module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.Convex.Cone.Extension | {
"line": 171,
"column": 4
} | {
"line": 171,
"column": 52
} | {
"line": 172,
"column": 2
} | [
{
"pp": "E : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\nf : E →ₗ.[ℝ] ℝ\nN : E → ℝ\nN_hom : ∀ (c : ℝ), 0 < c → ∀ (x : E), N (c • x) = c * N x\nN_add : ∀ (x y : E), N (x + y) ≤ N x + N y\nhf : ∀ (x : ↥f.domain), ↑f x ≤ N ↑x\nN_0 : N 0 = 0\ns : PointedCone ℝ (E × ℝ) := { carrier := {p | N p.1 ≤ p.2}, a... | [] | simpa [f'] using le_trans (hf ⟨x.1.1, x.2.1⟩) hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Gauge | {
"line": 411,
"column": 2
} | {
"line": 411,
"column": 50
} | {
"line": 413,
"column": 0
} | [
{
"pp": "E : Type u_2\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns : Set E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\nε : ℝ\nhs : ε • s ∈ 𝓝 0\nhε : 0 < ε\nx : E\nhx : x ∈ ε • s\n⊢ gauge s x ∈ Icc 0 ε",
"ppTerm": "?m.86",
"assigned": true,
"usedConstants": [
"Real",
"g... | [] | exact ⟨gauge_nonneg _, gauge_le_of_mem hε.le hx⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 25
} | {
"line": 73,
"column": 4
} | [
{
"pp": "case refine_2\nE : Type u_2\ninst✝⁴ : TopologicalSpace E\ninst✝³ : AddCommGroup E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : Module ℝ E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nhs₀ : 0 ∈ s\nhs₁ : Convex ℝ s\nhs₂ : IsOpen s\nx₀ : E\nhx₀ : x₀ ∉ s\nf : E →ₗ.[ℝ] ℝ := LinearPMap.mkSpanSingleton x₀ 1 ⋯\nφ : ... | [
"case refine_2\nE : Type u_2\ninst✝⁴ : TopologicalSpace E\ninst✝³ : AddCommGroup E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : Module ℝ E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nhs₀ : 0 ∈ s\nhs₁ : Convex ℝ s\nhs₂ : IsOpen s\nx₀ : E\nhx₀ : x₀ ∉ s\nf : E →ₗ.[ℝ] ℝ := LinearPMap.mkSpanSingleton x₀ 1 ⋯\nφ : E →ₗ[ℝ] ℝ\nh... | rw [map_add, map_neg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Convex.Gauge | {
"line": 519,
"column": 4
} | {
"line": 519,
"column": 54
} | {
"line": 520,
"column": 4
} | [
{
"pp": "case inl\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\np : Seminorm ℝ E\nx : E\nhp : {r | 0 < r ∧ x ∈ r • p.ball 0 1} = ∅\nh : ¬0 = p x\nhpx : 0 < p x\n⊢ False",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real.par... | [
"case inl\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\np : Seminorm ℝ E\nx : E\nhp : {r | 0 < r ∧ x ∈ r • p.ball 0 1} = ∅\nh : ¬0 = p x\nhpx : 0 < p x\nhpx₂ : 0 < 2 * p x\n⊢ False"
] | have hpx₂ : 0 < 2 * p x := mul_pos zero_lt_two hpx | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Convex.Gauge | {
"line": 533,
"column": 4
} | {
"line": 533,
"column": 34
} | {
"line": 535,
"column": 0
} | [
{
"pp": "case inr.refine_2\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\np : Seminorm ℝ E\nx : E\nhp : {r | 0 < r ∧ x ∈ r • p.ball 0 1}.Nonempty\nr : ℝ\nhr : r ∈ lowerBounds {r | 0 < r ∧ x ∈ r • p.ball 0 1}\nε : ℝ\nhε : 0 < ε\nhpε : 0 < p x + ε\n⊢ p x < p x + ε",
"ppTerm": "?inr.refine_2",
... | [] | exact lt_add_of_pos_right _ hε | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Convex.Gauge | {
"line": 560,
"column": 2
} | {
"line": 566,
"column": 91
} | {
"line": 568,
"column": 0
} | [
{
"pp": "E : Type u_2\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nx : E\n⊢ gauge (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] 0) x = 0 x",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"AddGroup.toSubtractionMonoi... | [] | simp only [← singleton_zero, gauge_def', mem_closure_zero_iff_norm, norm_smul, mul_eq_zero,
norm_eq_zero, inv_eq_zero]
rcases (norm_nonneg x).eq_or_lt' with hx | hx
· convert! csInf_Ioi (a := (0 : ℝ))
exact Set.ext fun r ↦ and_iff_left (.inr hx)
· convert! Real.sInf_empty
exact eq_empty_of_forall_notM... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Gauge | {
"line": 560,
"column": 2
} | {
"line": 566,
"column": 91
} | {
"line": 568,
"column": 0
} | [
{
"pp": "E : Type u_2\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nx : E\n⊢ gauge (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] 0) x = 0 x",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"AddGroup.toSubtractionMonoi... | [] | simp only [← singleton_zero, gauge_def', mem_closure_zero_iff_norm, norm_smul, mul_eq_zero,
norm_eq_zero, inv_eq_zero]
rcases (norm_nonneg x).eq_or_lt' with hx | hx
· convert! csInf_Ioi (a := (0 : ℝ))
exact Set.ext fun r ↦ and_iff_left (.inr hx)
· convert! Real.sInf_empty
exact eq_empty_of_forall_notM... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 475,
"column": 2
} | {
"line": 484,
"column": 10
} | {
"line": 485,
"column": 2
} | [
{
"pp": "case lt\nα : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : μ.WeaklyRegular\ninst✝ : SigmaFinite μ\nf : α → ℝ\nhf : Integrable f μ\nε : ℝ\nεpos : 0 < ε\nδ : ℝ≥0 := NNReal.mk (ε / 2) ⋯\nδpos : 0 < δ\nfp : α → ℝ≥0 := fun x ↦ (f x).toNNRea... | [
"case lsc\nα : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : μ.WeaklyRegular\ninst✝ : SigmaFinite μ\nf : α → ℝ\nhf : Integrable f μ\nε : ℝ\nεpos : 0 < ε\nδ : ℝ≥0 := NNReal.mk (ε / 2) ⋯\nδpos : 0 < δ\nfp : α → ℝ≥0 := fun x ↦ (f x).toNNReal\nint_fp :... | case lt =>
show ∀ x, (f x : EReal) < g x
intro x
rw [EReal.coe_real_ereal_eq_coe_toNNReal_sub_coe_toNNReal (f x)]
refine EReal.sub_lt_sub_of_lt_of_le ?_ ?_ ?_ ?_
· simp only [EReal.coe_ennreal_lt_coe_ennreal_iff]; exact fp_lt_gp x
· simp only [ENNReal.coe_le_coe, EReal.coe_ennreal_le_coe_ennreal... | Lean.Elab.Tactic.evalCase | Lean.Parser.Tactic.case |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 604,
"column": 4
} | {
"line": 605,
"column": 72
} | {
"line": 606,
"column": 4
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nhK : IsComplete K\nP : ∀ {n : ℕ}, 0 < (1 / 2) ^ n\nx : ℝ\nhx : x ∈ D f K\nn : ℕ → ℕ\nL : ℕ → ℕ → ℕ → F\nhn :\n ∀ (e p q : ℕ),\n n e ≤ p →\n n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / ... | [
"F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nhK : IsComplete K\nP : ∀ {n : ℕ}, 0 < (1 / 2) ^ n\nx : ℝ\nhx : x ∈ D f K\nn : ℕ → ℕ\nL : ℕ → ℕ → ℕ → F\nhn :\n ∀ (e p q : ℕ),\n n e ≤ p →\n n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A ... | obtain ⟨e, he⟩ : ∃ e : ℕ, (1 / 2) ^ e < ε / 12 :=
exists_pow_lt_of_lt_one (div_pos εpos (by norm_num)) (by norm_num) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap | {
"line": 60,
"column": 16
} | {
"line": 60,
"column": 21
} | {
"line": 60,
"column": 22
} | [
{
"pp": "case h_add\nX : Type u_1\nE : Type u_3\nF : Type u_4\ninst✝¹¹ : MeasurableSpace X\nμ : Measure X\n𝕜 : Type u_6\n𝕜' : Type u_7\ninst✝¹⁰ : RCLike 𝕜\ninst✝⁹ : RCLike 𝕜'\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜' F\ninst✝⁴ : Normed... | [
"case h_add\nX : Type u_1\nE : Type u_3\nF : Type u_4\ninst✝¹¹ : MeasurableSpace X\nμ : Measure X\n𝕜 : Type u_6\n𝕜' : Type u_7\ninst✝¹⁰ : RCLike 𝕜\ninst✝⁹ : RCLike 𝕜'\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜' F\ninst✝⁴ : NormedSpace ℝ F\nσ... | f_int | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 82
} | {
"line": 88,
"column": 2
} | [
{
"pp": "case pos\nX : Type u_1\nE : Type u_3\ninst✝⁷ : MeasurableSpace X\nμ : Measure X\n𝕜 : Type u_6\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedSpace ℝ E\nι : Type u_8\ninst✝² : Fintype ι\nM : ι → Type u_9\ninst✝¹ : (i : ι) → NormedAddCommGroup (M i)\ninst✝ ... | [
"case neg\nX : Type u_1\nE : Type u_3\ninst✝⁷ : MeasurableSpace X\nμ : Measure X\n𝕜 : Type u_6\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedSpace ℝ E\nι : Type u_8\ninst✝² : Fintype ι\nM : ι → Type u_9\ninst✝¹ : (i : ι) → NormedAddCommGroup (M i)\ninst✝ : (i : ι) → ... | · exact ((ContinuousMultilinearMap.apply 𝕜 M E m).integral_comp_comm φ_int).symm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap | {
"line": 130,
"column": 4
} | {
"line": 130,
"column": 22
} | {
"line": 132,
"column": 0
} | [
{
"pp": "case inr\nX : Type u_1\nE : Type u_3\nF : Type u_4\ninst✝⁷ : MeasurableSpace X\nμ : Measure X\n𝕜 : Type u_6\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : NormedSpace ℝ E\nL : E ... | [] | simp [integral, *] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap | {
"line": 130,
"column": 4
} | {
"line": 130,
"column": 22
} | {
"line": 132,
"column": 0
} | [
{
"pp": "case inr\nX : Type u_1\nE : Type u_3\nF : Type u_4\ninst✝⁷ : MeasurableSpace X\nμ : Measure X\n𝕜 : Type u_6\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : NormedSpace ℝ E\nL : E ... | [] | simp [integral, *] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap | {
"line": 130,
"column": 4
} | {
"line": 130,
"column": 22
} | {
"line": 132,
"column": 0
} | [
{
"pp": "case inr\nX : Type u_1\nE : Type u_3\nF : Type u_4\ninst✝⁷ : MeasurableSpace X\nμ : Measure X\n𝕜 : Type u_6\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : NormedSpace ℝ E\nL : E ... | [] | simp [integral, *] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap | {
"line": 205,
"column": 4
} | {
"line": 205,
"column": 22
} | {
"line": 207,
"column": 0
} | [
{
"pp": "case neg\nX : Type u_1\nE : Type u_3\nF : Type u_4\ninst✝⁵ : MeasurableSpace X\nμ : Measure X\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : X → E × F\nhf : Integrable f μ\nhE : ¬CompleteSpace E\nthis : ¬Co... | [] | simp [integral, *] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 800,
"column": 2
} | {
"line": 800,
"column": 49
} | {
"line": 801,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : LocallyCompactSpace E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nα : Type u_4\ninst✝ : TopologicalSpace α\nf : α → E → F\nr s : ℝ\nhf : Cont... | [
"𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : LocallyCompactSpace E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nα : Type u_4\ninst✝ : TopologicalSpace α\nf : α → E → F\nr s : ℝ\nhf : Continuous[instT... | rcases exists_between hrt with ⟨t', hrt', ht't⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 841,
"column": 2
} | {
"line": 841,
"column": 86
} | {
"line": 843,
"column": 0
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b : ℝ\nf : ℝ → E\nμ : Measure ℝ\nc : ℝ≥0∞\n⊢ ∫ (x : ℝ) in a..b, f x ∂c • μ = c.toReal • ∫ (x : ℝ) in a..b, f x ∂μ",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure.restrict_smul",
... | [] | simp only [intervalIntegral, Measure.restrict_smul, integral_smul_measure, smul_sub] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 841,
"column": 2
} | {
"line": 841,
"column": 86
} | {
"line": 843,
"column": 0
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b : ℝ\nf : ℝ → E\nμ : Measure ℝ\nc : ℝ≥0∞\n⊢ ∫ (x : ℝ) in a..b, f x ∂c • μ = c.toReal • ∫ (x : ℝ) in a..b, f x ∂μ",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure.restrict_smul",
... | [] | simp only [intervalIntegral, Measure.restrict_smul, integral_smul_measure, smul_sub] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 841,
"column": 2
} | {
"line": 841,
"column": 86
} | {
"line": 843,
"column": 0
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b : ℝ\nf : ℝ → E\nμ : Measure ℝ\nc : ℝ≥0∞\n⊢ ∫ (x : ℝ) in a..b, f x ∂c • μ = c.toReal • ∫ (x : ℝ) in a..b, f x ∂μ",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure.restrict_smul",
... | [] | simp only [intervalIntegral, Measure.restrict_smul, integral_smul_measure, smul_sub] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 446,
"column": 4
} | {
"line": 446,
"column": 14
} | {
"line": 447,
"column": 4
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\na b : ℝ\nμ : Measure ℝ\nf : ℝ → E\ninst✝ : NullSingletonClass μ\nh_int : IntegrableOn f (Icc a b) μ\nh : a ≤ b\nthis : ∀ x ∈ Icc a b, ∫ (t : ℝ) in Ioc a x, f t ∂μ = ∫ (t : ℝ) in a..x, f t ∂μ\n⊢ ContinuousOn (fun x ↦ ∫ (t :... | [
"case pos\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\na b : ℝ\nμ : Measure ℝ\nf : ℝ → E\ninst✝ : NullSingletonClass μ\nh_int : IntegrableOn f (Icc a b) μ\nh : a ≤ b\nthis : ∀ x ∈ Icc a b, ∫ (t : ℝ) in Ioc a x, f t ∂μ = ∫ (t : ℝ) in a..x, f t ∂μ\nx₀ : ℝ\na✝ : x₀ ∈ Icc a b\n⊢ ContinuousWit... | intro x₀ _ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 506,
"column": 2
} | {
"line": 506,
"column": 79
} | {
"line": 508,
"column": 2
} | [
{
"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... | have : IsCompact ({q} ×ˢ (Icc a b)) := isCompact_singleton.prod isCompact_Icc | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1130,
"column": 4
} | {
"line": 1130,
"column": 61
} | {
"line": 1130,
"column": 62
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c d : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhcd : IntervalIntegrable f μ c d\nhac : IntervalIntegrable f μ a c\n⊢ ∫ (x : ℝ) in a..b, f x ∂μ + ∫ (x : ℝ) in c..d, f x ∂μ =\n ∫ (x : ℝ) in c..d, f x ∂μ + (... | [
"E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c d : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhcd : IntervalIntegrable f μ c d\nhac : IntervalIntegrable f μ a c\n⊢ ∫ (x : ℝ) in a..b, f x ∂μ + ∫ (x : ℝ) in c..d, f x ∂μ = ∫ (x : ℝ) in c..d, f x ∂μ + ∫ (x : ℝ) in a..b,... | integral_add_adjacent_intervals hac (hac.symm.trans hab), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Haar.Quotient | {
"line": 133,
"column": 28
} | {
"line": 139,
"column": 41
} | {
"line": 141,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝¹⁰ : Group G\ninst✝⁹ : MeasurableSpace G\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : IsTopologicalGroup G\ninst✝⁶ : BorelSpace G\ninst✝⁵ : PolishSpace G\nΓ : Subgroup G\ninst✝⁴ : Γ.Normal\ninst✝³ : T2Space (G ⧸ Γ)\ninst✝² : SecondCountableTopology (G ⧸ Γ)\nμ : Measure (G ⧸ Γ)\nν : Measure... | [] | by
ext A hA
obtain ⟨x₁, h⟩ := @Quotient.exists_rep _ (QuotientGroup.leftRel Γ) x
convert! measure_preimage_smul μ x₁ A using 1
· rw [← h, Measure.map_apply (measurable_const_mul _) hA]
simp [← MulAction.Quotient.coe_smul_out, ← Quotient.mk''_eq_mk]
exact smulInvariantMeasure_quotient ν | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 1211,
"column": 4
} | {
"line": 1211,
"column": 45
} | {
"line": 1212,
"column": 2
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : IsScalarTower ℝ 𝕜 E\nf f' : 𝕜 → E\nz₀ z₁ : 𝕜\nhcont : ContinuousOn (fun t ↦ f' (z₀ + t • z₁)) (Icc 0 1)\nhderiv : ∀ t ∈ Icc 0 1, HasD... | [] | simp [hasDerivAt_iff_isLittleO, sub_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 488,
"column": 2
} | {
"line": 490,
"column": 47
} | {
"line": 492,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR C : ℝ\nhR : 0 ≤ R\nhf : ∀ z ∈ sphere c R, ‖f z‖ ≤ C\n⊢ ‖(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), f z‖ ≤ R * C",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.to... | [] | have : ‖(2 * π * I : ℂ)⁻¹‖ = (2 * π)⁻¹ := by simp [Real.pi_pos.le]
rw [norm_smul, this, ← div_eq_inv_mul, div_le_iff₀ Real.two_pi_pos, mul_comm (R * C), ← mul_assoc]
exact norm_integral_le_of_norm_le_const hR hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 488,
"column": 2
} | {
"line": 490,
"column": 47
} | {
"line": 492,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR C : ℝ\nhR : 0 ≤ R\nhf : ∀ z ∈ sphere c R, ‖f z‖ ≤ C\n⊢ ‖(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), f z‖ ≤ R * C",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.to... | [] | have : ‖(2 * π * I : ℂ)⁻¹‖ = (2 * π)⁻¹ := by simp [Real.pi_pos.le]
rw [norm_smul, this, ← div_eq_inv_mul, div_le_iff₀ Real.two_pi_pos, mul_comm (R * C), ← mul_assoc]
exact norm_integral_le_of_norm_le_const hR hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Prod | {
