J",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstant... | [
"case refine_1\nA : Type u_1\ninst✝ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nJ : Ideal A\nhIJ : hI.IsSubDPIdeal (J ⊓ I)\nn : ℕ\na b : A\nha : a ∈ I\nhb : b ∈ I\nhab : a - b ∈ J\nhab' : a - b ∈ I\n⊢ hI.dpow n (b + (a - b)) - hI.dpow n b ∈ J"
] | ← add_sub_cancel b a, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 429,
"column": 4
} | {
"line": 437,
"column": 37
} | {
"line": 439,
"column": 0
} | [
{
"pp": "case a\nA : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nS : Set A\nhS : S ⊆ ↑I\nJ : hI.SubDPIdeal := ⋯\n⊢ span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} ≤ ⨅ s ∈ insert ⊤ {J | S ⊆ ↑J.carrier}, s.carrier",
"ppTerm": "?a✝",
"assigned": true,
"usedCo... | [] | rw [le_iInf₂_iff]
intro K hK
have : S ≤ K := by
simp only [Set.mem_insert_iff, Set.mem_setOf_eq] at hK
rcases hK with rfl | hKS
exacts [hS, hKS]
rw [span_le]
rintro y ⟨n, hn, x, hx, rfl⟩
exact K.dpow_mem n hn x (this hx) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 429,
"column": 4
} | {
"line": 437,
"column": 37
} | {
"line": 439,
"column": 0
} | [
{
"pp": "case a\nA : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nS : Set A\nhS : S ⊆ ↑I\nJ : hI.SubDPIdeal := ⋯\n⊢ span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} ≤ ⨅ s ∈ insert ⊤ {J | S ⊆ ↑J.carrier}, s.carrier",
"ppTerm": "?a✝",
"assigned": true,
"usedCo... | [] | rw [le_iInf₂_iff]
intro K hK
have : S ≤ K := by
simp only [Set.mem_insert_iff, Set.mem_setOf_eq] at hK
rcases hK with rfl | hKS
exacts [hS, hKS]
rw [span_le]
rintro y ⟨n, hn, x, hx, rfl⟩
exact K.dpow_mem n hn x (this hx) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Frobenius | {
"line": 219,
"column": 91
} | {
"line": 237,
"column": 53
} | {
"line": 239,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nG : Type u_3\ninst✝⁶ : Group G\ninst✝⁵ : MulSemiringAction G S\ninst✝⁴ : SMulCommClass G R S\nQ : Ideal S\ninst✝³ : Finite G\ninst✝² : Algebra.IsInvariant R S G\ninst✝¹ : Q.IsPrime\ninst✝ : Finite (S ⧸ Q)\n⊢ ∃ σ... | [] | by
let P := Q.under R
have := Algebra.IsInvariant.isIntegral R S G
have : Q.IsMaximal := Ideal.Quotient.maximal_of_isField _ (Finite.isField_of_domain (S ⧸ Q))
obtain ⟨p, hc⟩ := CharP.exists (R ⧸ P)
have : Finite (R ⧸ P) := .of_injective _ Ideal.algebraMap_quotient_injective
cases nonempty_fintype (R ⧸ P)
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.FormalGroup.Basic | {
"line": 103,
"column": 43
} | {
"line": 103,
"column": 58
} | {
"line": 103,
"column": 59
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nS : Type u_2\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nσ✝ : Type u_3\nτ : Type u_4\nσ : Type\nF : FormalGroup R\naux₁ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\naux₂ : failed to pretty print expression (use 'set_... | [
"R : Type u_1\ninst✝² : CommRing R\nS : Type u_2\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nσ✝ : Type u_3\nτ : Type u_4\nσ : Type\nF : FormalGroup R\naux₁ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\naux₂ : failed to pretty print expression (use 'set_option pp.ra... | subst_add aux₂, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Grassmannian | {
"line": 170,
"column": 4
} | {
"line": 170,
"column": 57
} | {
"line": 171,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝⁸ : CommRing R\nM : Type v\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nk : ℕ\nA : Type w\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\nB : Type w\ninst✝³ : CommRing B\ninst✝² : Algebra R B\nf : A →ₐ[R] B\nC : Type w\ninst✝¹ : CommRing C\ninst✝ : Algebra R C\ng : B →ₐ[R] C\nN : G(k, A ... | [] | rw [map_toSubmodule g (map f N), map_toSubmodule f N] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.FormalGroup.Basic | {
"line": 121,
"column": 59
} | {
"line": 121,
"column": 74
} | {
"line": 121,
"column": 75
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nS : Type u_2\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nσ✝ : Type u_3\nτ : Type u_4\nσ : Type\nF : FormalGroup R\naux₁ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\naux₂ : failed to pretty print expression (use 'set_... | [
"R : Type u_1\ninst✝² : CommRing R\nS : Type u_2\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nσ✝ : Type u_3\nτ : Type u_4\nσ : Type\nF : FormalGroup R\naux₁ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\naux₂ : failed to pretty print expression (use 'set_option pp.ra... | subst_add aux₂, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.HahnSeries.HEval | {
"line": 193,
"column": 2
} | {
"line": 194,
"column": 66
} | {
"line": 195,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\nx : R⟦Γ⟧\nr : R\ng : Γ\n⊢ ((heval x) (C r)).coeff g = (r • 1).coeff g",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"WithT... | [
"Γ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\nx : R⟦Γ⟧\nr : R\ng : Γ\n⊢ ∑ᶠ (i : ℕ), (coeff i) (C r) * ((if 0 < x.orderTop then x else 0) ^ i).coeff g = if g = 0 then r else 0"
] | simp only [heval_apply, coeff_hsum, smulFamily_toFun, powers_toFun, HahnSeries.coeff_smul,
HahnSeries.coeff_one, smul_eq_mul, mul_ite, mul_one, mul_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 616,
"column": 4
} | {
"line": 616,
"column": 74
} | {
"line": 617,
"column": 4
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝¹ : PartialOrder Γ\ninst✝ : AddCommMonoid R\ns : SummableFamily Γ R α\nf : α ↪ β\n⊢ (⋃ a, (if h : a ∈ Set.range ⇑f then s (Classical.choose h) else 0).support).IsPWO",
"ppTerm": "?m.43",
"assigned": true,
... | [
"Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝¹ : PartialOrder Γ\ninst✝ : AddCommMonoid R\ns : SummableFamily Γ R α\nf : α ↪ β\nb : β\ng : Γ\nh : g ∈ (if h : b ∈ Set.range ⇑f then s (Classical.choose h) else 0).support\n⊢ g ∈ ⋃ a, (s a).support"
] | refine s.isPWO_iUnion_support.mono (Set.iUnion_subset fun b g h => ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Henselian | {
"line": 148,
"column": 6
} | {
"line": 149,
"column": 30
} | {
"line": 149,
"column": 30
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsLocalRing R\ntfae_3_to_2 :\n (∀ {K : Type u} [inst : Field K] (φ : R →+* K),\n Surjective ⇑φ →\n ∀ (f : R[X]),\n f.Monic → ∀ (a₀ : K), eval₂ φ a₀ f = 0 → eval₂ φ a₀ (derivative f) ≠ 0 → ∃ a, f.IsRoot a ∧ φ a = a₀) →\n ∀ (f : R[X]),\n ... | [] | rwa [← mem_nonunits_iff, ← mem_maximalIdeal, ← ker_eq_maximalIdeal φ hφ,
RingHom.mem_ker] at h₂ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 890,
"column": 21
} | {
"line": 891,
"column": 73
} | {
"line": 892,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝³ : AddCommGroup Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedAddMonoid Γ\ninst✝ : Field R\nq : ℚ≥0\n⊢ ↑q = ↑q.num / ↑q.den",
"ppTerm": "?m.101",
"assigned": true,
"usedConstants": [
"Semiring.to... | [] | by
simp [← single_zero_nnratCast, ← single_zero_natCast, NNRat.cast_def] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.IdealFilter.Topology | {
"line": 52,
"column": 4
} | {
"line": 52,
"column": 73
} | {
"line": 53,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝ : Ring A\nF : IdealFilter A\nI : Ideal A\nhI : I ∈ F\nJ : Ideal A\nhJ : J ∈ F\n⊢ ∃ z ∈ {x | ∃ I ∈ F, ↑I = x}, z ⊆ ↑I ∩ ↑J",
"ppTerm": "?m.80",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"Submodule.completeLattice",
"PartialOrder.toPreorde... | [] | exact ⟨I ⊓ J, ⟨I ⊓ J, Order.PFilter.inf_mem hI hJ, rfl⟩, fun _ h ↦ h⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 438,
"column": 6
} | {
"line": 438,
"column": 15
} | {
"line": 439,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : IsNoetherianRing R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsNoetherianRing S\np : Ideal R\ninst✝² : p.IsPrime\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : P.LiesOver p\ns : Finset R\nhp : p ∈ (span ↑s).minimalPrimes\nheq : ↑s.card = p... | [] | use y, hx | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 438,
"column": 6
} | {
"line": 438,
"column": 15
} | {
"line": 439,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : IsNoetherianRing R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsNoetherianRing S\np : Ideal R\ninst✝² : p.IsPrime\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : P.LiesOver p\ns : Finset R\nhp : p ∈ (span ↑s).minimalPrimes\nheq : ↑s.card = p... | [] | use y, hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 438,
"column": 6
} | {
"line": 438,
"column": 15
} | {
"line": 439,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : IsNoetherianRing R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsNoetherianRing S\np : Ideal R\ninst✝² : p.IsPrime\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : P.LiesOver p\ns : Finset R\nhp : p ∈ (span ↑s).minimalPrimes\nheq : ↑s.card = p... | [] | use y, hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 434,
"column": 2
} | {
"line": 439,
"column": 75
} | {
"line": 440,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : IsNoetherianRing R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsNoetherianRing S\np : Ideal R\ninst✝² : p.IsPrime\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : P.LiesOver p\ns : Finset R\nhp : p ∈ (span ↑s).minimalPrimes\nheq : ↑s.card = p... | [
"R : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : IsNoetherianRing R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsNoetherianRing S\np : Ideal R\ninst✝² : p.IsPrime\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : P.LiesOver p\ns : Finset R\nhp : p ∈ (span ↑s).minimalPrimes\nheq : ↑s.card = p.height\nP' ... | have : Set.SurjOn (Ideal.Quotient.mk (p.map (algebraMap R S))) P s' := by
refine Set.SurjOn.mono subset_rfl hsP'sub fun x hx ↦ ?_
obtain ⟨y, rfl⟩ := Ideal.Quotient.mk_surjective x
rw [SetLike.mem_coe, Ideal.mem_quotient_iff_mem] at hx
· use y, hx
· rw [Ideal.map_le_iff_le_comap, Ideal.LiesOver.over ... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 82,
"column": 32
} | {
"line": 82,
"column": 43
} | {
"line": 82,
"column": 44
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nM : Type u_3\nM₂ : Type u_4\nM₃ : Type u_5\nM₄ : Type u_6\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\nr : R\nrs : List R\n⊢ map (r • ⊤).mkQ (Ideal.ofList rs • ⊤) = Ideal.ofList rs • ⊤",
"ppTerm": "?m.... | [
"R : Type u_1\nS : Type u_2\nM : Type u_3\nM₂ : Type u_4\nM₃ : Type u_5\nM₄ : Type u_6\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\nr : R\nrs : List R\n⊢ Ideal.ofList rs • map (r • ⊤).mkQ ⊤ = Ideal.ofList rs • ⊤"
] | map_smul'', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.KrullDimension.Regular | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 48
} | {
"line": 156,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\nS : Finset R\nhS : ↑S ⊆ ↑(maximalIdeal R)\n⊢ ↑S ⊆ ↑(Ring.jacobson R)",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"Finset",
... | [] | rwa [IsLocalRing.ringJacobson_eq_maximalIdeal] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 562,
"column": 4
} | {
"line": 567,
"column": 65
} | {
"line": 571,
"column": 0
} | [
{
"pp": "case cons\nR : Type u_1\nM₄ : Type u_6\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M₄\ninst✝⁸ : Module R M₄\nrs✝ : List R\nM✝ : Type u_6\ninst✝⁷ : AddCommGroup M✝\ninst✝⁶ : Module R M✝\nr : R\nrs : List R\nh₄ : IsSMulRegular M✝ r\nh2✝ : IsWeaklyRegular (QuotSMulTop r M✝) rs\nih :\n ∀ {M : Type u_3} {... | [] | specialize ih
