module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.BoxIntegral.Box.SubboxInduction | {
"line": 132,
"column": 56
} | {
"line": 132,
"column": 81
} | {
"line": 132,
"column": 82
} | [
{
"pp": "ι : Type u_1\np : Box ι → Prop\nI : Box ι\nH_nhds :\n ∀ z ∈ Box.Icc I,\n ∃ U ∈ 𝓝[Box.Icc I] z,\n ∀ J ≤ I,\n ∀ (m : ℕ),\n z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J\nhpI : ¬p I\ns : Box ι → Set ι\nhs : ∀ J ≤ I, ¬p ... | [
"ι : Type u_1\np : Box ι → Prop\nI : Box ι\nH_nhds :\n ∀ z ∈ Box.Icc I,\n ∃ U ∈ 𝓝[Box.Icc I] z,\n ∀ J ≤ I,\n ∀ (m : ℕ),\n z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J\nhpI : ¬p I\ns : Box ι → Set ι\nhs : ∀ J ≤ I, ¬p J → ¬p (J.sp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 543,
"column": 8
} | {
"line": 543,
"column": 89
} | {
"line": 544,
"column": 6
} | [
{
"pp": "case ha\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nr : ℝ\nhr : r ≠ 0\nx y : E\ns t : Set E\n⊢ ENNReal.ofReal (|r| ^ finrank ℝ E) ≠ 0 ∨ μ t ≠ ∞",
... | [] | simp only [pow_pos (abs_pos.mpr hr), ENNReal.ofReal_eq_zero, not_le, Ne, true_or] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 543,
"column": 8
} | {
"line": 543,
"column": 89
} | {
"line": 544,
"column": 6
} | [
{
"pp": "case ha\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nr : ℝ\nhr : r ≠ 0\nx y : E\ns t : Set E\n⊢ ENNReal.ofReal (|r| ^ finrank ℝ E) ≠ 0 ∨ μ t ≠ ∞",
... | [] | simp only [pow_pos (abs_pos.mpr hr), ENNReal.ofReal_eq_zero, not_le, Ne, true_or] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 543,
"column": 8
} | {
"line": 543,
"column": 89
} | {
"line": 544,
"column": 6
} | [
{
"pp": "case ha\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nr : ℝ\nhr : r ≠ 0\nx y : E\ns t : Set E\n⊢ ENNReal.ofReal (|r| ^ finrank ℝ E) ≠ 0 ∨ μ t ≠ ∞",
... | [] | simp only [pow_pos (abs_pos.mpr hr), ENNReal.ofReal_eq_zero, not_le, Ne, true_or] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.BoxIntegral.Box.SubboxInduction | {
"line": 151,
"column": 72
} | {
"line": 151,
"column": 83
} | {
"line": 151,
"column": 84
} | [
{
"pp": "ι : Type u_1\np : Box ι → Prop\nI : Box ι\nH_nhds :\n ∀ z ∈ Box.Icc I,\n ∃ U ∈ 𝓝[Box.Icc I] z,\n ∀ J ≤ I,\n ∀ (m : ℕ),\n z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J\nJ : ℕ → Box ι\nhJmono : Antitone J\nhJle : ∀ (m ... | [
"ι : Type u_1\np : Box ι → Prop\nI : Box ι\nH_nhds :\n ∀ z ∈ Box.Icc I,\n ∃ U ∈ 𝓝[Box.Icc I] z,\n ∀ J ≤ I,\n ∀ (m : ℕ),\n z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J\nJ : ℕ → Box ι\nhJmono : Antitone J\nhJle : ∀ (m : ℕ), J m ≤ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Box.SubboxInduction | {
"line": 153,
"column": 4
} | {
"line": 153,
"column": 23
} | {
"line": 154,
"column": 6
} | [
{
"pp": "ι : Type u_1\np : Box ι → Prop\nI : Box ι\nH_nhds :\n ∀ z ∈ Box.Icc I,\n ∃ U ∈ 𝓝[Box.Icc I] z,\n ∀ J ≤ I,\n ∀ (m : ℕ),\n z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J\nJ : ℕ → Box ι\nhJmono : Antitone J\nhJle : ∀ (m ... | [
"ι : Type u_1\np : Box ι → Prop\nI : Box ι\nH_nhds :\n ∀ z ∈ Box.Icc I,\n ∃ U ∈ 𝓝[Box.Icc I] z,\n ∀ J ≤ I,\n ∀ (m : ℕ),\n z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J\nJ : ℕ → Box ι\nhJmono : Antitone J\nhJle : ∀ (m : ℕ), J m ≤ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Partition.Basic | {
"line": 173,
"column": 37
} | {
"line": 173,
"column": 59
} | {
"line": 173,
"column": 60
} | [
{
"pp": "ι : Type u_1\nI : Box ι\nπ : Prepartition I\nx : ι → ℝ\nJ₁ : Box ι\nh₁ : J₁ ∈ π\nhx₁ : x ∈ Box.Icc J₁\nJ₂ : Box ι\nh₂ : J₂ ∈ π\nhx₂ : x ∈ Box.Icc J₂\ni : ι\nH : ∀ (x_1 : ι), J₁.lower x_1 = x x_1 ↔ J₂.lower x_1 = x x_1\nhi₁ : J₁.lower i = x i\nhi₂ : J₂.lower i = x i\n⊢ x i < J₁.upper i",
"ppTerm": "... | [
