module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 300,
"column": 4
} | {
"line": 300,
"column": 53
} | [
{
"pp": "case pos\nε : Type u_3\ninst✝¹ : TopologicalSpace ε\ninst✝ : ENormedAddMonoid ε\nf : ℝ → ε\na b : ℝ\nμ : Measure ℝ\nh : IntervalIntegrable f μ a b\nhab : a ≤ b\n⊢ AEStronglyMeasurable f (μ.restrict (Ι a b))",
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"Real",
"congrArg",
... | rw [uIoc_of_le hab]; exact h.aestronglyMeasurable | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 300,
"column": 4
} | {
"line": 300,
"column": 53
} | [
{
"pp": "case pos\nε : Type u_3\ninst✝¹ : TopologicalSpace ε\ninst✝ : ENormedAddMonoid ε\nf : ℝ → ε\na b : ℝ\nμ : Measure ℝ\nh : IntervalIntegrable f μ a b\nhab : a ≤ b\n⊢ AEStronglyMeasurable f (μ.restrict (Ι a b))",
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"Real",
"congrArg",
... | rw [uIoc_of_le hab]; exact h.aestronglyMeasurable | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.NonIntegrable | {
"line": 65,
"column": 4
} | {
"line": 67,
"column": 84
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : CompleteSpace E\nf : ℝ → E\ng : ℝ → F\nk : Set ℝ\nl : Filter ℝ\ninst✝¹ : l.NeBot\ninst✝ : TendstoIxxClass Icc l l\nhl : k ∈ l\nhd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x\nhf : ... | have h : ∀ᶠ x : ℝ × ℝ in l ×ˢ l,
∀ y ∈ [[x.1, x.2]], (DifferentiableAt ℝ f y ∧ ‖deriv f y‖ ≤ C * ‖g y‖) ∧ y ∈ k :=
(tendsto_fst.uIcc tendsto_snd).eventually ((hd.and hC.bound).and hl).smallSets | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 450,
"column": 43
} | {
"line": 453,
"column": 6
} | [
{
"pp": "ε : Type u_3\ninst✝² : TopologicalSpace ε\ninst✝¹ : ENormedAddMonoid ε\nf : ℝ → ε\na b : ℝ\ninst✝ : PseudoMetrizableSpace ε\nc : ℝ\nh : ‖f (min a b)‖ₑ ≠ ∞\n⊢ IntervalIntegrable (fun x ↦ f (c + x)) volume (a - c) (b - c) ↔ IntervalIntegrable f volume a b",
"usedConstants": [
"Eq.mpr",
"R... | by
simp_rw [add_comm c]
rw [IntervalIntegrable.comp_add_right_iff (by grind)]
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 791,
"column": 57
} | {
"line": 792,
"column": 55
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\na b : ℝ\nμ : Measure ℝ\ninst✝³ : NormedDivisionRing 𝕜\ninst✝² : Module 𝕜 E\ninst✝¹ : NormSMulClass 𝕜 E\ninst✝ : SMulCommClass ℝ 𝕜 E\nr : 𝕜\nf : ℝ → E\n⊢ ∫ (x : ℝ) in a..b, r • f x ∂μ = r • ∫ (x : ℝ) in a..b, f x ... | by
simp only [intervalIntegral, integral_smul, smul_sub] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1000,
"column": 26
} | {
"line": 1000,
"column": 58
} | [
{
"pp": "case pos\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b : ℝ\nf : ℝ → E\nc d : ℝ\nhc : c = 0\n⊢ c • ∫ (x : ℝ) in a..b, f (d - c * x) = ∫ (x : ℝ) in d - c * b..d - c * a, f x",
"usedConstants": [
"Real",
"instHSMul",
"MeasureTheory.Measure",
"HMul.h... | simp [hc, integral_comp_sub_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1000,
"column": 26
} | {
"line": 1000,
"column": 58
} | [
{
"pp": "case neg\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b : ℝ\nf : ℝ → E\nc d : ℝ\nhc : ¬c = 0\n⊢ c • ∫ (x : ℝ) in a..b, f (d - c * x) = ∫ (x : ℝ) in d - c * b..d - c * a, f x",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"False",
"Real",
"... | simp [hc, integral_comp_sub_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 452,
"column": 4
} | {
"line": 464,
"column": 65
} | [
{
"pp": "case refine_3\nE : Type u_1\nX : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : TopologicalSpace X\nμ : Measure ℝ\ninst✝ : FirstCountableTopology X\nF : X → ℝ → E\nbound : ℝ → ℝ\na b a₀ b₀ : ℝ\nx₀ : X\nhF_meas : ∀ (x : X), AEStronglyMeasurable (F x) (μ.restrict (Ι a b))\nh_... | have : ∀ᶠ p : X × ℝ in 𝓝 (x₀, b₀),
‖∫ s in b₀..p.2, F p.1 s - F x₀ s ∂μ‖ ≤ |∫ s in b₀..p.2, 2 * bound s ∂μ| := by
rw [nhds_prod_eq]
refine (h_bound.prod_mk Ioo_nhds).mono ?_
rintro ⟨x, t⟩ ⟨hx : ∀ᵐ t ∂μ.restrict (Ι a b), ‖F x t‖ ≤ bound t, ht : t ∈ Ioo a b⟩
have H : ∀ᵐ t : ℝ ∂μ.restrict ... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 598,
"column": 2
} | {
"line": 598,
"column": 62
} | [
{
"pp": "E : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E\nca cb : E\na b : ℝ\nhf : IntervalIntegrable f volume a b\nhmeas_a : StronglyMeasurableAtFilter f (𝓝 a) volume\nhmeas_b : StronglyMeasurableAtFilter f (𝓝 b) volume\nha : Tendsto f (𝓝 a ⊓ ae volu... | refine .of_isLittleO <| (this.congr_left ?_).trans_isBigO ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 802,
"column": 2
} | {
"line": 802,
"column": 62
} | [
{
"pp": "E : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\nf : ℝ → E\nca cb : E\nla lb : Filter ℝ\na b : ℝ\nhf : IntervalIntegrable f volume a b\ns t : Set ℝ\ninst✝¹ : FTCFilter a (𝓝[s] a) la\ninst✝ : FTCFilter b (𝓝[t] b) lb\nhmeas_a : StronglyMeasurableAtFilter ... | refine .of_isLittleO <| (this.congr_left ?_).trans_isBigO ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1085,
"column": 6
} | {
"line": 1086,
"column": 51
} | [
{
"pp": "case refine_2\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nμ : Measure ℝ\na : ℕ → ℝ\nm n : ℕ\nhmn : m ≤ n\np : ℕ\nhmp : m ≤ p\nIH :\n (∀ k ∈ Ico m p, IntervalIntegrable f μ (a k) (a (k + 1))) →\n ∑ k ∈ Finset.Ico m p, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ = ∫ (x :... | intro k hk
exact h _ (Ico_subset_Ico_right p.le_succ hk) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1085,
"column": 6
} | {
"line": 1086,
"column": 51
} | [
{
"pp": "case refine_2\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nμ : Measure ℝ\na : ℕ → ℝ\nm n : ℕ\nhmn : m ≤ n\np : ℕ\nhmp : m ≤ p\nIH :\n (∀ k ∈ Ico m p, IntervalIntegrable f μ (a k) (a (k + 1))) →\n ∑ k ∈ Finset.Ico m p, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ = ∫ (x :... | intro k hk
exact h _ (Ico_subset_Ico_right p.le_succ hk) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Haar.Quotient | {
"line": 338,
"column": 2
} | {
"line": 338,
"column": 15
} | [
{
"pp": "G : Type u_1\ninst✝⁸ : Group G\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : IsTopologicalGroup G\ninst✝⁴ : BorelSpace G\nμ : Measure G\nΓ : Subgroup G\n𝓕 : Set G\nh𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 μ\ninst✝³ : Countable ↥Γ\ninst✝² : MeasurableSpace (G ⧸ Γ)\ninst✝¹ : BorelSpace ... | intro ⟨γ, hγ⟩ | Lean.Elab.Tactic.evalIntro | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1124,
"column": 46
} | {
"line": 1124,
"column": 92