(map_first_exact_on_four_term_exact_of_isSMulRegular_last h₁₂ h₂₃ h₄)
(map_exact r h₂₃ h₃) (map_surjective r h₃)
have H₁ := quotOfListConsSMulTopEquivQuotSMulTopInner_naturality r rs f₁
have H₂ := quotOfListConsSMulTopEquivQuotSMulTopInner_naturality r rs f₂
exact (Exact.iff_of_lad... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 562,
"column": 4
} | {
"line": 567,
"column": 65
} | {
"line": 571,
"column": 0
} | [
{
"pp": "case cons\nR : Type u_1\nM₄ : Type u_6\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M₄\ninst✝⁸ : Module R M₄\nrs✝ : List R\nM✝ : Type u_6\ninst✝⁷ : AddCommGroup M✝\ninst✝⁶ : Module R M✝\nr : R\nrs : List R\nh₄ : IsSMulRegular M✝ r\nh2✝ : IsWeaklyRegular (QuotSMulTop r M✝) rs\nih :\n ∀ {M : Type u_3} {... | [] | specialize ih
(map_first_exact_on_four_term_exact_of_isSMulRegular_last h₁₂ h₂₃ h₄)
(map_exact r h₂₃ h₃) (map_surjective r h₃)
have H₁ := quotOfListConsSMulTopEquivQuotSMulTopInner_naturality r rs f₁
have H₂ := quotOfListConsSMulTopEquivQuotSMulTopInner_naturality r rs f₂
exact (Exact.iff_of_lad... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Invariant.Profinite | {
"line": 80,
"column": 2
} | {
"line": 81,
"column": 68
} | {
"line": 82,
"column": 2
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹⁵ : CommRing A\ninst✝¹⁴ : CommRing B\ninst✝¹³ : Algebra A B\nG : Type u\ninst✝¹² : Group G\ninst✝¹¹ : MulSemiringAction G B\ninst✝¹⁰ : SMulCommClass G A B\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : CompactSpace G\ninst✝⁷ : TotallyDisconnectedSpace G\ninst✝⁶ : IsTopological... | [
"A : Type u_1\nB : Type u_2\ninst✝¹⁵ : CommRing A\ninst✝¹⁴ : CommRing B\ninst✝¹³ : Algebra A B\nG : Type u\ninst✝¹² : Group G\ninst✝¹¹ : MulSemiringAction G B\ninst✝¹⁰ : SMulCommClass G A B\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : CompactSpace G\ninst✝⁷ : TotallyDisconnectedSpace G\ninst✝⁶ : IsTopologicalGroup G\nins... | let a := (ProfiniteGrp.of G).isoLimittoFiniteQuotientFunctor.inv.hom
⟨fun N ↦ (s N).1, (fun {N N'} f ↦ congr_arg Subtype.val (hs f))⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.LocalIso | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 59
} | {
"line": 141,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁸ : CommSemiring R\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nT : Type u_3\ninst✝⁵ : CommSemiring T\ninst✝⁴ : Algebra S T\ninst✝³ : Algebra R T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalIso R S\ninst✝ : IsLocalIso S T\ns : Set S := {g | IsStandardOpenImmersion... | [] | exact .of_span_range_eq_top _ h fun i : ι ↦ T'' i.1 i.2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.LocalProperties.InjectiveDimension | {
"line": 49,
"column": 2
} | {
"line": 69,
"column": 70
} | {
"line": 71,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nn : ℕ\nS : Submonoid R\nM : ModuleCat R\ninst✝ : HasInjectiveDimensionLE M n\nthis : Small.{v, u} (Localization S)\n⊢ HasInjectiveDimensionLE (M.localizedModule S) n",
"ppTerm": "?m.23",
"assigned": true,
... | [] | induction n generalizing M with
| zero =>
have injle : HasInjectiveDimensionLE M 0 := ‹_›
simp only [HasInjectiveDimensionLE, zero_add, ← injective_iff_hasInjectiveDimensionLT_one]
at injle ⊢
rw [← Module.injective_iff_injective_object] at injle ⊢
exact Module.injective_of_isLocalizedModule S (M... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.RingTheory.LaurentSeries | {
"line": 612,
"column": 2
} | {
"line": 612,
"column": 26
} | {
"line": 613,
"column": 2
} | [
{
"pp": "case refine_1\nK : Type u_2\ninst✝ : Field K\nf : K⸨X⸩\nh : ∀ n < 0, f.coeff n = 0\n⊢ (ofPowerSeries ℤ K) (PowerSeries.mk fun n ↦ f.coeff ↑n) = f",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Int.instIsStrictOrderedRing",
"HahnSeries.instNonAssocSemiring",
... | [
"case refine_1.ofNat\nK : Type u_2\ninst✝ : Field K\nf : K⸨X⸩\nh : ∀ n < 0, f.coeff n = 0\na✝ : ℕ\n⊢ ((ofPowerSeries ℤ K) (PowerSeries.mk fun n ↦ f.coeff ↑n)).coeff (Int.ofNat a✝) = f.coeff (Int.ofNat a✝)",
"case refine_1.negSucc\nK : Type u_2\ninst✝ : Field K\nf : K⸨X⸩\nh : ∀ n < 0, f.coeff n = 0\nn : ℕ\n⊢ ((ofP... | on_goal 1 => ext (_ | n) | Batteries.Tactic.«_aux_Batteries_Tactic_PermuteGoals___elabRules_Batteries_Tactic_tacticOn_goal-_=>__1» | Batteries.Tactic.«tacticOn_goal-_=>_» |
Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem | {
"line": 194,
"column": 2
} | {
"line": 197,
"column": 75
} | {
"line": 198,
"column": 2
} | [
{
"pp": "case refine_2.refine_1\nR : Type u_3\nm : ℕ\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\ni : ℕ\nhim : i < m\nt : Finset (Fin m)\nht : t ∈ powersetCard (i + 1) univ\nht' : #t = #(Iic ⟨i, him⟩)\nhne : ∃ x, x ∈ Iic ⟨i, him⟩ \\ t\nhkm : (Iic ⟨i, him⟩ \\ t).min' hne ∈ Iic ⟨i, him⟩ ∧ (Iic ⟨i, him⟩ \\ t).m... | [
"case refine_2.refine_2\nR : Type u_3\nm : ℕ\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\ni : ℕ\nhim : i < m\nt : Finset (Fin m)\nht : t ∈ powersetCard (i + 1) univ\nht' : #t = #(Iic ⟨i, him⟩)\nhne : ∃ x, x ∈ Iic ⟨i, him⟩ \\ t\nhkm : (Iic ⟨i, him⟩ \\ t).min' hne ∈ Iic ⟨i, him⟩ ∧ (Iic ⟨i, him⟩ \\ t).min' hne ∉ t\... | · have hki := mem_Iic.2 (hk.le.trans <| mem_Iic.1 hkm.1)
rw [dif_pos hki, dif_pos]
by_contra h
exact lt_irrefl k <| ((lt_min'_iff _ _).1 hk) _ <| mem_sdiff.2 ⟨hki, h⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem | {
"line": 221,
"column": 33
} | {
"line": 221,
"column": 64
} | {
"line": 222,
"column": 8
} | [
{