"ι : Type u_1\nI : Box ι\nπ : Prepartition I\nx : ι → ℝ\nJ₁ : Box ι\nh₁ : J₁ ∈ π\nhx₁ : x ∈ Box.Icc J₁\nJ₂ : Box ι\nh₂ : J₂ ∈ π\nhx₂ : x ∈ Box.Icc J₂\ni : ι\nH : ∀ (x_1 : ι), J₁.lower x_1 = x x_1 ↔ J₂.lower x_1 = x x_1\nhi₁ : J₁.lower i = x i\nhi₂ : J₂.lower i = x i\n⊢ x i < J₁.upper i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Partition.Basic | {
"line": 174,
"column": 37
} | {
"line": 174,
"column": 59
} | {
"line": 174,
"column": 60
} | [
{
"pp": "ι : Type u_1\nI : Box ι\nπ : Prepartition I\nx : ι → ℝ\nJ₁ : Box ι\nh₁ : J₁ ∈ π\nhx₁ : x ∈ Box.Icc J₁\nJ₂ : Box ι\nh₂ : J₂ ∈ π\nhx₂ : x ∈ Box.Icc J₂\ni : ι\nH : ∀ (x_1 : ι), J₁.lower x_1 = x x_1 ↔ J₂.lower x_1 = x x_1\nhi₁ : J₁.lower i = x i\nhi₂ : J₂.lower i = x i\nH₁ : x i < J₁.upper i\n⊢ x i < J₂.up... | [
"ι : Type u_1\nI : Box ι\nπ : Prepartition I\nx : ι → ℝ\nJ₁ : Box ι\nh₁ : J₁ ∈ π\nhx₁ : x ∈ Box.Icc J₁\nJ₂ : Box ι\nh₂ : J₂ ∈ π\nhx₂ : x ∈ Box.Icc J₂\ni : ι\nH : ∀ (x_1 : ι), J₁.lower x_1 = x x_1 ↔ J₂.lower x_1 = x x_1\nhi₁ : J₁.lower i = x i\nhi₂ : J₂.lower i = x i\nH₁ : x i < J₁.upper i\n⊢ x i < J₂.upper i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Partition.Basic | {
"line": 188,
"column": 2
} | {
"line": 188,
"column": 13
} | {
"line": 188,
"column": 14
} | [
{
"pp": "ι : Type u_1\nI : Box ι\nπ : Prepartition I\ninst✝ : Fintype ι\nx : ι → ℝ\n⊢ InjOn (fun J ↦ {i | J.lower i = x i}) ↑({J ∈ π.boxes | x ∈ Box.Icc J})",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Finset.coe_filter",
"BoxIntegral.Prepartit... | [
"ι : Type u_1\nI : Box ι\nπ : Prepartition I\ninst✝ : Fintype ι\nx : ι → ℝ\n⊢ InjOn (fun J ↦ {i | J.lower i = x i}) {x_1 | x_1 ∈ π ∧ x ∈ Box.Icc x_1}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Partition.Basic | {
"line": 328,
"column": 4
} | {
"line": 328,
"column": 45
} | {
"line": 329,
"column": 2
} | [
{
"pp": "case pos\nι : Type u_1\nI : Box ι\nπ : Prepartition I\nπi : (J : Box ι) → Prepartition J\nJ : Box ι\nhJ : J ∈ π.biUnion πi\n⊢ π.biUnionIndex πi J ≤ I",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"BoxIntegral.Prepartition.le_of_mem",
"BoxIntegral.Prepartition.biUnion... | [] | exact π.le_of_mem (π.biUnionIndex_mem hJ) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.BoxIntegral.Partition.Basic | {
"line": 328,
"column": 4
} | {
"line": 328,
"column": 45
} | {
"line": 329,
"column": 2
} | [
{
"pp": "case pos\nι : Type u_1\nI : Box ι\nπ : Prepartition I\nπi : (J : Box ι) → Prepartition J\nJ : Box ι\nhJ : J ∈ π.biUnion πi\n⊢ π.biUnionIndex πi J ≤ I",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"BoxIntegral.Prepartition.le_of_mem",
"BoxIntegral.Prepartition.biUnion... | [] | exact π.le_of_mem (π.biUnionIndex_mem hJ) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.BoxIntegral.Partition.Basic | {
"line": 328,
"column": 4
} | {
"line": 328,
"column": 45
} | {
"line": 329,
"column": 2
} | [
{
"pp": "case pos\nι : Type u_1\nI : Box ι\nπ : Prepartition I\nπi : (J : Box ι) → Prepartition J\nJ : Box ι\nhJ : J ∈ π.biUnion πi\n⊢ π.biUnionIndex πi J ≤ I",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"BoxIntegral.Prepartition.le_of_mem",
"BoxIntegral.Prepartition.biUnion... | [] | exact π.le_of_mem (π.biUnionIndex_mem hJ) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.ZLattice.Basic | {
"line": 480,
"column": 6
} | {
"line": 481,
"column": 73
} | {
"line": 482,
"column": 6
} | [
{
"pp": "case refine_1.refine_2\nK : Type u_1\ninst✝⁹ : NormedField K\ninst✝⁸ : LinearOrder K\ninst✝⁷ : IsStrictOrderedRing K\ninst✝⁶ : HasSolidNorm K\ninst✝⁵ : FloorRing K\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace K E\ninst✝² : FiniteDimensional K E\ninst✝¹ : ProperSpace E\nL : Submodu... | [
"case refine_1.refine_2\nK : Type u_1\ninst✝⁹ : NormedField K\ninst✝⁸ : LinearOrder K\ninst✝⁷ : IsStrictOrderedRing K\ninst✝⁶ : HasSolidNorm K\ninst✝⁵ : FloorRing K\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace K E\ninst✝² : FiniteDimensional K E\ninst✝¹ : ProperSpace E\nL : Submodule ℤ E\ninst... | refine Submodule.add_mem _ h_mem