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na✝ b✝ : ℝ\nf : ℝ → E\nμ : Measure ℝ\na b : ℝ\nha : IntegrableOn f (Iic a) μ\nhb : IntegrableOn f (Iic b) μ\nhab : a ≤ b\n⊢ ∫ (x : ℝ) in Iic b, f x ∂μ = ∫ (x : ℝ) in Iic a, f x ∂μ + ∫ (x : ℝ) in Ioc a b, f x ∂μ",
"usedConstants": ... | ← setIntegral_union (Iic_disjoint_Ioc le_rfl), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Haar.Quotient | {
"line": 461,
"column": 2
} | {
"line": 463,
"column": 32
} | [
{
"pp": "G' : Type u_1\ninst✝¹⁰ : AddGroup G'\ninst✝⁹ : MeasurableSpace G'\ninst✝⁸ : TopologicalSpace G'\ninst✝⁷ : IsTopologicalAddGroup G'\ninst✝⁶ : BorelSpace G'\nμ' : Measure G'\nΓ' : AddSubgroup G'\n𝓕' : Set G'\nh𝓕 : IsAddFundamentalDomain (↥Γ'.op) 𝓕' μ'\ninst✝⁵ : Countable ↥Γ'\ninst✝⁴ : MeasurableSpace ... | have H₂ : AEStronglyMeasurable (QuotientAddGroup.automorphize ((g ∘ π) * f)) μ_𝓕 := by
simp_rw [H₀]
exact hg.mul F_ae_measurable | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 73,
"column": 6
} | {
"line": 73,
"column": 32
} | [
{
"pp": "T : ℝ\nhT : Fact (0 < T)\n⊢ ENNReal.ofReal T * (addHaarMeasure ⊤) univ = ENNReal.ofReal T",
"usedConstants": [
"Eq.mpr",
"TopologicalSpace.PositiveCompacts.coe_top",
"Real",
"MeasureTheory.Measure",
"HMul.hMul",
"TopologicalSpace.PositiveCompacts.instSetLike",
... | ← PositiveCompacts.coe_top | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 1084,
"column": 2
} | {
"line": 1091,
"column": 100
} | [
{
"pp": "case inr\ng' g φ : ℝ → ℝ\na b : ℝ\nhab : a ≤ b\nhcont : ContinuousOn g (Icc a b)\nhderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x\nφint : IntegrableOn φ (Icc a b) volume\nhφg : ∀ x ∈ Ioo a b, g' x ≤ φ x\na_lt_b : a < b\ns : Set ℝ := {t | g b - g t ≤ ∫ (u : ℝ) in t..b, φ u} ∩ Icc a b\ns_clos... | have A : closure (Ioc a b) ⊆ s := by
apply s_closed.closure_subset_iff.2
intro t ht
refine ⟨?_, ⟨ht.1.le, ht.2⟩⟩
exact
sub_le_integral_of_hasDeriv_right_of_le_Ico ht.2 (hcont.mono (Icc_subset_Icc ht.1.le le_rfl))
(fun x hx => hderiv x ⟨ht.1.trans_le hx.1, hx.2⟩)
(φint.mono_set (Icc... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 306,
"column": 37
} | {
"line": 306,
"column": 48
} | [
{
"pp": "E✝ : Type u_1\ninst✝¹ : NormedAddCommGroup E✝\nf✝ : ℝ → E✝\nT✝ : ℝ\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\nT t : ℝ\nh₁f : Periodic f T\nhT✝ : T ≠ 0\nh₂f : IntervalIntegrable f volume t (t + T)\na₁ a₂ : ℝ\nhT : 0 < T\nn₁ : ℕ\nhn₁ : (t - min a₁ a₂) / T ≤ ↑n₁\nn₂ : ℕ\nhn₂ : (max a₁ a₂ - t)... | by simp [a] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 562,
"column": 4
} | {
"line": 562,
"column": 83
} | [
{
"pp": "case inr.hf\nn : ℤ\nc w : ℂ\nR : ℝ\nhn : n < 0\nhw : w ∈ sphere c |R|\nh0 : R ≠ 0\n⊢ ¬CircleIntegrable (fun z ↦ (z - w) ^ n) c R",
"usedConstants": [
"circleIntegrable_sub_zpow_iff._simp_1",
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"False",
"... | simpa [circleIntegrable_sub_zpow_iff, *, not_or] using mem_sphere_iff_norm.1 hw | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 731,
"column": 60
} | {
"line": 733,
"column": 72
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nμ : Measure ℝ\nf : ℝ → E\ninst✝ : NoAtoms μ\na₀ : ℝ\nhf : IntegrableOn f (Ici a₀) μ\n⊢ ContinuousOn (fun b ↦ ∫ (x : ℝ) in Ici b, f x ∂μ) (Ici a₀)",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.integral_Ici_eq_integral... | by
simp_rw [integral_Ici_eq_integral_Ioi]
exact (hf.mono_set Ioi_subset_Ici_self).continuousOn_Ici_primitive_Ioi | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 211,
"column": 2
} | {
"line": 211,
"column": 61
} | [
{
"pp": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : s.Countable\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[w.im, z.im]])\nHd : ∀ x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min w.im z.im) (max w.im z.im) \\ s, HasFDerivAt f (f' ... | have htc : ContinuousOn F R := Hc.comp e.continuousOn hR.ge | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.Prod | {
"line": 269,
"column": 4
} | {
"line": 269,
"column": 55
} | [
{
"pp": "case h₁.h\nα : Type u_1\nβ : Type u_2\nE : Type u_3\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SFinite ν\nf : α × β → E\nh1f : AEStronglyMeasurable f (μ.prod ν)\n⊢ ∀ᵐ (x : α) ∂μ,\n HasFiniteIntegral (fun y ↦ AEStrongl... | filter_upwards [ae_ae_of_ae_prod h1f.ae_eq_mk.symm] | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Analysis.BoxIntegral.DivergenceTheorem | {
"line": 223,
"column": 67
} | {
"line": 224,
"column": 39
} | [
{
"pp": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nn : ℕ\ninst✝ : CompleteSpace E\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) → ℝ) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E\ns : Set (Fin (n + 1) → ℝ)\nhs : s.Countable\nHs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x\nHd : ... | by
gcongr with j _; exact this j | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Prod | {
"line": 344,
"column": 36
} | {
"line": 344,
"column": 63
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nE : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝³ : NormedAddCommGroup E\nR : Type u_4\ninst✝² : NormedRing R\ninst✝¹ : Module R E\ninst✝ : IsBoundedSMul R E\nf : α → R\ng : β → E\nhf : Integrable f μ\nhg : Integrable ... | by simpa using norm_smul_le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Prod | {
"line": 543,
"column": 2
} | {
"line": 545,
"column": 38
} | [
{
"pp": "case inl\nα : Type u_1\nE : Type u_3\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : SFinite μ\na b : ℝ\nf : ℝ → α → E\nh_int : Integrable (uncurry f) ((volume.restrict (uIoc a b)).prod μ)\nhab : a ≤ b\n⊢ ∫ (x : ℝ) in a..b, ∫ (y : α), f x y ∂... | · simp_rw [intervalIntegral.integral_of_le hab]
simp only [hab, Set.uIoc_of_le] at h_int
exact integral_integral_swap h_int | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Polynomial.Vieta | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 61
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\ns : Multiset R\nk : ℕ\nh : k ≤ s.card\n⊢ (map (fun x ↦ X + C (-x)) s).prod.coeff k = (-1) ^ (s.card - k) * s.esymm (s.card - k)",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"NegZeroClass.toNeg",
"HMul.hMul",
"Multiset.prod_X_add_C_co... | convert! prod_X_add_C_coeff (map (fun t => -t) s) _ using 1 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Topology.Algebra.Polynomial | {
"line": 193,
"column": 2
} | {
"line": 194,
"column": 32
} | [
{