"line": 278,
"column": 6
} | {
"line": 278,
"column": 75
} | {
"line": 279,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nE : Type u_3\ninst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝² : NormedAddCommGroup E\nF : Type u_4\nG : Type u_5\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nop : E → F → G\nhop : Continuous[instTopologicalSpacePr... | [] | simp [lintegral_const_mul', lintegral_mul_const', hg.2.ne, mul_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.Prod | {
"line": 278,
"column": 6
} | {
"line": 278,
"column": 75
} | {
"line": 279,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nE : Type u_3\ninst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝² : NormedAddCommGroup E\nF : Type u_4\nG : Type u_5\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nop : E → F → G\nhop : Continuous[instTopologicalSpacePr... | [] | simp [lintegral_const_mul', lintegral_mul_const', hg.2.ne, mul_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Prod | {
"line": 278,
"column": 6
} | {
"line": 278,
"column": 75
} | {
"line": 279,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nE : Type u_3\ninst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝² : NormedAddCommGroup E\nF : Type u_4\nG : Type u_5\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nop : E → F → G\nhop : Continuous[instTopologicalSpacePr... | [] | simp [lintegral_const_mul', lintegral_mul_const', hg.2.ne, mul_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Prod | {
"line": 355,
"column": 2
} | {
"line": 355,
"column": 75
} | {
"line": 357,
"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\ns : Set α\nt : Set β\nf : α × β → E\n⊢ ∫ (z : β × α) in t ×ˢ s, f z.swap ∂ν.pro... | [] | rw [← Measure.prod_restrict, ← Measure.prod_restrict, integral_prod_swap] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.Prod | {
"line": 355,
"column": 2
} | {
"line": 355,
"column": 75
} | {
"line": 357,
"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\ns : Set α\nt : Set β\nf : α × β → E\n⊢ ∫ (z : β × α) in t ×ˢ s, f z.swap ∂ν.pro... | [] | rw [← Measure.prod_restrict, ← Measure.prod_restrict, integral_prod_swap] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Prod | {
"line": 355,
"column": 2
} | {
"line": 355,
"column": 75
} | {
"line": 357,
"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\ns : Set α\nt : Set β\nf : α × β → E\n⊢ ∫ (z : β × α) in t ×ˢ s, f z.swap ∂ν.pro... | [] | rw [← Measure.prod_restrict, ← Measure.prod_restrict, integral_prod_swap] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Liouville | {
"line": 85,
"column": 6
} | {
"line": 86,
"column": 53
} | {
"line": 87,
"column": 4
} | [
{
"pp": "F : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nc : ℂ\nR C : ℝ\nf : ℂ → F\nhR : 0 < R\nhd : DiffContOnCl ℂ f (ball c R)\nhC : ∀ z ∈ sphere c R, ‖f z‖ ≤ C\ne : F →L[ℂ] F̂ := UniformSpace.Completion.toComplL\nthis : HasDerivAt (⇑e ∘ f) (e (deriv f c)) c\n⊢ ‖deriv f c‖ = ‖deriv (⇑e ∘ f... | [] | rw [this.deriv]
exact (UniformSpace.Completion.norm_coe _).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Liouville | {
"line": 85,
"column": 6
} | {
"line": 86,
"column": 53
} | {
"line": 87,
"column": 4
} | [
{
"pp": "F : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nc : ℂ\nR C : ℝ\nf : ℂ → F\nhR : 0 < R\nhd : DiffContOnCl ℂ f (ball c R)\nhC : ∀ z ∈ sphere c R, ‖f z‖ ≤ C\ne : F →L[ℂ] F̂ := UniformSpace.Completion.toComplL\nthis : HasDerivAt (⇑e ∘ f) (e (deriv f c)) c\n⊢ ‖deriv f c‖ = ‖deriv (⇑e ∘ f... | [] | rw [this.deriv]
exact (UniformSpace.Completion.norm_coe _).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 81
} | {
"line": 76,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nσ : Type u_2\nf : σ → R\ns : Finset σ\nn : ℕ\n⊢ (map f s.val).esymm n = ∑ t ∈ Finset.powersetCard n s, t.prod f",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Multiset.sum",
"Eq.mpr",
"Multiset.map",
"congrArg",
... | [
"R : Type u_1\ninst✝ : CommSemiring R\nσ : Type u_2\nf : σ → R\ns : Finset σ\nn : ℕ\n⊢ (map ((fun x ↦ x.prod) ∘ (fun x ↦ map f x) ∘ Finset.val) (Finset.powersetCard n s).val).sum =\n ∑ t ∈ Finset.powersetCard n s, t.prod f"
] | simp only [esymm, powersetCard_map, ← Finset.map_val_val_powersetCard, map_map] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 371,
"column": 4
} | {
"line": 371,
"column": 22
} | {
"line": 373,
"column": 0
} | [
{
"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\nIH : ‖∏ i ∈ s, f i‖₊ = s.sup fun i ↦ ‖f i‖₊\nhs : (↑(Finset.cons a s ha)).Pairwise fun i j ↦ ‖f i‖₊ ≠ ‖f j‖₊\nhs' : s.Nonempty\nj : ι\nhj : j ∈ s\nhj' : ‖∏... | [] | apply hs <;> grind | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.NumberTheory.NumberField.Completion.FinitePlace | {
"line": 124,
"column": 2
} | {
"line": 126,
"column": 56
} | {
"line": 127,
"column": 2
} | [
{
"pp": "R : Type u_2\ninst✝³ : CommRing R\ninst✝² : IsDedekindDomain R\nv : HeightOneSpectrum R\ninst✝¹ : Module.Finite ℤ R\ninst✝ : Module.Free ℤ R\nh : absNorm v.asIdeal ≤ 1\n⊢ absNorm v.asIdeal = 1",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroOn... | [
"R : Type u_2\ninst✝³ : CommRing R\ninst✝² : IsDedekindDomain R\nv : HeightOneSpectrum R\ninst✝¹ : Module.Finite ℤ R\ninst✝ : Module.Free ℤ R\nh : absNorm v.asIdeal ≤ 1\nthis : 0 < absNorm v.asIdeal\n⊢ absNorm v.asIdeal = 1"
] | have : 0 < absNorm v.asIdeal := by
rw [Nat.pos_iff_ne_zero, absNorm_ne_zero_iff]
exact v.asIdeal.finiteQuotientOfFreeOfNeBot v.ne_bot | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.NumberField.Completion.FinitePlace | {
"line": 280,
"column": 2
} | {
"line": 280,
"column": 61
} | {
"line": 282,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝⁶ : Field K\nR : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : Algebra R K\ninst✝³ : IsDedekindDomain R\ninst✝² : IsFractionRing R K\nv : HeightOneSpectrum R\ninst✝¹ : Module.Finite ℤ R\ninst✝ : Module.Free ℤ R\nx : R\n⊢ ‖(embedding v) ((algebraMap R K) x)‖ = ↑((toNNReal ⋯) (v.intValuation... | [] | simp [norm_embedding, adicAbv_def, valuation_of_algebraMap] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.NumberField.Completion.FinitePlace | {
"line": 280,
"column": 2
} | {
"line": 280,
"column": 61
} | {
"line": 282,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝⁶ : Field K\nR : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : Algebra R K\ninst✝³ : IsDedekindDomain R\ninst✝² : IsFractionRing R K\nv : HeightOneSpectrum R\ninst✝¹ : Module.Finite ℤ R\ninst✝ : Module.Free ℤ R\nx : R\n⊢ ‖(embedding v) ((algebraMap R K) x)‖ = ↑((toNNReal ⋯) (v.intValuation... | [] | simp [norm_embedding, adicAbv_def, valuation_of_algebraMap] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Completion.FinitePlace | {
"line": 280,
"column": 2
} | {
"line": 280,
"column": 61
} | {
"line": 282,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝⁶ : Field K\nR : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : Algebra R K\ninst✝³ : IsDedekindDomain R\ninst✝² : IsFractionRing R K\nv : HeightOneSpectrum R\ninst✝¹ : Module.Finite ℤ R\ninst✝ : Module.Free ℤ R\nx : R\n⊢ ‖(embedding v) ((algebraMap R K) x)‖ = ↑((toNNReal ⋯) (v.intValuation... | [] | simp [norm_embedding, adicAbv_def, valuation_of_algebraMap] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.Completion.FinitePlace | {
"line": 383,
"column": 81
} | {
"line": 395,
"column": 10
} | {
"line": 397,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nv₁ v₂ : HeightOneSpectrum (𝓞 K)\n⊢ mk v₁ = mk v₂ ↔ v₁ = v₂",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_exists._simp_1",
"Iff.mpr",
"Real.instIsOrderedRing",
"Norm.norm",
... | [] | by
refine ⟨?_, fun a ↦ by rw [a]⟩
contrapose!