(neg_mem (Set.mem_of_subset_of_mem ?_ (Subtype.mem (floor b x)))) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.BoxIntegral.Partition.Basic | {
"line": 364,
"column": 4
} | {
"line": 364,
"column": 62
} | {
"line": 364,
"column": 63
} | [
{
"pp": "ι : Type u_1\nI J✝ J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\nboxes : Finset (WithBot (Box ι))\nle_of_mem : ∀ J ∈ boxes, J ≤ ↑I\npairwise_disjoint : (↑boxes).Pairwise Disjoint\nJ : Box ι\nhJ : some J ∈ boxes\n⊢ J ≤ I",
"ppTerm": "?m.40",
"assi... | [
"ι : Type u_1\nI J✝ J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\nboxes : Finset (WithBot (Box ι))\nle_of_mem : ∀ J ∈ boxes, J ≤ ↑I\npairwise_disjoint : (↑boxes).Pairwise Disjoint\nJ : Box ι\nhJ : some J ∈ boxes\n⊢ J ≤ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Partition.Basic | {
"line": 390,
"column": 4
} | {
"line": 390,
"column": 41
} | {
"line": 390,
"column": 42
} | [
{
"pp": "ι : Type u_1\nI : Box ι\nπ : Prepartition I\nboxes : Finset (WithBot (Box ι))\nle_of_mem : ∀ J ∈ boxes, J ≤ ↑I\npairwise_disjoint : (↑boxes).Pairwise Disjoint\nH : ∀ J ∈ boxes, J ≠ ⊥ → ∃ J' ∈ π, J ≤ ↑J'\nJ : Box ι\nhJ : ↑J ∈ boxes\n⊢ ∃ J' ∈ π, J ≤ J'",
"ppTerm": "?m.52",
"assigned": false,
... | [
"ι : Type u_1\nI : Box ι\nπ : Prepartition I\nboxes : Finset (WithBot (Box ι))\nle_of_mem : ∀ J ∈ boxes, J ≤ ↑I\npairwise_disjoint : (↑boxes).Pairwise Disjoint\nH : ∀ J ∈ boxes, J ≠ ⊥ → ∃ J' ∈ π, J ≤ ↑J'\nJ : Box ι\nhJ : ↑J ∈ boxes\n⊢ ∃ J' ∈ π, J ≤ J'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Partition.Basic | {
"line": 541,
"column": 2
} | {
"line": 541,
"column": 13
} | {
"line": 541,
"column": 14
} | [
{
"pp": "ι : Type u_1\nI : Box ι\nπ : Prepartition I\np : Box ι → Prop\nhp : ∀ J ∈ π, p J\nJ : Box ι\n⊢ J ∈ π.filter p ↔ J ∈ π",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"BoxIntegral.Prepartition.filter",
"BoxIntegral.Prepartition",
"congrArg",
... | [
"ι : Type u_1\nI : Box ι\nπ : Prepartition I\np : Box ι → Prop\nhp : ∀ J ∈ π, p J\nJ : Box ι\n⊢ J ∈ π → p J"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 505,
"column": 41
} | {
"line": 505,
"column": 52
} | {
"line": 505,
"column": 53
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_3\ninst✝³ : RCLike 𝕜\nE : Type u_4\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : Fintype ι\nb : OrthonormalBasis ι 𝕜 E\nv : EuclideanSpace 𝕜 ι\n⊢ ∑ i, v.ofLp i • b i = b.repr.symm v",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants... | [
"ι : Type u_1\n𝕜 : Type u_3\ninst✝³ : RCLike 𝕜\nE : Type u_4\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : Fintype ι\nb : OrthonormalBasis ι 𝕜 E\nv : EuclideanSpace 𝕜 ι\n⊢ ∑ x, v.ofLp x • b x = b.repr.symm v"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Partition.Tagged | {
"line": 143,
"column": 8
} | {
"line": 143,
"column": 34
} | {
"line": 143,
"column": 34
} | [
{
"pp": "case refine_2\nι : Type u_1\nI : Box ι\np : (ι → ℝ) → Box ι → Prop\nπ : Prepartition I\nπi : (J : Box ι) → TaggedPrepartition J\nH : ∀ J ∈ π, ∀ J' ∈ πi J, p ((πi J).tag J') J'\nJ' : Box ι\nx✝ : ∃ J'_1 ∈ π, J' ∈ πi J'_1\nJ : Box ι\nhJ : J ∈ π\nhJ' : J' ∈ πi J\n⊢ p ((π.biUnionTagged πi).tag J') J'",
... | [
"case refine_2\nι : Type u_1\nI : Box ι\np : (ι → ℝ) → Box ι → Prop\nπ : Prepartition I\nπi : (J : Box ι) → TaggedPrepartition J\nH : ∀ J ∈ π, ∀ J' ∈ πi J, p ((πi J).tag J') J'\nJ' : Box ι\nx✝ : ∃ J'_1 ∈ π, J' ∈ πi J'_1\nJ : Box ι\nhJ : J ∈ π\nhJ' : J' ∈ πi J\n⊢ p ((πi J).tag J') J'"
] | π.tag_biUnionTagged hJ hJ' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.BoxIntegral.Partition.Tagged | {
"line": 143,
"column": 2
} | {
"line": 144,
"column": 23
} | {
"line": 146,
"column": 0
} | [
{