"pp": "F : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\ni : ℕ\nh1 : p.Monic\nh2 : (map f p).Splits\nhi : i ≤ (map f p).natDegree\ns : Multiset K\nhs : s ≤ (map f p).roots ∧ s.card = (map f p).natDegree - i\nB : ℝ≥0\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ ↑B\n⊢ ‖s.prod‖... | rw [← coe_nnnorm, ← NNReal.coe_pow, NNReal.coe_le_coe, ← nnnormHom_apply, ← MonoidHom.coe_coe,
MonoidHom.map_multiset_prod] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Valuation.Archimedean | {
"line": 42,
"column": 69
} | {
"line": 44,
"column": 45
} | [
{
"pp": "F : Type u_1\nΓ₀ : Type u_2\nO : Type u_3\ninst✝³ : Field F\ninst✝² : LinearOrderedCommGroupWithZero Γ₀\ninst✝¹ : CommRing O\ninst✝ : Algebra O F\nv : Valuation F Γ₀\nhv : v.Integers O\n⊢ WfDvdMonoid O ↔ WellFounded ((fun x1 x2 ↦ x1 > x2) on ⇑v ∘ ⇑(algebraMap O F))",
"usedConstants": [
"Linea... | by
refine ⟨fun _ ↦ wellFounded_dvdNotUnit.mono ?_, fun h ↦ ⟨h.mono ?_⟩⟩ <;>
simp [Function.onFun, hv.dvdNotUnit_iff_lt] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Real.Embedding | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 7
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : M\nn : ℕ\nhn : x ≤ n • 1\n⊢ BddAbove (ratLt x)",
"usedConstants": [
"Archimedean.ratLt",
"Rat",
"M... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Analysis.Complex.Polynomial.Basic | {
"line": 187,
"column": 2
} | {
"line": 194,
"column": 10
} | [
{
"pp": "p : ℝ[X]\nz : ℂ\nh0 : (aeval z) p = 0\nhz : z.im ≠ 0\n⊢ X ^ 2 - C (2 * z.re) * X + C (‖z‖ ^ 2) ∣ p",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Norm.norm",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"Polynomial.C",
"NegZeroClass.toNeg",
... | rw [← map_dvd_map' (algebraMap ℝ ℂ)]
convert! p.mul_star_dvd_of_aeval_eq_zero_im_ne_zero h0 hz
calc
map (algebraMap ℝ ℂ) (X ^ 2 - C (2 * z.re) * X + C (‖z‖ ^ 2))
_ = X ^ 2 - C (↑(2 * z.re) : ℂ) * X + C (‖z‖ ^ 2 : ℂ) := by simp
_ = (X - C (conj z)) * (X - C z) := by
rw [← add_conj, map_add, ← mul_c... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Polynomial.Basic | {
"line": 187,
"column": 2
} | {
"line": 194,
"column": 10
} | [
{
"pp": "p : ℝ[X]\nz : ℂ\nh0 : (aeval z) p = 0\nhz : z.im ≠ 0\n⊢ X ^ 2 - C (2 * z.re) * X + C (‖z‖ ^ 2) ∣ p",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Norm.norm",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"Polynomial.C",
"NegZeroClass.toNeg",
... | rw [← map_dvd_map' (algebraMap ℝ ℂ)]
convert! p.mul_star_dvd_of_aeval_eq_zero_im_ne_zero h0 hz
calc
map (algebraMap ℝ ℂ) (X ^ 2 - C (2 * z.re) * X + C (‖z‖ ^ 2))
_ = X ^ 2 - C (↑(2 * z.re) : ℂ) * X + C (‖z‖ ^ 2 : ℂ) := by simp
_ = (X - C (conj z)) * (X - C z) := by
rw [← add_conj, map_add, ← mul_c... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Ultra.Basic | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 39
} | [
{
"pp": "X : Type u_1\ninst✝¹ : PseudoMetricSpace X\ninst✝ : IsUltrametricDist X\nx : X\nr : ℝ\nhr : r ≠ 0\n⊢ IsOpen (sphere x r)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"PseudoMetricSpace.toUniformSpace",
"BooleanAlgebra.toCompl",
"DistribLattice.toLatt... | rw [← closedBall_diff_ball, sdiff_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Valuation.RankOne | {
"line": 192,
"column": 4
} | {
"line": 194,
"column": 10
} | [
{
"pp": "R : Type u_1\nΓ₀ : Type u_2\ninst✝³ : Ring R\ninst✝² : LinearOrderedCommGroupWithZero Γ₀\nK : Type u_3\ninst✝¹ : DivisionRing K\nv : Valuation K Γ₀\ninst✝ : v.RankLeOne\nH' : ∀ (x : K), v x ≠ 0 → v x = 1\nx : (ValueGroup₀ v)ˣ\nhx : x ≠ 1\nk : K\nhk : (restrict₀ v) k = ↑x\n⊢ False",
"usedConstants":... | have h0 : v k ≠ 0 := by
rw [ne_eq, ← restrict₀_eq_zero_iff, hk]
simp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.Real.Embedding | {
"line": 180,
"column": 6
} | {
"line": 180,
"column": 19
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx y : M\n⊢ embedRealFun (x + y) = embedRealFun x + embedRealFun y",
"usedConstants": [
"Eq.mpr",
"Real",
... | embedRealFun, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Real.Embedding | {
"line": 192,
"column": 4
} | {
"line": 192,
"column": 79
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx y : M\nh : x < y\nhyz : 0 < y - x\nhy : y = y - x + x\nn : ℕ\nhn : 1 ≤ n • (y - x)\n⊢ ↑{ num := 1, den := n + 1, den_nz :=... | simpa using hn.trans_lt <| nsmul_lt_nsmul_left hyz (show n < n + 1 by simp) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.MvPowerSeries.LexOrder | {
"line": 115,
"column": 9
} | {
"line": 115,
"column": 13
} | [
{
"pp": "case mpr\nσ : Type u_1\nR : Type u_2\ninst✝² : Semiring R\ninst✝¹ : LinearOrder σ\ninst✝ : WellFoundedGT σ\nφ : MvPowerSeries σ R\nw : WithTop (Lex (σ →₀ ℕ))\nh : ∀ (d : σ →₀ ℕ), ↑(toLex d) < w → (coeff d) φ = 0\nh' : φ.lexOrder < w\nhφ : φ ≠ 0\nd : σ →₀ ℕ\nhd : φ.lexOrder = ↑(toLex d)\n⊢ ↑(toLex d) < ... | ← hd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 370,
"column": 4
} | {
"line": 370,
"column": 12
} | [
{
"pp": "case cons.inr\nM : Type u_1\nι : Type u_2\ninst✝¹ : SeminormedCommGroup M\ninst✝ : IsUltrametricDist M\nf : ι → M\na : ι\ns : Finset ι\nha : a ∉ s\nhs : (↑(Finset.cons a s ha)).Pairwise fun i j ↦ ‖f i‖₊ ≠ ‖f j‖₊\nhs' : s.Nonempty\nIH : ‖∏ i ∈ s, f i‖₊ = s.sup fun i ↦ ‖f i‖₊\nj : ι\nhj : j ∈ s\nhj' : ‖∏... | rw [hj'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.MvPowerSeries.LexOrder | {
"line": 162,
"column": 2
} | {
"line": 172,
"column": 78
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝³ : Semiring R\ninst✝² : LinearOrder σ\ninst✝¹ : WellFoundedGT σ\ninst✝ : NoZeroDivisors R\nφ ψ : MvPowerSeries σ R\n⊢ (φ * ψ).lexOrder = φ.lexOrder + ψ.lexOrder",
"usedConstants": [
"Eq.mpr",
"MvPowerSeries.coeff_ne_zero_of_lexOrder",
"WithTop.ins... | obtain rfl | hφ := eq_or_ne φ 0
· simp
obtain rfl | hψ := eq_or_ne ψ 0
· simp
rcases exists_finsupp_eq_lexOrder_of_ne_zero hφ with ⟨p, hp⟩
rcases exists_finsupp_eq_lexOrder_of_ne_zero hψ with ⟨q, hq⟩
apply le_antisymm _ (lexOrder_mul_ge φ ψ)
rw [hp, hq]
apply lexOrder_le_of_coeff_ne_zero (d := p + q)
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.LexOrder | {
"line": 162,
"column": 2
} | {
"line": 172,
"column": 78
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝³ : Semiring R\ninst✝² : LinearOrder σ\ninst✝¹ : WellFoundedGT σ\ninst✝ : NoZeroDivisors R\nφ ψ : MvPowerSeries σ R\n⊢ (φ * ψ).lexOrder = φ.lexOrder + ψ.lexOrder",
"usedConstants": [
"Eq.mpr",
"MvPowerSeries.coeff_ne_zero_of_lexOrder",
"WithTop.ins... | obtain rfl | hφ := eq_or_ne φ 0
· simp
obtain rfl | hψ := eq_or_ne ψ 0