intro h
rw [DFunLike.ne_iff]
have ⟨x, hx1, hx2⟩ : ∃ x : 𝓞 K, x ∈ v₁.asIdeal ∧ x ∉ v₂.asIdeal := by
by_contra! H
exact h <| HeightOneSpectrum.ext_iff.mpr <| IsMaximal.eq_of_le (isMaximal v₁) IsPrime.ne_top' H
use x
simp only [mk_apply]
rw [← norm_lt_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point | {
"line": 143,
"column": 65
} | {
"line": 143,
"column": 70
} | {
"line": 143,
"column": 71
} | [
{
"pp": "case a\nR : Type r\ninst✝ : CommRing R\nW' : Projective R\nP : Fin 3 → R\nhP : W'.Equation P\n⊢ -3 * P x ^ 2 * P x * W'.negY P + 3 * P x * P x ^ 2 * P y - P y ^ 2 * W'.negY P * P z + P y * W'.negY P ^ 2 * P z +\n W'.a₁ * P x * W'.negY P ^ 2 * P z -\n ... | [
"case a\nR : Type r\ninst✝ : CommRing R\nW' : Projective R\nP : Fin 3 → R\nhP : W'.Equation P\n⊢ -3 * P x ^ 2 * P x * W'.negY P + 3 * P x * P x ^ 2 * P y - P y ^ 2 * W'.negY P * P z + P y * W'.negY P ^ 2 * P z +\n W'.a₁ * P x * W'.negY P ^ 2 * P z -\n ... | dblZ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point | {
"line": 203,
"column": 6
} | {
"line": 203,
"column": 64
} | {
"line": 203,
"column": 65
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Projective R\nP Q : Fin 3 → R\nh : ¬P ≈ Q\nu v : R\nhu : IsUnit u\nhv : IsUnit v\n⊢ W'.add (u • P) (v • Q) = (u * v) ^ 2 • W'.add P Q",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"instHSMul",
"instS... | [
"R : Type r\ninst✝ : CommRing R\nW' : Projective R\nP Q : Fin 3 → R\nh : ¬P ≈ Q\nu v : R\nhu : IsUnit u\nhv : IsUnit v\n⊢ W'.addXYZ (u • P) (v • Q) = (u * v) ^ 2 • W'.add P Q"
] | add_of_not_equiv <| h.comp (smul_equiv_smul P Q hu hv).mp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point | {
"line": 436,
"column": 16
} | {
"line": 436,
"column": 51
} | {
"line": 436,
"column": 51
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nW : Projective F\nP : Fin 3 → F\nhP : ¬W.Nonsingular P\n⊢ (if hP : W.Nonsingular P ∧ P z ≠ 0 then Affine.Point.some (P x / P z) (P y / P z) ⋯ else 0) = 0",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"WeierstrassCurve.Projective.neg._proof_1",
... | [
"F : Type u\ninst✝ : Field F\nW : Projective F\nP : Fin 3 → F\nhP : ¬W.Nonsingular P\n⊢ 0 = 0"
] | dif_neg <| not_and_of_not_left _ hP | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic | {
"line": 402,
"column": 2
} | {
"line": 403,
"column": 80
} | {
"line": 405,
"column": 0
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Projective R\nP : Fin 3 → R\nhPz : P z = 0\n⊢ W'.Nonsingular P ↔ W'.Equation P ∧ (3 * P x ^ 2 ≠ 0 ∨ P y ^ 2 + W'.a₁ * P x * P y - W'.a₂ * P x ^ 2 ≠ 0)",
"ppTerm": "?m.101",
"assigned": true,
"usedConstants": [
"False",
"WeierstrassCurve.Proje... | [] | simp only [nonsingular_iff, hPz, add_zero, zero_sub, mul_zero,
zero_pow <| OfNat.ofNat_ne_zero _, neg_ne_zero, ne_self_iff_false, false_or] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic | {
"line": 402,
"column": 2
} | {
"line": 403,
"column": 80
} | {
"line": 405,
"column": 0
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Projective R\nP : Fin 3 → R\nhPz : P z = 0\n⊢ W'.Nonsingular P ↔ W'.Equation P ∧ (3 * P x ^ 2 ≠ 0 ∨ P y ^ 2 + W'.a₁ * P x * P y - W'.a₂ * P x ^ 2 ≠ 0)",
"ppTerm": "?m.101",
"assigned": true,
"usedConstants": [
"False",
"WeierstrassCurve.Proje... | [] | simp only [nonsingular_iff, hPz, add_zero, zero_sub, mul_zero,
zero_pow <| OfNat.ofNat_ne_zero _, neg_ne_zero, ne_self_iff_false, false_or] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic | {
"line": 402,
"column": 2
} | {
"line": 403,
"column": 80
} | {
"line": 405,
"column": 0
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Projective R\nP : Fin 3 → R\nhPz : P z = 0\n⊢ W'.Nonsingular P ↔ W'.Equation P ∧ (3 * P x ^ 2 ≠ 0 ∨ P y ^ 2 + W'.a₁ * P x * P y - W'.a₂ * P x ^ 2 ≠ 0)",
"ppTerm": "?m.101",
"assigned": true,
"usedConstants": [
"False",
"WeierstrassCurve.Proje... | [] | simp only [nonsingular_iff, hPz, add_zero, zero_sub, mul_zero,
zero_pow <| OfNat.ofNat_ne_zero _, neg_ne_zero, ne_self_iff_false, false_or] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Preserves | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 98
} | {
"line": 116,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nα : Type u_1\ninst✝² : Small.{w, u_1} α\nX : α → C\nc : Cofan X\nhc : IsColimit c\ninst✝¹ : (ofArrows X c.inj).HasPairwisePullbacks\ninst✝ : PreservesLimit (Discrete.functor fun x ↦ op (X x)) F\nthis : HasCoproduct X\nhi :\n ∀ (b : (fun X ↦ X) ... | [
"C : Type u\ninst✝³ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nα : Type u_1\ninst✝² : Small.{w, u_1} α\nX : α → C\nc : Cofan X\nhc : IsColimit c\ninst✝¹ : (ofArrows X c.inj).HasPairwisePullbacks\ninst✝ : PreservesLimit (Discrete.functor fun x ↦ op (X x)) F\nthis : HasCoproduct X\nhi :\n ∀ (b : (fun X ↦ X) (∏ᶜ fun b ↦ ... | apply_fun F.map ((opCoproductIsoProduct' hc (productIsProduct _)).hom) using injective_of_mono _ | Mathlib.Tactic._aux_Mathlib_Tactic_ApplyFun___elabRules_Mathlib_Tactic_applyFun_1 | Mathlib.Tactic.applyFun |
Mathlib.RingTheory.RingHom.Smooth | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 39
} | {
"line": 67,
"column": 2
} | [
{
"pp": "R S T : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nH : (algebraMap R T).FormallySmooth\n⊢ (algebraMap S (S ⊗[R] T)).FormallySmooth",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Algebr... | [
"R S T : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nH : Algebra.FormallySmooth R T\n⊢ Algebra.FormallySmooth S (S ⊗[R] T)"
] | rw [formallySmooth_algebraMap] at H ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 205,
"column": 6
} | {
"line": 205,
"column": 11
} | {
"line": 205,
"column": 12
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Projective R\nP : Fin 3 → R\nhPz : P z = 0\n⊢ W'.dblZ P = 0",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"WeierstrassCurve.Projective.negY",
"Eq.mpr",
"HMul.hMul",
"AddGroupWithOne.toAddGroup",
"congrArg",
... | [
"R : Type r\ninst✝ : CommRing R\nW' : Projective R\nP : Fin 3 → R\nhPz : P z = 0\n⊢ P z * (P y - W'.negY P) ^ 3 = 0"
] | dblZ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 209,
"column": 6
} | {
"line": 209,
"column": 11
} | {
"line": 209,
"column": 12
} | [
{
"pp": "R : Type r\ninst✝¹ : CommRing R\nW' : Projective R\ninst✝ : NoZeroDivisors R\nP Q : Fin 3 → R\nhQz : Q z ≠ 0\nhx : P x * Q z = Q x * P z\nhy : P y * Q z = Q y * P z\nhy' : P y * Q z = W'.negY Q * P z\n⊢ W'.dblZ P = 0",
"ppTerm": "?m.92",
"assigned": true,
"usedConstants": [
"Weierstra... | [
"R : Type r\ninst✝¹ : CommRing R\nW' : Projective R\ninst✝ : NoZeroDivisors R\nP Q : Fin 3 → R\nhQz : Q z ≠ 0\nhx : P x * Q z = Q x * P z\nhy : P y * Q z = Q y * P z\nhy' : P y * Q z = W'.negY Q * P z\n⊢ P z * (P y - W'.negY P) ^ 3 = 0"
] | dblZ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Smooth.Fiber | {
"line": 90,
"column": 4
} | {
"line": 90,
"column": 46
} | {
"line": 91,
"column": 2
} | [
{
"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... | [] | simp [RingHom.algebraMap_toAlgebra, ψ, Sp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Smooth.Fiber | {
