module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 273,
"column": 52
} | {
"line": 273,
"column": 60
} | [
{
"pp": "case funProp.discharger\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nn✝ : ℕ\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : UniqueDiffOn 𝕜 s\n𝔸 : Type u_5\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕜 𝔸\ninst✝² : Module 𝔸 F\n... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 273,
"column": 52
} | {
"line": 273,
"column": 60
} | [
{
"pp": "case funProp.discharger\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nn✝ : ℕ\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : UniqueDiffOn 𝕜 s\n𝔸 : Type u_5\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕜 𝔸\ninst✝² : Module 𝔸 F\n... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 273,
"column": 52
} | {
"line": 273,
"column": 60
} | [
{
"pp": "case funProp.discharger\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nn✝ : ℕ\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : UniqueDiffOn 𝕜 s\n𝔸 : Type u_5\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕜 𝔸\ninst✝² : Module 𝔸 F\n... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 273,
"column": 52
} | {
"line": 273,
"column": 60
} | [
{
"pp": "case funProp.discharger\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nn✝ : ℕ\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : UniqueDiffOn 𝕜 s\n𝔸 : Type u_5\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕜 𝔸\ninst✝² : Module 𝔸 F\n... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 273,
"column": 52
} | {
"line": 273,
"column": 60
} | [
{
"pp": "case funProp.discharger\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nn✝ : ℕ\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : UniqueDiffOn 𝕜 s\n𝔸 : Type u_5\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕜 𝔸\ninst✝² : Module 𝔸 F\n... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 273,
"column": 52
} | {
"line": 273,
"column": 60
} | [
{
"pp": "case funProp.discharger\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nn✝ : ℕ\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : UniqueDiffOn 𝕜 s\n𝔸 : Type u_5\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕜 𝔸\ninst✝² : Module 𝔸 F\n... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 273,
"column": 52
} | {
"line": 273,
"column": 60
} | [
{
"pp": "case funProp.discharger\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nn✝ : ℕ\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : UniqueDiffOn 𝕜 s\n𝔸 : Type u_5\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕜 𝔸\ninst✝² : Module 𝔸 F\n... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 273,
"column": 52
} | {
"line": 273,
"column": 60
} | [
{
"pp": "case funProp.discharger\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nn✝ : ℕ\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : UniqueDiffOn 𝕜 s\n𝔸 : Type u_5\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕜 𝔸\ninst✝² : Module 𝔸 F\n... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 273,
"column": 52
} | {
"line": 273,
"column": 60
} | [
{
"pp": "case funProp.discharger\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nn✝ : ℕ\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : UniqueDiffOn 𝕜 s\n𝔸 : Type u_5\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕜 𝔸\ninst✝² : Module 𝔸 F\n... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 273,
"column": 52
} | {
"line": 273,
"column": 60
} | [
{
"pp": "case funProp.discharger\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nn✝ : ℕ\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : UniqueDiffOn 𝕜 s\n𝔸 : Type u_5\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕜 𝔸\ninst✝² : Module 𝔸 F\n... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 273,
"column": 52
} | {
"line": 273,
"column": 60
} | [
{
"pp": "case funProp.discharger\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nn✝ : ℕ\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : UniqueDiffOn 𝕜 s\n𝔸 : Type u_5\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕜 𝔸\ninst✝² : Module 𝔸 F\n... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 273,
"column": 52
} | {
"line": 273,
"column": 60
} | [
{
"pp": "case funProp.discharger\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nn✝ : ℕ\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : UniqueDiffOn 𝕜 s\n𝔸 : Type u_5\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕜 𝔸\ninst✝² : Module 𝔸 F\n... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 273,
"column": 52
} | {
"line": 273,
"column": 60
} | [
{
"pp": "case funProp.discharger\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nn✝ : ℕ\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : UniqueDiffOn 𝕜 s\n𝔸 : Type u_5\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕜 𝔸\ninst✝² : Module 𝔸 F\n... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Log.Deriv | {
"line": 292,
"column": 41
} | {
"line": 315,
"column": 49
} | [
{
"pp": "x : ℝ\nh : |x| < 1\nn : ℕ\n⊢ |1 / 2 * log ((1 + x) / (1 - x)) - ∑ i ∈ Finset.range n, x ^ (2 * i + 1) / (2 * ↑i + 1)| ≤\n |x| ^ (2 * n + 1) / (1 - x ^ 2)",
"usedConstants": [
"abs_nonneg._simp_1",
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"neg_lt_neg_iff._simp_1",