· simp
rcases exists_finsupp_eq_lexOrder_of_ne_zero hφ with ⟨p, hp⟩
rcases exists_finsupp_eq_lexOrder_of_ne_zero hψ with ⟨q, hq⟩
apply le_antisymm _ (lexOrder_mul_ge φ ψ)
rw [hp, hq]
apply lexOrder_le_of_coeff_ne_zero (d := p + q)
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Valued.NormedValued | {
"line": 158,
"column": 59
} | {
"line": 158,
"column": 92
} | [
{
"pp": "L : Type u_1\ninst✝¹ : Field L\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nval : Valued L Γ₀\nhv : v.RankOne\nx y : L\n⊢ ↑((RankOne.hom v) (v.restrict (x - y))) = ↑((RankOne.hom v) (v.restrict (y - x)))",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
... | rw [← neg_sub, Valuation.map_neg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyOpen | {
"line": 155,
"column": 4
} | {
"line": 157,
"column": 64
} | [
{
"pp": "case inr\nX Y✝ : Scheme\nf✝ : X ⟶ Y✝\nY : Scheme\ninst✝¹ : IsIntegral Y\ninst✝ : Subsingleton ↥Y\nS : CommRingCat\nf : Spec S ⟶ Y\nthis :\n ∀ {Y : Scheme} [IsIntegral Y] [Subsingleton ↥Y] (f : Spec S ⟶ Y), (∃ K, Y = Spec K ∧ IsField ↑K) → UniversallyOpen f\nhY : ¬∃ K, Y = Spec K ∧ IsField ↑K\n⊢ Univer... | have inst : Subsingleton (Spec Γ(Y, ⊤)) := Y.isoSpec.inv.homeomorph.subsingleton
exact (MorphismProperty.cancel_right_of_respectsIso _ _ Y.isoSpec.hom).mp
(this _ ⟨_, rfl, isField_of_isIntegral_of_subsingleton _⟩) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyOpen | {
"line": 155,
"column": 4
} | {
"line": 157,
"column": 64
} | [
{
"pp": "case inr\nX Y✝ : Scheme\nf✝ : X ⟶ Y✝\nY : Scheme\ninst✝¹ : IsIntegral Y\ninst✝ : Subsingleton ↥Y\nS : CommRingCat\nf : Spec S ⟶ Y\nthis :\n ∀ {Y : Scheme} [IsIntegral Y] [Subsingleton ↥Y] (f : Spec S ⟶ Y), (∃ K, Y = Spec K ∧ IsField ↑K) → UniversallyOpen f\nhY : ¬∃ K, Y = Spec K ∧ IsField ↑K\n⊢ Univer... | have inst : Subsingleton (Spec Γ(Y, ⊤)) := Y.isoSpec.inv.homeomorph.subsingleton
exact (MorphismProperty.cancel_right_of_respectsIso _ _ Y.isoSpec.hom).mp
(this _ ⟨_, rfl, isField_of_isIntegral_of_subsingleton _⟩) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.CoproductSheafCondition | {
"line": 94,
"column": 2
} | {
"line": 95,
"column": 46
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\nS : C\nι : Type u_3\nX : ι → C\nf : (i : ι) → X i ⟶ S\ninst✝⁴ : (ofArrows X f).HasPairwisePullbacks\nc : Cofan X\nhc : IsColimit c\nhc' : IsUniversalColimit c\ninst✝³ : HasPullback (Cofan.IsColimit.desc hc f) (Cofan.IsColimit.desc hc f)\ninst✝² : ∀ (i : ι),... | have (i : E.I₀) : HasPullback (E.f i) ((E.sigmaOfIsColimit hc).f PUnit.unit) := by
dsimp [sigmaOfIsColimit_f]; infer_instance | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Unramified.Basic | {
"line": 222,
"column": 49
} | {
"line": 222,
"column": 98
} | [
{
"pp": "R : Type u_1\ninst✝¹⁰ : CommRing R\nA : Type u_2\ninst✝⁹ : CommRing A\ninst✝⁸ : Algebra R A\nB : Type u_3\ninst✝⁷ : CommRing B\ninst✝⁶ : Algebra R B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsScalarTower R A B\ninst✝³ : FormallyUnramified R A\ninst✝² : FormallyUnramified A B\nC : Type u_3\ninst✝¹ : CommRing C\n... | by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Unramified.Basic | {
"line": 234,
"column": 2
} | {
"line": 234,
"column": 60
} | [
{
"pp": "R : Type u_1\ninst✝⁹ : CommRing R\nA : Type u_2\ninst✝⁸ : CommRing A\ninst✝⁷ : Algebra R A\nB : Type u_3\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R B\ninst✝⁴ : Algebra A B\ninst✝³ : IsScalarTower R A B\ninst✝² : FormallyUnramified R B\nQ : Type u_3\ninst✝¹ : CommRing Q\ninst✝ : Algebra A Q\nI : Ideal Q\n... | letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.RingTheory.Smooth.Basic | {
"line": 174,
"column": 2
} | {
"line": 174,
"column": 65
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing A\ninst✝⁵ : Algebra R A\nB : Type u_1\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R B\nC : Type u_4\ninst✝² : CommRing C\ninst✝¹ : Algebra R C\ninst✝ : FormallySmooth R A\nf : A →ₐ[R] C\ng : B →ₐ[R] C\nhg : Function.Surjective ⇑g\nhg' : IsNilpot... | apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Smooth.Kaehler | {
"line": 166,
"column": 4
} | {
"line": 168,
"column": 70
} | [
{
"pp": "R : Type u_1\nP : Type u_2\nS : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing P\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R P\ninst✝² : Algebra P S\nl : S ⊗[P] Ω[P⁄R] →ₗ[P] ↥(RingHom.ker (algebraMap P S))\nhl : l ∘ₗ kerToTensor R P S = LinearMap.id\nσ : S → P\nhσ : ∀ (x : S), (algebraMap P S) (σ x) = x... | simp only [sectionOfRetractionKerToTensorAux_prop l hl (σ (a * b)) (σ a * σ b) (by simp [hσ]),
Derivation.leibniz, tmul_add, tmul_smul, map_add, map_smul, Submodule.coe_add,
SetLike.val_smul, smul_eq_mul, mul_sub, sub_mul, this, sub_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Smooth.Kaehler | {
"line": 358,
"column": 4
} | {
"line": 363,
"column": 65
} | [
{
"pp": "case refine_4\nR : Type u_1\nP : Type u_2\nS : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing P\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R P\ninst✝² : Algebra P S\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R P S\nhf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥\nhf : Surjective ⇑(algebraMap P S)\nP' : Type... | intro f
ext x
simp only [AlgebraTensorModule.curry_apply, Derivation.coe_comp, LinearMap.coe_comp,
LinearMap.coe_restrictScalars, Derivation.coeFn_coe, Function.comp_apply, curry_apply,
LinearEquiv.coe_coe, LinearMap.coe_mk, AddHom.coe_coe,
LinearEquiv.apply_symm_apply, LinearEquiv.symm_apply_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Smooth.Kaehler | {
"line": 358,
"column": 4
} | {
"line": 363,
"column": 65
} | [
{
"pp": "case refine_4\nR : Type u_1\nP : Type u_2\nS : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing P\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R P\ninst✝² : Algebra P S\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R P S\nhf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥\nhf : Surjective ⇑(algebraMap P S)\nP' : Type... | intro f
ext x
simp only [AlgebraTensorModule.curry_apply, Derivation.coe_comp, LinearMap.coe_comp,
LinearMap.coe_restrictScalars, Derivation.coeFn_coe, Function.comp_apply, curry_apply,
LinearEquiv.coe_coe, LinearMap.coe_mk, AddHom.coe_coe,
LinearEquiv.apply_symm_apply, LinearEquiv.symm_apply_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 573,
"column": 2
} | {
"line": 575,
"column": 74