"line": 90,
"column": 4
} | {
"line": 90,
"column": 46
} | {
"line": 91,
"column": 2
} | [
{
"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... | [] | simp [RingHom.algebraMap_toAlgebra, ψ, Sp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Smooth.Fiber | {
"line": 90,
"column": 4
} | {
"line": 90,
"column": 46
} | {
"line": 91,
"column": 2
} | [
{
"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... | [] | simp [RingHom.algebraMap_toAlgebra, ψ, Sp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Kaehler.TensorProduct | {
"line": 280,
"column": 2
} | {
"line": 280,
"column": 9
} | {
"line": 282,
"column": 0
} | [
{
"pp": "case tmul.tmul.tmul\nR : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra R B\ninst✝³ : Algebra A B\ninst✝² : Algebra S B\ninst✝¹ : IsScalarTower R A B\ni... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 308,
"column": 22
} | {
"line": 308,
"column": 27
} | {
"line": 308,
"column": 28
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Projective F\ninst✝ : DecidableEq 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\nhy : P y * Q z ≠ W.negY Q * P z\n⊢ ((eval P) W.polynomialX ^ 2 - W.a₁ * (eval P) W.polynomialX * P z * (P y - W.negY P)... | [
"F : Type u\ninst✝¹ : Field F\nW : Projective F\ninst✝ : DecidableEq 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\nhy : P y * Q z ≠ W.negY Q * P z\n⊢ ((eval P) W.polynomialX ^ 2 - W.a₁ * (eval P) W.polynomialX * P z * (P y - W.negY P) -\n ... | dblZ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Extension.Presentation.Submersive | {
"line": 161,
"column": 66
} | {
"line": 163,
"column": 49
} | {
"line": 165,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\nι : Type w\nσ : Type t\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nP : PreSubmersivePresentation R S ι σ\ninst✝ : Finite σ\n⊢ IsUnit P.jacobian ↔ Function.Bijective ⇑P.aevalDifferential",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
... | [] | by
rw [P.jacobian_eq_det_aevalDifferential, ← LinearMap.isUnit_iff_isUnit_det]
exact Module.End.isUnit_iff P.aevalDifferential | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.AdicCompletion.Functoriality | {
"line": 449,
"column": 2
} | {
"line": 451,
"column": 7
} | {
"line": 452,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nS : Type u_6\ninst✝¹ : CommRing S\nf : R →+* S\ninst✝ : IsPrecomplete I R\nhaus : IsHausdorff (Ideal.map f I) S\nx✝ : Algebra R S := f.toAlgebra\nfₗ : R →ₗ[R] S := (Algebra.ofId R S).toLinearMap\nh : Function.Surjective ⇑((restrictScalars R (Ideal.map f I... | [
"R : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nS : Type u_6\ninst✝¹ : CommRing S\nf : R →+* S\ninst✝ : IsPrecomplete I R\nhaus : IsHausdorff (Ideal.map f I) S\nx✝ : Algebra R S := f.toAlgebra\nfₗ : R →ₗ[R] S := (Algebra.ofId R S).toLinearMap\nh : Function.Surjective ⇑((restrictScalars R (Ideal.map f I)).mkQ ∘ₗ fₗ... | have : I • ⊤ = restrictScalars R (Ideal.map f I) := by
simp only [Ideal.smul_top_eq_map, restrictScalars_inj]
rfl | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Extension.Presentation.Submersive | {
"line": 551,
"column": 21
} | {
"line": 553,
"column": 75
} | {
"line": 555,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\nι : Type w\nσ : Type t\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Finite σ\nr : R\ninst✝ : IsLocalization.Away r S\n⊢ IsUnit (PreSubmersivePresentation.localizationAway S r).jacobian",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants... | [] | by
rw [localizationAway_jacobian]
exact IsLocalization.map_units _ (⟨r, 1, by simp⟩ : Submonoid.powers r) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Smooth.StandardSmoothCotangent | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 37
} | {
"line": 88,
"column": 4
} | [
{
"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\nx : ↥P.ker\nhx : ∀ (i : σ), (aeval P.val) ((pderiv (P.map i)) ↑x) = 0\n⊢ ↑x ∈ Ideal.span (Set.range P.relation)",
"ppTerm": "... | [
"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\nx : ↥P.ker\nhx : ∀ (i : σ), (aeval P.val) ((pderiv (P.map i)) ↑x) = 0\n⊢ ↑x ∈ P.ker"
] | rw [P.span_range_relation_eq_ker] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Smooth.StandardSmoothCotangent | {
"line": 173,
"column": 2
} | {
"line": 175,
"column": 6
} | {
"line": 177,
"column": 0
} | [
{
"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\nx : P.toExtension.CotangentSpace\ny : P.toExtension.Cotangent\n⊢ P.sectionCotangent x = y ↔ ∀ (i : σ), (P.cotangentSpaceBasis.rep... | [] | simp only [sectionCotangent, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply]
rw [← (cotangentEquiv P).injective.eq_iff, funext_iff, LinearEquiv.apply_symm_apply]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Smooth.StandardSmoothCotangent | {
"line": 173,
"column": 2
} | {
"line": 175,
"column": 6
} | {
"line": 177,
"column": 0
} | [
{
"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\nx : P.toExtension.CotangentSpace\ny : P.toExtension.Cotangent\n⊢ P.sectionCotangent x = y ↔ ∀ (i : σ), (P.cotangentSpaceBasis.rep... | [] | simp only [sectionCotangent, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply]
rw [← (cotangentEquiv P).injective.eq_iff, funext_iff, LinearEquiv.apply_symm_apply]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 393,
"column": 25
} | {
"line": 393,
"column": 30
} | {
"line": 393,
"column": 31
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Projective F\ninst✝ : DecidableEq 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\nhy : P y * Q z ≠ W.negY Q * P z\n⊢ (-(eval P) W.polynomialX *\n ((eval P) W.polynomialX ^ 2 - W.a₁ * (eval P... | [
"F : Type u\ninst✝¹ : Field F\nW : Projective F\ninst✝ : DecidableEq 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\nhy : P y * Q z ≠ W.negY Q * P z\n⊢ (-(eval P) W.polynomialX *\n ((eval P) W.polynomialX ^ 2 - W.a₁ * (eval P) W.polynomi... | dblZ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Extension.Cotangent.LocalizationAway | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 42
} | {
"line": 133,
"column": 2
} | [
{
"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⊢ Function.Injective\n ... | [
"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 := localizatio... | set Q := Generators.localizationAway T g | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.RingTheory.Extension.Cotangent.LocalizationAway | {
"line": 174,
"column": 39
} | {
"line": 174,
"column": 60
} | {
"line": 174,
"column": 61
} | [
{
"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\nx : ((localizationAway T g... | [
"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\nx : ((localizationAway T g).comp P).to... | cotangentCompAwaySec, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Extension.Cotangent.LocalizationAway | {
"line": 174,
"column": 61
} | {
"line": 174,
"column": 86
} | {
"line": 174,
"column": 86