... | by
let F (x : ℝ) : ℝ :=
1 / 2 * log ((1 + x) / (1 - x)) - (∑ i ∈ range n, x ^ (2 * i + 1) / (2 * i + 1))
let F' (y : ℝ) : ℝ := (y ^ 2) ^ n / (1 - y ^ 2)
have hI : Icc (-|x|) |x| ⊆ Ioo (-1 : ℝ) 1 := Icc_subset_Ioo (by simp [h]) h
-- First step: compute the derivative of `F`
have A : ∀ y ∈ Ioo (-1 : ℝ) 1, H... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 211,
"column": 59
} | {
"line": 211,
"column": 67
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nx : E\ni : ℕ\nf g : E → F\nhf : ContDiffWithinAt 𝕜 (↑i) f s x\nhg : ContDiffWithinAt 𝕜 (↑i) g s x... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 211,
"column": 59
} | {
"line": 211,
"column": 67
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nx : E\ni : ℕ\nf g : E → F\nhf : ContDiffWithinAt 𝕜 (↑i) f s x\nhg : ContDiffWithinAt 𝕜 (↑i) g s x... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 211,
"column": 59
} | {
"line": 211,
"column": 67
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nx : E\ni : ℕ\nf g : E → F\nhf : ContDiffWithinAt 𝕜 (↑i) f s x\nhg : ContDiffWithinAt 𝕜 (↑i) g s x... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 173,
"column": 28
} | {
"line": 173,
"column": 59
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nε : ℝ\nhε : 0 < ε\nx : E\nhx : DifferentiableAt 𝕜 f x\nδ : ℝ := ε / 2 / 2\nR : ℝ\nR_pos : R > 0\... | rw [mem_ball_iff_norm] at hz hy | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 212,
"column": 59
} | {
"line": 212,
"column": 67
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nx : E\ni : ℕ\nf g : E → F\nhf : ContDiffWithinAt 𝕜 (↑i) f s x\nhg : ContDiffWithinAt 𝕜 (↑i) g s x... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 212,
"column": 59
} | {
"line": 212,
"column": 67
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nx : E\ni : ℕ\nf g : E → F\nhf : ContDiffWithinAt 𝕜 (↑i) f s x\nhg : ContDiffWithinAt 𝕜 (↑i) g s x... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 212,
"column": 59
} | {
"line": 212,
"column": 67
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nx : E\ni : ℕ\nf g : E → F\nhf : ContDiffWithinAt 𝕜 (↑i) f s x\nhg : ContDiffWithinAt 𝕜 (↑i) g s x... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 782,
"column": 2
} | {
"line": 784,
"column": 56
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nn : ℕ∞ω\nR : Type u_3\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : HasSummableGeomSeries R\nx : Rˣ\n⊢ ContDiffAt 𝕜 n Ring.inverse ↑x",
"usedConstants": [
"Units.val",
"NormedCommRing.toSeminormedCommRing",
"analy... | have := AnalyticOnNhd.contDiffOn (analyticOnNhd_inverse (𝕜 := 𝕜) (A := R)) (n := n)
Units.isOpen.uniqueDiffOn x x.isUnit
exact this.contDiffAt (Units.isOpen.mem_nhds x.isUnit) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 782,
"column": 2
} | {
"line": 784,
"column": 56
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nn : ℕ∞ω\nR : Type u_3\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : HasSummableGeomSeries R\nx : Rˣ\n⊢ ContDiffAt 𝕜 n Ring.inverse ↑x",
"usedConstants": [
"Units.val",
"NormedCommRing.toSeminormedCommRing",
"analy... | have := AnalyticOnNhd.contDiffOn (analyticOnNhd_inverse (𝕜 := 𝕜) (A := R)) (n := n)
Units.isOpen.uniqueDiffOn x x.isUnit
exact this.contDiffAt (Units.isOpen.mem_nhds x.isUnit) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 264,
"column": 8
} | {
"line": 264,
"column": 21
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nhK : IsComplete K\nP : ∀ {n : ℕ}, 0 < (1 / 2) ^ n\nc : 𝕜\nhc : 1 < ‖c‖\nx ... | congr 1; abel | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 264,
"column": 8
} | {
"line": 264,
"column": 21
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nhK : IsComplete K\nP : ∀ {n : ℕ}, 0 < (1 / 2) ^ n\nc : 𝕜\nhc : 1 < ‖c‖\nx ... | congr 1; abel | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Cone.Extension | {
"line": 181,
"column": 2
} | {
"line": 181,
"column": 54
} | [
{
"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\ns : ConvexCone ℝ (E × ℝ) := { carrier := {p | N p.1 ≤ p.2}, smul_mem' := ⋯, a... | refine ⟨-g.comp (inl ℝ E ℝ), fun x ↦ ?_, fun x ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.LocallyConvex.AbsConvex | {
"line": 282,
"column": 54
} | {
"line": 282,
"column": 62
} | [
{
"pp": "case h\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\nx✝¹ : E\nx✝ : x✝¹ ∈ -s\n⊢ ‖-1‖ ≤ 1 ∧ x✝¹ ∈ -1 • s",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"NegZeroClass.toNeg",
"NormedCommRing.toSemino... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 454,
"column": 2
} | {
"line": 454,
"column": 13
} | [
{
"pp": "case h\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nL : F\nr ε x r' : ℝ\nrr' : r' ∈ Ioc (r / 2) r\nhr' : ∀ y ∈ Icc x (x + r'), ∀ z ∈ Icc x (x + r'), ‖f z - f y - (z - y) • L‖ ≤ ε * r\ns : ℝ\ns_gt : r / 2 < s\ns_lt : s < r'\nthis : s ∈ Ioc (r / 2) r\nx' : ℝ\nhx' : x'... | use s, this | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Analysis.Convex.Gauge | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 35
} | [
{
"pp": "E : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\nx : E\n⊢ gauge (-s) x = gauge s (-x)",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"SubtractionMonoid.toInvolutiveNeg",
"Real",
"gauge",
"congrArg",
"neg_neg",
"id",
