module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 504,
"column": 6
} | {
"line": 504,
"column": 16
} | {
"line": 504,
"column": 16
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : TopologicalSpace X\nμ : Measure ℝ\ninst✝¹ : NullSingletonClass μ\ninst✝ : IsLocallyFiniteMeasure μ\nf : X → ℝ → E\na₀ : ℝ\nhf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologic... | [
"E : Type u_1\nX : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : TopologicalSpace X\nμ : Measure ℝ\ninst✝¹ : NullSingletonClass μ\ninst✝ : IsLocallyFiniteMeasure μ\nf : X → ℝ → E\na₀ : ℝ\nhf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Fu... | lt_min_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.NonIntegrable | {
"line": 149,
"column": 6
} | {
"line": 149,
"column": 28
} | {
"line": 150,
"column": 6
} | [
{
"pp": "case inl\nE : Type u_1\nF : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedAddCommGroup F\nf : ℝ → E\ng : ℝ → F\na b c : ℝ\nhne : a ≠ b\nhc : c ∈ [[a, b]]\nh_deriv : ∀ᶠ (x : ℝ) in 𝓝[[[a, b]] \\ {c}] c, DifferentiableAt ℝ f x\nh_infty : Tendsto (fun x ↦ ‖f x‖) (𝓝[[[a,... | [
"case inl\nE : Type u_1\nF : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedAddCommGroup F\nf : ℝ → E\ng : ℝ → F\na b c : ℝ\nhne : a ≠ b\nhc : c ∈ [[a, b]]\nh_deriv : ∀ᶠ (x : ℝ) in 𝓝[[[a, b]] \\ {c}] c, DifferentiableAt ℝ f x\nh_infty : Tendsto (fun x ↦ ‖f x‖) (𝓝[[[a, b]] \\ {c}]... | rw [← Iic_sdiff_right] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Haar.Quotient | {
"line": 174,
"column": 2
} | {
"line": 174,
"column": 50
} | {
"line": 175,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝¹⁴ : Group G\ninst✝¹³ : MeasurableSpace G\ninst✝¹² : TopologicalSpace G\ninst✝¹¹ : IsTopologicalGroup G\ninst✝¹⁰ : BorelSpace G\ninst✝⁹ : PolishSpace G\nΓ : Subgroup G\ninst✝⁸ : Γ.Normal\ninst✝⁷ : T2Space (G ⧸ Γ)\ninst✝⁶ : SecondCountableTopology (G ⧸ Γ)\nμ : Measure (G ⧸ Γ)\nν : Mea... | [
"G : Type u_1\ninst✝¹⁴ : Group G\ninst✝¹³ : MeasurableSpace G\ninst✝¹² : TopologicalSpace G\ninst✝¹¹ : IsTopologicalGroup G\ninst✝¹⁰ : BorelSpace G\ninst✝⁹ : PolishSpace G\nΓ : Subgroup G\ninst✝⁸ : Γ.Normal\ninst✝⁷ : T2Space (G ⧸ Γ)\ninst✝⁶ : SecondCountableTopology (G ⧸ Γ)\nμ : Measure (G ⧸ Γ)\nν : Measure G\ninst... | rw [measure_eq_div_smul μ' μ neZeroV neTopV, hV] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Haar.Quotient | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 6
} | {
"line": 176,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝¹⁴ : Group G\ninst✝¹³ : MeasurableSpace G\ninst✝¹² : TopologicalSpace G\ninst✝¹¹ : IsTopologicalGroup G\ninst✝¹⁰ : BorelSpace G\ninst✝⁹ : PolishSpace G\nΓ : Subgroup G\ninst✝⁸ : Γ.Normal\ninst✝⁷ : T2Space (G ⧸ Γ)\ninst✝⁶ : SecondCountableTopology (G ⧸ Γ)\nμ : Measure (G ⧸ Γ)\nν : Mea... | [
"G : Type u_1\ninst✝¹⁴ : Group G\ninst✝¹³ : MeasurableSpace G\ninst✝¹² : TopologicalSpace G\ninst✝¹¹ : IsTopologicalGroup G\ninst✝¹⁰ : BorelSpace G\ninst✝⁹ : PolishSpace G\nΓ : Subgroup G\ninst✝⁸ : Γ.Normal\ninst✝⁷ : T2Space (G ⧸ Γ)\ninst✝⁶ : SecondCountableTopology (G ⧸ Γ)\nμ : Measure (G ⧸ Γ)\nν : Measure G\ninst... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Order.Filter.AtTopBot.Floor | {
"line": 34,
"column": 2
} | {
"line": 34,
"column": 40
} | {
"line": 35,
"column": 0
} | [
{
"pp": "case calc_2\nK : Type u_1\ninst✝³ : Ring K\ninst✝² : LinearOrder K\ninst✝¹ : IsStrictOrderedRing K\ninst✝ : FloorSemiring K\na c : K\nd n : ℕ\nh : ⌈|a|⌉₊ * ⌈|c|⌉₊ ^ n < (n - d)!\n⊢ ↑⌈|a|⌉₊ * ↑⌈|c|⌉₊ ^ n = ↑(⌈|a|⌉₊ * ⌈|c|⌉₊ ^ n)",
"ppTerm": "?calc_2",
"assigned": true,
"usedConstants": [
... | [] | · simp_rw [Nat.cast_mul, Nat.cast_pow] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 331,
"column": 4
} | {
"line": 331,
"column": 24
} | {
"line": 333,
"column": 0
} | [
{
"pp": "case neg.refine_2\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ\nh₀ : ¬R = 0\nh : IntegrableOn (fun θ ↦ (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π)) volume\n⊢ ∀ᵐ (a : ℝ) ∂volume.restrict (Ι 0 (2 * π)),\n ‖f (circleMap c R a)‖ ≤ |R|⁻¹ * ‖... | [] | simp [norm_smul, h₀] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 331,
"column": 4
} | {
"line": 331,
"column": 24
} | {
"line": 333,
"column": 0
} | [
{
"pp": "case neg.refine_2\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ\nh₀ : ¬R = 0\nh : IntegrableOn (fun θ ↦ (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π)) volume\n⊢ ∀ᵐ (a : ℝ) ∂volume.restrict (Ι 0 (2 * π)),\n ‖f (circleMap c R a)‖ ≤ |R|⁻¹ * ‖... | [] | simp [norm_smul, h₀] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 331,
"column": 4
} | {
"line": 331,
"column": 24
} | {
"line": 333,
"column": 0
} | [
{
"pp": "case neg.refine_2\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ\nh₀ : ¬R = 0\nh : IntegrableOn (fun θ ↦ (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π)) volume\n⊢ ∀ᵐ (a : ℝ) ∂volume.restrict (Ι 0 (2 * π)),\n ‖f (circleMap c R a)‖ ≤ |R|⁻¹ * ‖... | [] | simp [norm_smul, h₀] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 282,
"column": 41
} | {
"line": 312,
"column": 8
} | {
"line": 314,
"column": 0
} | [
{
"pp": "E : 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\n⊢ IntervalIntegrable f volume a₁ a₂",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Left.neg_pos_iff._simp_1",
"Math... | [] | by
wlog hT : 0 < T
· rcases (not_lt.1 hT).eq_or_lt with h | h
· tauto
· have hnT : 0 < -T := by aesop
nth_rw 1 [(by ring : t = (t + T) + (-T))] at h₂f
apply this h₁f.neg hnT.ne' h₂f.symm _ _ hnT
-- Replace [a₁, a₂] by [t - n₁ * T, t + n₂ * T], where n₁ and n₂ are natural numbers
obtain ⟨n₁, ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 569,
"column": 51
} | {
"line": 569,
"column": 57
} | {
"line": 569,
"column": 57
} | [
{
"pp": "n : ℤ\nhn✝ : n ≠ -1\nc w : ℂ\nR : ℝ\nhw : w ∈ sphere c |R|\nhn : n < -1\n⊢ -1 < 0",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"of_decide_eq_true",
"PartialOrder.toPreorder",
"Int.instConditionallyCompleteLinearOrder",
"id",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 569,
"column": 51
} | {
"line": 569,
"column": 57
} | {
"line": 569,
"column": 57
} | [
{
"pp": "n : ℤ\nhn✝ : n ≠ -1\nc w : ℂ\nR : ℝ\nhw : w ∈ sphere c |R|\nhn : n < -1\n⊢ -1 < 0",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"of_decide_eq_true",
"PartialOrder.toPreorder",
"Int.instConditionallyCompleteLinearOrder",
"id",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 569,
"column": 51
} | {
"line": 569,
"column": 57
} | {
"line": 569,
"column": 57
} | [
{
"pp": "n : ℤ\nhn✝ : n ≠ -1\nc w : ℂ\nR : ℝ\nhw : w ∈ sphere c |R|\nhn : n < -1\n⊢ -1 < 0",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"of_decide_eq_true",
"PartialOrder.toPreorder",
"Int.instConditionallyCompleteLinearOrder",
"id",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Prod | {
"line": 434,
"column": 4
} | {
"line": 434,
"column": 17
} | {
"line": 434,
"column": 17
} | [
{
"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\ninst✝² : SFinite ν\ninst✝¹ : NormedSpace ℝ E\ninst✝ : SFinite μ\ng : ↥(Lp E 1 (μ.prod ν))\nthis : ∀ (i : ↥(Lp E 1 (μ.prod ν))), Measurable fun z... | [
"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\ninst✝² : SFinite ν\ninst✝¹ : NormedSpace ℝ E\ninst✝ : SFinite μ\ng : ↥(Lp E 1 (μ.prod ν))\nthis : ∀ (i : ↥(Lp E 1 (μ.prod ν))), Measurable fun z ↦ ‖↑↑i z - ... | ← ofReal_zero | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Integral.DivergenceTheorem | {
"line": 214,
"column": 2
} | {
"line": 216,
"column": 22
} | {
"line": 217,
"column": 2
} | [
{
"pp": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nn : ℕ\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : s.Countable\nHc : ContinuousOn f (Box.Icc I)\nHd : ∀ x ∈ Box.Ioo I... | [
"E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nn : ℕ\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : s.Countable\nHc : ContinuousOn f (Box.Icc I)\nHd : ∀ x ∈ Box.Ioo I \\ s, HasFD... | rcases Metric.uniformContinuousOn_iff_le.1 (I.isCompact_Icc.uniformContinuousOn_of_continuous Hc)
(ε / ∏ j, ((I.face i).upper j - (I.face i).lower j)) (div_pos εpos (hvol_pos (I.face i)))
with ⟨δ, δpos, hδ⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.Integral.Prod | {
"line": 510,
"column": 2
} | {
"line": 511,
"column": 26
} | {
"line": 513,
"column": 0
} | [
{