"pp": "case refine_2\nι : Type u_1\nI : Box ι\np : (ι → ℝ) → Box ι → Prop\nπ : Prepartition I\nπi : (J : Box ι) → TaggedPrepartition J\nH : ∀ J ∈ π, ∀ J' ∈ πi J, p ((πi J).tag J') J'\nJ' : Box ι\nx✝ : ∃ J'_1 ∈ π, J' ∈ πi J'_1\nJ : Box ι\nhJ : J ∈ π\nhJ' : J' ∈ πi J\n⊢ p ((π.biUnionTagged πi).tag J') J'",
... | [] | · rw [π.tag_biUnionTagged hJ hJ']
exact H J hJ J' hJ' | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 706,
"column": 14
} | {
"line": 706,
"column": 49
} | {
"line": 706,
"column": 50
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : Set E\nx : E\nh : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)\nt : ... | [
"E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : Set E\nx : E\nh : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)\nt : Set E\nht : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Partition.Tagged | {
"line": 352,
"column": 4
} | {
"line": 352,
"column": 15
} | {
"line": 352,
"column": 16
} | [
{
"pp": "ι : Type u_1\nI✝ J✝ : Box ι\nπ π₁ π₂ : TaggedPrepartition I✝\nx : ι → ℝ\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nI J : Box ι\nh : I ≤ J\nt₁ : Box ι → ι → ℝ\nht₁ : ∀ (J : Box ι), t₁ J ∈ Box.Icc I\nb₁ : Finset (Box ι)\nh₁le : ∀ J ∈ b₁, J ≤ I\nh₁d : (↑b₁).Pairwise (Disjoint on Box.toSet)\nt₂ : Box ι → ι → ℝ\nht... | [
"ι : Type u_1\nI✝ J✝ : Box ι\nπ π₁ π₂ : TaggedPrepartition I✝\nx : ι → ℝ\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nI J : Box ι\nh : I ≤ J\nt₁ : Box ι → ι → ℝ\nht₁ : ∀ (J : Box ι), t₁ J ∈ Box.Icc I\nb₁ : Finset (Box ι)\nh₁le : ∀ J ∈ b₁, J ≤ I\nh₁d : (↑b₁).Pairwise (Disjoint on Box.toSet)\nt₂ : Box ι → ι → ℝ\nht₂ : ∀ (J : B... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 559,
"column": 2
} | {
"line": 559,
"column": 86
} | {
"line": 560,
"column": 4
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_3\ninst✝⁴ : RCLike 𝕜\nE : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : Fintype ι\nU : Submodule 𝕜 E\ninst✝ : U.HasOrthogonalProjection\nb : OrthonormalBasis ι 𝕜 ↥U\nx : E\n⊢ U.orthogonalProjectionOnto x = ∑ i, ⟪↑(b i), x⟫ • b i",
"p... | [
"ι : Type u_1\n𝕜 : Type u_3\ninst✝⁴ : RCLike 𝕜\nE : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : Fintype ι\nU : Submodule 𝕜 E\ninst✝ : U.HasOrthogonalProjection\nb : OrthonormalBasis ι 𝕜 ↥U\nx : E\n⊢ U.orthogonalProjectionOnto x = ∑ i, ⟪↑(b i), x⟫ • b i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Partition.Split | {
"line": 159,
"column": 6
} | {
"line": 159,
"column": 51
} | {
"line": 159,
"column": 51
} | [
{
"pp": "case inl\nι : Type u_1\nM : Type u_2\nn : ℕ\nI✝ J : Box ι\ni✝ : ι\nx✝ : ℝ\nI : Box ι\ni : ι\nx : ℝ\n⊢ I.splitLower i x ≤ ↑I",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"BoxIntegral.Box.splitLower_le"
],
"usedFVars": [
"ι",
"I",
"i",
"x"
... | [] | exacts [Box.splitLower_le, Box.splitUpper_le] | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_exacts_1 | Batteries.Tactic.exacts |
Mathlib.Analysis.BoxIntegral.Partition.Split | {
"line": 243,
"column": 4
} | {
"line": 243,
"column": 50
} | {
"line": 243,
"column": 51
} | [
{
"pp": "ι : Type u_1\nI : Box ι\ns✝ : Finset (ι × ℝ)\na : ι × ℝ\ns : Finset (ι × ℝ)\nx✝ : a ∉ s\nhs : (splitMany I s).IsPartition\n⊢ (splitMany I (insert a s)).IsPartition",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"BoxIntegral.Prepartition",
... | [
"ι : Type u_1\nI : Box ι\ns✝ : Finset (ι × ℝ)\na : ι × ℝ\ns : Finset (ι × ℝ)\nx✝ : a ∉ s\nhs : (splitMany I s).IsPartition\n⊢ ((splitMany I s).biUnion fun J ↦ split J a.1 a.2).IsPartition"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 768,
"column": 8
} | {
"line": 768,
"column": 23
} | {
"line": 768,
"column": 24
} | [
{
"pp": "case e_a\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedB... | [
"case e_a\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (�... | inter_comm _ u, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 768,