"S... | rw [← gauge_neg_set_neg, neg_neg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Convex.Gauge | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 35
} | [
{
"pp": "E : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\nx : E\n⊢ gauge (-s) x = gauge s (-x)",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"SubtractionMonoid.toInvolutiveNeg",
"Real",
"gauge",
"congrArg",
"neg_neg",
"id",
"S... | rw [← gauge_neg_set_neg, neg_neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Gauge | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 35
} | [
{
"pp": "E : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\nx : E\n⊢ gauge (-s) x = gauge s (-x)",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"SubtractionMonoid.toInvolutiveNeg",
"Real",
"gauge",
"congrArg",
"neg_neg",
"id",
"S... | rw [← gauge_neg_set_neg, neg_neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Gauge | {
"line": 324,
"column": 2
} | {
"line": 324,
"column": 42
} | [
{
"pp": "case h\nE : Type u_2\ninst✝² : AddCommGroup E\ninst✝¹ : Module ℝ E\ns : Set E\ninst✝ : TopologicalSpace E\nha : Absorbent ℝ s\nhb : Bornology.IsVonNBounded ℝ s\nu : Set E\nhu : u ∈ 𝓝 0\nr : ℝ\nhr₀ : r > 0\nhr : ∀ (c : ℝ), r ≤ ‖c‖ → s ⊆ c • u\nc : ℝ\nhc₀ : 0 < c\nhcr : c < r⁻¹\ny : E\nhy : y ∈ s\nhx : ... | have hrc := (lt_inv_comm₀ hr₀ hc₀).2 hcr | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 593,
"column": 8
} | {
"line": 593,
"column": 21
} | [
{
"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 / ... | congr 1; abel | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 593,
"column": 8
} | {
"line": 593,
"column": 21
} | [
{
"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 / ... | congr 1; abel | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.LocallyConvex.SeparatingDual | {
"line": 142,
"column": 54
} | {
"line": 142,
"column": 62
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝¹² : Field R\ninst✝¹¹ : AddCommGroup V\ninst✝¹⁰ : TopologicalSpace R\ninst✝⁹ : TopologicalSpace V\ninst✝⁸ : IsTopologicalRing R\ninst✝⁷ : Module R V\ninst✝⁶ : SeparatingDual R V\nW : Type u_3\ninst✝⁵ : AddCommGroup W\ninst✝⁴ : TopologicalSpace W\ninst✝³ : Module R W\nin... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.LocallyConvex.SeparatingDual | {
"line": 142,
"column": 54
} | {
"line": 142,
"column": 62
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝¹² : Field R\ninst✝¹¹ : AddCommGroup V\ninst✝¹⁰ : TopologicalSpace R\ninst✝⁹ : TopologicalSpace V\ninst✝⁸ : IsTopologicalRing R\ninst✝⁷ : Module R V\ninst✝⁶ : SeparatingDual R V\nW : Type u_3\ninst✝⁵ : AddCommGroup W\ninst✝⁴ : TopologicalSpace W\ninst✝³ : Module R W\nin... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.LocallyConvex.SeparatingDual | {
"line": 142,
"column": 54
} | {
"line": 142,
"column": 62
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝¹² : Field R\ninst✝¹¹ : AddCommGroup V\ninst✝¹⁰ : TopologicalSpace R\ninst✝⁹ : TopologicalSpace V\ninst✝⁸ : IsTopologicalRing R\ninst✝⁷ : Module R V\ninst✝⁶ : SeparatingDual R V\nW : Type u_3\ninst✝⁵ : AddCommGroup W\ninst✝⁴ : TopologicalSpace W\ninst✝³ : Module R W\nin... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Constructions.Polish.StronglyMeasurable | {
"line": 39,
"column": 4
} | {
"line": 39,
"column": 12
} | [
{
"pp": "case inl\nι : Type u_1\nX : Type u_2\nE : Type u_3\ninst✝⁴ : MeasurableSpace X\ninst✝³ : TopologicalSpace E\ninst✝² : Countable ι\nf : ι → X → E\ninst✝¹ : IsCompletelyPseudoMetrizableSpace E\nhf : ∀ (i : ι), StronglyMeasurable (f i)\ninst✝ : ⊥.IsCountablyGenerated\n⊢ MeasurableSet {x | ∃ c, Tendsto (fu... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Constructions.Polish.StronglyMeasurable | {
"line": 39,
"column": 4
} | {
"line": 39,
"column": 12
} | [
{
"pp": "case inl\nι : Type u_1\nX : Type u_2\nE : Type u_3\ninst✝⁴ : MeasurableSpace X\ninst✝³ : TopologicalSpace E\ninst✝² : Countable ι\nf : ι → X → E\ninst✝¹ : IsCompletelyPseudoMetrizableSpace E\nhf : ∀ (i : ι), StronglyMeasurable (f i)\ninst✝ : ⊥.IsCountablyGenerated\n⊢ MeasurableSet {x | ∃ c, Tendsto (fu... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Constructions.Polish.StronglyMeasurable | {
"line": 39,
"column": 4
} | {
"line": 39,
"column": 12
} | [
{
"pp": "case inl\nι : Type u_1\nX : Type u_2\nE : Type u_3\ninst✝⁴ : MeasurableSpace X\ninst✝³ : TopologicalSpace E\ninst✝² : Countable ι\nf : ι → X → E\ninst✝¹ : IsCompletelyPseudoMetrizableSpace E\nhf : ∀ (i : ι), StronglyMeasurable (f i)\ninst✝ : ⊥.IsCountablyGenerated\n⊢ MeasurableSet {x | ∃ c, Tendsto (fu... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 209,
"column": 2
} | {
"line": 209,
"column": 36
} | [
{
"pp": "α : 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≥0\nfmeas : Measurable f\nε : ℝ≥0∞\nε0 : ε ≠ 0\nthis : ε / 2 ≠ 0\nw : α → ℝ≥0\nwpos : ∀ (x : α), 0 < w x\nwmeas : Measurable w\nwint : ∫... | refine ⟨g, fun x => ?_, gcont, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 289,
"column": 44
} | {
"line": 289,
"column": 78
} | [
{
"pp": "case refine_2\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≥0\nfint : Integrable (fun x ↦ ↑(f x)) μ\nfmeas : AEMeasurable f μ\nε : ℝ≥0\nεpos : 0 < ↑ε\nδ : ℝ≥0\nδpos : 0 < δ\nhδε : ... | integral_eq_lintegral_of_nonneg_ae | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 291,