} | [
{
"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\nx : ((localizationAway T g... | [
"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\nx : ((localizationAway T g).comp P).to... | Module.Basis.constr_basis | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 258,
"column": 19
} | {
"line": 258,
"column": 48
} | {
"line": 258,
"column": 48
} | [
{
"pp": "case inl\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 : Unit\n⊢ D.cotangentEquivProd.symm (cMulXSubOneCotangent S D.gbar, ... | [
"case inl\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 : Unit\n⊢ (cotangentCompLocalizationAwayEquiv D.gbar D.presLeft.toGenerators ⋯).... | cotangentEquivProd_symm_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Flat.FaithfullyFlat.Descent | {
"line": 46,
"column": 2
} | {
"line": 46,
"column": 96
} | {
"line": 47,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nT : Type u_3\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\ninst✝ : FaithfullyFlat R S\nH : Function.Surjective ⇑(algebraMap S (S ⊗[R] T))\nthis :\n LinearMap.lTensor S (Algebra.linearMap R T) = ↑R (Algebra.linear... | [
"R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nT : Type u_3\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\ninst✝ : FaithfullyFlat R S\nH : Function.Surjective ⇑(algebraMap S (S ⊗[R] T))\nthis :\n LinearMap.lTensor S (Algebra.linearMap R T) = ↑R (Algebra.linearMap S (S ⊗[R... | apply (Module.FaithfullyFlat.lTensor_surjective_iff_surjective R S (Algebra.linearMap R T)).mp | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.AlgebraicGeometry.Morphisms.Descent | {
"line": 66,
"column": 26
} | {
"line": 89,
"column": 18
} | {
"line": 91,
"column": 0
} | [
{
"pp": "P P' : MorphismProperty Scheme\ninst✝¹ : IsZariskiLocalAtTarget P\ninst✝ : P'.IsStableUnderBaseChange\nH : ∀ {R : CommRingCat} {X Y : Scheme} (f : X ⟶ Spec R) (g : Y ⟶ Spec R), P' f → P (pullback.fst f g) → P g\n⊢ P.DescendsAlong P'",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
... | [] | by
apply MorphismProperty.DescendsAlong.mk'
introv h hf
wlog hZ : ∃ R, Z = Spec R generalizing X Y Z
· rw [IsZariskiLocalAtTarget.iff_of_openCover (P := P) Z.affineCover]
intro i
let ι := Z.affineCover.f i
let e : pullback (pullback.snd f ι) (pullback.snd g ι) ≅
pullback (pullback.fst f g) (... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Conductor | {
"line": 75,
"column": 6
} | {
"line": 75,
"column": 27
} | {
"line": 76,
"column": 6
} | [
{
"pp": "case mp\nR : Type u_1\nS : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\nL : Type u_3\ninst✝⁵ : CommRing L\ninst✝⁴ : Algebra S L\ninst✝³ : Algebra R L\ninst✝² : IsScalarTower R S L\ninst✝¹ : IsDomain S\ninst✝ : IsTorsionFree S L\nx : S\ny : L\nh✝ : Nontrivial L\n⊢ (∃ y_1 ∈ c... | [
"case mp\nR : Type u_1\nS : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\nL : Type u_3\ninst✝⁵ : CommRing L\ninst✝⁴ : Algebra S L\ninst✝³ : Algebra R L\ninst✝² : IsScalarTower R S L\ninst✝¹ : IsDomain S\ninst✝ : IsTorsionFree S L\nx : S\nh✝ : Nontrivial L\ny : S\nhy : y ∈ conductor R x\n... | rintro ⟨y, hy, rfl⟩ z | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.RingTheory.IntegralClosure.GoingDown | {
"line": 33,
"column": 2
} | {
"line": 38,
"column": 53
} | {
"line": 39,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhp : p.Monic\nhq : q.Monic\nH : q ∣ p\ni : ℕ\nhi : i ≠ q.natDegree\nP : Ideal R\nhPJ : Ideal.span {x | ∃ i < p.natDegree, p.coeff i = x} ≤ P\nhP : P.IsPrime\n⊢ q.coeff i ∈ P",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [
"R : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhp : p.Monic\nhq : q.Monic\nH : q ∣ p\ni : ℕ\nhi : i ≠ q.natDegree\nP : Ideal R\nhPJ : Ideal.span {x | ∃ i < p.natDegree, p.coeff i = x} ≤ P\nhP : P.IsPrime\nthis : map (Ideal.Quotient.mk P) p = X ^ p.natDegree\n⊢ q.coeff i ∈ P"
] | have : p.map (Ideal.Quotient.mk P) = X ^ p.natDegree := by
ext i
obtain hi | rfl | hi := lt_trichotomy i p.natDegree
· simpa [hi.ne, Ideal.Quotient.eq_zero_iff_mem] using hPJ (Ideal.subset_span ⟨_, hi, rfl⟩)
· simp [hp]
· simp [coeff_eq_zero_of_natDegree_lt hi, hi.ne'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.IntegralClosure.GoingDown | {
"line": 31,
"column": 2
} | {
"line": 45,
"column": 71
} | {
"line": 47,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhp : p.Monic\nhq : q.Monic\nH : q ∣ p\ni : ℕ\nhi : i ≠ q.natDegree\n⊢ q.coeff i ∈ (Ideal.span {x | ∃ i < p.natDegree, p.coeff i = x}).radical",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Polynomial.natDegree_mul_X_pow",
... | [] | rw [Ideal.radical_eq_sInf, Ideal.mem_sInf]
rintro P ⟨hPJ, hP⟩
have : p.map (Ideal.Quotient.mk P) = X ^ p.natDegree := by
ext i
obtain hi | rfl | hi := lt_trichotomy i p.natDegree
· simpa [hi.ne, Ideal.Quotient.eq_zero_iff_mem] using hPJ (Ideal.subset_span ⟨_, hi, rfl⟩)
· simp [hp]
· simp [coeff_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.IntegralClosure.GoingDown | {
"line": 31,
"column": 2
} | {
"line": 45,
"column": 71
} | {
"line": 47,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhp : p.Monic\nhq : q.Monic\nH : q ∣ p\ni : ℕ\nhi : i ≠ q.natDegree\n⊢ q.coeff i ∈ (Ideal.span {x | ∃ i < p.natDegree, p.coeff i = x}).radical",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Polynomial.natDegree_mul_X_pow",
... | [] | rw [Ideal.radical_eq_sInf, Ideal.mem_sInf]
rintro P ⟨hPJ, hP⟩
have : p.map (Ideal.Quotient.mk P) = X ^ p.natDegree := by
ext i
obtain hi | rfl | hi := lt_trichotomy i p.natDegree
· simpa [hi.ne, Ideal.Quotient.eq_zero_iff_mem] using hPJ (Ideal.subset_span ⟨_, hi, rfl⟩)
· simp [hp]
· simp [coeff_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 98,
"column": 8
} | {
"line": 98,
"column": 30
} | {
"line": 98,
"column": 30
} | [
{
"pp": "case h\nR : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsDomain R\ninst✝ : FaithfulSMul R S\np✝ : S[X]\nhp✝ : IsAlmostIntegral R[X] p✝\ni : ℕ\nq : S[X]\np : R[X]\nhp : p ∈ R[X]⁰\nhp' : ∀ (n : ℕ), p • q ^ n ∈ (algebraMap R[X] S[X]).range\nn : ℕ\nr : ... | [
"case h\nR : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsDomain R\ninst✝ : FaithfulSMul R S\np✝ : S[X]\nhp✝ : IsAlmostIntegral R[X] p✝\ni : ℕ\nq : S[X]\np : R[X]\nhp : p ∈ R[X]⁰\nhp' : ∀ (n : ℕ), p • q ^ n ∈ (algebraMap R[X] S[X]).range\nn : ℕ\nr : R[X]\nhr : m... | ← leadingCoeff_pow' h' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 99,
"column": 4
} | {
"line": 99,
"column": 28
} | {
"line": 100,
"column": 2
} | [
{
"pp": "case h\nR : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsDomain R\ninst✝ : FaithfulSMul R S\np✝ : S[X]\nhp✝ : IsAlmostIntegral R[X] p✝\ni : ℕ\nq : S[X]\np : R[X]\nhp : p ∈ R[X]⁰\nhp' : ∀ (n : ℕ), p • q ^ n ∈ (algebraMap R[X] S[X]).range\nn : ℕ\nr : ... | [] | rw [leadingCoeff_mul' h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 185,
"column": 2
} | {
"line": 186,
"column": 28
} | {
"line": 188,
"column": 0
} | [
{