"column": 24
} | {
"line": 768,
"column": 39
} | {
"line": 768,
"column": 40
} | [
{
"pp": "case e_a\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedB... | [
"case e_a\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (�... | inter_comm _ u, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 1022,
"column": 4
} | {
"line": 1022,
"column": 15
} | {
"line": 1023,
"column": 6
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\n𝕜 : Type u_3\ninst✝⁹ : RCLike 𝕜\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\nF : Type u_5\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace ℝ F\nF' : Type u_6\ninst✝⁴ : NormedAddCommGroup F'\ninst✝³ : InnerProductSpace ℝ F'\ninst... | [
"ι : Type u_1\nι' : Type u_2\n𝕜 : Type u_3\ninst✝⁹ : RCLike 𝕜\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\nF : Type u_5\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace ℝ F\nF' : Type u_6\ninst✝⁴ : NormedAddCommGroup F'\ninst✝³ : InnerProductSpace ℝ F'\ninst✝² : Fintype... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Partition.Filter | {
"line": 349,
"column": 4
} | {
"line": 349,
"column": 37
} | {
"line": 350,
"column": 6
} | [
{
"pp": "case inr\nι : Type u_1\ninst✝ : Fintype ι\nI : Box ι\nc₁ c₂ : ℝ≥0\nl : IntegrationParams\nr₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ r₁ π₁\nh₂ : l.MemBaseSet I c₂ r₂ π₂\nhU : π₁.iUnion = π₂.iUnion\nH :\n ∀ {ι : Type u_1} [inst : Fintype ι] {I : Box ι} {c₁ c₂ :... | [
"case inr\nι : Type u_1\ninst✝ : Fintype ι\nI : Box ι\nc₁ c₂ : ℝ≥0\nl : IntegrationParams\nr₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ r₁ π₁\nh₂ : l.MemBaseSet I c₂ r₂ π₂\nhU : π₁.iUnion = π₂.iUnion\nH :\n ∀ {ι : Type u_1} [inst : Fintype ι] {I : Box ι} {c₁ c₂ : ℝ≥0} {l : I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Partition.Filter | {
"line": 376,
"column": 4
} | {
"line": 376,
"column": 26
} | {
"line": 376,
"column": 27
} | [
{
"pp": "ι : Type u_1\ninst✝ : Fintype ι\nI : Box ι\nc : ℝ≥0\nl : IntegrationParams\nπ : TaggedPrepartition I\nr : (ι → ℝ) → ↑(Set.Ioi 0)\nhπ : l.MemBaseSet I c r π\np : Box ι → Prop\nhD : l.bDistortion = true\nπ₁ : Prepartition I\nhπ₁U : π₁.iUnion = ↑I \\ π.iUnion\nhc : π₁.distortion ≤ c\nπ₂ : TaggedPrepartiti... | [
"ι : Type u_1\ninst✝ : Fintype ι\nI : Box ι\nc : ℝ≥0\nl : IntegrationParams\nπ : TaggedPrepartition I\nr : (ι → ℝ) → ↑(Set.Ioi 0)\nhπ : l.MemBaseSet I c r π\np : Box ι → Prop\nhD : l.bDistortion = true\nπ₁ : Prepartition I\nhπ₁U : π₁.iUnion = ↑I \\ π.iUnion\nhc : π₁.distortion ≤ c\nπ₂ : TaggedPrepartition I := π.fi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Partition.Filter | {
"line": 451,
"column": 2
} | {
"line": 451,
"column": 61
} | {
"line": 452,
"column": 2
} | [
{
"pp": "ι : Type u_1\ninst✝ : Fintype ι\nl : IntegrationParams\nI : Box ι\nπ₀ : Prepartition I\n⊢ (toFilteriUnion I π₀).HasBasis (fun r ↦ ∀ (c : ℝ≥0), l.RCond (r c)) fun r ↦\n {π | ∃ c, l.MemBaseSet I c (r c) π ∧ π.iUnion = π₀.iUnion}",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
... | [
"ι : Type u_1\ninst✝ : Fintype ι\nl : IntegrationParams\nI : Box ι\nπ₀ : Prepartition I\nthis :\n ∀ (c : ℝ≥0),\n (l.toFilterDistortioniUnion I c π₀).HasBasis l.RCond fun r ↦ {π | l.MemBaseSet I c r π ∧ π.iUnion = π₀.iUnion}\n⊢ (toFilteriUnion I π₀).HasBasis (fun r ↦ ∀ (c : ℝ≥0), l.RCond (r c)) fun r ↦\n {π |... | have := fun c => l.hasBasis_toFilterDistortioniUnion I c π₀ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.BoxIntegral.Partition.Filter | {
"line": 457,
"column": 2
} | {
"line": 457,
"column": 90
} | {
"line": 458,
"column": 4
} | [
{
"pp": "ι : Type u_1\ninst✝ : Fintype ι\nl : IntegrationParams\nI : Box ι\n⊢ (toFilteriUnion I ⊤).HasBasis (fun r ↦ ∀ (c : ℝ≥0), l.RCond (r c)) fun r ↦\n {π | ∃ c, l.MemBaseSet I c (r c) π ∧ π.IsPartition}",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"BoxIntegr... | [