"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\ninst✝² : SFinite ν\ninst✝¹ : NormedSpace ℝ E\ninst✝ : SFinite μ\nf : α × β → E\ns : Set α\nt : Set β\nhf : IntegrableOn f (s ×ˢ t) (μ.prod ν)\n⊢... | [] | simp only [← Measure.prod_restrict s t, IntegrableOn] at hf ⊢
exact integral_prod f hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Prod | {
"line": 510,
"column": 2
} | {
"line": 511,
"column": 26
} | {
"line": 513,
"column": 0
} | [
{
"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\ninst✝² : SFinite ν\ninst✝¹ : NormedSpace ℝ E\ninst✝ : SFinite μ\nf : α × β → E\ns : Set α\nt : Set β\nhf : IntegrableOn f (s ×ˢ t) (μ.prod ν)\n⊢... | [] | simp only [← Measure.prod_restrict s t, IntegrableOn] at hf ⊢
exact integral_prod f hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.BoxIntegral.DivergenceTheorem | {
"line": 275,
"column": 39
} | {
"line": 278,
"column": 90
} | {
"line": 280,
"column": 0
} | [
{
"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) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : s.Countable\nHs : ∀ x ∈ s, ContinuousWit... | [] | by
refine HasIntegral.sum fun i _ => ?_
simp only [hasFDerivWithinAt_pi', continuousWithinAt_pi] at Hd Hs
exact hasIntegral_GP_pderiv I _ _ s hs (fun x hx => Hs x hx i) (fun x hx => Hd x hx i) i | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.PolynomialGaloisGroup | {
"line": 67,
"column": 7
} | {
"line": 67,
"column": 18
} | {
"line": 67,
"column": 19
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\np : F[X]\nσ τ : p.Gal\nh : ∀ x ∈ p.rootSet p.SplittingField, σ x = τ x\nx : p.SplittingField\n⊢ (↑σ).equalizer ↑τ = ⊤",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"eq_top_iff",
"AlgEqui... | [
"F : Type u_1\ninst✝ : Field F\np : F[X]\nσ τ : p.Gal\nh : ∀ x ∈ p.rootSet p.SplittingField, σ x = τ x\nx : p.SplittingField\n⊢ ⊤ ≤ (↑σ).equalizer ↑τ"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.PolynomialGaloisGroup | {
"line": 201,
"column": 2
} | {
"line": 201,
"column": 12
} | {
"line": 202,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝³ : Field F\np : F[X]\nE : Type u_2\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : Fact (map (algebraMap F E) p).Splits\nϕ : p.Gal\nhϕ : (galActionHom p E) ϕ = 1\n⊢ ϕ = 1",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Polynomial.SplittingField.instComm... | [
"F : Type u_1\ninst✝³ : Field F\np : F[X]\nE : Type u_2\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : Fact (map (algebraMap F E) p).Splits\nϕ : p.Gal\nhϕ : (galActionHom p E) ϕ = 1\nx : p.SplittingField\nhx : x ∈ p.rootSet p.SplittingField\n⊢ ϕ x = 1 x"
] | ext (x hx) | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.FieldTheory.PolynomialGaloisGroup | {
"line": 253,
"column": 2
} | {
"line": 253,
"column": 12
} | {
"line": 254,
"column": 2
} | [
{
"pp": "case neg\nF : Type u_1\ninst✝ : Field F\np q : F[X]\nhpq : ¬p * q = 0\nf g : (p * q).Gal\nhfg :\n (restrict p (p * q).SplittingField) f = (restrict p (p * q).SplittingField) g ∧\n (restrict q (p * q).SplittingField) f = (restrict q (p * q).SplittingField) g\n⊢ f = g",
"ppTerm": "?neg✝",
"as... | [
"case neg\nF : Type u_1\ninst✝ : Field F\np q : F[X]\nhpq : ¬p * q = 0\nf g : (p * q).Gal\nhfg :\n (restrict p (p * q).SplittingField) f = (restrict p (p * q).SplittingField) g ∧\n (restrict q (p * q).SplittingField) f = (restrict q (p * q).SplittingField) g\nx : (p * q).SplittingField\nhx : x ∈ (p * q).rootSet... | ext (x hx) | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.FieldTheory.PolynomialGaloisGroup | {
"line": 254,
"column": 2
} | {
"line": 254,
"column": 40
} | {
"line": 255,
"column": 2
} | [
{
"pp": "case neg\nF : Type u_1\ninst✝ : Field F\np q : F[X]\nhpq : ¬p * q = 0\nf g : (p * q).Gal\nhfg :\n (restrict p (p * q).SplittingField) f = (restrict p (p * q).SplittingField) g ∧\n (restrict q (p * q).SplittingField) f = (restrict q (p * q).SplittingField) g\nx : (p * q).SplittingField\nhx : x ∈ (p ... | [
"case neg\nF : Type u_1\ninst✝ : Field F\np q : F[X]\nhpq : ¬p * q = 0\nf g : (p * q).Gal\nhfg :\n (restrict p (p * q).SplittingField) f = (restrict p (p * q).SplittingField) g ∧\n (restrict q (p * q).SplittingField) f = (restrict q (p * q).SplittingField) g\nx : (p * q).SplittingField\nhx : x ∈ ↑(p.aroots (p *... | rw [rootSet_def, aroots_mul hpq] at hx | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 315,
"column": 2
} | {
"line": 315,
"column": 6
} | {
"line": 316,
"column": 2
} | [
{
"pp": "σ : Type u_5\nR : Type u_6\ninst✝² : CommSemiring R\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\n⊢ hsymm σ R 1 = ∑ i, X i",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"Finset.univ",
"CommSemiring.toSemiring",
"MvPolynomial.X",
... | [
"σ : Type u_5\nR : Type u_6\ninst✝² : CommSemiring R\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\n⊢ ∑ i, X i = hsymm σ R 1"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 381,
"column": 54
} | {
"line": 383,
"column": 6
} | {
"line": 385,
"column": 0
} | [
{
"pp": "σ : Type u_5\nR : Type u_6\ninst✝² : CommSemiring R\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\n⊢ msymm σ R (Nat.Partition.indiscrete 0) = 1",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Nat.instMulZeroClass",
"Finset.univ",
... | [] | by
rw [msymm, Fintype.sum_subsingleton _ ⟨(Sym.nil : Sym σ 0), rfl⟩]
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 390,
"column": 2
} | {
"line": 390,
"column": 6
} | {
"line": 391,
"column": 2
} | [
{
"pp": "σ : Type u_5\nR : Type u_6\ninst✝² : CommSemiring R\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nthis : (fun x ↦ x ∈ Set.univ) = fun x ↦ Nat.Partition.ofSym x = Nat.Partition.indiscrete 1\n⊢ msymm σ R (Nat.Partition.indiscrete 1) = ∑ i, X i",
"ppTerm": "?m.39",
"assigned": true,
"usedConstan... | [
"σ : Type u_5\nR : Type u_6\ninst✝² : CommSemiring R\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nthis : (fun x ↦ x ∈ Set.univ) = fun x ↦ Nat.Partition.ofSym x = Nat.Partition.indiscrete 1\n⊢ ∑ i, X i = msymm σ R (Nat.Partition.indiscrete 1)"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.MeasureTheory.Integral.DivergenceTheorem | {
"line": 346,
"column": 6
} | {
"line": 350,
"column": 76
} | {
"line": 351,
"column": 6
} | [
{
"pp": "case refine_2\nE : Type u\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\nn : ℕ\nF : Type u_1\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : Preorder F\ninst✝¹ : MeasureSpace F\ninst✝ : BorelSpace F\neL : F ≃L[ℝ] Fin (n + 1) → ℝ\nhe_ord : ∀ (x y : F), eL x ≤ eL y ↔ x ≤ y\n... | [
"case refine_3\nE : Type u\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\nn : ℕ\nF : Type u_1\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : Preorder F\ninst✝¹ : MeasureSpace F\ninst✝ : BorelSpace F\neL : F ≃L[ℝ] Fin (n + 1) → ℝ\nhe_ord : ∀ (x y : F), eL x ≤ eL y ↔ x ≤ y\nhe_vol : Mea... | · refine fun x hx i => (Hd (eL.symm x) ⟨?_, hx.2⟩ i).comp x eL.symm.hasFDerivAt
rw [← hIcc]
refine preimage_interior_subset_interior_preimage eL.continuous ?_
simpa only [Set.mem_preimage, eL.apply_symm_apply, ← pi_univ_Icc,
interior_pi_set (@finite_univ (Fin _) _), interior_Icc] using... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Algebra.Polynomial | {
"line": 105,
"column": 28
} | {
"line": 105,
"column": 42
} | {
"line": 105,
"column": 42
} | [
{
"pp": "case refine_1\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝⁵ : Semiring R\ninst✝⁴ : Ring S\ninst✝³ : Field k\ninst✝² : LinearOrder k\ninst✝¹ : IsStrictOrderedRing k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\na✝ : R\n... | [
"case refine_1\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝⁵ : Semiring R\ninst✝⁴ : Ring S\ninst✝³ : Field k\ninst✝² : LinearOrder k\ninst✝¹ : IsStrictOrderedRing k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\na✝ : R\nhc : f a✝ ≠ ... | leadingCoeff_C | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Polynomial.Basic | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 37
} | {
"line": 89,
"column": 2
} | [
{
"pp": "case neg.refine_2.refine_2\np : ℚ[X]\nhp : ¬p = 0\ninj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)\na : Finset ℂ := ?neg.refine_1✝\nb : Finset ℂ := ?neg.refine_2.refine_1✝\n⊢ (p.rootSet ℂ).toFinset.card =\n (Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset).card +\n ... | [