"column": 8
} | {
"line": 299,
"column": 70
} | [] | ENNReal.toReal (∫⁻ a : α, ENNReal.ofReal (g a).toReal ∂μ) =
ENNReal.toReal (∫⁻ a : α, g a ∂μ) := by congr 1
_ ≤ ENNReal.toReal ((∫⁻ a : α, f a ∂μ) + δ) := by
apply ENNReal.toReal_mono _ gint
simpa using int_f_ne_top
_ = ENNReal.toReal (∫⁻ a : α, f a ∂μ) + δ := by
... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 694,
"column": 9
} | {
"line": 694,
"column": 65
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nhK : IsComplete K\n⊢ MeasurableSet (⋂ e, ⋃ n, ⋂ p, ⋂ (_ : p ≥ n), ⋂ q, ⋂ (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e))",
"usedConstants": [
"Real",
"instHDiv",
"Set.iInter",... | apply_rules [MeasurableSet.iUnion, MeasurableSet.iInter] | Lean.Elab.Tactic.SolveByElim.evalApplyRules | Lean.Parser.Tactic.applyRules |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 694,
"column": 9
} | {
"line": 694,
"column": 65
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nhK : IsComplete K\nb✝ : ℕ\n⊢ MeasurableSet (⋃ n, ⋂ p, ⋂ (_ : p ≥ n), ⋂ q, ⋂ (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ b✝))",
"usedConstants": [
"Real",
"instHDiv",
"Set.iInt... | apply_rules [MeasurableSet.iUnion, MeasurableSet.iInter] | Lean.Elab.Tactic.SolveByElim.evalApplyRules | Lean.Parser.Tactic.applyRules |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 694,
"column": 9
} | {
"line": 694,
"column": 65
} | [
{
"pp": "case h\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nhK : IsComplete K\nb✝¹ b✝ : ℕ\n⊢ MeasurableSet (⋂ p, ⋂ (_ : p ≥ b✝), ⋂ q, ⋂ (_ : q ≥ b✝), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ b✝¹))",
"usedConstants": [
"Real",
"instHDiv",
... | apply_rules [MeasurableSet.iUnion, MeasurableSet.iInter] | Lean.Elab.Tactic.SolveByElim.evalApplyRules | Lean.Parser.Tactic.applyRules |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 694,
"column": 9
} | {
"line": 694,
"column": 65
} | [
{
"pp": "case h\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nhK : IsComplete K\nb✝² b✝¹ b✝ : ℕ\n⊢ MeasurableSet (⋂ (_ : b✝ ≥ b✝¹), ⋂ q, ⋂ (_ : q ≥ b✝¹), B f K ((1 / 2) ^ b✝) ((1 / 2) ^ q) ((1 / 2) ^ b✝²))",
"usedConstants": [
"Real",
"instHDiv",
... | apply_rules [MeasurableSet.iUnion, MeasurableSet.iInter] | Lean.Elab.Tactic.SolveByElim.evalApplyRules | Lean.Parser.Tactic.applyRules |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 694,
"column": 9
} | {
"line": 694,
"column": 65
} | [
{
"pp": "case h\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nhK : IsComplete K\nb✝³ b✝² b✝¹ : ℕ\nb✝ : b✝¹ ≥ b✝²\n⊢ MeasurableSet (⋂ q, ⋂ (_ : q ≥ b✝²), B f K ((1 / 2) ^ b✝¹) ((1 / 2) ^ q) ((1 / 2) ^ b✝³))",
"usedConstants": [
"Real",
"instHDiv",
... | apply_rules [MeasurableSet.iUnion, MeasurableSet.iInter] | Lean.Elab.Tactic.SolveByElim.evalApplyRules | Lean.Parser.Tactic.applyRules |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 694,
"column": 9
} | {
"line": 694,
"column": 65
} | [
{
"pp": "case h\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nhK : IsComplete K\nb✝⁴ b✝³ b✝² : ℕ\nb✝¹ : b✝² ≥ b✝³\nb✝ : ℕ\n⊢ MeasurableSet (⋂ (_ : b✝ ≥ b✝³), B f K ((1 / 2) ^ b✝²) ((1 / 2) ^ b✝) ((1 / 2) ^ b✝⁴))",
"usedConstants": [
"Real",
"instHDi... | apply_rules [MeasurableSet.iUnion, MeasurableSet.iInter] | Lean.Elab.Tactic.SolveByElim.evalApplyRules | Lean.Parser.Tactic.applyRules |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 694,
"column": 9
} | {
"line": 694,
"column": 65
} | [
{
"pp": "case h\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nhK : IsComplete K\nb✝⁵ b✝⁴ b✝³ : ℕ\nb✝² : b✝³ ≥ b✝⁴\nb✝¹ : ℕ\nb✝ : b✝¹ ≥ b✝⁴\n⊢ MeasurableSet (B f K ((1 / 2) ^ b✝³) ((1 / 2) ^ b✝¹) ((1 / 2) ^ b✝⁵))",
"usedConstants": [
"Real",
"instHDi... | apply_rules [MeasurableSet.iUnion, MeasurableSet.iInter] | Lean.Elab.Tactic.SolveByElim.evalApplyRules | Lean.Parser.Tactic.applyRules |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 411,
"column": 44
} | {
"line": 411,
"column": 78
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nf : α → ℝ≥0\nfint : Integrable (fun x ↦ ↑(f x)) μ\nε : ℝ≥0\nεpos : 0 < ↑ε\nIf : ∫⁻ (x : α), ↑(f x) ∂μ < ∞\ng : α → ℝ≥0\ngf : ∀ (x : α), g x ≤ f x\ngcont :... | integral_eq_lintegral_of_nonneg_ae | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 517,
"column": 4
} | {
"line": 519,
"column": 34
} | [
{
"pp": "case refine_2\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\ng : α → EReal\ng_lt_f : ∀ (x : α), ↑(-f x) < g x\ngcont : LowerSemicontinuous... | exact
continuous_neg.comp_lowerSemicontinuous_antitone gcont fun x y hxy =>
EReal.neg_le_neg_iff.2 hxy | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 517,
"column": 4
} | {
"line": 519,
"column": 34
} | [
{
"pp": "case refine_2\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\ng : α → EReal\ng_lt_f : ∀ (x : α), ↑(-f x) < g x\ngcont : LowerSemicontinuous... | exact
continuous_neg.comp_lowerSemicontinuous_antitone gcont fun x y hxy =>
EReal.neg_le_neg_iff.2 hxy | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 517,
"column": 4
} | {
"line": 519,
"column": 34
} | [
{
"pp": "case refine_2\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\ng : α → EReal\ng_lt_f : ∀ (x : α), ↑(-f x) < g x\ngcont : LowerSemicontinuous... | exact