"ι : Type u_1\ninst✝ : Fintype ι\nl : IntegrationParams\nI : Box ι\n⊢ (toFilteriUnion I ⊤).HasBasis (fun r ↦ ∀ (c : ℝ≥0), l.RCond (r c)) fun r ↦\n {π | ∃ c, l.MemBaseSet I c (r c) π ∧ π.iUnion = ↑I}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 1153,
"column": 2
} | {
"line": 1153,
"column": 13
} | {
"line": 1153,
"column": 14
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_3\ninst✝⁵ : RCLike 𝕜\nE : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : Fintype ι\ninst✝¹ : FiniteDimensional 𝕜 E\nn : ℕ\nhn : finrank 𝕜 E = n\ninst✝ : DecidableEq ι\nV : ι → Submodule 𝕜 E\nhV : IsInternal V\nhV' : OrthogonalFamily 𝕜 (... | [
"ι : Type u_1\n𝕜 : Type u_3\ninst✝⁵ : RCLike 𝕜\nE : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : Fintype ι\ninst✝¹ : FiniteDimensional 𝕜 E\nn : ℕ\nhn : finrank 𝕜 E = n\ninst✝ : DecidableEq ι\nV : ι → Submodule 𝕜 E\nhV : IsInternal V\nhV' : OrthogonalFamily 𝕜 (fun i ↦ ↥(V ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Partition.Filter | {
"line": 492,
"column": 2
} | {
"line": 492,
"column": 41
} | {
"line": 492,
"column": 42
} | [
{
"pp": "ι : Type u_1\ninst✝ : Fintype ι\nc : ℝ≥0\nl : IntegrationParams\nI : Box ι\nhc : ⊤.distortion ≤ c\nr : (ι → ℝ) → ↑(Set.Ioi 0)\nhc' : ⊤.compl.distortion ≤ c\n⊢ ∃ π, l.MemBaseSet I c r π ∧ π.IsPartition",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
... | [
"ι : Type u_1\ninst✝ : Fintype ι\nc : ℝ≥0\nl : IntegrationParams\nI : Box ι\nhc : ⊤.distortion ≤ c\nr : (ι → ℝ) → ↑(Set.Ioi 0)\nhc' : ⊤.compl.distortion ≤ c\n⊢ ∃ π, l.MemBaseSet I c r π ∧ π.iUnion = ↑I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Partition.Filter | {
"line": 506,
"column": 2
} | {
"line": 506,
"column": 13
} | {
"line": 506,
"column": 14
} | [
{
"pp": "ι : Type u_1\ninst✝ : Fintype ι\nI✝ J : Box ι\nc c₁ c₂ : ℝ≥0\nl✝ l₁ l₂ : IntegrationParams\nr₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁ π₂ : TaggedPrepartition I✝\nr : (ι → ℝ) → ↑(Set.Ioi 0)\nl : IntegrationParams\nI : Box ι\n⊢ (l.toFilterDistortion I I.distortion).NeBot",
"ppTerm": "?m.21",
"assigned... | [
"ι : Type u_1\ninst✝ : Fintype ι\nI✝ J : Box ι\nc c₁ c₂ : ℝ≥0\nl✝ l₁ l₂ : IntegrationParams\nr₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁ π₂ : TaggedPrepartition I✝\nr : (ι → ℝ) → ↑(Set.Ioi 0)\nl : IntegrationParams\nI : Box ι\n⊢ (l.toFilterDistortion I I.distortion).NeBot"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Partition.Additive | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 16
} | {
"line": 140,
"column": 2
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\nn : ℕ\nN : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : AddCommMonoid N\nI₀✝ : WithTop (Box ι)\nI : Box ι\ni : ι\ninst✝ : Finite ι\nf : Box ι → M\nI₀ : WithTop (Box ι)\nhf :\n ∀ (I : Box ι),\n ↑I ≤ I₀ →\n ∀ {i : ι} {x : ℝ},\n x ∈ Set.Ioo (I.lower i) (I.upper ... | [
"ι : Type u_1\nM : Type u_2\nn : ℕ\nN : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : AddCommMonoid N\nI₀✝ : WithTop (Box ι)\nI : Box ι\ni : ι\ninst✝ : Finite ι\nf : Box ι → M\nI₀ : WithTop (Box ι)\nhf :\n ∀ (I : Box ι),\n ↑I ≤ I₀ →\n ∀ {i : ι} {x : ℝ},\n x ∈ Set.Ioo (I.lower i) (I.upper i) →\n ... | refine ⟨f, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 1220,
"column": 4
} | {
"line": 1220,
"column": 91
} | {
"line": 1221,
"column": 4
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\n𝕜 : Type u_3\ninst✝¹⁰ : RCLike 𝕜\nE✝ : Type u_4\ninst✝⁹ : NormedAddCommGroup E✝\ninst✝⁸ : InnerProductSpace 𝕜 E✝\nF : Type u_5\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : InnerProductSpace ℝ F\nF' : Type u_6\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : InnerProductSpace ℝ F'\n... | [