"case neg.refine_2.refine_2.refine_2\np : ℚ[X]\nhp : ¬p = 0\ninj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)\na : Finset ℂ := ?neg.refine_1✝\nb : Finset ℂ := ?neg.refine_2.refine_1✝\nc : Finset ℂ := ?neg.refine_2.refine_2.refine_1✝\n⊢ (p.rootSet ℂ).toFinset.card =\n (Finset.image (⇑(IsScalarTower.toAlgH... | on_goal 1 => let c : Finset ℂ := ?_ | Batteries.Tactic.«_aux_Batteries_Tactic_PermuteGoals___elabRules_Batteries_Tactic_tacticOn_goal-_=>__1» | Batteries.Tactic.«tacticOn_goal-_=>_» |
Mathlib.Data.Real.Embedding | {
"line": 91,
"column": 13
} | {
"line": 91,
"column": 28
} | {
"line": 91,
"column": 28
} | [
{
"pp": "case h\nM : 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⊢ ↑n ∈ upperBounds {r | r.num • 1 < r.den • x}",
"ppTerm": "?h",
"assigned": t... | [
"case h\nM : 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⊢ ∀ x_1 ∈ {r | r.num • 1 < r.den • x}, x_1 ≤ ↑n"
] | mem_upperBounds | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Polynomial.Basic | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 38
} | {
"line": 190,
"column": 2
} | [
{
"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",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Polynomial.C",
"NormedCommRing.toSeminormedCommRing",
"Real",
"Dvd.dvd",
... | [
"p : ℝ[X]\nz : ℂ\nh0 : (aeval z) p = 0\nhz : z.im ≠ 0\n⊢ map (algebraMap ℝ ℂ) (X ^ 2 - C (2 * z.re) * X + C (‖z‖ ^ 2)) ∣ map (algebraMap ℝ ℂ) p"
] | rw [← map_dvd_map' (algebraMap ℝ ℂ)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Valuation.Discrete.RankOne | {
"line": 58,
"column": 27
} | {
"line": 58,
"column": 40
} | {
"line": 58,
"column": 41
} | [
{
"pp": "Γ : Type u_1\ninst✝¹ : LinearOrderedCommGroupWithZero Γ\nR : Type u_2\ninst✝ : Ring R\nv : Valuation R Γ\nhv : v.IsRankOneDiscrete\nk : ℤ\n⊢ ↑((intEquivOfZPowersEqTop (generator' v)⁻¹ ⋯).symm (generator' v) ^ k) = exp (-k)",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Mul... | [
"Γ : Type u_1\ninst✝¹ : LinearOrderedCommGroupWithZero Γ\nR : Type u_2\ninst✝ : Ring R\nv : Valuation R Γ\nhv : v.IsRankOneDiscrete\nk : ℤ\n⊢ ↑((intEquivOfZPowersEqTop (generator' v)⁻¹ ⋯).symm (generator' v) ^ k) = ↑(Multiplicative.ofAdd (-k))"
] | WithZero.exp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Real.Embedding | {
"line": 170,
"column": 8
} | {
"line": 170,
"column": 23
} | {
"line": 170,
"column": 23
} | [
{
"pp": "case a\nM : 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 : ℝ\n⊢ x ∈ upperBounds (ratLt' 0) → 0 ≤ x",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
... | [
"case a\nM : 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 : ℝ\n⊢ (∀ x_1 ∈ ratLt' 0, x_1 ≤ x) → 0 ≤ x"
] | mem_upperBounds | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Real.Embedding | {
"line": 220,
"column": 13
} | {
"line": 220,
"column": 28
} | {
"line": 220,
"column": 28
} | [
{
"pp": "case a\nM : 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\n⊢ ∀ b ∈ upperBounds (ratLt' 1), 1 ≤ b",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.... | [
"case a\nM : 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\n⊢ ∀ (b : ℝ), (∀ x ∈ ratLt' 1, x ≤ b) → 1 ≤ b"
] | mem_upperBounds | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 310,
"column": 33
} | {
"line": 310,
"column": 51
} | {
"line": 310,
"column": 51
} | [
{
"pp": "case pos\nM✝ : Type u_1\nι : Type u_2\ninst✝² : SeminormedCommGroup M✝\ninst✝¹ : IsUltrametricDist M✝\ninst✝ : Nonempty ι\nf : ι → M✝\ninhabited_h : Inhabited ι\na : ι\nt : Multiset ι\nM : ι\nhMs : t ≠ 0 → M ∈ t\nhM : ‖(Multiset.map f t).prod‖ ≤ ‖f M‖\nhMa : ‖f M‖ ≤ ‖f a‖\n⊢ ‖(f a ::ₘ Multiset.map f t)... | [
"case pos\nM✝ : Type u_1\nι : Type u_2\ninst✝² : SeminormedCommGroup M✝\ninst✝¹ : IsUltrametricDist M✝\ninst✝ : Nonempty ι\nf : ι → M✝\ninhabited_h : Inhabited ι\na : ι\nt : Multiset ι\nM : ι\nhMs : t ≠ 0 → M ∈ t\nhM : ‖(Multiset.map f t).prod‖ ≤ ‖f M‖\nhMa : ‖f M‖ ≤ ‖f a‖\n⊢ ‖f a * (Multiset.map f t).prod‖ ≤ ‖f a‖... | Multiset.prod_cons | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 316,
"column": 33
} | {
"line": 316,
"column": 51
} | {
"line": 316,
"column": 51
} | [
{
"pp": "case neg.inr.refine_2\nM✝ : Type u_1\nι : Type u_2\ninst✝² : SeminormedCommGroup M✝\ninst✝¹ : IsUltrametricDist M✝\ninst✝ : Nonempty ι\nf : ι → M✝\ninhabited_h : Inhabited ι\na : ι\nt : Multiset ι\nM : ι\nhMs : t ≠ 0 → M ∈ t\nhM : ‖(Multiset.map f t).prod‖ ≤ ‖f M‖\nhMa : ‖f a‖ < ‖f M‖\nht : t ≠ 0\n⊢ ‖(... | [
"case neg.inr.refine_2\nM✝ : Type u_1\nι : Type u_2\ninst✝² : SeminormedCommGroup M✝\ninst✝¹ : IsUltrametricDist M✝\ninst✝ : Nonempty ι\nf : ι → M✝\ninhabited_h : Inhabited ι\na : ι\nt : Multiset ι\nM : ι\nhMs : t ≠ 0 → M ∈ t\nhM : ‖(Multiset.map f t).prod‖ ≤ ‖f M‖\nhMa : ‖f a‖ < ‖f M‖\nht : t ≠ 0\n⊢ ‖f a * (Multis... | Multiset.prod_cons | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPowerSeries.LexOrder | {
"line": 81,
"column": 6
} | {
"line": 81,
"column": 9
} | {
"line": 81,
"column": 9
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝² : Semiring R\ninst✝¹ : LinearOrder σ\ninst✝ : WellFoundedGT σ\nφ : MvPowerSeries σ R\nd : σ →₀ ℕ\nh : ↑(toLex d) = φ.lexOrder\nhφ : φ ≠ 0\nne : (⇑toLex '' Function.support φ).Nonempty\nhφ' : toLex d = ⋯.min (⇑toLex '' Function.support φ) ne\n⊢ toLex d ∈ ⇑toLex '' Func... | [
"σ : Type u_1\nR : Type u_2\ninst✝² : Semiring R\ninst✝¹ : LinearOrder σ\ninst✝ : WellFoundedGT σ\nφ : MvPowerSeries σ R\nd : σ →₀ ℕ\nh : ↑(toLex d) = φ.lexOrder\nhφ : φ ≠ 0\nne : (⇑toLex '' Function.support φ).Nonempty\nhφ' : toLex d = ⋯.min (⇑toLex '' Function.support φ) ne\n⊢ ⋯.min (⇑toLex '' Function.support φ)... | hφ' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPowerSeries.LexOrder | {
"line": 162,
"column": 2
} | {
"line": 162,
"column": 33
} | {
"line": 163,
"column": 2
} | [
{
"pp": "case inr\nσ : 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\nhφ : φ ≠ 0\n⊢ (φ * ψ).lexOrder = φ.lexOrder + ψ.lexOrder",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Nat.instMu... | [
"case inr.inl\nσ : Type u_1\nR : Type u_2\ninst✝³ : Semiring R\ninst✝² : LinearOrder σ\ninst✝¹ : WellFoundedGT σ\ninst✝ : NoZeroDivisors R\nφ : MvPowerSeries σ R\nhφ : φ ≠ 0\n⊢ (φ * 0).lexOrder = φ.lexOrder + lexOrder 0",
"case inr.inr\nσ : Type u_1\nR : Type u_2\ninst✝³ : Semiring R\ninst✝² : LinearOrder σ\ninst... | obtain rfl | hψ := eq_or_ne ψ 0 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.PowerSeries.Inverse | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 6
} | {
"line": 67,
"column": 2
} | [
{
"pp": "case neg.e_a\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : R⟦X⟧\nh✝ : ¬n = 0\n⊢ (∑ x ∈ antidiagonal (single () n),\n if x.2 < single () n then (MvPowerSeries.coeff x.1) φ * (MvPowerSeries.coeff x.2) (MvPowerSeries.inv.aux a φ)\n else 0) =\n ∑ x ∈ antidiagonal n, if x.2 < n then (coeff x... | [
"case neg.e_a\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : R⟦X⟧\nh✝ : ¬n = 0\n⊢ (∑ x ∈ antidiagonal n, if x.2 < n then (coeff x.1) φ * (coeff x.2) (MvPowerSeries.inv.aux a φ) else 0) =\n ∑ x ∈ antidiagonal (single () n),\n if x.2 < single () n then (MvPowerSeries.coeff x.1) φ * (MvPowerSeries.coeff x.2)... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.PowerSeries.Inverse | {
"line": 340,
"column": 2
} | {
"line": 346,
"column": 22
} | {
"line": 348,
"column": 0
} | [
{
"pp": "k : Type u_2\ninst✝ : Field k\nP : k[X]\nhP : P ≠ 0\n⊢ Multiset.count X (normalizedFactors ↑P) = Multiset.count Polynomial.X (normalizedFactors P)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Polynomial.instNormalizationMonoid",
"UniqueFactorizationMonoid.normalize... | [] | apply eq_of_forall_le_iff
simp only [← Nat.cast_le (α := ℕ∞)]
rw [X_eq_normalize, PowerSeries.X_eq_normalizeX, ← emultiplicity_eq_count_normalizedFactors
irreducible_X hP, ← emultiplicity_eq_count_normalizedFactors X_irreducible] <;>
simp only [← pow_dvd_iff_le_emultiplicity, Polynomial.X_pow_dvd_iff,
Pow... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.Inverse | {
"line": 340,
"column": 2
} | {
"line": 346,
"column": 22
} | {
"line": 348,
"column": 0
} | [
{
"pp": "k : Type u_2\ninst✝ : Field k\nP : k[X]\nhP : P ≠ 0\n⊢ Multiset.count X (normalizedFactors ↑P) = Multiset.count Polynomial.X (normalizedFactors P)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Polynomial.instNormalizationMonoid",