"pp": "K : Type u\ninst✝¹ : CommRing K\nR : Type u_1\ninst✝ : SMul R (FractionRing K[X])\nc : R\np : K⟮X⟯\n⊢ (c • p).toFractionRing = c • p.toFractionRing",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"congrArg",
"CommSemiring.toSemiring",... | [] | cases p
rw [← ofFractionRing_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 185,
"column": 2
} | {
"line": 186,
"column": 28
} | {
"line": 188,
"column": 0
} | [
{
"pp": "K : Type u\ninst✝¹ : CommRing K\nR : Type u_1\ninst✝ : SMul R (FractionRing K[X])\nc : R\np : K⟮X⟯\n⊢ (c • p).toFractionRing = c • p.toFractionRing",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"congrArg",
"CommSemiring.toSemiring",... | [] | cases p
rw [← ofFractionRing_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski | {
"line": 196,
"column": 2
} | {
"line": 197,
"column": 25
} | {
"line": 199,
"column": 0
} | [
{
"pp": "X : Scheme\nU : X.AffineZariskiSite\nS : Sieve U\n⊢ S ∈ (grothendieckTopology X) U ↔ Ideal.span (sectionsOfPresieve S.arrows) = ⊤",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Scheme.AffineZariskiSite",
"Semiring.toModule",
"O... | [] | rw [← generate_presieveOfSections_mem_grothendieckTopology, presieveOfSections_sectionsOfPresieve,
Sieve.generate_sieve] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski | {
"line": 196,
"column": 2
} | {
"line": 197,
"column": 25
} | {
"line": 199,
"column": 0
} | [
{
"pp": "X : Scheme\nU : X.AffineZariskiSite\nS : Sieve U\n⊢ S ∈ (grothendieckTopology X) U ↔ Ideal.span (sectionsOfPresieve S.arrows) = ⊤",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Scheme.AffineZariskiSite",
"Semiring.toModule",
"O... | [] | rw [← generate_presieveOfSections_mem_grothendieckTopology, presieveOfSections_sectionsOfPresieve,
Sieve.generate_sieve] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski | {
"line": 196,
"column": 2
} | {
"line": 197,
"column": 25
} | {
"line": 199,
"column": 0
} | [
{
"pp": "X : Scheme\nU : X.AffineZariskiSite\nS : Sieve U\n⊢ S ∈ (grothendieckTopology X) U ↔ Ideal.span (sectionsOfPresieve S.arrows) = ⊤",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Scheme.AffineZariskiSite",
"Semiring.toModule",
"O... | [] | rw [← generate_presieveOfSections_mem_grothendieckTopology, presieveOfSections_sectionsOfPresieve,
Sieve.generate_sieve] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 792,
"column": 6
} | {
"line": 792,
"column": 90
} | {
"line": 793,
"column": 4
} | [
{
"pp": "k : Type u_3\nK : Type u_4\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\ninst✝ : Algebra.IsAlgebraic k K\nhf : ¬Module.Finite k⟮X⟯ K⟮X⟯\n⊢ ¬Module.Finite k K",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"Module.rank_lt_aleph0_iff",
"Eq.mpr",
"IsNo... | [] | rwa [← Module.rank_lt_aleph0_iff, ← rank_ratFunc_ratFunc, Module.rank_lt_aleph0_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.RingTheory.Unramified.LocalStructure | {
"line": 218,
"column": 2
} | {
"line": 219,
"column": 40
} | {
"line": 220,
"column": 2
} | [
{
"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... | #adaptation_note /-- After nightly-2026-04-06, typeclass synthesis fails to find these
instances; provide them explicitly. -/ | «_aux_Mathlib_Tactic_AdaptationNote___elabRules_tactic#adaptation_note__1» | «tactic#adaptation_note_» |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 330,
"column": 8
} | {
"line": 330,
"column": 46
} | {
"line": 330,
"column": 47
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : FaithfulSMul R S\ninst✝³ : IsIntegrallyClosed R\ninst✝² : IsDomain S\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R P\nthis✝² : IsDomain R\nf : R[X] →ₐ[R] S\nhx' : f.Finite\nhf' : Funct... | [
"R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : FaithfulSMul R S\ninst✝³ : IsIntegrallyClosed R\ninst✝² : IsDomain S\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R P\nthis✝² : IsDomain R\nf : R[X] →ₐ[R] S\nhx' : f.Finite\nhf' : Function.Injectiv... | faithfulSMul_iff_algebraMap_injective, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Smooth.IntegralClosure | {
"line": 339,
"column": 43
} | {
"line": 339,
"column": 50
} | {
"line": 339,
"column": 50
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nB : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : Algebra.IsStandardEtale R S\n𝓟 : StandardEtalePresentation R S\nn : ℕ\n𝓟' : StandardEtalePresentation B (B ⊗[R] S) := 𝓟.baseChange\ne : B ⊗[R] ... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 472,
"column": 4
} | {
"line": 473,
"column": 82
} | {
"line": 474,
"column": 4
} | [
{
"pp": "case inr\nR S : Type u\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\np : Ideal S\ninst✝² : p.IsPrime\ninst✝¹ : WeaklyQuasiFiniteAt R p\nx : S\nhx : R[x] = ⊤\nthis :\n ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (p : Ideal S) [inst_3 : p.IsPrime]... | [
"case inr\nR S : Type u\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\np : Ideal S\ninst✝² : p.IsPrime\ninst✝¹ : WeaklyQuasiFiniteAt R p\nx : S\nhx : R[x] = ⊤\nthis :\n ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (p : Ideal S) [inst_3 : p.IsPrime]\n [Weakl... | suffices ⊤ ≤ (Algebra.adjoin (integralClosure R S) {x}).restrictScalars R from
top_le_iff.mp fun x _ ↦ (Subalgebra.mem_restrictScalars _).mp (this trivial) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 494,
"column": 6
} | {
"line": 495,
"column": 89
} | {
"line": 496,
"column": 6
} | [
{
"pp": "case pos.refine_3\nR✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\ninst✝⁵ : Algebra R✝ S✝\nR S : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\np : Ideal S\ninst✝¹ : p.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R p\nx : S\nhx : R[x] = ⊤\nH : integralClosure R S = ⊥\nH₀ : ... | [
"case pos.refine_3\nR✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\ninst✝⁵ : Algebra R✝ S✝\nR S : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\np : Ideal S\ninst✝¹ : p.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R p\nx : S\nhx : R[x] = ⊤\nH : integralClosure R S = ⊥\nH₀ : Function.Sur... | suffices algebraMap R S (f.coeff n) + algebraMap R S (if n = m then a else 0) ∉ p by
simpa [eraseLead_coeff, show n ≠ f.natDegree by rintro rfl; exact hfn (by simpa)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 591,
"column": 42
} | {
"line": 591,
"column": 62
} | {
"line": 591,
"column": 62
} | [
{
"pp": "n : ℕ\nIH :\n ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (p : Ideal S) [inst_3 : p.IsPrime]\n [WeaklyQuasiFiniteAt R p] (f : MvPolynomial (Fin n) R →ₐ[R] S), f.Finite → ZariskisMainProperty R p\nR S : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : ... | [] | by exact ⟨hrp, hrR'⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 349,
"column": 2
} | {
"line": 349,
"column": 9
} | {
"line": 350,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nm n k : ℕ\nhg : g.natDegree ≤ n\n⊢ (-1) ^ (m * (n + k + (k + n))) * f.coeff m ^ k * f.resultant g m n = f.coeff m ^ k * f.resultant g m n",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
... | [