"ι : Type u_1\nι' : Type u_2\n𝕜 : Type u_3\ninst✝¹⁰ : RCLike 𝕜\nE✝ : Type u_4\ninst✝⁹ : NormedAddCommGroup E✝\ninst✝⁸ : InnerProductSpace 𝕜 E✝\nF : Type u_5\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : InnerProductSpace ℝ F\nF' : Type u_6\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : InnerProductSpace ℝ F'\ninst✝³ : Fin... | rw [norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero (L (p1 x)) (L3 (p2 x)) Mx_orth] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.BoxIntegral.Partition.Additive | {
"line": 151,
"column": 2
} | {
"line": 151,
"column": 94
} | {
"line": 152,
"column": 2
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\nn : ℕ\nN : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : AddCommMonoid N\nI₀✝ : WithTop (Box ι)\nI✝ : Box ι\ni : ι\ninst✝ : Finite ι\nf : Box ι → M\nI₀ : WithTop (Box ι)\nhf : ∀ (I : Box ι), ↑I ≤ I₀ → ∀ (s : Finset (ι × ℝ)), ∑ J ∈ (splitMany I s).boxes, f J = f I\nI : Box ι\n... | [
"ι : Type u_1\nM : Type u_2\nn : ℕ\nN : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : AddCommMonoid N\nI₀✝ : WithTop (Box ι)\nI✝ : Box ι\ni : ι\ninst✝ : Finite ι\nf : Box ι → M\nI₀ : WithTop (Box ι)\nhf : ∀ (I : Box ι), ↑I ≤ I₀ → ∀ (s : Finset (ι × ℝ)), ∑ J ∈ (splitMany I s).boxes, f J = f I\nI : Box ι\nhI : ↑I ≤ I₀... | have Hle : ∀ J ∈ π, ↑J ≤ I₀ := fun J hJ => (WithTop.coe_le_coe.2 <| π.le_of_mem hJ).trans hI | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Module.ZLattice.Basic | {
"line": 745,
"column": 20
} | {
"line": 745,
"column": 51
} | {
"line": 745,
"column": 52
} | [
{
"pp": "K : Type u_1\ninst✝⁴ : NormedField K\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace K E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace K F\nL : Submodule ℤ E\ne : F ≃ₗ[K] E\nx✝ : E\nh : x✝ ∈ L\n⊢ (↑ℤ ↑e.symm) x✝ ∈ ZLattice.comap K L ↑e",
"ppTerm": "?m.198",
... | [
"K : Type u_1\ninst✝⁴ : NormedField K\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace K E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace K F\nL : Submodule ℤ E\ne : F ≃ₗ[K] E\nx✝ : E\nh : x✝ ∈ L\n⊢ x✝ ∈ ↑L"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Partition.Additive | {
"line": 167,
"column": 2
} | {
"line": 181,
"column": 55
} | {
"line": 183,
"column": 0
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\nI₀ : WithTop (Box ι)\nI : Box ι\ninst✝ : Finite ι\nf : ι →ᵇᵃ[I₀] M\nhI : ↑I ≤ I₀\nπ₁ π₂ : Prepartition I\nh : π₁.iUnion = π₂.iUnion\n⊢ ∑ J ∈ π₁.boxes, f J = ∑ J ∈ π₂.boxes, f J",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
... | [] | rcases exists_splitMany_inf_eq_filter_of_finite {π₁, π₂} ((finite_singleton _).insert _) with
⟨s, hs⟩
simp only [inf_splitMany] at hs
rcases hs _ (Or.inl rfl), hs _ (Or.inr rfl) with ⟨h₁, h₂⟩; clear hs
rw [h] at h₁
calc
∑ J ∈ π₁.boxes, f J = ∑ J ∈ π₁.boxes, ∑ J' ∈ (splitMany J s).boxes, f J' :=
Fi... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.BoxIntegral.Partition.Additive | {
"line": 167,
"column": 2
} | {
"line": 181,
"column": 55
} | {
"line": 183,
"column": 0
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\nI₀ : WithTop (Box ι)\nI : Box ι\ninst✝ : Finite ι\nf : ι →ᵇᵃ[I₀] M\nhI : ↑I ≤ I₀\nπ₁ π₂ : Prepartition I\nh : π₁.iUnion = π₂.iUnion\n⊢ ∑ J ∈ π₁.boxes, f J = ∑ J ∈ π₂.boxes, f J",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
... | [] | rcases exists_splitMany_inf_eq_filter_of_finite {π₁, π₂} ((finite_singleton _).insert _) with
⟨s, hs⟩
simp only [inf_splitMany] at hs
rcases hs _ (Or.inl rfl), hs _ (Or.inr rfl) with ⟨h₁, h₂⟩; clear hs
rw [h] at h₁
calc
∑ J ∈ π₁.boxes, f J = ∑ J ∈ π₁.boxes, ∑ J' ∈ (splitMany J s).boxes, f J' :=
Fi... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.BoxIntegral.Integrability | {
"line": 85,
"column": 4
} | {
"line": 85,
"column": 55
} | {
"line": 85,
"column": 56
} | [
{