continuous_neg.comp_lowerSemicontinuous_antitone gcont fun x y hxy =>
EReal.neg_le_neg_iff.2 hxy | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 71
} | [
{
"pp": "α : Type u_1\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\nm : MeasurableSpace α\nμ : Measure α\nι : Type u_4\ninst✝ : Countable ι\nF : ι → α → G\nf : α → G\nbound : ι → α → ℝ\nhF_meas : ∀ (n : ι), AEStronglyMeasurable (F n) μ\nh_bound : ∀ (n : ι), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bou... | simp only [HasSum, ← integral_finsetSum _ fun n _ => hF_integrable n] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 682,
"column": 2
} | {
"line": 684,
"column": 88
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nμ : Measure ℝ\n⊢ ∫ (x : ℝ) in a..b, f x ∂μ = (if a ≤ b then 1 else -1) • ∫ (x : ℝ) in Ι a b, f x ∂μ",
"usedConstants": [
"not_le",
"Eq.mpr",
"Set.Ioc",
"NegZeroClass.toNeg",
"Real... | split_ifs with h
· simp only [integral_of_le h, uIoc_of_le h, one_smul]
· simp only [integral_of_ge (not_le.1 h).le, uIoc_of_ge (not_le.1 h).le, neg_one_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 682,
"column": 2
} | {
"line": 684,
"column": 88
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nμ : Measure ℝ\n⊢ ∫ (x : ℝ) in a..b, f x ∂μ = (if a ≤ b then 1 else -1) • ∫ (x : ℝ) in Ι a b, f x ∂μ",
"usedConstants": [
"not_le",
"Eq.mpr",
"Set.Ioc",
"NegZeroClass.toNeg",
"Real... | split_ifs with h
· simp only [integral_of_le h, uIoc_of_le h, one_smul]
· simp only [integral_of_ge (not_le.1 h).le, uIoc_of_ge (not_le.1 h).le, neg_one_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Haar.Quotient | {
"line": 177,
"column": 6
} | {
"line": 177,
"column": 38
} | [
{
"pp": "case h\nG : 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)\... | Measure.map_apply meas_π meas_V, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 991,
"column": 4
} | {
"line": 991,
"column": 29
} | [
{
"pp": "g' g φ : ℝ → ℝ\na b : ℝ\nhab : a ≤ b\nhcont : ContinuousOn g (Icc a b)\nhderiv : ∀ x ∈ Ico a b, HasDerivWithinAt g (g' x) (Ioi x) x\nφint : IntegrableOn φ (Icc a b) volume\nhφg : ∀ x ∈ Ico a b, g' x ≤ φ x\nε : ℝ\nεpos : 0 < ε\nG' : ℝ → EReal\nf_lt_G' : ∀ (x : ℝ), ↑(φ x) < G' x\nG'cont : LowerSemicontin... | simp only [s, inter_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 594,
"column": 69
} | {
"line": 594,
"column": 82
} | [
{
"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 : NoAtoms μ\ninst✝ : IsLocallyFiniteMeasure μ\nf : X → ℝ → E\na₀ : ℝ\nhf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (F... | congr 1; abel | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 594,
"column": 69
} | {
"line": 594,
"column": 82
} | [
{
"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 : NoAtoms μ\ninst✝ : IsLocallyFiniteMeasure μ\nf : X → ℝ → E\na₀ : ℝ\nhf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (F... | congr 1; abel | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1154,
"column": 2
} | {
"line": 1154,
"column": 52
} | [
{
"pp": "case inr\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhf : IntegrableOn f (Ioi a) μ\nhg : IntegrableOn f (Ioi b) μ\nthis :\n ∀ {a b : ℝ},\n IntegrableOn f (Ioi a) μ →\n IntegrableOn f (Ioi b) μ →\n a ≤ b → ∫ (x : ℝ) in Ioi a, ... | · rw [integral_symm, ← this hg hf hab.le, neg_sub] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1168,
"column": 2
} | {
"line": 1168,
"column": 52
} | [
{
"pp": "case inr\nE : Type u_5\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\na b : ℝ\nf : ℝ → E\nμ : Measure ℝ\ninst✝ : NoAtoms μ\nhf : IntegrableOn f (Iio b) μ\nhg : IntegrableOn f (Iio a) μ\nthis :\n ∀ {a b : ℝ},\n IntegrableOn f (Iio b) μ →\n IntegrableOn f (Iio a) μ →\n a ≤ b → ... | · rw [integral_symm, ← this hg hf hab.le, neg_sub] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1182,
"column": 2
} | {
"line": 1182,
"column": 52
} | [
{
"pp": "case inr\nE : Type u_5\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\na b : ℝ\nf : ℝ → E\nμ : Measure ℝ\ninst✝ : NoAtoms μ\nhf : IntegrableOn f (Ici a) μ\nhg : IntegrableOn f (Ici b) μ\nthis :\n ∀ {a b : ℝ},\n IntegrableOn f (Ici a) μ →\n IntegrableOn f (Ici b) μ →\n a ≤ b → ... | · rw [integral_symm, ← this hg hf hab.le, neg_sub] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Algebra.Order.Floor | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 73
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : Ring α\ninst✝⁴ : LinearOrder α\ninst✝³ : FloorRing α\ninst✝² : TopologicalSpace α\ninst✝¹ : IsStrictOrderedRing α\ninst✝ : OrderClosedTopology α\nn : ℤ\n⊢ Tendsto floor (𝓝[≥] ↑n) (pure n)",
"usedConstants": [
"Pure.pure",
"Int.cast",
"Set.Ici",
"Int.f... | simpa only [floor_intCast] using tendsto_floor_right_pure_floor (n : α) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.Algebra.Order.Floor | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 73
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : Ring α\ninst✝⁴ : LinearOrder α\ninst✝³ : FloorRing α\ninst✝² : TopologicalSpace α\ninst✝¹ : IsStrictOrderedRing α\ninst✝ : OrderClosedTopology α\nn : ℤ\n⊢ Tendsto floor (𝓝[≥] ↑n) (pure n)",
"usedConstants": [
"Pure.pure",
"Int.cast",
"Set.Ici",
"Int.f... | simpa only [floor_intCast] using tendsto_floor_right_pure_floor (n : α) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Order.Floor | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 73