"pp": "case add_single\nR : Type u_3\nn m : ℕ\ninst✝ : CommSemiring R\nr : R\nhnm : n ≤ m\ni : Fin n\nb✝ : ℕ\nf✝ : Fin n →₀ ℕ\na✝¹ : i ∉ f✝.support\na✝ : b✝ ≠ 0\nih : leadingCoeff (⇑toLex) (esymmAlgHomMonomial (Fin m) f✝ r) = r\n⊢ leadingCoeff (⇑toLex) (esymmAlgHomMonomial (Fin m) f✝ r * esymmAlgHomMonomial (... | [
"case add_single\nR : Type u_3\nn m : ℕ\ninst✝ : CommSemiring R\nr : R\nhnm : n ≤ m\ni : Fin n\nb✝ : ℕ\nf✝ : Fin n →₀ ℕ\na✝¹ : i ∉ f✝.support\na✝ : b✝ ≠ 0\nih : leadingCoeff (⇑toLex) (esymmAlgHomMonomial (Fin m) f✝ r) = r\n⊢ leadingCoeff (⇑toLex) (esymmAlgHomMonomial (Fin m) f✝ r * esymm (Fin m) R (↑i + 1) ^ b✝) = ... | esymmAlgHomMonomial_single_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem | {
"line": 234,
"column": 33
} | {
"line": 234,
"column": 64
} | {
"line": 235,
"column": 8
} | [
{
"pp": "case add_single\nR : Type u_3\nn m : ℕ\ninst✝¹ : CommSemiring R\nr : R\nhr : r ≠ 0\nhnm : n ≤ m\ninst✝ : Nontrivial R\ni : Fin n\nb✝ : ℕ\nf✝ : Fin n →₀ ℕ\na✝¹ : i ∉ f✝.support\na✝ : b✝ ≠ 0\nih : ⇑(ofLex (supDegree (⇑toLex) (esymmAlgHomMonomial (Fin m) f✝ r))) = (accumulate n m) ⇑f✝\nthis : ↑i < m\n⊢ ⇑(... | [
"case add_single\nR : Type u_3\nn m : ℕ\ninst✝¹ : CommSemiring R\nr : R\nhr : r ≠ 0\nhnm : n ≤ m\ninst✝ : Nontrivial R\ni : Fin n\nb✝ : ℕ\nf✝ : Fin n →₀ ℕ\na✝¹ : i ∉ f✝.support\na✝ : b✝ ≠ 0\nih : ⇑(ofLex (supDegree (⇑toLex) (esymmAlgHomMonomial (Fin m) f✝ r))) = (accumulate n m) ⇑f✝\nthis : ↑i < m\n⊢ ⇑(ofLex (supDe... | esymmAlgHomMonomial_single_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem | {
"line": 241,
"column": 6
} | {
"line": 241,
"column": 94
} | {
"line": 243,
"column": 0
} | [
{
"pp": "case add_single.hp\nR : Type u_3\nn m : ℕ\ninst✝¹ : CommSemiring R\nr : R\nhr : r ≠ 0\nhnm : n ≤ m\ninst✝ : Nontrivial R\ni : Fin n\nb✝ : ℕ\nf✝ : Fin n →₀ ℕ\na✝¹ : i ∉ f✝.support\na✝ : b✝ ≠ 0\nih : ⇑(ofLex (supDegree (⇑toLex) (esymmAlgHomMonomial (Fin m) f✝ r))) = (accumulate n m) ⇑f✝\nthis : ↑i < m\n⊢... | [] | rwa [Ne, ← leadingCoeff_eq_zero toLex.injective, leadingCoeff_esymmAlgHomMonomial _ hnm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem | {
"line": 241,
"column": 6
} | {
"line": 241,
"column": 94
} | {
"line": 243,
"column": 0
} | [
{
"pp": "case add_single.hp\nR : Type u_3\nn m : ℕ\ninst✝¹ : CommSemiring R\nr : R\nhr : r ≠ 0\nhnm : n ≤ m\ninst✝ : Nontrivial R\ni : Fin n\nb✝ : ℕ\nf✝ : Fin n →₀ ℕ\na✝¹ : i ∉ f✝.support\na✝ : b✝ ≠ 0\nih : ⇑(ofLex (supDegree (⇑toLex) (esymmAlgHomMonomial (Fin m) f✝ r))) = (accumulate n m) ⇑f✝\nthis : ↑i < m\n⊢... | [] | rwa [Ne, ← leadingCoeff_eq_zero toLex.injective, leadingCoeff_esymmAlgHomMonomial _ hnm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem | {
"line": 241,
"column": 6
} | {
"line": 241,
"column": 94
} | {
"line": 243,
"column": 0
} | [
{
"pp": "case add_single.hp\nR : Type u_3\nn m : ℕ\ninst✝¹ : CommSemiring R\nr : R\nhr : r ≠ 0\nhnm : n ≤ m\ninst✝ : Nontrivial R\ni : Fin n\nb✝ : ℕ\nf✝ : Fin n →₀ ℕ\na✝¹ : i ∉ f✝.support\na✝ : b✝ ≠ 0\nih : ⇑(ofLex (supDegree (⇑toLex) (esymmAlgHomMonomial (Fin m) f✝ r))) = (accumulate n m) ⇑f✝\nthis : ↑i < m\n⊢... | [] | rwa [Ne, ← leadingCoeff_eq_zero toLex.injective, leadingCoeff_esymmAlgHomMonomial _ hnm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities | {
"line": 109,
"column": 4
} | {
"line": 109,
"column": 77
} | {
"line": 110,
"column": 4
} | [
{
"pp": "case inl\nσ : Type u_1\ninst✝¹ : DecidableEq σ\ninst✝ : Fintype σ\nk : ℕ\nt : Finset σ × σ\nh1 : t.2 ∈ t.1\nh : #t.1 ≤ k\n⊢ #(t.1.erase t.2, t.2).1 ≤ k ∧ (#(t.1.erase t.2, t.2).1 = k → (t.1.erase t.2, t.2).2 ∈ (t.1.erase t.2, t.2).1)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
... | [
"case inl\nσ : Type u_1\ninst✝¹ : DecidableEq σ\ninst✝ : Fintype σ\nk : ℕ\nt : Finset σ × σ\nh1 : t.2 ∈ t.1\nh : #t.1 ≤ k\n⊢ #t.1 ≤ k + 1 ∧ (#t.1 - 1 = k → ¬True ∧ True)"
] | simp only [card_erase_of_mem h1, tsub_le_iff_right, mem_erase, ne_eq, h1] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.MvPowerSeries.Expand | {
"line": 39,
"column": 18
} | {
"line": 39,
"column": 34
} | {
"line": 39,
"column": 35
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝ : CommRing R\np : ℕ\nhp : p ≠ 0\nr : R\n| (expand p hp) (C r)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"MvPowerSeries.expand",
"MulOne.toOne",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"AlgHom",... | [
"σ : Type u_1\nR : Type u_3\ninst✝ : CommRing R\np : ℕ\nhp : p ≠ 0\nr : R\n| (expand p hp) (C r * 1)"
] | ← mul_one (C r), | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.RingTheory.MvPowerSeries.Expand | {
"line": 56,
"column": 2
} | {
"line": 57,
"column": 27
} | {
"line": 59,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝ : CommRing R\n⊢ expand 1 ⋯ = AlgHom.id R (MvPowerSeries σ R)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"MvPowerSeries.expand",
"Nat.instMulZeroClass",
"Nat.instOne",
"congrArg",
"CommSemiring.toSemiring",
... | [] | ext1 i
simp [expand, subst_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.Expand | {
"line": 56,
"column": 2
} | {
"line": 57,
"column": 27
} | {
"line": 59,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝ : CommRing R\n⊢ expand 1 ⋯ = AlgHom.id R (MvPowerSeries σ R)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"MvPowerSeries.expand",
"Nat.instMulZeroClass",
"Nat.instOne",
"congrArg",
"CommSemiring.toSemiring",
... | [] | ext1 i
simp [expand, subst_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Expand | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 41
} | {
"line": 156,
"column": 2