"UniqueFactorizationMonoid.normalize... | [] | apply eq_of_forall_le_iff
simp only [← Nat.cast_le (α := ℕ∞)]
rw [X_eq_normalize, PowerSeries.X_eq_normalizeX, ← emultiplicity_eq_count_normalizedFactors
irreducible_X hP, ← emultiplicity_eq_count_normalizedFactors X_irreducible] <;>
simp only [← pow_dvd_iff_le_emultiplicity, Polynomial.X_pow_dvd_iff,
Pow... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point | {
"line": 118,
"column": 80
} | {
"line": 120,
"column": 12
} | {
"line": 122,
"column": 0
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nW : Projective F\nP : Fin 3 → F\nhPz : P z ≠ 0\n⊢ W.neg P = P z • ![P x / P z, W.toAffine.negY (P x / P z) (P y / P z), 1]",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"WeierstrassCurve.Projective.neg._proof_1",
"WeierstrassCurve.Project... | [] | by
erw [neg, smul_fin3, mul_div_cancel₀ _ hPz, ← negY_of_Z_ne_zero hPz, mul_div_cancel₀ _ hPz,
mul_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point | {
"line": 208,
"column": 2
} | {
"line": 208,
"column": 69
} | {
"line": 209,
"column": 2
} | [
{
"pp": "case pos\nR : Type r\ninst✝ : CommRing R\nW' : Projective R\nP Q : Fin 3 → R\nu v : R\nhu : IsUnit u\nhv : IsUnit v\nh : P ≈ Q\n⊢ W'.add (u • P) (v • Q) ≈ W'.add P Q",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Units.instMulAction",
"instHSMul",
... | [
"case neg\nR : Type r\ninst✝ : CommRing R\nW' : Projective R\nP Q : Fin 3 → R\nu v : R\nhu : IsUnit u\nhv : IsUnit v\nh : ¬P ≈ Q\n⊢ W'.add (u • P) (v • Q) ≈ W'.add P Q"
] | · exact ⟨hu.unit ^ 4, by convert! (add_smul_of_equiv h hu hv).symm⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic | {
"line": 300,
"column": 2
} | {
"line": 300,
"column": 30
} | {
"line": 301,
"column": 2
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Projective R\n⊢ W'.polynomialX = C W'.a₁ * Y * Z - (C 3 * X ^ 2 + C (2 * W'.a₂) * X * Z + C W'.a₄ * Z ^ 2)",
"ppTerm": "?m.139",
"assigned": true,
"usedConstants": [
"Derivation",
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Nat.instMul... | [
"R : Type r\ninst✝ : CommRing R\nW' : Projective R\n⊢ (pderiv x)\n (Y ^ 2 * Z + C W'.a₁ * X * Y * Z + C W'.a₃ * Y * Z ^ 2 -\n (X ^ 3 + C W'.a₂ * X ^ 2 * Z + C W'.a₄ * X * Z ^ 2 + C W'.a₆ * Z ^ 3)) =\n C W'.a₁ * Y * Z - (C 3 * X ^ 2 + C (2 * W'.a₂) * X * Z + C W'.a₄ * Z ^ 2)"
] | rw [polynomialX, polynomial] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point | {
"line": 279,
"column": 8
} | {
"line": 280,
"column": 85
} | {
"line": 281,
"column": 8
} | [
{
"pp": "case pos\nF : Type u\ninst✝ : Field F\nW : Projective F\nP Q : Fin 3 → F\nhP : W.Nonsingular P\nhQ : W.Nonsingular Q\nhPz : ¬P z = 0\nhQz : ¬Q z = 0\nhxy : P x * Q z = Q x * P z ∧ P y * Q z = W.negY Q * P z\nhy : P y * Q z = Q y * P z\n⊢ W.Nonsingular (W.add P Q)",
"ppTerm": "?pos✝",
"assigned"... | [
"case neg\nF : Type u\ninst✝ : Field F\nW : Projective F\nP Q : Fin 3 → F\nhP : W.Nonsingular P\nhQ : W.Nonsingular Q\nhPz : ¬P z = 0\nhQz : ¬Q z = 0\nhxy : P x * Q z = Q x * P z ∧ P y * Q z = W.negY Q * P z\nhy : ¬P y * Q z = Q y * P z\n⊢ W.Nonsingular (W.add P Q)"
] | · simp only [add_of_Y_eq hP.left hPz hQz hxy.left hy hxy.right, nonsingular_smul _ <|
isUnit_dblU_of_Y_eq hP hPz hQz hxy.left hy hxy.right, nonsingular_zero] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Sites.Preserves | {
"line": 57,
"column": 2
} | {
"line": 59,
"column": 62
} | {
"line": 61,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nI : C\nF : Cᵒᵖ ⥤ Type w\nhF : IsSheafFor F (ofArrows Empty.elim fun a ↦ Empty.instIsEmpty.elim a)\n⊢ IsTerminal (F.obj (op I))",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"False",
"CategoryTheory.Presieve.isTerminal_of_isSheaf... | [] | refine @IsTerminal.ofUnique _ _ _ fun Y ↦ ?_
choose t h using hF (by tauto) (by tauto)
exact ⟨⟨↾fun _ ↦ t⟩, fun a ↦ by ext; exact h.2 _ (by tauto)⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Preserves | {
"line": 57,
"column": 2
} | {
"line": 59,
"column": 62
} | {
"line": 61,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nI : C\nF : Cᵒᵖ ⥤ Type w\nhF : IsSheafFor F (ofArrows Empty.elim fun a ↦ Empty.instIsEmpty.elim a)\n⊢ IsTerminal (F.obj (op I))",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"False",
"CategoryTheory.Presieve.isTerminal_of_isSheaf... | [] | refine @IsTerminal.ofUnique _ _ _ fun Y ↦ ?_
choose t h using hF (by tauto) (by tauto)
exact ⟨⟨↾fun _ ↦ t⟩, fun a ↦ by ext; exact h.2 _ (by tauto)⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic | {
"line": 427,
"column": 33
} | {
"line": 430,
"column": 90
} | {
"line": 432,
"column": 0
} | [
{
"pp": "R : Type r\ninst✝¹ : CommRing R\nW' : Projective R\ninst✝ : NoZeroDivisors R\nP : Fin 3 → R\nhP : W'.Nonsingular P\nhPz : P z = 0\n⊢ P y ≠ 0",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"False",
"Nat.instMulZeroClass",
"Weiers... | [] | by
intro hPy
simp only [nonsingular_of_Z_eq_zero hPz, X_eq_zero_of_Z_eq_zero hP.left hPz, hPy, add_zero,
sub_zero, mul_zero, zero_pow two_ne_zero, or_self, ne_self_iff_false, and_false] at hP | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Category.TopCat.ULift | {
"line": 77,
"column": 6
} | {
"line": 77,
"column": 88
} | {
"line": 78,
"column": 6
} | [
{
"pp": "case refine_1\nJ✝ : Type w\ninst✝ : Category.{w', w} J✝\nK : J✝ ⥤ TopCat\nc : Cone K\nhc : IsLimit c\nj : J✝\nt : Set ↑((K ⋙ uliftFunctor).obj j)\nht : IsOpen t\n⊢ IsOpen (⇑(ConcreteCategory.hom ((uliftFunctor.mapCone c).π.app j)) ⁻¹' t)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConst... | [
"case refine_1\nJ✝ : Type w\ninst✝ : Category.{w', w} J✝\nK : J✝ ⥤ TopCat\nc : Cone K\nhc : IsLimit c\nj : J✝\nt : Set ↑((K ⋙ uliftFunctor).obj j)\nht : IsOpen t\n⊢ IsOpen (⇑Homeomorph.ulift.symm ⁻¹' ⇑(ConcreteCategory.hom (uliftFunctor.map (c.π.app j))) ⁻¹' t)"
] | refine ⟨Homeomorph.ulift.symm ⁻¹' ((uliftFunctor.map (c.π.app j)) ⁻¹' t), ?_, rfl⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.AlgebraicGeometry.Sites.BigZariski | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 19
} | {
"line": 73,
"column": 4
} | [
{
"pp": "case refine_2\nX Y : Scheme\n𝓤 : Y.OpenCover\nhS : Presieve.ofArrows 𝓤.X 𝓤.f ∈ (precoverage IsOpenImmersion).coverings Y\nx : Presieve.FamilyOfElements (yoneda.obj X) (Presieve.ofArrows 𝓤.X 𝓤.f)\nhx : x.Compatible\ne : Y ⟶ X := Cover.glueMorphisms 𝓤 (fun j ↦ x (𝓤.f j) ⋯) ⋯\ne' : (yoneda.obj X).o... | [
"case refine_2\nX Y : Scheme\n𝓤 : Y.OpenCover\nhS : Presieve.ofArrows 𝓤.X 𝓤.f ∈ (precoverage IsOpenImmersion).coverings Y\nx : Presieve.FamilyOfElements (yoneda.obj X) (Presieve.ofArrows 𝓤.X 𝓤.f)\nhx : x.Compatible\ne : Y ⟶ X := ⋯\ne' : (yoneda.obj X).obj (op Y)\nh : x.IsAmalgamation e'\n⊢ ∀ (x : 𝓤.I₀), 𝓤.f ... | apply 𝓤.hom_ext | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.RingHom.Unramified | {
"line": 72,
"column": 2
} | {
"line": 74,
"column": 28
} | {
"line": 76,
"column": 0
} | [
{
"pp": "⊢ HoldsForLocalization fun {R S} [CommRing R] [CommRing S] ↦ FormallyUnramified",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CommRing",
"IsLocalization",
"Algebra.algebraMap",
"congrArg",
"CommSemiring.toSemiring",
"Algebra",
... | [] | intro R S _ _ _ M _
rw [formallyUnramified_algebraMap]
exact .of_isLocalization M | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.RingHom.Unramified | {
"line": 72,
"column": 2
} | {
"line": 74,
"column": 28
} | {
"line": 76,
"column": 0
} | [
{
"pp": "⊢ HoldsForLocalization fun {R S} [CommRing R] [CommRing S] ↦ FormallyUnramified",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CommRing",
"IsLocalization",
"Algebra.algebraMap",
"congrArg",
"CommSemiring.toSemiring",
"Algebra",
... | [] | intro R S _ _ _ M _
rw [formallyUnramified_algebraMap]
exact .of_isLocalization M | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Smooth.Pi | {
"line": 44,
"column": 6
} | {
"line": 44,
"column": 98
} | {
"line": 45,
"column": 2
} | [
{
"pp": "case g.map_mul\nR : Type u_1\nI : Type u_2\nA : I → Type u_3\ninst✝³ : CommRing R\ninst✝² : (i : I) → CommRing (A i)\ninst✝¹ : (i : I) → Algebra R (A i)\ninst✝ : FormallySmooth R ((i : I) → A i)\ni : I\nx y : A i\n⊢ (Ideal.Quotient.mk (RingHom.ker (Pi.evalAlgHom R A i).toRingHom ^ 2)) ((LinearMap.singl... | [] | simp +instances only [AlgHom.toRingHom_eq_coe, LinearMap.coe_single, Pi.single_mul, map_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Unramified.Pi | {