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : Ring α\ninst✝⁴ : LinearOrder α\ninst✝³ : FloorRing α\ninst✝² : TopologicalSpace α\ninst✝¹ : IsStrictOrderedRing α\ninst✝ : OrderClosedTopology α\nn : ℤ\n⊢ Tendsto floor (𝓝[≥] ↑n) (pure n)",
"usedConstants": [
"Pure.pure",
"Int.cast",
"Set.Ici",
"Int.f... | simpa only [floor_intCast] using tendsto_floor_right_pure_floor (n : α) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1259,
"column": 4
} | {
"line": 1259,
"column": 73
} | [
{
"pp": "case inl\nf : ℝ → ℝ\na b : ℝ\nμ : Measure ℝ\nhf : 0 ≤ᶠ[ae (μ.restrict (Ioc a b ∪ Ioc b a))] f\nhfi : IntervalIntegrable f μ a b\nhab : a ≤ b\n⊢ ∫ (x : ℝ) in a..b, f x ∂μ = 0 ↔ f =ᶠ[ae (μ.restrict (Ioc a b ∪ Ioc b a))] 0",
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"Set.Ioc",... | simp only [Ioc_eq_empty hab.not_gt, empty_union, union_empty] at hf ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1259,
"column": 4
} | {
"line": 1259,
"column": 73
} | [
{
"pp": "case inr\nf : ℝ → ℝ\na b : ℝ\nμ : Measure ℝ\nhf : 0 ≤ᶠ[ae (μ.restrict (Ioc a b ∪ Ioc b a))] f\nhfi : IntervalIntegrable f μ a b\nhab : b ≤ a\n⊢ ∫ (x : ℝ) in a..b, f x ∂μ = 0 ↔ f =ᶠ[ae (μ.restrict (Ioc a b ∪ Ioc b a))] 0",
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"Set.Ioc",... | simp only [Ioc_eq_empty hab.not_gt, empty_union, union_empty] at hf ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Algebra.Order.Floor | {
"line": 184,
"column": 2
} | {
"line": 184,
"column": 57
} | [
{
"pp": "α : Type u_1\ninst✝⁶ : Ring α\ninst✝⁵ : LinearOrder α\ninst✝⁴ : FloorRing α\ninst✝³ : TopologicalSpace α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : OrderClosedTopology α\ninst✝ : IsTopologicalAddGroup α\nn : ℤ\n⊢ Tendsto fract (𝓝[<] ↑n) (𝓝 (↑n - (↑n - 1)))",
"usedConstants": [
"Int.cast",
... | refine (tendsto_id.mono_left nhdsWithin_le_nhds).sub ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1275,
"column": 4
} | {
"line": 1275,
"column": 40
} | [
{
"pp": "case inr\nf : ℝ → ℝ\na b : ℝ\nμ : Measure ℝ\nhf : 0 ≤ᶠ[ae (μ.restrict (Ι a b))] f\nhfi : IntervalIntegrable f μ a b\nhba : b ≤ a\n⊢ 0 ≤ ∫ (x : ℝ) in Ioc b a, f x ∂μ",
"usedConstants": [
"MeasureTheory.ae",
"Real.instLE",
"Real",
"MeasureTheory.Measure",
"Real.instZero"... | rw [uIoc_comm, uIoc_of_le hba] at hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 1082,
"column": 4
} | {
"line": 1082,
"column": 29
} | [
{
"pp": "g' g φ : ℝ → ℝ\na b : ℝ\nhab : a ≤ b\nhcont : ContinuousOn g (Icc a b)\nhderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x\nφint : IntegrableOn φ (Icc a b) volume\nhφg : ∀ x ∈ Ioo a b, g' x ≤ φ x\na_lt_b : a < b\ns : Set ℝ := {t | g b - g t ≤ ∫ (u : ℝ) in t..b, φ u} ∩ Icc a b\nthis : Continuou... | simp only [s, inter_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 325,
"column": 4
} | {
"line": 325,
"column": 12
} | [
{
"pp": "case inr\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\nT t₁ t₂ : ℝ\nhf : Periodic f T\nthis :\n ∀ {E : Type u_1} [inst : NormedAddCommGroup E] {f : ℝ → E} {T t₁ t₂ : ℝ},\n Periodic f T → T ≠ 0 → (IntervalIntegrable f volume t₁ (t₁ + T) ↔ IntervalIntegrable f volume t₂ (t₂ + T))\nhT : ¬T ≠... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 325,
"column": 4
} | {
"line": 325,
"column": 12
} | [
{
"pp": "case inr\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\nT t₁ t₂ : ℝ\nhf : Periodic f T\nthis :\n ∀ {E : Type u_1} [inst : NormedAddCommGroup E] {f : ℝ → E} {T t₁ t₂ : ℝ},\n Periodic f T → T ≠ 0 → (IntervalIntegrable f volume t₁ (t₁ + T) ↔ IntervalIntegrable f volume t₂ (t₂ + T))\nhT : ¬T ≠... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 325,
"column": 4
} | {
"line": 325,
"column": 12
} | [
{
"pp": "case inr\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\nT t₁ t₂ : ℝ\nhf : Periodic f T\nthis :\n ∀ {E : Type u_1} [inst : NormedAddCommGroup E] {f : ℝ → E} {T t₁ t₂ : ℝ},\n Periodic f T → T ≠ 0 → (IntervalIntegrable f volume t₁ (t₁ + T) ↔ IntervalIntegrable f volume t₂ (t₂ + T))\nhT : ¬T ≠... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 689,
"column": 6
} | {
"line": 689,
"column": 43
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nμ : Measure ℝ\nf : ℝ → E\ninst✝ : NoAtoms μ\na₀ : ℝ\nhf : IntegrableOn f (Ioi a₀) μ\na : ℝ\nha : a₀ ≤ a\nh_int : IntervalIntegrable f μ a₀ a\nb : ℝ\nhb : b ∈ Icc a₀ a\n⊢ ∫ (x : ℝ) in Ioi b, f x ∂μ = ∫ (x : ℝ) in Ioi a₀, f x ∂μ - ∫ (... | simp [← integral_Ioi_sub_Ioi hf hb.1] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 715,
"column": 10
} | {
"line": 715,
"column": 23
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nμ : Measure ℝ\nf : ℝ → E\ninst✝ : NoAtoms μ\na₀ : ℝ\nhf : IntegrableOn f (Iio a₀) μ\na : ℝ\n⊢ a ∈ Iic a₀ → ContinuousWithinAt (fun b ↦ ∫ (x : ℝ) in Iio b, f x ∂μ) (Iic a₀) a",
"usedConstants": [
"Real",
"Membership.m... | (ha : a ≤ a₀) | Lean.Elab.Tactic.evalIntro | Lean.Parser.Term.typeAscription |
Mathlib.Analysis.BoxIntegral.DivergenceTheorem | {
"line": 215,
"column": 6
} | {
"line": 215,
"column": 70
} | [
{
"pp": "case refine_3.refine_1\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nn : ℕ\ninst✝ : CompleteSpace E\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) → ℝ) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E\ns : Set (Fin (n + 1) → ℝ)\nhs : s.Countable\nHs : ∀ x ∈ s, ContinuousWithinA... | refine (mul_le_mul_of_nonneg_right ?_ <| norm_nonneg _).trans hδ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 537,