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Projective R\nP Q : Fin 3 → R\nhP : W'.Equation P\nhQ : W'.Equation Q\n⊢ W'.addX P Q * (P z * Q z) ^ 2 =\n ((P y * Q z - Q y * P z) ^ 2 * P z * Q z + W'.a₁ * (P y * Q z - Q y * P z) * P z * Q z * (P x * Q z - Q x * P z) -\n W'.a₂ * P z * Q z * (P x * Q... | linear_combination (norm := (rw [addX]; ring1))
(2 * Q x * P z * Q z ^ 3 - P x * Q z ^ 4) * (equation_iff _).mp hP
+ (Q x * P z ^ 4 - 2 * P x * P z ^ 3 * Q z) * (equation_iff _).mp hQ | Mathlib.Tactic.LinearCombination._aux_Mathlib_Tactic_LinearCombination___elabRules_Mathlib_Tactic_LinearCombination_linearCombination_1 | Mathlib.Tactic.LinearCombination.linearCombination |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 573,
"column": 2
} | {
"line": 575,
"column": 74
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Projective R\nP Q : Fin 3 → R\nhP : W'.Equation P\nhQ : W'.Equation Q\n⊢ W'.addX P Q * (P z * Q z) ^ 2 =\n ((P y * Q z - Q y * P z) ^ 2 * P z * Q z + W'.a₁ * (P y * Q z - Q y * P z) * P z * Q z * (P x * Q z - Q x * P z) -\n W'.a₂ * P z * Q z * (P x * Q... | linear_combination (norm := (rw [addX]; ring1))
(2 * Q x * P z * Q z ^ 3 - P x * Q z ^ 4) * (equation_iff _).mp hP
+ (Q x * P z ^ 4 - 2 * P x * P z ^ 3 * Q z) * (equation_iff _).mp hQ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 573,
"column": 2
} | {
"line": 575,
"column": 74
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Projective R\nP Q : Fin 3 → R\nhP : W'.Equation P\nhQ : W'.Equation Q\n⊢ W'.addX P Q * (P z * Q z) ^ 2 =\n ((P y * Q z - Q y * P z) ^ 2 * P z * Q z + W'.a₁ * (P y * Q z - Q y * P z) * P z * Q z * (P x * Q z - Q x * P z) -\n W'.a₂ * P z * Q z * (P x * Q... | linear_combination (norm := (rw [addX]; ring1))
(2 * Q x * P z * Q z ^ 3 - P x * Q z ^ 4) * (equation_iff _).mp hP
+ (Q x * P z ^ 4 - 2 * P x * P z ^ 3 * Q z) * (equation_iff _).mp hQ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Unramified.Finite | {
"line": 67,
"column": 40
} | {
"line": 67,
"column": 45
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : EssFiniteType R S\nthis : ∀ (t : S ⊗[R] S), (TensorProduct.lmul' R) t = 1 ↔ 1 - t ∈ KaehlerDifferential.ideal R S\ne : S ⊗[R] S\nhe₁ : IsIdempotentElem e\nhe₂ : Ideal.span (Set.range fun s ↦ 1 ⊗ₜ[R] s - ... | ← he₂ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Etale.Field | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 96
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : Algebra.IsSeparable K L\ninst✝ : EssFiniteType K L\nthis✝ : FormallyUnramified K L\nthis : Module.Finite K L\nB : Type (max u_1 u_2)\nx✝¹ : CommRing B\nx✝ : Algebra K B\nI : Ideal B\nh : I ^ 2 = ⊥\nf : L →ₐ[K... | rw [← helper, ← aeval_algHom_apply, helper, hx, aeval_algHom_apply, minpoly.aeval, map_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Etale.Locus | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 60
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : FinitePresentation R A\nf : A\n⊢ FormallyEtale R (Localization.Away f) ↔ Etale R (Localization.Away f)",
"usedConstants": [
"Algebra.Etale.mk",
"OreLocalization.instAlgebra",
"CommS... | refine ⟨fun H ↦ ⟨H, inferInstance⟩, fun _ ↦ inferInstance⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Etale.Locus | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 60
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : FinitePresentation R A\n⊢ FormallyEtale R A ↔ Etale R A",
"usedConstants": [
"Algebra.Etale.mk",
"CommSemiring.toSemiring",
"Algebra.FormallyEtale",
"Algebra.Etale.formallyEta... | refine ⟨fun H ↦ ⟨H, inferInstance⟩, fun _ ↦ inferInstance⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Etale.Field | {
"line": 133,
"column": 4
} | {
"line": 133,
"column": 67
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Algebra.IsSeparable K L\nB : Type (max u_1 u_2)\nx✝¹ : CommRing B\nx✝ : Algebra K B\nI : Ideal B\nh : I ^ 2 = ⊥\nf : L →ₐ[K] B ⧸ I\ng : (k : L) → ↥K⟮k⟯ →ₐ[K] B\nhg₁ : ∀ (k : L), (fun g ↦ (Ideal.Quotient.mkₐ K ... | obtain ⟨⟨α, hα⟩, e⟩ := Field.exists_primitive_element K K⟮x, y⟯ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.AdicCompletion.Functoriality | {
"line": 396,
"column": 2
} | {
"line": 398,
"column": 23
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M →ₗ[R] N\nh : Function.Surjective ⇑((I • ⊤).mkQ ∘ₗ f)\nx : M\nn : ℕ\ny : N ⧸ I ^ n • ⊤\ny' : N ⧸ I ^ (n + 1) • ⊤\nhyy' : (factor ⋯)... | have : f x ≡ y0 [SMOD (I ^ n • ⊤ : Submodule R N)] := by
rw [SModEq, ← mkQ_apply, ← mkQ_apply, ← factor_mk (pow_smul_top_le I N n.le_succ) y0,
hy0, hyy', hxy] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.AdicCompletion.Functoriality | {
"line": 408,
"column": 53
} | {
"line": 413,
"column": 15
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M →ₗ[R] N\nh✝ : Function.Surjective ⇑((I • ⊤).mkQ ∘ₗ f)\ny : AdicCompletion I N\nh : ∃ x, ∀ (n : ℕ), x n ≡ x (n + 1) [SMOD I ^ n • ⊤... | by
obtain ⟨x, hx⟩ := h
use AdicCompletion.mk I M ⟨x, fun h ↦
eq_factor_of_eq_factor_succ (fun _ _ ↦ pow_smul_top_le I M) _ (fun n ↦ (hx n).1) h⟩
ext n
simp [hx n] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.AdicCompletion.Exactness | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 87
} | [
{
"pp": "R : Type u\ninst✝⁶ : CommRing R\nI : Ideal R\nM : Type u\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nN : Type u\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : IsNoetherianRing R\ninst✝ : Module.Finite R N\nf : M →ₗ[R] N\nhf : Function.Injective ⇑f\nk : ℕ\nhk : ∀ n ≥ k, I ^ n • ⊤ ⊓ f.range =... | refine AdicCompletion.mk_zero_of _ _ _ ⟨42, fun n _ ↦ ⟨n + k, by lia, n, by lia, ?_⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.AdicCompletion.Exactness | {
"line": 172,
"column": 4
} | {
"line": 173,
"column": 22
} | [
{
"pp": "R : Type u\ninst✝⁸ : CommRing R\nI : Ideal R\nM : Type u\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nN : Type u\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : Type u\ninst✝³ : AddCommGroup P\ninst✝² : Module R P\ninst✝¹ : IsNoetherianRing R\ninst✝ : Module.Finite R N\nf : M →ₗ[R] N\ng : N →ₗ[R] ... | rw [map_smul'', Submodule.map_top, range_eq_top.mpr hg]
exact hker (k + n) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.AdicCompletion.Exactness | {
"line": 172,
"column": 4
} | {
"line": 173,
"column": 22
} | [
{