"line": 72,
"column": 6
} | {
"line": 78,
"column": 39
} | {
"line": 79,
"column": 4
} | [
{
"pp": "R : Type u_1\nI : Type u_2\ninst✝⁵ : Finite I\nf : I → Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : (i : I) → CommRing (f i)\ninst✝² : (i : I) → Algebra R (f i)\nval✝ : Fintype I\nH✝ : ∀ (i : I), FormallyUnramified R (f i)\nB : Type (max u_2 u_3)\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nJ : Ideal B\nhJ : ... | [] | apply AlgHom.ofLinearMap
(((Ideal.Quotient.mkₐ R J').comp f₂).toLinearMap.comp (LinearMap.single _ _ x))
· simp only [AlgHom.comp_toLinearMap, LinearMap.coe_comp, LinearMap.coe_single,
Function.comp_apply, AlgHom.toLinearMap_apply, Ideal.Quotient.mkₐ_eq_mk]
rw [eq_comm, ← sub_eq_zero, ← ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Unramified.Pi | {
"line": 72,
"column": 6
} | {
"line": 78,
"column": 39
} | {
"line": 79,
"column": 4
} | [
{
"pp": "R : Type u_1\nI : Type u_2\ninst✝⁵ : Finite I\nf : I → Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : (i : I) → CommRing (f i)\ninst✝² : (i : I) → Algebra R (f i)\nval✝ : Fintype I\nH✝ : ∀ (i : I), FormallyUnramified R (f i)\nB : Type (max u_2 u_3)\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nJ : Ideal B\nhJ : ... | [] | apply AlgHom.ofLinearMap
(((Ideal.Quotient.mkₐ R J').comp f₂).toLinearMap.comp (LinearMap.single _ _ x))
· simp only [AlgHom.comp_toLinearMap, LinearMap.coe_comp, LinearMap.coe_single,
Function.comp_apply, AlgHom.toLinearMap_apply, Ideal.Quotient.mkₐ_eq_mk]
rw [eq_comm, ← sub_eq_zero, ← ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Unramified.Pi | {
"line": 93,
"column": 6
} | {
"line": 97,
"column": 33
} | {
"line": 98,
"column": 6
} | [
{
"pp": "case intro.mpr.a\nR : Type u_1\nI : Type u_2\ninst✝⁵ : Finite I\nf : I → Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : (i : I) → CommRing (f i)\ninst✝² : (i : I) → Algebra R (f i)\nval✝ : Fintype I\nH✝ : ∀ (i : I), FormallyUnramified R (f i)\nB : Type (max u_2 u_3)\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\n... | [
"case intro.mpr.a\nR : Type u_1\nI : Type u_2\ninst✝⁵ : Finite I\nf : I → Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : (i : I) → CommRing (f i)\ninst✝² : (i : I) → Algebra R (f i)\nval✝ : Fintype I\nH✝ : ∀ (i : I), FormallyUnramified R (f i)\nB : Type (max u_2 u_3)\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nJ : Ideal B\... | simp only [Ideal.Quotient.algebraMap_eq, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
Function.comp_apply, ← map_sub, Ideal.Quotient.eq_zero_iff_mem, f₁', f₂',
AlgHom.comp_toLinearMap, AlgHom.ofLinearMap_apply, LinearMap.coe_comp,
LinearMap.coe_single, Function.comp_apply, AlgHom.toLinearMap_apply... | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Smooth.Local | {
"line": 41,
"column": 2
} | {
"line": 43,
"column": 54
} | {
"line": 44,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : IsLocalRing S\ninst✝³ : Algebra R S\nP : Extension R S\ninst✝² : FormallySmooth R P.Ring\ninst✝¹ : Module.Free P.Ring Ω[P.Ring⁄R]\ninst✝ : Module.Finite P.Ring Ω[P.Ring⁄R]\nh' : P.ker.FG\n⊢ FormallySmooth R S ↔ Function.Inje... | [
"R : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : IsLocalRing S\ninst✝³ : Algebra R S\nP : Extension R S\ninst✝² : FormallySmooth R P.Ring\ninst✝¹ : Module.Free P.Ring Ω[P.Ring⁄R]\ninst✝ : Module.Finite P.Ring Ω[P.Ring⁄R]\nh' : P.ker.FG\nthis : Module.Finite P.Ring P.Cotangent\n⊢ Forma... | have : Module.Finite P.Ring P.Cotangent :=
have : Module.Finite P.Ring P.ker := .of_fg h'
.of_surjective _ Extension.Cotangent.mk_surjective | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Smooth.Local | {
"line": 90,
"column": 78
} | {
"line": 90,
"column": 90
} | {
"line": 91,
"column": 6
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹⁴ : CommRing R\ninst✝¹³ : CommRing S\ninst✝¹² : IsLocalRing S\ninst✝¹¹ : Algebra R S\nP : Type u_3\nK : Type u_4\ninst✝¹⁰ : Field K\ninst✝⁹ : CommRing P\ninst✝⁸ : Algebra R P\ninst✝⁷ : Algebra P S\ninst✝⁶ : IsScalarTower R P S\ninst✝⁵ : Algebra S K\ninst✝⁴ : Algebra P ... | [
"R : Type u_1\nS : Type u_2\ninst✝¹⁴ : CommRing R\ninst✝¹³ : CommRing S\ninst✝¹² : IsLocalRing S\ninst✝¹¹ : Algebra R S\nP : Type u_3\nK : Type u_4\ninst✝¹⁰ : Field K\ninst✝⁹ : CommRing P\ninst✝⁸ : Algebra R P\ninst✝⁷ : Algebra P S\ninst✝⁶ : IsScalarTower R P S\ninst✝⁵ : Algebra S K\ninst✝⁴ : Algebra P K\ninst✝³ : ... | curry_apply, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Smooth.Fiber | {
"line": 174,
"column": 71
} | {
"line": 174,
"column": 84
} | {
"line": 174,
"column": 84
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nP : Type u_3\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Flat R S\ninst✝⁹ : CommRing P\ninst✝⁸ : Algebra R P\ninst✝⁷ : Algebra P S\ninst✝⁶ : IsScalarTower R P S\ninst✝⁵ : IsLocalRing R\ninst✝⁴ : IsLocalRing S\ninst✝³ : IsLocalHom (alg... | [] | simp_all [fP] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Smooth.Fiber | {
"line": 174,
"column": 71
} | {
"line": 174,
"column": 84
} | {
"line": 174,
"column": 84
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nP : Type u_3\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Flat R S\ninst✝⁹ : CommRing P\ninst✝⁸ : Algebra R P\ninst✝⁷ : Algebra P S\ninst✝⁶ : IsScalarTower R P S\ninst✝⁵ : IsLocalRing R\ninst✝⁴ : IsLocalRing S\ninst✝³ : IsLocalHom (alg... | [] | simp_all [fP] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Smooth.Fiber | {
"line": 174,
"column": 71
} | {
"line": 174,
"column": 84
} | {
"line": 174,
"column": 84
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nP : Type u_3\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Flat R S\ninst✝⁹ : CommRing P\ninst✝⁸ : Algebra R P\ninst✝⁷ : Algebra P S\ninst✝⁶ : IsScalarTower R P S\ninst✝⁵ : IsLocalRing R\ninst✝⁴ : IsLocalRing S\ninst✝³ : IsLocalHom (alg... | [] | simp_all [fP] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Smooth.Fiber | {
"line": 182,
"column": 2
} | {
"line": 182,
"column": 27
} | {
"line": 183,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nP : Type u_3\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Flat R S\ninst✝⁹ : CommRing P\ninst✝⁸ : Algebra R P\ninst✝⁷ : Algebra P S\ninst✝⁶ : IsScalarTower R P S\ninst✝⁵ : IsLocalRing R\ninst✝⁴ : IsLocalRing S\ninst✝³ : IsLocalHom (alg... | [
"R : Type u_1\nS : Type u_2\nP : Type u_3\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Flat R S\ninst✝⁹ : CommRing P\ninst✝⁸ : Algebra R P\ninst✝⁷ : Algebra P S\ninst✝⁶ : IsScalarTower R P S\ninst✝⁵ : IsLocalRing R\ninst✝⁴ : IsLocalRing S\ninst✝³ : IsLocalHom (algebraMap R S)... | algebraize [fP.toRingHom] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.RingTheory.AdicCompletion.Functoriality | {
"line": 112,
"column": 8
} | {
"line": 112,
"column": 31
} | {
"line": 112,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nN : Type u_3\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : Type u_4\ninst✝³ : AddCommGroup P\ninst✝² : Module R P\nT : Type u_5\ninst✝¹ : AddCommGroup T\ninst✝ : Module (AdicCompletion I R) ... | [
"R : Type u_1\ninst✝⁸ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nN : Type u_3\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : Type u_4\ninst✝³ : AddCommGroup P\ninst✝² : Module R P\nT : Type u_5\ninst✝¹ : AddCommGroup T\ninst✝ : Module (AdicCompletion I R) T\nf : M →ₗ[... | val_smul_eq_evalₐ_smul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AdicCompletion.Functoriality | {
"line": 112,
"column": 32
} | {
"line": 112,
"column": 55
} | {
"line": 112,
"column": 56
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nN : Type u_3\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : Type u_4\ninst✝³ : AddCommGroup P\ninst✝² : Module R P\nT : Type u_5\ninst✝¹ : AddCommGroup T\ninst✝ : Module (AdicCompletion I R) ... | [
"R : Type u_1\ninst✝⁸ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nN : Type u_3\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : Type u_4\ninst✝³ : AddCommGroup P\ninst✝² : Module R P\nT : Type u_5\ninst✝¹ : AddCommGroup T\ninst✝ : Module (AdicCompletion I R) T\nf : M →ₗ[... | val_smul_eq_evalₐ_smul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AdicCompletion.AsTensorProduct | {
"line": 205,
"column": 2
} | {
"line": 205,
"column": 51
} | {