"column": 2
} | {
"line": 538,
"column": 32
} | [
{
"pp": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : s.Countable\nhw : w ∈ ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ x ∈ ball c R \\ s, DifferentiableAt ℂ f x\n⊢ ∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z = (2 *... | rw [← two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable
hs hw hc hd, smul_inv_smul₀] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 265,
"column": 2
} | {
"line": 265,
"column": 10
} | [
{
"pp": "σ : Type u_5\nR : Type u_6\ninst✝³ : CommSemiring R\ninst✝² : Fintype σ\ninst✝¹ : DecidableEq σ\ninst✝ : Nontrivial R\nn : ℕ\ns t : Finset σ\nhst : s ≠ t\nh : ∑ i ∈ t, Finsupp.single i 1 = ∑ i ∈ s, Finsupp.single i 1\nthis : (t.biUnion fun i ↦ (Finsupp.single i 1).support) = s.biUnion fun i ↦ (Finsupp.... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Integral.DivergenceTheorem | {
"line": 336,
"column": 6
} | {
"line": 336,
"column": 43
} | [
{
"pp": "E : 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 : Measur... | simp only [hIcc, eL.symm_apply_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Polynomial.Vieta | {
"line": 52,
"column": 4
} | {
"line": 52,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\ns✝ : Multiset R\nx✝ : ℕ\na✝ : x✝ ∈ range (s✝.card + 1)\ns : Multiset R\nht : s ≤ s✝ ∧ s.card = x✝\n⊢ (map (fun r ↦ X) (s✝ - s)).prod * C (map (fun i ↦ i) s).prod = C s.prod * X ^ (s✝.card - x✝)",
"usedConstants": [
"Multiset.prod_replicate",
"Polyno... | simp [ht, prod_replicate, map_id', card_sub] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Polynomial.Vieta | {
"line": 154,
"column": 11
} | {
"line": 154,
"column": 13
} | [
{
"pp": "case h.e'_2.h.e'_3\nR : Type u_1\nσ : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : Fintype σ\ns : Multiset (MvPolynomial σ R) := Multiset.map (fun i ↦ X i) univ.val\nthis : Fintype.card σ = s.card\n⊢ Multiset.map (fun i ↦ Polynomial.X + Polynomial.C (X i)) univ.val =\n Multiset.map (fun r ↦ Polynomial... | s, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Polynomial.Vieta | {
"line": 166,
"column": 11
} | {
"line": 166,
"column": 13
} | [
{
"pp": "case h.e'_2.h.e'_3.h.h.e'_3\nR : Type u_1\nσ : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : Fintype σ\nk : ℕ\ns : Multiset (MvPolynomial σ R) := Multiset.map (fun i ↦ X i) univ.val\nh : k ≤ s.card\nthis : Fintype.card σ = s.card\ne_2✝ : AddMonoidAlgebra.semiring = AddMonoidAlgebra.commSemiring.toSemiring... | s, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Complex.Polynomial.Basic | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 64
} | [
{
"pp": "p : ℚ[X]\np_irr : Irreducible p\np_deg : Nat.Prime p.natDegree\np_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2\n⊢ Fintype.card ↑(p.rootSet ℂ) = p.natDegree",
"usedConstants": [
"Multiset.toFinset",
"Eq.mpr",
"Complex.commRing",
"congrArg",
"Fin... | simp_rw [rootSet_def, Finset.coe_sort_coe, Fintype.card_coe] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.NumberTheory.NumberField.InfinitePlace.Embeddings | {
"line": 174,
"column": 2
} | {
"line": 174,
"column": 21
} | [
{
"pp": "K : Type u_1\ninst✝³ : Field K\nk : Type u_2\ninst✝² : Field k\ninst✝¹ : Algebra k K\ninst✝ : Algebra.IsAlgebraic k K\nφ : k →+* ℂ\n⊢ K →+* ℂ",
"usedConstants": [
"Field.toSemifield",
"Complex.instCommSemiring",
"Semifield.toCommSemiring",
"Complex",
"RingHom.toAlgebra... | letI := φ.toAlgebra | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.NumberTheory.NumberField.InfinitePlace.Embeddings | {
"line": 181,
"column": 2
} | {
"line": 181,
"column": 21
} | [
{
"pp": "K : Type u_1\ninst✝³ : Field K\nk : Type u_2\ninst✝² : Field k\ninst✝¹ : Algebra k K\ninst✝ : Algebra.IsAlgebraic k K\nφ : k →+* ℂ\n⊢ IsAlgClosed.lift.comp (algebraMap k K) = φ",
"usedConstants": [
"Field.toSemifield",
"Complex.instCommSemiring",
"Semifield.toCommSemiring",
... | letI := φ.toAlgebra | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.NumberTheory.NumberField.InfinitePlace.Embeddings | {
"line": 247,
"column": 2
} | {
"line": 247,
"column": 21
} | [
{
"pp": "K : Type u_1\ninst✝³ : Field K\nk : Type u_2\ninst✝² : Field k\ninst✝¹ : Algebra k K\ninst✝ : IsGalois k K\nφ ψ : K →+* ℂ\nh : φ.comp (algebraMap k K) = ψ.comp (algebraMap k K)\nthis : Algebra k ℂ := (φ.comp (algebraMap k K)).toAlgebra\n⊢ ∃ σ, φ.comp ↑σ.symm = ψ",
"usedConstants": [
"Field.to... | letI := φ.toAlgebra | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.Data.Real.Embedding | {
"line": 137,
"column": 6
} | {
"line": 138,
"column": 88
} | [
{
"pp": "case h.mp.refine_2\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 y : M\na : ℚ\nh : a.num • 1 < a.den • (x + y)\nk : ℕ\nhk : 1 + 1 ≤ k • (a.den • (x + y) - a.num • 1)\n... | have : k • a.num • 1 - k • a.den • y < m • 1 :=
lt_of_lt_of_le (lt_add_of_pos_left _ zero_lt_one) (by simpa using hk'.trans hm1) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Valuation.RankOne | {
"line": 136,
"column": 4
} | {
"line": 137,
"column": 60
} | [
{