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝ : CommRing R\np : ℕ\nhp : p ≠ 0\nφ : MvPowerSeries σ R\nd n : σ →₀ ℕ\nhn₁ : n ∈ Function.support φ\nhn₂ : (fun x ↦ p • x) n = d\n⊢ d ∈ Function.support ((expand p hp) φ)",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"MvPowerSeries.expand",... | [
"σ : Type u_1\nR : Type u_3\ninst✝ : CommRing R\np : ℕ\nhp : p ≠ 0\nφ : MvPowerSeries σ R\nd n : σ →₀ ℕ\nhn₁ : n ∈ Function.support φ\nhn₂ : (fun x ↦ p • x) n = d\n⊢ (expand p hp) φ (p • n) ≠ 0"
] | simp only [← hn₂, Function.mem_support] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Morita.Matrix | {
"line": 96,
"column": 43
} | {
"line": 96,
"column": 70
} | {
"line": 96,
"column": 70
} | [
{
"pp": "R : Type u\nι : Type v\ninst✝² : Ring R\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : ModuleCat (Matrix ι ι R)\nthis : Module R ↑M := Module.compHom (↑M) (Matrix.scalar ι)\nr : R\nm : Matrix ι ι R\nx : ↑M\n⊢ (r • m) • x = ((Matrix.diagonal fun x ↦ r) * m) • x",
"ppTerm": "?m.68",
"assigned": ... | [
"R : Type u\nι : Type v\ninst✝² : Ring R\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : ModuleCat (Matrix ι ι R)\nthis : Module R ↑M := Module.compHom (↑M) (Matrix.scalar ι)\nr : R\nm : Matrix ι ι R\nx : ↑M\n⊢ ((Matrix.diagonal fun x ↦ r) * m) • x = ((Matrix.diagonal fun x ↦ r) * m) • x"
] | Matrix.smul_eq_diagonal_mul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPowerSeries.Expand | {
"line": 233,
"column": 51
} | {
"line": 245,
"column": 73
} | {
"line": 247,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝¹ : CommRing R\np : ℕ\nhp : p ≠ 0\ninst✝ : ExpChar R p\nf : MvPowerSeries σ R\n⊢ (map (frobenius R p)) ((expand p hp) f) = f ^ p",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"MvPowerSeries.expand",
"Iff.mpr",
"zero_le",
"... | [] | by
classical
rw [eq_iff_frequently_trunc'_eq, Filter.frequently_atTop]
intro n
use (p • n)
refine ⟨le_self_nsmul zero_le hp, ?_⟩
· have : (((trunc' R (p • n) f).expand p).map (frobenius R p)).toMvPowerSeries =
MvPowerSeries.map (frobenius R p) ((trunc' R (p • n) f).expand p) := by
simp only [MvP... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.WittVector.Basic | {
"line": 78,
"column": 14
} | {
"line": 78,
"column": 94
} | {
"line": 78,
"column": 94
} | [
{
"pp": "p : ℕ\nα : Type u_3\nβ : Type u_4\nf : α → β\nhf : Surjective f\nx : 𝕎 β\nn : ℕ\n⊢ (mapFun f (mk p fun n ↦ Classical.choose ⋯)).coeff n = x.coeff n",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"congrArg",
"WittVector.mk",
"Classical.choose_spec",
"Nat",... | [] | simp only [mapFun, coeff_mk, comp_apply, Classical.choose_spec (hf (x.coeff n))] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Perfection | {
"line": 159,
"column": 50
} | {
"line": 159,
"column": 65
} | {
"line": 159,
"column": 66
} | [
{
"pp": "M✝ : Type u_1\ninst✝³ : CommMonoid M✝\np✝ p : ℕ\nM : Type u_2\ninst✝² : CommMonoid M\ninst✝¹ : PerfectRing M p\nN : Type u_3\ninst✝ : CommMonoid N\nf : M →* N\nr : M\nn : ℕ\n⊢ f ((powMulEquiv M p ^ n * powMulEquiv M p).symm r ^ p) = (fun n ↦ f ((powMulEquiv M (p ^ n)).symm r)) n",
"ppTerm": "?m.108... | [
"M✝ : Type u_1\ninst✝³ : CommMonoid M✝\np✝ p : ℕ\nM : Type u_2\ninst✝² : CommMonoid M\ninst✝¹ : PerfectRing M p\nN : Type u_3\ninst✝ : CommMonoid N\nf : M →* N\nr : M\nn : ℕ\n⊢ f (((powMulEquiv M p).trans (powMulEquiv M p ^ n)).symm r ^ p) = (fun n ↦ f ((powMulEquiv M (p ^ n)).symm r)) n"
] | MulAut.mul_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Perfection | {
"line": 179,
"column": 61
} | {
"line": 179,
"column": 98
} | {
"line": 179,
"column": 98
} | [
{
"pp": "M✝ : Type u_1\ninst✝² : CommMonoid M✝\np✝ p : ℕ\nM : Type u_2\nN : Type u_3\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nφ : M →* N\nf : Perfection M p\nn : ℕ\n⊢ (fun n ↦ φ ((coeffMonoidHom M p n) f)) (n + 1) ^ p = (fun n ↦ φ ((coeffMonoidHom M p n) f)) n",
"ppTerm": "?m.27",
"assigned": true,... | [] | rw [← map_pow, coeffMonoidHom_pow_p'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Perfection | {
"line": 179,
"column": 61
} | {
"line": 179,
"column": 98
} | {
"line": 179,
"column": 98
} | [
{
"pp": "M✝ : Type u_1\ninst✝² : CommMonoid M✝\np✝ p : ℕ\nM : Type u_2\nN : Type u_3\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nφ : M →* N\nf : Perfection M p\nn : ℕ\n⊢ (fun n ↦ φ ((coeffMonoidHom M p n) f)) (n + 1) ^ p = (fun n ↦ φ ((coeffMonoidHom M p n) f)) n",
"ppTerm": "?m.27",
"assigned": true,... | [] | rw [← map_pow, coeffMonoidHom_pow_p'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Perfection | {
"line": 179,
"column": 61
} | {
"line": 179,
"column": 98
} | {
"line": 179,
"column": 98
} | [
{
"pp": "M✝ : Type u_1\ninst✝² : CommMonoid M✝\np✝ p : ℕ\nM : Type u_2\nN : Type u_3\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nφ : M →* N\nf : Perfection M p\nn : ℕ\n⊢ (fun n ↦ φ ((coeffMonoidHom M p n) f)) (n + 1) ^ p = (fun n ↦ φ ((coeffMonoidHom M p n) f)) n",
"ppTerm": "?m.27",
"assigned": true,... | [] | rw [← map_pow, coeffMonoidHom_pow_p'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Perfection | {
"line": 571,
"column": 6
} | {
"line": 571,
"column": 23
} | {
"line": 571,
"column": 24
} | [
{
"pp": "K : Type u₁\ninst✝² : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝¹ : CommRing O\ninst✝ : Algebra O K\nhv : v.Integers O\np : ℕ\nr : O\nhx0 : (Ideal.Quotient.mk (Ideal.span {↑p})) r ≠ 0\ns : O\nhy0 : (Ideal.Quotient.mk (Ideal.span {↑p})) s ≠ 0\nhxy0 : (Ideal.Quotient.mk (Ideal.span {↑p})) (r * s) ≠... | [
"K : Type u₁\ninst✝² : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝¹ : CommRing O\ninst✝ : Algebra O K\nhv : v.Integers O\np : ℕ\nr : O\nhx0 : (Ideal.Quotient.mk (Ideal.span {↑p})) r ≠ 0\ns : O\nhy0 : (Ideal.Quotient.mk (Ideal.span {↑p})) s ≠ 0\nhxy0 : (Ideal.Quotient.mk (Ideal.span {↑p})) (r * s) ≠ 0\n⊢ preVal... | preVal_mk hv hx0, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Perfection | {