"R : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nm n k : ℕ\nhg : g.natDegree ≤ n\n⊢ f.coeff m ^ k * f.resultant g m n * (-1) ^ (m * n * 2) * (-1) ^ (m * k * 2) = f.coeff m ^ k * f.resultant g m n"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 561,
"column": 4
} | {
"line": 561,
"column": 11
} | {
"line": 562,
"column": 2
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝ : CommRing R\nf g₁ g₂ : R[X]\nthis :\n ∀ {R : Type u_1} [inst : CommRing R] (f g₁ g₂ : R[X]) (m : ℕ),\n f.natDegree ≤ m →\n m = f.natDegree → f.resultant (g₁ * g₂) m (g₁.natDegree + g₂.natDegree) = f.resultant g₁ m * f.resultant g₂ m\nc : ℕ\nhm : f.natDegree ≤ f.n... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 566,
"column": 4
} | {
"line": 566,
"column": 11
} | {
"line": 567,
"column": 2
} | [
{
"pp": "case Splits\nR✝ : Type u_1\ninst✝¹ : CommRing R✝\nR : Type u_1\ninst✝ : Field R\nf : R[X]\nhff : f.Splits\ng₁ g₂ : R[X]\n⊢ f.leadingCoeff ^ (g₁.natDegree + g₂.natDegree) *\n ((Multiset.map (fun i ↦ eval i g₁) f.roots).prod * (Multiset.map (fun i ↦ eval i g₂) f.roots).prod) =\n f.leadingCoeff ^ ... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 576,
"column": 8
} | {
"line": 576,
"column": 14
} | {
"line": 576,
"column": 15
} | [
{
"pp": "case surjective\nR✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Surjective ⇑φ\nf : S[X]\nIH :\n ∀ (q g₁ g₂ : R[X]), q.resultant (g₁ * g₂) q.natDegree (g₁.natDegree + g₂.natDegree) = q.resultant g₁ * q.resultant g₂\ng₁ g₂ : S[X]\... | [
"case surjective\nR✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Surjective ⇑φ\nf : S[X]\nIH :\n ∀ (q g₁ g₂ : R[X]), q.resultant (g₁ * g₂) q.natDegree (g₁.natDegree + g₂.natDegree) = q.resultant g₁ * q.resultant g₂\ng₁ g₂ : S[X]\nf' : R[X]\n... | ← hg₁, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 585,
"column": 2
} | {
"line": 585,
"column": 9
} | {
"line": 586,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf₁ f₂ g : R[X]\nn : ℕ\nhn : g.natDegree ≤ n\n⊢ (-1) ^ ((f₁.natDegree + f₂.natDegree) * n) *\n ((-1) ^ (n * f₁.natDegree) * f₁.resultant g f₁.natDegree n * g.resultant f₂ n) =\n f₁.resultant g f₁.natDegree n * ((-1) ^ (f₂.natDegree * n) * g.resultant f₂ n)",
... | [
"R : Type u_1\ninst✝ : CommRing R\nf₁ f₂ g : R[X]\nn : ℕ\nhn : g.natDegree ≤ n\n⊢ f₁.resultant g f₁.natDegree n * g.resultant f₂ n * (-1) ^ (f₁.natDegree * n * 2) * (-1) ^ (f₂.natDegree * n) =\n f₁.resultant g f₁.natDegree n * g.resultant f₂ n * (-1) ^ (f₂.natDegree * n)"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 375,
"column": 24
} | {
"line": 375,
"column": 64
} | {
"line": 376,
"column": 4
} | [
{
"pp": "case tmul.refine_1.mul_X\nR : Type u_1\ninst✝ : CommRing R\nn m k : ℕ\nhn : n = m + k\nthis✝¹ : Algebra (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) :=\n (universalFactorizationMap R n m k hn).toAlgebra\nthis✝ : IsDomain (MvPolynomial (Fin m) ℤ ⊗[ℤ] MvPolynomial (Fin k... | [] | simpa [← map_mul] using! IH.mul _ (H₁ i) | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 375,
"column": 24
} | {
"line": 375,
"column": 64
} | {
"line": 376,
"column": 4
} | [
{
"pp": "case tmul.refine_1.mul_X\nR : Type u_1\ninst✝ : CommRing R\nn m k : ℕ\nhn : n = m + k\nthis✝¹ : Algebra (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) :=\n (universalFactorizationMap R n m k hn).toAlgebra\nthis✝ : IsDomain (MvPolynomial (Fin m) ℤ ⊗[ℤ] MvPolynomial (Fin k... | [] | simpa [← map_mul] using! IH.mul _ (H₁ i) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 375,
"column": 24
} | {
"line": 375,
"column": 64
} | {
"line": 376,
"column": 4
} | [
{
"pp": "case tmul.refine_1.mul_X\nR : Type u_1\ninst✝ : CommRing R\nn m k : ℕ\nhn : n = m + k\nthis✝¹ : Algebra (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) :=\n (universalFactorizationMap R n m k hn).toAlgebra\nthis✝ : IsDomain (MvPolynomial (Fin m) ℤ ⊗[ℤ] MvPolynomial (Fin k... | [] | simpa [← map_mul] using! IH.mul _ (H₁ i) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 700,
"column": 4
} | {
"line": 700,
"column": 35
} | {
"line": 701,
"column": 4
} | [
{
"pp": "case neg\nR✝ : Type u_1\ninst✝¹ : CommRing R✝\nR : Type u_1\ninst✝ : Field R\nf : R[X]\nhf' : f.Splits\ng : R[X]\nr : R\nhf : f ≠ 0\nhg : g ≠ 0\nhf0 : ¬f.natDegree = 0\nhg0 : ¬g.natDegree = 0\n⊢ (f.scaleRoots r).resultant (g.scaleRoots r) f.natDegree g.natDegree = r ^ (f.natDegree * g.natDegree) * f.re... | [
"case neg.inl\nR✝ : Type u_1\ninst✝¹ : CommRing R✝\nR : Type u_1\ninst✝ : Field R\nf : R[X]\nhf' : f.Splits\ng : R[X]\nhf : f ≠ 0\nhg : g ≠ 0\nhf0 : ¬f.natDegree = 0\nhg0 : ¬g.natDegree = 0\n⊢ (f.scaleRoots 0).resultant (g.scaleRoots 0) f.natDegree g.natDegree = 0 ^ (f.natDegree * g.natDegree) * f.resultant g",
"... | obtain rfl | hr := eq_or_ne r 0 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 891,
"column": 4
} | {
"line": 892,
"column": 61
} | {
"line": 893,
"column": 2
} | [
{
"pp": "case neg.refine_1\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.Monic\nhf0 : ¬f.natDegree = 0\nH : IsUnit (f.resultant g)\np q : R[X]\nhp : p.degree < ↑g.natDegree\nhq : q.degree < ↑f.natDegree\ne : f * p + g * q = C (f.resultant g)\n⊢ IsCoprime f g",
"ppTerm": "?neg.refine_1✝",
"assign... | [] | exact ⟨C (H.unit⁻¹).1 * p, C (H.unit⁻¹).1 * q, by simp only [mul_assoc, ← mul_add, mul_comm p,
mul_comm q, e, ← map_mul, IsUnit.val_inv_mul, map_one]⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 967,
"column": 2
} | {
"line": 967,
"column": 9
} | {
"line": 969,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf : R[X]\nhf : f.natDegree = 2\ne : Fin (f.natDegree - 1 + f.natDegree) ≃ Fin 3 := finCongr ⋯\nthis : (Matrix.reindex e e) f.sylvesterDeriv = !![f.coeff 0, f.coeff 1, 0; f.coeff 1, 2 * f.coeff 2, f.coeff 1; 1, 0, 2]\n⊢ (f.coeff 0 * (2 * f.coeff 2) * 2 - f.coeff 0 * f.c... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 979,
"column": 2
} | {
"line": 979,
"column": 9
} | {
"line": 980,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf : R[X]\nhf : 0 < f.natDegree\n⊢ (-1) ^ ((f.natDegree - 1) * f.natDegree) * (f.leadingCoeff * f.sylvesterDeriv.det) =\n (-1) ^ ((f.natDegree - 1) * f.natDegree / 2) * f.leadingCoeff *\n (f.sylvesterDeriv.det * (-1) ^ ((f.natDegree - 1) * f.natDegree / 2))",
... | [
"R : Type u_1\ninst✝ : CommRing R\nf : R[X]\nhf : 0 < f.natDegree\n⊢ f.leadingCoeff * f.sylvesterDeriv.det * (-1) ^ ((f.natDegree - 1) * f.natDegree) =\n f.leadingCoeff * f.sylvesterDeriv.det * (-1) ^ ((f.natDegree - 1) * f.natDegree / 2 * 2)"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 1021,
"column": 2
} | {
"line": 1021,
"column": 9
} | {
"line": 1023,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf : R[X]\nhf : f.natDegree = 3\ne : Fin (f.natDegree - 1 + f.natDegree) ≃ Fin 5 := finCongr ⋯\n⊢ (f.coeff 0 *\n (f.coeff 0 * (3 * f.coeff 3 * (3 * f.coeff 3) * 3) +\n -(2 * f.coeff 2 *\n (f.coeff 1 * (3 * f.coeff 3) * 3 - 2 * f.co... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
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