"pp": "R : Type u\ninst✝⁸ : CommRing R\nI : Ideal R\nM : Type u\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nN : Type u\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : Type u\ninst✝³ : AddCommGroup P\ninst✝² : Module R P\ninst✝¹ : IsNoetherianRing R\ninst✝ : Module.Finite R N\nf : M →ₗ[R] N\ng : N →ₗ[R] ... | rw [map_smul'', Submodule.map_top, range_eq_top.mpr hg]
exact hker (k + n) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Smooth.Fiber | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 98
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nP : Type u_3\ninst✝¹⁴ : CommRing R\ninst✝¹³ : CommRing S\ninst✝¹² : Algebra R S\ninst✝¹¹ : Module.Flat R S\ninst✝¹⁰ : CommRing P\ninst✝⁹ : Algebra R P\ninst✝⁸ : Algebra P S\ninst✝⁷ : IsScalarTower R P S\ninst✝⁶ : IsLocalRing R\ninst✝⁵ : IsLocalRing S\ninst✝⁴ : IsLocalHom (al... | rw [Algebra.FormallySmooth.iff_injective_cotangentComplexBaseChange_residueField (P := P) h₁ h₂] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Smooth.Fiber | {
"line": 133,
"column": 19
} | {
"line": 133,
"column": 72
} | [
{
"pp": "case h.e'_3.h.add\nR : Type u_1\nS : Type u_2\nP : Type u_3\ninst✝¹⁴ : CommRing R\ninst✝¹³ : CommRing S\ninst✝¹² : Algebra R S\ninst✝¹¹ : Module.Flat R S\ninst✝¹⁰ : CommRing P\ninst✝⁹ : Algebra R P\ninst✝⁸ : Algebra P S\ninst✝⁷ : IsScalarTower R P S\ninst✝⁶ : IsLocalRing R\ninst✝⁵ : IsLocalRing S\ninst... | simp only [LinearEquiv.map_add, LinearMap.map_add, *] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Smooth.Fiber | {
"line": 133,
"column": 19
} | {
"line": 133,
"column": 72
} | [
{
"pp": "case h.e'_3.h.add\nR : Type u_1\nS : Type u_2\nP : Type u_3\ninst✝¹⁴ : CommRing R\ninst✝¹³ : CommRing S\ninst✝¹² : Algebra R S\ninst✝¹¹ : Module.Flat R S\ninst✝¹⁰ : CommRing P\ninst✝⁹ : Algebra R P\ninst✝⁸ : Algebra P S\ninst✝⁷ : IsScalarTower R P S\ninst✝⁶ : IsLocalRing R\ninst✝⁵ : IsLocalRing S\ninst... | simp only [LinearEquiv.map_add, LinearMap.map_add, *] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Smooth.Fiber | {
"line": 133,
"column": 19
} | {
"line": 133,
"column": 72
} | [
{
"pp": "case h.e'_3.h.add\nR : Type u_1\nS : Type u_2\nP : Type u_3\ninst✝¹⁴ : CommRing R\ninst✝¹³ : CommRing S\ninst✝¹² : Algebra R S\ninst✝¹¹ : Module.Flat R S\ninst✝¹⁰ : CommRing P\ninst✝⁹ : Algebra R P\ninst✝⁸ : Algebra P S\ninst✝⁷ : IsScalarTower R P S\ninst✝⁶ : IsLocalRing R\ninst✝⁵ : IsLocalRing S\ninst... | simp only [LinearEquiv.map_add, LinearMap.map_add, *] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Extension.Presentation.Submersive | {
"line": 612,
"column": 4
} | {
"line": 613,
"column": 77
} | [
{
"pp": "case h.e'_3\nR : Type u\nS : Type v\nι : Type w\nσ : Type t\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Finite σ\nP : SubmersivePresentation R S ι σ\nthis : Fintype σ\n⊢ ((LinearMap.toMatrix (Pi.basisFun S σ) (Pi.basisFun S σ)) P.aevalDifferential).det = P.jacobian",
"u... | rw [LinearMap.toMatrix_eq_toMatrix', jacobian_eq_jacobiMatrix_det,
aevalDifferential_toMatrix'_eq_mapMatrix_jacobiMatrix, P.algebraMap_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Extension.Cotangent.LocalizationAway | {
"line": 140,
"column": 4
} | {
"line": 150,
"column": 18
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\nι : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\ninst✝¹ : IsScalarTower R S T\ng : S\ninst✝ : IsLocalization.Away g T\nP : Generators R S ι\nQ : Generators S T Unit :=... | rw [IsScalarTower.algebraMap_apply _ (P.Ring ⧸ P.ker ^ 2) _,
IsLocalization.map_eq_zero_iff (Submonoid.powers f) (Localization.Away f)] at h
obtain ⟨⟨m, ⟨n, rfl⟩⟩, hm⟩ := h
rw [IsLocalizedModule.eq_zero_iff (Submonoid.powers g)]
use ⟨g ^ n, n, rfl⟩
dsimp [f] at hm
rw [← map_pow, ← map_mul, Ide... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Extension.Cotangent.LocalizationAway | {
"line": 140,
"column": 4
} | {
"line": 150,
"column": 18
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\nι : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\ninst✝¹ : IsScalarTower R S T\ng : S\ninst✝ : IsLocalization.Away g T\nP : Generators R S ι\nQ : Generators S T Unit :=... | rw [IsScalarTower.algebraMap_apply _ (P.Ring ⧸ P.ker ^ 2) _,
IsLocalization.map_eq_zero_iff (Submonoid.powers f) (Localization.Away f)] at h
obtain ⟨⟨m, ⟨n, rfl⟩⟩, hm⟩ := h
rw [IsLocalizedModule.eq_zero_iff (Submonoid.powers g)]
use ⟨g ^ n, n, rfl⟩
dsimp [f] at hm
rw [← map_pow, ← map_mul, Ide... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.RingHom.StandardSmooth | {
"line": 135,
"column": 6
} | {
"line": 135,
"column": 88
} | [
{
"pp": "R S T : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nh : Algebra.IsStandardSmooth R T\nthis : Algebra.IsStandardSmooth S (S ⊗[R] T)\n⊢ (algebraMap S (S ⊗[R] T)).IsStandardSmooth",
"usedConstants": [
"Eq.mpr",
"CommRi... | rw [RingHom.IsStandardSmooth]; convert! this; ext; simp_rw [Algebra.smul_def]; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.RingHom.StandardSmooth | {
"line": 135,
"column": 6
} | {
"line": 135,
"column": 88
} | [
{
"pp": "R S T : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nh : Algebra.IsStandardSmooth R T\nthis : Algebra.IsStandardSmooth S (S ⊗[R] T)\n⊢ (algebraMap S (S ⊗[R] T)).IsStandardSmooth",
"usedConstants": [
"Eq.mpr",
"CommRi... | rw [RingHom.IsStandardSmooth]; convert! this; ext; simp_rw [Algebra.smul_def]; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.RingHom.StandardSmooth | {
"line": 205,
"column": 2
} | {
"line": 206,
"column": 78
} | [
{
"pp": "n : ℕ\nR : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Algebra.IsStandardSmoothOfRelativeDimension n R S\nthis : (α : Type ?u.6766.11) → [_root_.Finite α] → Fintype α := Fintype.ofFinite\n⊢ ∃ g, g.Etale",
"usedConstants": [
"Algebra.IsStandardSm... | obtain ⟨ι, σ, _, _, P, e⟩ :=
Algebra.IsStandardSmoothOfRelativeDimension.out (R := R) (S := S) (n := n) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.Morphisms.Smooth | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 50
} | [
{
"pp": "n : ℕ\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : SmoothOfRelativeDimension n f\nx : ↥X\nU : Y.Opens\nhU : IsAffineOpen U\nV : X.Opens\nhV : IsAffineOpen V\nhx : x ∈ V\ne : V ≤ f ⁻¹ᵁ U\nhf : IsStandardSmoothOfRelativeDimension n (CommRingCat.Hom.hom (Scheme.Hom.appLE f U V e))\n⊢ ∃ U,\n ∃ (_ : IsAffineOpen U)... | exact ⟨U, hU, V, hV, hx, e, hf.isStandardSmooth⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Smooth.NoetherianDescent | {
"line": 268,
"column": 31
} | {
"line": 268,
"column": 40