"line": 206,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Module.Finite R M\nn : ℕ\np : (Fin n → R) →ₗ[R] M\nhp : Function.Surjective ⇑p\nf : AdicCompletion I R ⊗[R] (Fin n → R) →ₗ[AdicCompletion I R] AdicCompletion I M :=\n ofTensorProduct I M... | [
"R : Type u_1\ninst✝³ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Module.Finite R M\nn : ℕ\np : (Fin n → R) →ₗ[R] M\nhp : Function.Surjective ⇑p\nf : AdicCompletion I R ⊗[R] (Fin n → R) →ₗ[AdicCompletion I R] AdicCompletion I M :=\n ofTensorProduct I M ∘ₗ LinearMa... | let g := map I p ∘ₗ ofTensorProduct I (Fin n → R) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.AdicCompletion.AsTensorProduct | {
"line": 371,
"column": 2
} | {
"line": 374,
"column": 26
} | {
"line": 376,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝⁷ : CommRing R\nI : Ideal R\ninst✝⁶ : IsNoetherianRing 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\nf : M →ₗ[R] N\ninst✝¹ : Module.Finite R M\ninst✝ : Module.Finite R N\nhf : Function.Injective ⇑f\n⊢ Function.Inj... | [] | rw [tensor_map_id_left_eq_map I f]
simp only [LinearMap.coe_comp, LinearEquiv.coe_coe, EmbeddingLike.comp_injective,
EquivLike.injective_comp]
exact map_injective I hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.AdicCompletion.AsTensorProduct | {
"line": 371,
"column": 2
} | {
"line": 374,
"column": 26
} | {
"line": 376,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝⁷ : CommRing R\nI : Ideal R\ninst✝⁶ : IsNoetherianRing 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\nf : M →ₗ[R] N\ninst✝¹ : Module.Finite R M\ninst✝ : Module.Finite R N\nhf : Function.Injective ⇑f\n⊢ Function.Inj... | [] | rw [tensor_map_id_left_eq_map I f]
simp only [LinearMap.coe_comp, LinearEquiv.coe_coe, EmbeddingLike.comp_injective,
EquivLike.injective_comp]
exact map_injective I hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.AdicCompletion.Functoriality | {
"line": 385,
"column": 2
} | {
"line": 385,
"column": 11
} | {
"line": 386,
"column": 2
} | [
{
"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\ny : N\nn : ℕ\nhxy : f x ≡ y [SMOD I ^ n • ⊤]\nz : M\nhz : z ∈ I ^ n •... | [
"case h\nR : 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\ny : N\nn : ℕ\nhxy : f x ≡ y [SMOD I ^ n • ⊤]\nz : M\nhz : z ∈ I ^ n • ⊤\n... | use x + z | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.Smooth.AdicCompletion | {
"line": 106,
"column": 2
} | {
"line": 110,
"column": 70
} | {
"line": 112,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\nS : Type u_3\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : FormallySmooth R A\nf : S →ₐ[R] A\nhf : Function.Surjective ⇑f\n⊢ ∃ g, (AdicCompletion.kerProj hf).comp g = AlgHom.id R A",
"ppTerm": "?m.65",... | [] | obtain ⟨g, hg⟩ := exists_adicCompletionEvalOneₐ_comp_eq
(Ideal.quotientKerAlgEquivOfSurjective hf).symm.toAlgHom
use g
ext x
simpa using! congr(Ideal.quotientKerAlgEquivOfSurjective hf ($hg x)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Smooth.AdicCompletion | {
"line": 106,
"column": 2
} | {
"line": 110,
"column": 70
} | {
"line": 112,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\nS : Type u_3\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : FormallySmooth R A\nf : S →ₐ[R] A\nhf : Function.Surjective ⇑f\n⊢ ∃ g, (AdicCompletion.kerProj hf).comp g = AlgHom.id R A",
"ppTerm": "?m.65",... | [] | obtain ⟨g, hg⟩ := exists_adicCompletionEvalOneₐ_comp_eq
(Ideal.quotientKerAlgEquivOfSurjective hf).symm.toAlgHom
use g
ext x
simpa using! congr(Ideal.quotientKerAlgEquivOfSurjective hf ($hg x)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Extension.Presentation.Submersive | {
"line": 316,
"column": 2
} | {
"line": 317,
"column": 49
} | {
"line": 318,
"column": 2
} | [
{
"pp": "R : Type u\nS : Type v\nι : Type w\nσ : Type t\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\nι' : Type u_1\nσ' : Type u_2\nT : Type u_3\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\nQ : PreSubmersivePresentation S T ι' σ'\nP : PreSubm... | [
"R : Type u\nS : Type v\nι : Type w\nσ : Type t\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\nι' : Type u_1\nσ' : Type u_2\nT : Type u_3\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\nQ : PreSubmersivePresentation S T ι' σ'\nP : PreSubmersivePresen... | rw [jacobiMatrix_apply, jacobiMatrix_apply, comp_map, Sum.elim_inl,
← Q.comp_aeval_relation_inl P.toPresentation] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Extension.Presentation.Submersive | {
"line": 415,
"column": 28
} | {
"line": 415,
"column": 56
} | {
"line": 415,
"column": 57
} | [
{
"pp": "case intro\nR : Type u\nS : Type v\nι : Type w\nσ : Type t\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\nP : PreSubmersivePresentation R S ι σ\ninst✝ : Finite σ\nval✝ : Fintype σ\nh : (baseChange T P).jacobiMatrix = (MvPolynomi... | [
"case intro\nR : Type u\nS : Type v\nι : Type w\nσ : Type t\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\nP : PreSubmersivePresentation R S ι σ\ninst✝ : Finite σ\nval✝ : Fintype σ\nh : (baseChange T P).jacobiMatrix = (MvPolynomial.map (alge... | Generators.algebraMap_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Smooth.StandardSmoothCotangent | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 6
} | {
"line": 150,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nι : Type u_3\nσ : Type u_4\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Finite σ\nP : SubmersivePresentation R S ι σ\nr : σ\n⊢ P.basisCotangent r = Cotangent.mk ⟨P.relation r, ⋯⟩",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": ... | [
"R : Type u_1\nS : Type u_2\nι : Type u_3\nσ : Type u_4\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Finite σ\nP : SubmersivePresentation R S ι σ\nr : σ\n⊢ Cotangent.mk ⟨P.relation r, ⋯⟩ = P.basisCotangent r"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.Extension.Cotangent.LocalizationAway | {
"line": 91,
"column": 6
} | {
"line": 91,
"column": 49
} | {
"line": 91,
"column": 50
} | [
{
"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\n⊢ (compLocalizationAwayAlg... | [
"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\n⊢ (algebraMap P.Ring (Localization.Awa... | compLocalizationAwayAlgHom_toAlgHom_toComp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.RingHom.StandardSmooth | {
"line": 266,
"column": 2
} | {
"line": 267,
"column": 27
} | {
"line": 269,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nw✝² w✝¹ : Type\nw✝ : _root_.Finite w✝¹\nleft✝ : _root_.Finite w✝²\nP : Algebra.SubmersivePresentation R S w✝² w✝¹\nthis : Algebra R S := f.toAlgebra\n⊢ ∃ n g, g.comp MvPolynomial.C = f ∧ g.Etale",
"ppTerm": "?m.76",
"... | [] | exact ⟨_, RingHom.IsStandardSmoothOfRelativeDimension.exists_etale_mvPolynomial
⟨_, _, _, ‹_›, P, rfl⟩⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 262,
"column": 47
} | {
"line": 262,
"column": 73
} | {
"line": 262,
"column": 73
} | [
{
"pp": "case inr\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\nr : σ\n⊢ (Extension.Cotangent.map ((localizationAway S D.gbar).toComp ... | [
"case inr\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\nr : σ\n⊢ Extension.Cotangent.mk\n ⟨((localizationAway S D.gbar).toComp D.pres... | Extension.Cotangent.map_mk | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 684,
"column": 2
} | {
"line": 686,
"column": 68
} | {
"line": 688,
"column": 0
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nW : Projective F\nP Q : Fin 3 → F\nhP : W.Equation P\nhQ : W.Equation Q\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z = Q x * P z\n⊢ W.negAddY P Q = -addU P Q",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"mul_div_mul_right",
... | [] | rw [addU, neg_div, neg_neg, ← mul_div_mul_right _ _ <| mul_ne_zero hPz hQz,
← negAddY_of_X_eq' hP hQ hx, ← sq,
mul_div_cancel_right₀ _ <| pow_ne_zero 2 <| mul_ne_zero hPz hQz] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 684,
"column": 2
} | {
"line": 686,
"column": 68
} | {
"line": 688,
"column": 0
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nW : Projective F\nP Q : Fin 3 → F\nhP : W.Equation P\nhQ : W.Equation Q\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z = Q x * P z\n⊢ W.negAddY P Q = -addU P Q",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"mul_div_mul_right",