"line": 584,
"column": 6
} | {
"line": 584,
"column": 23
} | {
"line": 584,
"column": 24
} | [
{
"pp": "case neg\nK : Type u₁\ninst✝² : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝¹ : CommRing O\ninst✝ : Algebra O K\nhv : v.Integers O\np : ℕ\nr : O\nhx0 : (Ideal.Quotient.mk (Ideal.span {↑p})) r ≠ 0\ns : O\nhy0 : (Ideal.Quotient.mk (Ideal.span {↑p})) s ≠ 0\nhxy0 : ¬(Ideal.Quotient.mk (Ideal.span {↑p})... | [
"case neg\nK : Type u₁\ninst✝² : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝¹ : CommRing O\ninst✝ : Algebra O K\nhv : v.Integers O\np : ℕ\nr : O\nhx0 : (Ideal.Quotient.mk (Ideal.span {↑p})) r ≠ 0\ns : O\nhy0 : (Ideal.Quotient.mk (Ideal.span {↑p})) s ≠ 0\nhxy0 : ¬(Ideal.Quotient.mk (Ideal.span {↑p})) (r + s) = ... | preVal_mk hv hx0, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.WittVector.Verschiebung | {
"line": 179,
"column": 6
} | {
"line": 184,
"column": 55
} | {
"line": 186,
"column": 0
} | [] | [] | _ = ghostComponent (n + 1) (verschiebung <| mk p x) := by
apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl
funext k
simp only [← aeval_verschiebungPoly]
exact eval₂Hom_congr (RingHom.ext_int _ _) rfl rfl
_ = _ := by rw [ghostComponent_verschiebung]; rfl | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.RingTheory.WittVector.Frobenius | {
"line": 233,
"column": 4
} | {
"line": 233,
"column": 58
} | {
"line": 234,
"column": 4
} | [
{
"pp": "p : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\n⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"WittVector.instOne",
"RingHom.instRingHom... | [
"p : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\n⊢ ∀ (R : Type u_1) [_Rcr : CommRing R] (n : ℕ), (ghostComponent n) (frobeniusFun 1) = 1"
] | simp only [Function.comp_apply, map_one, forall_const] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Polynomial.Dickson | {
"line": 148,
"column": 58
} | {
"line": 148,
"column": 78
} | {
"line": 149,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\n⊢ X * dickson 1 1 (n + 1) - 1 * dickson 1 1 n = Chebyshev.C R (↑n + 2)",
"ppTerm": "?m.74",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.instOne",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"congrArg",
... | [
"R : Type u_1\ninst✝ : CommRing R\nn : ℕ\n⊢ X * dickson 1 1 (n + 1) - 1 * dickson 1 1 n = X * Chebyshev.C R (↑n + 1) - Chebyshev.C R ↑n"
] | Chebyshev.C_add_two, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Dickson | {
"line": 256,
"column": 12
} | {
"line": 256,
"column": 31
} | {
"line": 256,
"column": 32
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nK : Type\nw✝¹ : Field K\nw✝ : CharP K p\nH : Set.univ.Infinite\nh : {x | ∃ y, x = y + y⁻¹ ∧ y ≠ 0}.Finite\nx : K\nx✝ : x ∈ {x | ∃ y, x = y + y⁻¹ ∧ y ≠ 0}\nφ : K[X] := ⋯\nhφ : φ ≠ 0\ny : K\nhy : ¬y = 0\n⊢ x = y + y⁻¹ ↔ y ^ 2 - x * y + 1 = 0",
"ppTerm": "?m.419",
... | [
"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nK : Type\nw✝¹ : Field K\nw✝ : CharP K p\nH : Set.univ.Infinite\nh : {x | ∃ y, x = y + y⁻¹ ∧ y ≠ 0}.Finite\nx : K\nx✝ : x ∈ {x | ∃ y, x = y + y⁻¹ ∧ y ≠ 0}\nφ : K[X] := X ^ 2 - C x * X + 1\nhφ : φ ≠ 0\ny : K\nhy : ¬y = 0\n⊢ x * y = (y + y⁻¹) * y ↔ y ^ 2 - x * y + 1 = 0"
] | ← mul_left_inj' hy, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Eisenstein.Distinguished | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 32
} | {
"line": 46,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nI : Ideal R\ndistinguish : f.IsDistinguishedAt I\ni : ℕ\nne : i = f.natDegree\n⊢ (map (Ideal.Quotient.mk I) f).coeff i = (X ^ f.natDegree).coeff i",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCom... | [
"case neg\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nI : Ideal R\ndistinguish : f.IsDistinguishedAt I\ni : ℕ\nne : ¬i = f.natDegree\n⊢ (map (Ideal.Quotient.mk I) f).coeff i = (X ^ f.natDegree).coeff i"
] | · simp [ne, distinguish.monic] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Polynomial.Hermite.Basic | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 61
} | {
"line": 200,
"column": 0
} | [
{
"pp": "case neg\nn k : ℕ\nh : ¬Even (n + k)\n⊢ (hermite n).coeff k = 0",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Odd",
"Nat.not_even_iff_odd",
"instHAdd",
"HAdd.hAdd",
"Nat",
"Even",
"Polynomial.coeff_hermite_of_odd_add",
"Iff.mp",
... | [] | · exact coeff_hermite_of_odd_add (Nat.not_even_iff_odd.1 h) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Polynomial.Opposites | {
"line": 95,
"column": 2
} | {
"line": 96,
"column": 44
} | {
"line": 98,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : Semiring R\np : R[X]ᵐᵒᵖ\np0 : ¬p = 0\n⊢ ((opRingEquiv R) p).natDegree = (unop p).natDegree",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"False",
"eq_false",
"Finset.max'.congr_simp",
"congrArg",
"Rin... | [] | · simp only [p0, natDegree_eq_support_max', Ne, EmbeddingLike.map_eq_zero_iff, not_false_iff,
support_opRingEquiv, unop_eq_zero_iff] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.PowerSeries.Expand | {
"line": 40,
"column": 18
} | {
"line": 40,
"column": 34
} | {
"line": 40,
"column": 35
} | [
{
"pp": "R : Type u_2\ninst✝ : CommRing R\np : ℕ\nhp : p ≠ 0\nr : R\n| (expand p hp) (C r)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"AlgHom",
"AlgHom.funLike",
"RingHom",
... | [
"R : Type u_2\ninst✝ : CommRing R\np : ℕ\nhp : p ≠ 0\nr : R\n| (expand p hp) (C r * 1)"
] | ← mul_one (C r), | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.RingTheory.Radical.NatInt | {
"line": 41,
"column": 39
} | {