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\nA : Type u\nB : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\nn : ℕ\ninst✝ : IsStandardSmoothOfRelativeDimension n A B\nι σ : Type\nw✝¹ : Finite σ\nw✝ : Finite ι\nP : SubmersivePresentation A B ι σ\nhP : P.dimension = ... | simp [A₀] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Smooth.NoetherianDescent | {
"line": 268,
"column": 31
} | {
"line": 268,
"column": 40
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\nA : Type u\nB : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\nn : ℕ\ninst✝ : IsStandardSmoothOfRelativeDimension n A B\nι σ : Type\nw✝¹ : Finite σ\nw✝ : Finite ι\nP : SubmersivePresentation A B ι σ\nhP : P.dimension = ... | simp [A₀] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Smooth.NoetherianDescent | {
"line": 268,
"column": 31
} | {
"line": 268,
"column": 40
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\nA : Type u\nB : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\nn : ℕ\ninst✝ : IsStandardSmoothOfRelativeDimension n A B\nι σ : Type\nw✝¹ : Finite σ\nw✝ : Finite ι\nP : SubmersivePresentation A B ι σ\nhP : P.dimension = ... | simp [A₀] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Smooth.NoetherianDescent | {
"line": 270,
"column": 34
} | {
"line": 270,
"column": 43
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\nA : Type u\nB : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\nn : ℕ\ninst✝ : IsStandardSmoothOfRelativeDimension n A B\nι σ : Type\nw✝¹ : Finite σ\nw✝ : Finite ι\nP : SubmersivePresentation A B ι σ\nhP : P.dimension = ... | simp [A₀] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Smooth.NoetherianDescent | {
"line": 270,
"column": 34
} | {
"line": 270,
"column": 43
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\nA : Type u\nB : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\nn : ℕ\ninst✝ : IsStandardSmoothOfRelativeDimension n A B\nι σ : Type\nw✝¹ : Finite σ\nw✝ : Finite ι\nP : SubmersivePresentation A B ι σ\nhP : P.dimension = ... | simp [A₀] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Smooth.NoetherianDescent | {
"line": 270,
"column": 34
} | {
"line": 270,
"column": 43
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\nA : Type u\nB : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\nn : ℕ\ninst✝ : IsStandardSmoothOfRelativeDimension n A B\nι σ : Type\nw✝¹ : Finite σ\nw✝ : Finite ι\nP : SubmersivePresentation A B ι σ\nhP : P.dimension = ... | simp [A₀] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.Smooth | {
"line": 353,
"column": 2
} | {
"line": 353,
"column": 76
} | [
{
"pp": "X : Scheme\nK : Type u\ninst✝³ : Field K\ninst✝² : PerfectField K\ninst✝¹ : IsIntegral X\nf : X ⟶ Spec (CommRingCat.of K)\ninst✝ : LocallyOfFinitePresentation f\nthis✝ : (CommRingCat.Hom.hom (stalkMap f (genericPoint ↥X))).EssFiniteType\nalgInst✝ : Algebra ↑((Spec (CommRingCat.of K)).presheaf.stalk (f ... | let : Field K' := (e.toRingEquiv.symm.isField (Field.toIsField K)).toField | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 99,
"column": 4
} | {
"line": 99,
"column": 17
} | [
{
"pp": "case refine_3\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nσ✝ : Type u_3\nι : Type u_4\nP : Generators R S ι\nσ : Type u_5\nb : Module.Basis σ S P.toExtension.Cotangent\nD : Aux P b\n⊢ D.g - 1 ∈ RingHom.ker (algebraMap P.Ring S)",
"usedConstants": [
... | exact D.hgmem | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 207,
"column": 2
} | {
"line": 208,
"column": 8
} | [
{
"pp": "case h.e_6.h.e_val.h1\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nι : Type u_4\nP : Generators R S ι\nσ : Type u_5\nb : Module.Basis σ S P.toExtension.Cotangent\nD : Aux P b\ninst✝ : Nontrivial S\nthis : Nontrivial (MvPolynomial ι R ⧸ Ideal.span (Set.ran... | · rw [Presentation.naive, Generators.naive_σ];
simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 301,
"column": 4
} | {
"line": 301,
"column": 35
} | [
{
"pp": "case refine_2\nk : Type u_1\nK : Type u_2\ninst✝⁴ : Field k\ninst✝³ : Field K\ninst✝² : Algebra k K\np : ℕ\nhp : Nat.Prime p\nH : ∀ (s : Finset K), LinearIndepOn k id ↑s → LinearIndepOn k (fun x ↦ x ^ p) ↑s\ninst✝¹ : ExpChar k p\ninst✝ : Algebra.EssFiniteType k K\ns : Finset K\nhs : (fun t ↦ IsTranscen... | obtain rfl | ne := eq_or_ne n i | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.Morphisms.Etale | {
"line": 137,
"column": 51
} | {
"line": 137,
"column": 82
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\n⊢ (∀ {U : Y.Opens},\n IsAffineOpen U →\n ∀ {V : X.Opens}, IsAffineOpen V → ∀ (e : V ≤ f ⁻¹ᵁ U), (CommRingCat.Hom.hom (Scheme.Hom.appLE f U V e)).Etale) ↔\n (∀ {U : Y.Opens},\n IsAffineOpen U →\n ∀ {V : X.Opens},\n IsAffineOpen V → ∀ (e ... | locallyOfFinitePresentation_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.MorphismProperty.Descent | {
"line": 112,
"column": 4
} | {
"line": 112,
"column": 54
} | [
{
"pp": "case H.hfst\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\nP✝ Q✝ W : MorphismProperty C\nA X✝ Y✝ Z✝ : C\nfst : A ⟶ X✝\nsnd : A ⟶ Y✝\nf✝ : X✝ ⟶ Z✝\ng✝ : Y✝ ⟶ Z✝\ninst✝⁴ : HasPullbacks C\nP Q : MorphismProperty C\ninst✝³ : P.DescendsAlong Q\ninst✝² : P.RespectsIso\ninst✝¹ : Q.IsStableUnderBaseChange\nX Y... | iterate 4 rw [cancel_left_of_respectsIso (P := P)] | Lean.Parser.Tactic._aux_Init_TacticsExtra___macroRules_Lean_Parser_Tactic_tacticIterate_____1 | Lean.Parser.Tactic.tacticIterate____ |
Mathlib.CategoryTheory.MorphismProperty.Descent | {
"line": 196,
"column": 4
} | {
"line": 196,
"column": 82
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP Q : MorphismProperty C\ninst✝ : P.RespectsIso\nH : ∀ {X Y Z : C} {f : Z ⟶ X} {g : Z ⟶ Y} [inst : HasPushout f g], Q f → P (pushout.inl f g) → P g\nA X Y Z : C\nf : A ⟶ X\ng : A ⟶ Y\ninl : X ⟶ Z\ninr : Y ⟶ Z\nh : IsPushout f g inl inr\nhf : Q f\nhfst : P i... | rwa [← P.cancel_right_of_respectsIso _ h.isoPushout.inv, h.inl_isoPushout_inv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.RingTheory.Finiteness.NilpotentKer | {
"line": 47,
"column": 4
} | {
"line": 48,
"column": 28
} | [
{
"pp": "case succ.refine_1\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : Algebra R S\ninst✝¹ : Algebra R T\ninst✝ : Module.Finite R T\nf : S →ₐ[R] T\nhf₁ : Function.Surjective ⇑f\nI : Ideal S\nhI : RingHom.ker f = I\nhf₂ : I ≤ nilradical S\nh... | · simpa [LinearMap.range_eq_top_of_surjective (φ.toLinearMap.restrictScalars R) hφ] using
Module.Finite.fg_top | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.Morphisms.FlatDescent | {