... | [] | rw [addU, neg_div, neg_neg, ← mul_div_mul_right _ _ <| mul_ne_zero hPz hQz,
← negAddY_of_X_eq' hP hQ hx, ← sq,
mul_div_cancel_right₀ _ <| pow_ne_zero 2 <| mul_ne_zero hPz hQz] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 684,
"column": 2
} | {
"line": 686,
"column": 68
} | {
"line": 688,
"column": 0
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nW : Projective F\nP Q : Fin 3 → F\nhP : W.Equation P\nhQ : W.Equation Q\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z = Q x * P z\n⊢ W.negAddY P Q = -addU P Q",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"mul_div_mul_right",
... | [] | rw [addU, neg_div, neg_neg, ← mul_div_mul_right _ _ <| mul_ne_zero hPz hQz,
← negAddY_of_X_eq' hP hQ hx, ← sq,
mul_div_cancel_right₀ _ <| pow_ne_zero 2 <| mul_ne_zero hPz hQz] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 52
} | {
"line": 258,
"column": 0
} | [
{
"pp": "k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\np : ℕ\nhp : Nat.Prime p\nH : ∀ (s : Finset K), LinearIndepOn k id ↑s → LinearIndepOn k (fun x ↦ x ^ p) ↑s\na : ι → K\ninst✝ : ExpChar k p\ns : Set ι\nn : ι\nha : IsTranscendenceBasis k fun i ↦ a ↑i\nhn :... | [] | refine ⟨i, hi.comp_equiv e₂.symm, by convert! hi'⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 275,
"column": 25
} | {
"line": 275,
"column": 36
} | {
"line": 275,
"column": 37
} | [
{
"pp": "k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\np : ℕ\nhp : Nat.Prime p\nH : ∀ (s : Finset K), LinearIndepOn k id ↑s → LinearIndepOn k (fun x ↦ x ^ p) ↑s\na : ι → K\nn : ι\ninst✝ : ExpChar k p\nha : adjoin k (Set.range a) = ⊤\nha' : IsTranscendenceBas... | [
"k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\np : ℕ\nhp : Nat.Prime p\nH : ∀ (s : Finset K), LinearIndepOn k id ↑s → LinearIndepOn k (fun x ↦ x ^ p) ↑s\na : ι → K\nn : ι\ninst✝ : ExpChar k p\nha : adjoin k (Set.range a) = ⊤\nha' : IsTranscendenceBasis k fun i ↦... | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 300,
"column": 2
} | {
"line": 300,
"column": 26
} | {
"line": 301,
"column": 2
} | [
{
"pp": "k : 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 ↦ IsTranscendenceBasis k Su... | [
"k : 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 ↦ IsTranscendenceBasis k Subtype.val) ↑... | rw [Set.image_id] at hi₂ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.MorphismProperty.Descent | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 37
} | {
"line": 140,
"column": 38
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nP : MorphismProperty C\ninst✝³ : (isomorphisms C).DescendsAlong P\ninst✝² : P.IsStableUnderBaseChange\ninst✝¹ : HasPullbacks C\ninst✝ : HasEqualizers C\nS T : C\nf : T ⟶ S\nhf : P f\nX Y : Over S\na b : X ⟶ Y\nhab : (Over.pullback f).map a = (Over.pullback ... | [
"case e'_2.h₀\nC : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nP : MorphismProperty C\ninst✝³ : (isomorphisms C).DescendsAlong P\ninst✝² : P.IsStableUnderBaseChange\ninst✝¹ : HasPullbacks C\ninst✝ : HasEqualizers C\nS T : C\nf : T ⟶ S\nhf : P f\nX Y : Over S\na b : X ⟶ Y\nhab : (Over.pullback f).map a = (Over.pullbac... | convert! congr($(hab).left) <;> ext | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.RingTheory.IntegralClosure.IsIntegral.AlmostIntegral | {
"line": 36,
"column": 4
} | {
"line": 36,
"column": 35
} | {
"line": 37,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\na b : S\nr : R\nhr : r ∈ R⁰\nhr' : ∀ (n : ℕ), r • a ^ n ∈ (algebraMap R S).range\ns : R\nhs : s ∈ R⁰\nhs' : ∀ (n : ℕ), s • b ^ n ∈ (algebraMap R S).range\nn : ℕ\n⊢ (r * s) • (a * b) ^ n ∈ (algebraMap R S).range",... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\na b : S\nr : R\nhr : r ∈ R⁰\nhr' : ∀ (n : ℕ), r • a ^ n ∈ (algebraMap R S).range\ns : R\nhs : s ∈ R⁰\nhs' : ∀ (n : ℕ), s • b ^ n ∈ (algebraMap R S).range\nn : ℕ\n⊢ r • a ^ n * s • b ^ n ∈ (algebraMap R S).range"
] | rw [mul_pow, mul_smul_mul_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.IntegralClosure.IsIntegral.AlmostIntegral | {
"line": 38,
"column": 14
} | {
"line": 43,
"column": 71
} | {
"line": 44,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\n⊢ ∀ {a b : S}, a ∈ {s | IsAlmostIntegral R s} → b ∈ {s | IsAlmostIntegral R s} → a + b ∈ {s | IsAlmostIntegral R s}",
"ppTerm": "?m.121",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"No... | [] | by
rintro a b ⟨r, hr, hr'⟩ ⟨s, hs, hs'⟩
refine ⟨r * s, mul_mem hr hs, fun n ↦ ?_⟩
simp only [add_pow, Finset.smul_sum, ← smul_mul_assoc _ (_ * _),
← smul_mul_smul_comm _ (a ^ _)]
exact sum_mem fun i _ ↦ mul_mem (mul_mem (hr' _) (hs' _)) (by simp) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Valuation.IsTrivialOn | {
"line": 57,
"column": 2
} | {
"line": 58,
"column": 48
} | {
"line": 60,
"column": 0
} | [
{
"pp": "case neg\nΓ : Type u_1\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nA : Type u_2\ninst✝² : CommSemiring A\nB : Type u_3\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nv : Valuation B Γ\nhv : Valuation.IsTrivialOn A v\nw : B\nhpos : 1 < v w\np : A[X]\nhp : p ≠ 0\ni : ℕ\nhi : i < p.natDegree\nh0 : ¬p.coeff i = 0\... | [] | · simp [hv.eq_one p.leadingCoeff (leadingCoeff_ne_zero.mpr hp),
hv.eq_one _ h0, pow_lt_pow_right₀ hpos hi] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Valuation.IsTrivialOn | {
"line": 77,
"column": 8
} | {
"line": 77,
"column": 23
} | {
"line": 77,
"column": 24
} | [
{
"pp": "case inr\nΓ : Type u_1\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nA : Type u_2\ninst✝² : CommRing A\nK : Type u_3\ninst✝¹ : Field K\ninst✝ : Algebra A K\nv : Valuation K Γ\nhv : IsTrivialOn A v\ny : K\nh0 : y ≠ 0\nhy : v y ≠ 1\nthis : ∀ (y : K), y ≠ 0 → v y ≠ 1 → 1 < v y → Transcendental A y\nhlt : v ... | [
"case inr\nΓ : Type u_1\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nA : Type u_2\ninst✝² : CommRing A\nK : Type u_3\ninst✝¹ : Field K\ninst✝ : Algebra A K\nv : Valuation K Γ\nhv : IsTrivialOn A v\ny : K\nh0 : y ≠ 0\nhy : v y ≠ 1\nthis : ∀ (y : K), y ≠ 0 → v y ≠ 1 → 1 < v y → Transcendental A y\nhlt : v y ≤ 1\n⊢ ¬Is... | Transcendental, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 198,
"column": 4
} | {
"line": 198,
"column": 39
} | {
"line": 199,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nσ : Type w\nf : MvPolynomial σ S\nH : ∀ (n : σ →₀ ℕ), IsIntegral R (coeff n f)\n⊢ IsIntegral (MvPolynomial σ R) f",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.i... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nσ : Type w\nf : MvPolynomial σ S\nH : ∀ (n : σ →₀ ℕ), IsIntegral R (coeff n f)\n⊢ IsIntegral (MvPolynomial σ R) (∑ v ∈ f.support, (monomial v) (coeff v f))"
] | rw [← f.support_sum_monomial_coeff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.QuasiFinite.Weakly | {
"line": 185,
"column": 4
} | {
"line": 185,
"column": 97
} | {
"line": 186,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal S\ninst✝⁴ : p.IsPrime\ninst✝³ : QuasiFiniteAt R (Ideal.map (Ideal.Quotient.mk (Ideal.map (algebraMap R S) (Ideal.under R p))) p)\nA : Type u_4\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nq : Ideal (A ⊗... | [
"R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal S\ninst✝⁴ : p.IsPrime\ninst✝³ : QuasiFiniteAt R (Ideal.map (Ideal.Quotient.mk (Ideal.map (algebraMap R S) (Ideal.under R p))) p)\nA : Type u_4\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nq : Ideal (A ⊗[R] S)\ninst... | simp only [← TensorProduct.includeLeft.comp_algebraMap, ← Ideal.map_map, ← Ideal.comap_comap] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.QuasiFinite.Weakly | {
"line": 216,
"column": 4
} | {
"line": 217,
"column": 86
} | {
"line": 218,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\nq : Ideal T\ninst✝¹ : q.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R q\nthis : QuasiFiniteAt S (Ideal.map (I... | [] | simpa [Ideal.under, IsScalarTower.algebraMap_eq R S T, ← Ideal.map_map,
← Ideal.comap_comap, -Ideal.under_under] using Ideal.map_mono Ideal.map_comap_le | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.LocalRing.ResidueField.Polynomial | {
"line": 70,
"column": 2
} | {
"line": 76,
"column": 59
} | {
"line": 77,
"column": 2
} | [
{
"pp": "case refine_2\nR : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nI : Ideal R\ninst✝⁴ : I.IsPrime\nJ : Ideal R[X]\ninst✝³ : J.IsPrime\ninst✝² : J.LiesOver I\ninst✝¹ : Algebra (Localization.AtPrime I) (Localization.AtPrime J)\ninst✝ : Localization.AtPrime.IsLiesO... | [