"line": 45,
"column": 30
} | {
"line": 47,
"column": 0
} | [
{
"pp": "⊢ primeFactors = Nat.primeFactors",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"UniqueFactorizationMonoid.normalizedFactors",
"Multiset.toFinset",
"Eq.mpr",
"instNormalizedGCDMonoidNat",
"NormalizationMonoid.ofUniqueUnits",
"congrArg",
"... | [] | by
ext n : 1
rw [primeFactors, Nat.factors_eq, Nat.primeFactors]
-- this convert is necessary because of the different DecidableEq instances
convert! List.toFinset_coe _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.PolynomialLaw.Basic | {
"line": 385,
"column": 2
} | {
"line": 416,
"column": 19
} | {
"line": 418,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝⁶ : CommSemiring R\nM : Type u_1\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nN : Type u_2\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\nS : Type v\ninst✝¹ : CommSemiring S\ninst✝ : Algebra R S\nf : M →ₚₗ[R] N\n⊢ Function.FactorsThrough (toFunLifted S f) (π R M S)",
"ppTerm": ... | [] | rintro ⟨s, p⟩ ⟨s', p'⟩ h
simp only [toFunLifted]
set u := rTensor M (φ R s).rangeRestrict.toLinearMap p with hu
have uFG : Subalgebra.FG (R := R) (φ R s).range := by
rw [← Algebra.map_top]
exact Subalgebra.FG.map _ Algebra.FiniteType.out
set u' := rTensor M (φ R s').rangeRestrict.toLinearMap p' with hu'... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PolynomialLaw.Basic | {
"line": 385,
"column": 2
} | {
"line": 416,
"column": 19
} | {
"line": 418,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝⁶ : CommSemiring R\nM : Type u_1\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nN : Type u_2\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\nS : Type v\ninst✝¹ : CommSemiring S\ninst✝ : Algebra R S\nf : M →ₚₗ[R] N\n⊢ Function.FactorsThrough (toFunLifted S f) (π R M S)",
"ppTerm": ... | [] | rintro ⟨s, p⟩ ⟨s', p'⟩ h
simp only [toFunLifted]
set u := rTensor M (φ R s).rangeRestrict.toLinearMap p with hu
have uFG : Subalgebra.FG (R := R) (φ R s).range := by
rw [← Algebra.map_top]
exact Subalgebra.FG.map _ Algebra.FiniteType.out
set u' := rTensor M (φ R s').rangeRestrict.toLinearMap p' with hu'... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.SimpleModule.Isotypic | {
"line": 119,
"column": 23
} | {
"line": 119,
"column": 77
} | {
"line": 119,
"column": 77
} | [
{
"pp": "R : Type u_2\nM : Type u\nS : Type u_4\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup S\ninst✝¹ : Module R M\ninst✝ : Module R S\nN : Submodule R M\n⊢ IsIsotypicOfType R (↥N) S ↔ ∀ (x : { a // a ≤ N }) [IsSimpleModule R ↥↑x], Nonempty (↥↑x ≃ₗ[R] S)",
"ppTerm": "?m.39",
"assign... | [
"R : Type u_2\nM : Type u\nS : Type u_4\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup S\ninst✝¹ : Module R M\ninst✝ : Module R S\nN : Submodule R M\n⊢ IsIsotypicOfType R (↥N) S ↔\n ∀ (a : Submodule R ↥N) [IsSimpleModule R ↥↑((Submodule.MapSubtype.orderIso N).toEquiv a)],\n Nonempty (↥↑((S... | ← (Submodule.MapSubtype.orderIso N).forall_congr_right | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.SimpleModule.Isotypic | {
"line": 127,
"column": 23
} | {
"line": 127,
"column": 77
} | {
"line": 127,
"column": 77
} | [
{
"pp": "R : Type u_2\nM : Type u\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nN : Submodule R M\n⊢ IsIsotypic R ↥N ↔ ∀ (x : { a // a ≤ N }) [IsSimpleModule R ↥↑x], IsIsotypicOfType R ↥N ↥↑x",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule"... | [
"R : Type u_2\nM : Type u\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nN : Submodule R M\n⊢ IsIsotypic R ↥N ↔\n ∀ (a : Submodule R ↥N) [IsSimpleModule R ↥↑((Submodule.MapSubtype.orderIso N).toEquiv a)],\n IsIsotypicOfType R ↥N ↥↑((Submodule.MapSubtype.orderIso N).toEquiv a)"
] | ← (Submodule.MapSubtype.orderIso N).forall_congr_right | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.SimpleModule.Isotypic | {
"line": 419,
"column": 74
} | {
"line": 419,
"column": 90
} | {
"line": 419,
"column": 90
} | [
{
"pp": "R : Type u_2\nM : Type u\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsSemisimpleModule R M\nm : Submodule R M\nh : m.IsFullyInvariant\nS : Submodule R M\nle : S ≤ m\nx✝¹ : IsSimpleModule R ↥S\nS' : Submodule R M\nx✝ : S' ∈ {m | Nonempty (↥m ≃ₗ[R] ↥S)}\ne : ↥S' ≃ₗ[R] ↥S\np :... | [
"R : Type u_2\nM : Type u\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsSemisimpleModule R M\nm : Submodule R M\nh : m.IsFullyInvariant\nS : Submodule R M\nle : S ≤ m\nx✝¹ : IsSimpleModule R ↥S\nS' : Submodule R M\nx✝ : S' ∈ {m | Nonempty (↥m ≃ₗ[R] ↥S)}\ne : ↥S' ≃ₗ[R] ↥S\np : M →ₗ[R] M\n... | S'.range_subtype | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Spectrum.Prime.IsOpenComapC | {
"line": 69,
"column": 2
} | {
"line": 73,
"column": 47
} | {
"line": 75,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\n⊢ IsOpenMap (PrimeSpectrum.comap C)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Set.compl_iInter",
"PrimeSpectrum.zeroLocus",
"compl_compl",
"congrArg",
"CommSemiring.toSemir... | [] | rintro U ⟨s, z⟩
rw [← compl_compl U, ← z, ← iUnion_of_singleton_coe s, zeroLocus_iUnion, compl_iInter,
image_iUnion]
simp_rw [← imageOfDf_eq_comap_C_compl_zeroLocus]
exact isOpen_iUnion fun f => isOpen_imageOfDf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Spectrum.Prime.IsOpenComapC | {
"line": 69,
"column": 2
} | {
"line": 73,
"column": 47
} | {
"line": 75,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\n⊢ IsOpenMap (PrimeSpectrum.comap C)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Set.compl_iInter",
"PrimeSpectrum.zeroLocus",
"compl_compl",
"congrArg",
"CommSemiring.toSemir... | [] | rintro U ⟨s, z⟩