"line": 150,
"column": 4
} | {
"line": 151,
"column": 93
} | [
{
"pp": "X Y Z : Scheme\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : HasPullback f g\nhf : (@Surjective ⊓ @Flat ⊓ @QuasiCompact) f\nhg : IsOpenImmersion (pullback.fst f g)\nthis✝¹ : UniversallyOpen g\nU : Z.Opens := { carrier := Set.range ⇑g, is_open' := ⋯ }\nf' : pullback f U.ι ⟶ ↑U := pullback.snd f U.ι\ng' : Y ⟶ ↑U := IsO... | apply MorphismProperty.of_pullback_fst_of_descendsAlong
(P := isomorphisms Scheme) (Q := @Surjective ⊓ @Flat ⊓ @QuasiCompact) (f := f') ?_ this | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Conductor | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 66
} | [
{
"pp": "case refine_2\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p ∈ comap (algebraMap R S) (conductor R x)\nl : R →₀ S\nH : l ∈ Finsupp.supported S S ↑I\nH' : (l.sum fun i a ↦ a • (algebraMap R S) i) = z\nlem :\n ∀ {a : R... | exact ⟨0, SetLike.mem_coe.mpr <| Ideal.zero_mem _, map_zero _⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Conductor | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 66
} | [
{
"pp": "case refine_2\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p ∈ comap (algebraMap R S) (conductor R x)\nl : R →₀ S\nH : l ∈ Finsupp.supported S S ↑I\nH' : (l.sum fun i a ↦ a • (algebraMap R S) i) = z\nlem :\n ∀ {a : R... | exact ⟨0, SetLike.mem_coe.mpr <| Ideal.zero_mem _, map_zero _⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Conductor | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 66
} | [
{
"pp": "case refine_2\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p ∈ comap (algebraMap R S) (conductor R x)\nl : R →₀ S\nH : l ∈ Finsupp.supported S S ↑I\nH' : (l.sum fun i a ↦ a • (algebraMap R S) i) = z\nlem :\n ∀ {a : R... | exact ⟨0, SetLike.mem_coe.mpr <| Ideal.zero_mem _, map_zero _⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.IntegralClosure.Algebra.Ideal | {
"line": 44,
"column": 35
} | {
"line": 44,
"column": 49
} | [
{
"pp": "R : Type u_3\ninst✝ : CommRing R\nI : Ideal R\nP x y : R[X]\nhx✝ : x ∈ Algebra.adjoin R {x | ∃ r ∈ I, C r * X = x}\nhy✝ : y ∈ Algebra.adjoin R {x | ∃ r ∈ I, C r * X = x}\nhx : ∀ (i : ℕ), x.coeff i ∈ I ^ i\nhy : ∀ (i : ℕ), y.coeff i ∈ I ^ i\ni : ℕ\nx✝ : ℕ × ℕ\nj₁ j₂ : ℕ\nhj : (j₁, j₂) ∈ Finset.antidiago... | simpa using hj | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.IntegralClosure.Algebra.Ideal | {
"line": 44,
"column": 35
} | {
"line": 44,
"column": 49
} | [
{
"pp": "R : Type u_3\ninst✝ : CommRing R\nI : Ideal R\nP x y : R[X]\nhx✝ : x ∈ Algebra.adjoin R {x | ∃ r ∈ I, C r * X = x}\nhy✝ : y ∈ Algebra.adjoin R {x | ∃ r ∈ I, C r * X = x}\nhx : ∀ (i : ℕ), x.coeff i ∈ I ^ i\nhy : ∀ (i : ℕ), y.coeff i ∈ I ^ i\ni : ℕ\nx✝ : ℕ × ℕ\nj₁ j₂ : ℕ\nhj : (j₁, j₂) ∈ Finset.antidiago... | simpa using hj | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.IntegralClosure.Algebra.Ideal | {
"line": 44,
"column": 35
} | {
"line": 44,
"column": 49
} | [
{
"pp": "R : Type u_3\ninst✝ : CommRing R\nI : Ideal R\nP x y : R[X]\nhx✝ : x ∈ Algebra.adjoin R {x | ∃ r ∈ I, C r * X = x}\nhy✝ : y ∈ Algebra.adjoin R {x | ∃ r ∈ I, C r * X = x}\nhx : ∀ (i : ℕ), x.coeff i ∈ I ^ i\nhy : ∀ (i : ℕ), y.coeff i ∈ I ^ i\ni : ℕ\nx✝ : ℕ × ℕ\nj₁ j₂ : ℕ\nhj : (j₁, j₂) ∈ Finset.antidiago... | simpa using hj | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.IntegralClosure.Algebra.Ideal | {
"line": 67,
"column": 4
} | {
"line": 67,
"column": 55
} | [
{
"pp": "case inr.refine_2\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal R\nx : S\nh✝ : Nontrivial R\np : (↥(Algebra.adjoin R {x | ∃ r ∈ I, C r * X = x}))[X]\nhp : p.Monic\nq : R[X] := ∑ i ∈ Finset.range (p.natDegree + 1), C ((↑(p.coeff i)).coeff (p.natDeg... | simp only [Finset.mem_range, Nat.lt_succ_iff] at hi | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.QuasiFinite.Basic | {
"line": 404,
"column": 6
} | {
"line": 405,
"column": 92
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP Q : Ideal S\ninst✝² : P.IsPrime\ninst✝¹ : Q.IsPrime\nh₁ : P ≤ Q\ninst✝ : QuasiFiniteAt R Q\n⊢ ∀ (y : ↥Q.primeCompl), IsUnit ((IsScalarTower.toAlgHom R S (Localization.AtPrime P)) ↑y)",
"usedConstants": [
... | simp only [IsScalarTower.coe_toAlgHom', Subtype.forall, Ideal.mem_primeCompl_iff]
exact fun a ha ↦ IsLocalization.map_units (M := P.primeCompl) _ ⟨a, fun h ↦ ha (h₁ h)⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.QuasiFinite.Basic | {
"line": 404,
"column": 6
} | {
"line": 405,
"column": 92
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP Q : Ideal S\ninst✝² : P.IsPrime\ninst✝¹ : Q.IsPrime\nh₁ : P ≤ Q\ninst✝ : QuasiFiniteAt R Q\n⊢ ∀ (y : ↥Q.primeCompl), IsUnit ((IsScalarTower.toAlgHom R S (Localization.AtPrime P)) ↑y)",
"usedConstants": [
... | simp only [IsScalarTower.coe_toAlgHom', Subtype.forall, Ideal.mem_primeCompl_iff]
exact fun a ha ↦ IsLocalization.map_units (M := P.primeCompl) _ ⟨a, fun h ↦ ha (h₁ h)⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.QuasiFinite.Basic | {
"line": 416,
"column": 2
} | {
"line": 416,
"column": 95
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP Q : Ideal S\ninst✝² : P.IsPrime\ninst✝¹ : Q.IsPrime\nh₁ : P ≤ Q\nh₂ : Ideal.under R P = Ideal.under R Q\ninst✝ : QuasiFiniteAt R Q\nthis : Disjoint ↑Q.primeCompl ↑P\n⊢ P = Q",
"usedConstants": [
"Alg... | have inst := IsLocalization.isPrime_of_isPrime_disjoint _ (Localization.AtPrime Q) P ‹_› this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.QuasiFinite.Basic | {
"line": 445,
"column": 33
} | {
"line": 445,
"column": 73
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\np : Ideal S\ninst✝³ : p.IsPrime\ninst✝² : IsArtinianRing R\ninst✝¹ : EssFiniteType R S\ninst✝ : QuasiFiniteAt R p\n⊢ ∀ {x₁ x₂ : Localization.AtPrime p}, LinearMap.id x₁ = LinearMap.id x₂ → ∃ c, c • x₁ = c • x₂",... | by simpa using ⟨1, p.primeCompl.one_mem⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.QuasiFinite.Weakly | {
"line": 158,
"column": 6
} | {
"line": 158,
"column": 37
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP Q : Ideal S\ninst✝² : P.IsPrime\ninst✝¹ : Q.IsPrime\nh₁ : P ≤ Q\nh₂ : Ideal.comap (algebraMap R S) P = Ideal.comap (algebraMap R S) Q\ninst✝ : WeaklyQuasiFiniteAt R Q\nthis : (Ideal.map (Ideal.Quotient.mk (Ide... | simp [Ideal.map_comap_le, ← h₂] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
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