"case refine_3\nR : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nI : Ideal R\ninst✝⁴ : I.IsPrime\nJ : Ideal R[X]\ninst✝³ : J.IsPrime\ninst✝² : J.LiesOver I\ninst✝¹ : Algebra (Localization.AtPrime I) (Localization.AtPrime J)\ninst✝ : Localization.AtPrime.IsLiesOverAlgebra I... | · apply AlgHom.coe_ringHom_injective
apply IsFractionRing.injective_comp_algebraMap (A := I.ResidueField[X])
dsimp [RatFunc.liftAlgHom]
simp only [AlgHom.comp_toRingHom, AlgHom.coe_ringHom_mk, RingHom.comp_assoc,
RatFunc.liftRingHom_comp_algebraMap, RingHomCompTriple.comp_eq, f]
ext <;> simp [← Is... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.LocalRing.ResidueField.Polynomial | {
"line": 141,
"column": 2
} | {
"line": 142,
"column": 69
} | {
"line": 143,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nP : Ideal R\ninst✝ : P.IsPrime\nI : Ideal R[X]\np : P.ResidueField[X]\nhp : Ideal.map (mapRingHom (algebraMap R P.ResidueField)) I = Ideal.span {p}\nthis✝¹ : Algebra (R ⧸ P)[X] P.ResidueField[X] := (mapRingHom (algebraMap (R ⧸ P) P.ResidueField)).toAlgebra\nthis✝ : Is... | [
"R : Type u_1\ninst✝¹ : CommRing R\nP : Ideal R\ninst✝ : P.IsPrime\nI : Ideal R[X]\np : P.ResidueField[X]\nhp : Ideal.map (mapRingHom (algebraMap R P.ResidueField)) I = Ideal.span {p}\nthis✝¹ : Algebra (R ⧸ P)[X] P.ResidueField[X] := (mapRingHom (algebraMap (R ⧸ P) P.ResidueField)).toAlgebra\nthis✝ : IsLocalization... | obtain ⟨r, hr', rfl⟩ := (Ideal.mem_map_iff_of_surjective _
(Polynomial.map_surjective _ Ideal.Quotient.mk_surjective)).mp hr | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.Morphisms.QuasiFinite | {
"line": 393,
"column": 4
} | {
"line": 393,
"column": 54
} | {
"line": 394,
"column": 4
} | [
{
"pp": "case inr\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : LocallyOfFiniteType f\nx : ↥X\nthis :\n ∀ {X Y : Scheme} {f : X ⟶ Y} [LocallyOfFiniteType f] {x : ↥X},\n (∃ R, Y = Spec R) → (QuasiFiniteAt f x ↔ IsOpen {⟨x, ⋯⟩})\nhY : ¬∃ R, Y = Spec R\n⊢ QuasiFiniteAt f x ↔ IsOpen {⟨x, ⋯⟩}",
"ppTerm": "?inr",
"as... | [
"case inr\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : LocallyOfFiniteType f\nx : ↥X\nthis :\n ∀ {X Y : Scheme} {f : X ⟶ Y} [LocallyOfFiniteType f] {x : ↥X},\n (∃ R, Y = Spec R) → (QuasiFiniteAt f x ↔ IsOpen {⟨x, ⋯⟩})\nhY : ¬∃ R, Y = Spec R\ni : Y.affineCover.I₀\ny : ↥(Y.affineCover.X i)\nhy : (Y.affineCover.f i) y = f x\... | obtain ⟨i, y, hy⟩ := Y.affineCover.exists_eq (f x) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 207,
"column": 4
} | {
"line": 207,
"column": 50
} | {
"line": 208,
"column": 2
} | [
{
"pp": "case refine_2\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nP : StandardEtalePair R\n⊢ IsUnit ((algebraMap (AdjoinRoot P.f) (Localization.Away ((AdjoinRoot.mk P.f) P.g))) ((AdjoinRoot.mk P.f) P.g))",
... | [] | exact IsLocalization.Away.algebraMap_isUnit .. | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 291,
"column": 4
} | {
"line": 294,
"column": 92
} | {
"line": 295,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nP✝ : StandardEtalePair R\nP : StandardEtalePresentation R S\n⊢ Ideal.span (Set.range ![(Bivariate.equivMvPolynomial R) (C P.f), (Bivariate.equivMvPolynomia... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nP✝ : StandardEtalePair R\nP : StandardEtalePresentation R S\n⊢ Ideal.span (Set.range ![(Bivariate.equivMvPolynomial R) (C P.f), (Bivariate.equivMvPolynomial R) (Y * C ... | rw [Algebra.Generators.ker_ofAlgHom, AlgHom.toRingHom_eq_coe, AlgHom.comp_toRingHom,
AlgHom.comp_toRingHom,
RingHom.ker_comp_of_injective _ (by exact P.lift_bijective.injective),
RingHom.ker_comp_of_injective _ (by exact P.equivMvPolynomialQuotient.symm.injective)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Smooth.IntegralClosure | {
"line": 151,
"column": 2
} | {
"line": 155,
"column": 50
} | {
"line": 156,
"column": 2
} | [
{
"pp": "R : Type u_1\nB : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nσ : Type u_4\nx : MvPolynomial σ R ⊗[R] B\nhx : x ∈ integralClosure (MvPolynomial σ R) (MvPolynomial σ R ⊗[R] B)\ne₀ : MvPolynomial σ R ⊗[R] B ≃ₐ[R] MvPolynomial σ B :=\n (Algebra.TensorProduct.comm R (MvPolynom... | [
"R : Type u_1\nB : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nσ : Type u_4\nx : MvPolynomial σ R ⊗[R] B\nhx : x ∈ integralClosure (MvPolynomial σ R) (MvPolynomial σ R ⊗[R] B)\ne₀ : MvPolynomial σ R ⊗[R] B ≃ₐ[R] MvPolynomial σ B :=\n (Algebra.TensorProduct.comm R (MvPolynomial σ R) B).... | have : e₀.toAlgHom.comp
(Algebra.TensorProduct.map (AlgHom.id R (MvPolynomial σ R)) (integralClosure R B).val) =
(MvPolynomial.mapAlgHom (integralClosure R B).val).comp e₁.toAlgHom := by
ext <;> simp [e₀, e₁, MvPolynomial.coeff_map, MvPolynomial.coeff_one,
apply_ite ((↑) : (integralClosure R B) → ... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Unramified.LocalStructure | {
"line": 258,
"column": 11
} | {
"line": 258,
"column": 26
} | {
"line": 258,
"column": 26
} | [
{
"pp": "R : Type u_1\nS : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nQ : Ideal S\ninst✝⁴ : Q.IsPrime\ninst✝³ : Module.Finite R S\ninst✝² : IsUnramifiedAt R Q\ninst✝¹ : Algebra (Localization.AtPrime (Ideal.under R Q)) (Localization.AtPrime Q)\ninst✝ : Localization.AtPrime.IsLiesOv... | [
"R : Type u_1\nS : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nQ : Ideal S\ninst✝⁴ : Q.IsPrime\ninst✝³ : Module.Finite R S\ninst✝² : IsUnramifiedAt R Q\ninst✝¹ : Algebra (Localization.AtPrime (Ideal.under R Q)) (Localization.AtPrime Q)\ninst✝ : Localization.AtPrime.IsLiesOverAlgebra (I... | ← Q'.over_def p | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Unramified.LocalStructure | {
"line": 263,
"column": 4
} | {
"line": 263,
"column": 48
} | {
"line": 264,
"column": 4
} | [
{
"pp": "case refine_2\nR : Type u_1\nS : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nQ : Ideal S\ninst✝⁴ : Q.IsPrime\ninst✝³ : Module.Finite R S\ninst✝² : IsUnramifiedAt R Q\ninst✝¹ : Algebra (Localization.AtPrime (Ideal.under R Q)) (Localization.AtPrime Q)\ninst✝ : Localization.A... | [
"case refine_2\nR : Type u_1\nS : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nQ : Ideal S\ninst✝⁴ : Q.IsPrime\ninst✝³ : Module.Finite R S\ninst✝² : IsUnramifiedAt R Q\ninst✝¹ : Algebra (Localization.AtPrime (Ideal.under R Q)) (Localization.AtPrime Q)\ninst✝ : Localization.AtPrime.IsLie... | rw [← Ideal.algebraMap_residueField_eq_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 185,
"column": 77
} | {
"line": 185,
"column": 91
} | {
"line": 187,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nm n : ℕ\n⊢ resultant 0 0 m n = 0 ^ (m + n)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Polynomial.resultant",
"Polynomial.resultant_zero_right",
"Monoid.toMulOneClass",
"congrArg",
"CommSemirin... | [] | simp [pow_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 185,
"column": 77
} | {
"line": 185,
"column": 91
} | {
"line": 187,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nm n : ℕ\n⊢ resultant 0 0 m n = 0 ^ (m + n)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Polynomial.resultant",
"Polynomial.resultant_zero_right",
"Monoid.toMulOneClass",
"congrArg",
"CommSemirin... | [] | simp [pow_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 185,
"column": 77
} | {
"line": 185,
"column": 91
} | {
"line": 187,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nm n : ℕ\n⊢ resultant 0 0 m n = 0 ^ (m + n)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Polynomial.resultant",
"Polynomial.resultant_zero_right",
"Monoid.toMulOneClass",
"congrArg",
"CommSemirin... | [] | simp [pow_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 59,
"column": 8
} | {
"line": 59,
"column": 72
} | {
"line": 60,
"column": 6
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nU V : Y.Opens\nx : ↑Γ(X, f ⁻¹ᵁ Opposite.unop (Opposite.op V))\nhx : x ∈ integralClosure ↑Γ(Y, Opposite.unop (Opposite.op V)) ↑Γ(X, f ⁻¹ᵁ Opposite.unop (Opposite.op V))\ni : Opposite.op V ⟶ Opposite.op U\nalgInst✝⁴ : Algebra ↑Γ(Y, U) ↑Γ(X, f ⁻¹ᵁ U) := (CommRingCat.Hom.hom (app f... | [] | simp [RingHom.algebraMap_toAlgebra, ← CommRingCat.hom_comp]; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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