rw [← compl_compl U, ← z, ← iUnion_of_singleton_coe s, zeroLocus_iUnion, compl_iInter,
image_iUnion]
simp_rw [← imageOfDf_eq_comap_C_compl_zeroLocus]
exact isOpen_iUnion fun f => isOpen_imageOfDf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.WittVector.DiscreteValuationRing | {
"line": 151,
"column": 4
} | {
"line": 153,
"column": 25
} | {
"line": 153,
"column": 25
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : Field k\ninst✝¹ : CharP k p\ninst✝ : PerfectRing k p\n⊢ IsDiscreteValuationRing.HasUnitMulPowIrreducibleFactorization (𝕎 k)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Units.val",
"HMul.hMul",
"WittVecto... | [] | refine ⟨p, irreducible p, fun {x} hx => ?_⟩
obtain ⟨n, b, hb⟩ := exists_eq_pow_p_mul' x hx
exact ⟨n, b, hb.symm⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.WittVector.DiscreteValuationRing | {
"line": 151,
"column": 4
} | {
"line": 153,
"column": 25
} | {
"line": 153,
"column": 25
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : Field k\ninst✝¹ : CharP k p\ninst✝ : PerfectRing k p\n⊢ IsDiscreteValuationRing.HasUnitMulPowIrreducibleFactorization (𝕎 k)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Units.val",
"HMul.hMul",
"WittVecto... | [] | refine ⟨p, irreducible p, fun {x} hx => ?_⟩
obtain ⟨n, b, hb⟩ := exists_eq_pow_p_mul' x hx
exact ⟨n, b, hb.symm⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 143,
"column": 6
} | {
"line": 143,
"column": 19
} | {
"line": 143,
"column": 20
} | [
{
"pp": "p n : ℕ\nmvpz : ↑p ^ (n + 1) = C (↑p ^ (n + 1))\n⊢ wittPolyProd p (n + 1) =\n -(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +\n ↑p ^ (n + 1) * X (0, n + 1) * (rename (Prod.mk 1)) (wittPolynomial p ℤ (n + 1)) +\n ↑p ^ (n + 1) * X (1, n + 1) * (rename (Prod.mk 0)) (wit... | [
"p n : ℕ\nmvpz : ↑p ^ (n + 1) = C (↑p ^ (n + 1))\n⊢ (rename (Prod.mk 0)) (wittPolynomial p ℤ (n + 1)) * (rename (Prod.mk 1)) (wittPolynomial p ℤ (n + 1)) =\n -(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +\n ↑p ^ (n + 1) * X (0, n + 1) * (rename (Prod.mk 1)) (wittPolynomial p ℤ (n + 1))... | wittPolyProd, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 251,
"column": 2
} | {
"line": 251,
"column": 12
} | {
"line": 252,
"column": 2
} | [
{
"pp": "case h\np : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝¹ : CommRing k\ninst✝ : CharP k p\nn : ℕ\nf₀ : (↑↑(univ ×ˢ range (n + 1)) → k) → k\nhf₀ : ∀ (x : Fin 2 × ℕ → k), f₀ (x ∘ Subtype.val) = (aeval x) (polyOfInterest p n)\nf : TruncatedWittVector p (n + 1) k → TruncatedWittVector p (n + 1) k → k :=... | [
"case h\np : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝¹ : CommRing k\ninst✝ : CharP k p\nn : ℕ\nf₀ : (↑↑(univ ×ˢ range (n + 1)) → k) → k\nhf₀ : ∀ (x : Fin 2 × ℕ → k), f₀ (x ∘ Subtype.val) = (aeval x) (polyOfInterest p n)\nf : TruncatedWittVector p (n + 1) k → TruncatedWittVector p (n + 1) k → k :=\n fun x y ... | rw [← hf₀] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.SetTheory.Cardinal.Cofinality.Club | {
"line": 144,
"column": 6
} | {
"line": 145,
"column": 51
} | {
"line": 146,
"column": 6
} | [
{
"pp": "case inr.refine_2.inr\nα : Type v\ninst✝¹ : LinearOrder α\ninst✝ : WellFoundedLT α\nf : α → α\nhα : cof α ≠ ℵ₀\nhf : IsNormal f\nh✝ : Nonempty α\na : α\nh : NoMaxOrder α\n⊢ BddAbove (Set.range fun n ↦ f^[n] a)",
"ppTerm": "?inr.refine_2.inr",
"assigned": true,
"usedConstants": [
"Card... | [
"case inr.refine_2.inr\nα : Type v\ninst✝¹ : LinearOrder α\ninst✝ : WellFoundedLT α\nf : α → α\nhα : cof α ≠ ℵ₀\nhf : IsNormal f\nh✝¹ : Nonempty α\na : α\nh✝ : NoMaxOrder α\nh : IsCofinal (Set.range fun n ↦ f^[n] a)\n⊢ #↑(Set.range fun n ↦ f^[n] a) ≤ ℵ₀"
] | refine .of_not_isCofinal fun h ↦ (cof_le h).not_gt
((aleph0_le_cof.lt_of_ne' hα).trans_le' ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.SetTheory.Descriptive.Tree | {
"line": 32,
"column": 32
} | {
"line": 32,
"column": 53
} | {
"line": 32,
"column": 53
} | [
{
"pp": "A : Type u_1\nS : Set (Set (List A))\nhS : S ⊆ {T | ∀ ⦃x : List A⦄ ⦃a : A⦄, x ++ [a] ∈ T → x ∈ T}\nx : List A\na : A\nh : x ++ [a] ∈ sInf S\nT : Set (List A)\nhT : T ∈ S\n⊢ x ∈ T",
"ppTerm": "?m.74",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"hS",
"T",
"hT"... | [] | exact hS hT <| h T hT | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.SetTheory.Lists | {
"line": 189,
"column": 4
} | {
"line": 189,
"column": 44
} | {
"line": 190,
"column": 4
} | [
{
"pp": "α : Type u_1\nl₁ l₂ : Lists' α true\nH : ∀ (a : Lists α), a ∈ l₁.toList → a ∈ l₂\n⊢ l₁ ⊆ l₂",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Lists'.toList",
"Lists'.recOfList",
"Lists",
"Membership.mem",
"Lists'.instMembershipLists",
"HasSubset.... | [
"case ofList\nα : Type u_1\nl₂ : Lists' α true\nl₁ : List (Lists α)\nH : ∀ (a : Lists α), a ∈ (ofList l₁).toList → a ∈ l₂\n⊢ ofList l₁ ⊆ l₂"
] | induction l₁ using recOfList with | _ l₁ | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Topology.Order.SuccPred | {
"line": 80,
"column": 2
} | {
"line": 80,
"column": 27
} | {
"line": 81,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace α\ninst✝² : OrderTopology α\ninst✝¹ : SuccOrder α\ninst✝ : NoMaxOrder α\na : α\ns : Set α\n⊢ AccPt a (𝓟 s) ↔ ¬IsMin a ∧ ∀ b < a, (s ∩ Ioo b a).Nonempty",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"α : Type u_1\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace α\ninst✝² : OrderTopology α\ninst✝¹ : SuccOrder α\ninst✝ : NoMaxOrder α\na : α\ns : Set α\n⊢ (∃ᶠ (y : α) in 𝓝 a, y ≠ a ∧ y ∈ s) ↔ ¬IsMin a ∧ ∀ b < a, (s ∩ Ioo b a).Nonempty"
] | rw [accPt_iff_frequently] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
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