"pp": "R : Type u_1\nΓ₀ : Type u_2\ninst✝² : Ring R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nhv : v.RankOne\ninst✝ : v.IsNontrivial\n⊢ ∃ x, v.restrict x ≠ 0 ∧ v.restrict x ≠ 1",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"LinearOrderedCommGroupWithZero.toLinea... | obtain ⟨x, ⟨hx0, hx1⟩⟩ := IsNontrivial.exists_val_nontrivial (v := v)
exact ⟨x, by simp [hx0], by grind [restrict_eq_one_iff]⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Valuation.RankOne | {
"line": 136,
"column": 4
} | {
"line": 137,
"column": 60
} | [
{
"pp": "R : Type u_1\nΓ₀ : Type u_2\ninst✝² : Ring R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nhv : v.RankOne\ninst✝ : v.IsNontrivial\n⊢ ∃ x, v.restrict x ≠ 0 ∧ v.restrict x ≠ 1",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"LinearOrderedCommGroupWithZero.toLinea... | obtain ⟨x, ⟨hx0, hx1⟩⟩ := IsNontrivial.exists_val_nontrivial (v := v)
exact ⟨x, by simp [hx0], by grind [restrict_eq_one_iff]⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Valuation.RankOne | {
"line": 261,
"column": 8
} | {
"line": 261,
"column": 16
} | [
{
"pp": "case neg.inl\nR : Type u_3\ninst✝¹ : CommRing R\ninst✝ : ValuativeRel R\nh : MulArchimedean (ValueGroupWithZero R)\nb : ValueGroupWithZero R\nH : 0 < b\nH' : 1 b ≤ 1 0\n⊢ IsNontrivial R",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"LinearOrderedCommGroupWithZero.toLinearOrdere... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Valuation.RankOne | {
"line": 261,
"column": 8
} | {
"line": 261,
"column": 16
} | [
{
"pp": "case neg.inl\nR : Type u_3\ninst✝¹ : CommRing R\ninst✝ : ValuativeRel R\nh : MulArchimedean (ValueGroupWithZero R)\nb : ValueGroupWithZero R\nH : 0 < b\nH' : 1 b ≤ 1 0\n⊢ IsNontrivial R",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"LinearOrderedCommGroupWithZero.toLinearOrdere... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Valuation.RankOne | {
"line": 261,
"column": 8
} | {
"line": 261,
"column": 16
} | [
{
"pp": "case neg.inl\nR : Type u_3\ninst✝¹ : CommRing R\ninst✝ : ValuativeRel R\nh : MulArchimedean (ValueGroupWithZero R)\nb : ValueGroupWithZero R\nH : 0 < b\nH' : 1 b ≤ 1 0\n⊢ IsNontrivial R",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"LinearOrderedCommGroupWithZero.toLinearOrdere... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors | {
"line": 67,
"column": 22
} | {
"line": 67,
"column": 63
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nφ : MvPowerSeries σ R\nhφ : constantCoeff φ ∈ nonZeroDivisorsRight R\nx : MvPowerSeries σ R\nhx : x * φ = 0\nd e : σ →₀ ℕ\nhe : ∀ y < e, (coeff y) x = (coeff y) 0\nu snd✝ : σ →₀ ℕ\nhuv : (u, snd✝) ∈ antidiagonal e\na✝ : (u, snd✝) ≠ (e, 0)\nthis : u < e\n⊢... | simp only [he u this, zero_mul, map_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors | {
"line": 67,
"column": 22
} | {
"line": 67,
"column": 63
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nφ : MvPowerSeries σ R\nhφ : constantCoeff φ ∈ nonZeroDivisorsRight R\nx : MvPowerSeries σ R\nhx : x * φ = 0\nd e : σ →₀ ℕ\nhe : ∀ y < e, (coeff y) x = (coeff y) 0\nu snd✝ : σ →₀ ℕ\nhuv : (u, snd✝) ∈ antidiagonal e\na✝ : (u, snd✝) ≠ (e, 0)\nthis : u < e\n⊢... | simp only [he u this, zero_mul, map_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors | {
"line": 67,
"column": 22
} | {
"line": 67,
"column": 63
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nφ : MvPowerSeries σ R\nhφ : constantCoeff φ ∈ nonZeroDivisorsRight R\nx : MvPowerSeries σ R\nhx : x * φ = 0\nd e : σ →₀ ℕ\nhe : ∀ y < e, (coeff y) x = (coeff y) 0\nu snd✝ : σ →₀ ℕ\nhuv : (u, snd✝) ∈ antidiagonal e\na✝ : (u, snd✝) ≠ (e, 0)\nthis : u < e\n⊢... | simp only [he u this, zero_mul, map_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors | {
"line": 71,
"column": 6
} | {
"line": 71,
"column": 14
} | [
{
"pp": "case h.e'_2.h₀.a\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nφ : MvPowerSeries σ R\nhφ : constantCoeff φ ∈ nonZeroDivisorsRight R\nx : MvPowerSeries σ R\nhx : x * φ = 0\nd u snd✝ : σ →₀ ℕ\nhe : ∀ y < u, (coeff y) x = (coeff y) 0\nhuv : (u, snd✝) ∈ antidiagonal u\na✝ : (u, snd✝) ≠ (u, 0)\n⊢ False",... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors | {
"line": 92,
"column": 6
} | {
"line": 92,
"column": 14
} | [
{
"pp": "case h.e'_2.h₀.a\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nφ : MvPowerSeries σ R\nhφ : constantCoeff φ ∈ nonZeroDivisorsLeft R\nx : MvPowerSeries σ R\nhx : φ * x = 0\nd fst✝ u : σ →₀ ℕ\nhe : ∀ y < u, (coeff y) x = (coeff y) 0\nhuv : (fst✝, u) ∈ antidiagonal u\na✝ : (fst✝, u) ≠ (0, u)\n⊢ False",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors | {
"line": 161,
"column": 8
} | {
"line": 164,
"column": 56
} | [
{
"pp": "case inl\nσ : Type u_1\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\nw : σ → ℕ\nf g : MvPowerSeries σ R\nhf : weightedOrder w f < ⊤\nhg : weightedOrder w g < ⊤\np : ℕ := (weightedOrder w f).toNat\nhp : ↑p = weightedOrder w f\nq : ℕ := (weightedOrder w g).toNat\nhq : ↑q = weightedOrder w... | · refine weightedHomogeneousComponent_of_weightedOrder ?_ H
simp only [ENat.coe_toNat_eq_self, ne_eq, weightedOrder_eq_top_iff, p, q]
rw [← ne_eq, ne_zero_iff_weightedOrder_finite w]
exact ENat.coe_toNat (ne_top_of_lt (by simpa)) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
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