"pp": "ι : Type u\nE : Type v\ninst✝³ : Fintype ι\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nl : IntegrationParams\nhl : l.bRiemann = false\ns : Set (ι → ℝ)\nhs : MeasurableSet s\nI : Box ι\ny : E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nε : ℝ≥0\nε0 : 0 < ε\nA : μ (s ∩ Box.Icc I)... | [
"ι : Type u\nE : Type v\ninst✝³ : Fintype ι\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nl : IntegrationParams\nhl : l.bRiemann = false\ns : Set (ι → ℝ)\nhs : MeasurableSet s\nI : Box ι\ny : E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nε : ℝ≥0\nε0 : 0 < ε\nA : μ (s ∩ Box.Icc I) ≠ ∞\nB : μ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Integrability | {
"line": 101,
"column": 4
} | {
"line": 101,
"column": 58
} | {
"line": 101,
"column": 59
} | [
{
"pp": "case hbc.refine_2\nι : Type u\nE : Type v\ninst✝³ : Fintype ι\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nl : IntegrationParams\nhl : l.bRiemann = false\ns : Set (ι → ℝ)\nhs : MeasurableSet s\nI : Box ι\ny : E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nε : ℝ≥0\nε0 : 0 < ε\nA ... | [
"case hbc.refine_2\nι : Type u\nE : Type v\ninst✝³ : Fintype ι\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nl : IntegrationParams\nhl : l.bRiemann = false\ns : Set (ι → ℝ)\nhs : MeasurableSet s\nI : Box ι\ny : E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nε : ℝ≥0\nε0 : 0 < ε\nA : μ (s ∩ Box... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho | {
"line": 124,
"column": 25
} | {
"line": 124,
"column": 36
} | {
"line": 124,
"column": 37
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : WellFoundedLT ι\nv : ι → E\ni j : ι\nhij : i < j\nb : ι → E := gramSchmidt 𝕜 v\nk : ι\nhki' : k ∈ Iio i\n⊢ k ... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : WellFoundedLT ι\nv : ι → E\ni j : ι\nhij : i < j\nb : ι → E := gramSchmidt 𝕜 v\nk : ι\nhki' : k ∈ Iio i\n⊢ k < i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho | {
"line": 205,
"column": 4
} | {
"line": 205,
"column": 15
} | {
"line": 205,
"column": 16
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : WellFoundedLT ι\nf : ι → E\nn : ι\nh₀ : LinearIndependent 𝕜 (f ∘ Subtype.val)\nh : gramSchmidt 𝕜 f n = 0\nh₁... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : WellFoundedLT ι\nf : ι → E\nn : ι\nh₀ : LinearIndependent 𝕜 (f ∘ Subtype.val)\nh : gramSchmidt 𝕜 f n = 0\nh₁ : f n ∈ spa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho | {
"line": 257,
"column": 2
} | {
"line": 257,
"column": 13
} | {
"line": 257,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : WellFoundedLT ι\nf : ι → E\nn : ι\nhn : (↑‖gramSchmidt 𝕜 f n‖)⁻¹ • gramSchmidt 𝕜 f n ≠ 0\n⊢ gramSchmidt 𝕜 f... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : WellFoundedLT ι\nf : ι → E\nn : ι\nhn : (↑‖gramSchmidt 𝕜 f n‖)⁻¹ • gramSchmidt 𝕜 f n ≠ 0\n⊢ ¬gramSchmidt 𝕜 f n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho | {
"line": 299,
"column": 2
} | {
"line": 299,
"column": 36
} | {
"line": 299,
"column": 37
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : WellFoundedLT ι\nf : ι → E\n⊢ span 𝕜 (Set.range (gramSchmidtNormed 𝕜 f)) = span 𝕜 (Set.range (gramSchmidt �... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : WellFoundedLT ι\nf : ι → E\n⊢ span 𝕜 (gramSchmidtNormed 𝕜 f '' Set.univ) = span 𝕜 (gramSchmidt 𝕜 f '' Set.univ)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho | {
"line": 334,
"column": 2
} | {
"line": 334,
"column": 17
} | {
"line": 334,
"column": 18
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝⁴ : LinearOrder ι\ninst✝³ : LocallyFiniteOrderBot ι\ninst✝² : WellFoundedLT ι\ninst✝¹ : Fintype ι\ninst✝ : FiniteDimensional 𝕜 E\nh : finrank 𝕜 E = Fintype.card ι\nf : ι... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝⁴ : LinearOrder ι\ninst✝³ : LocallyFiniteOrderBot ι\ninst✝² : WellFoundedLT ι\ninst✝¹ : Fintype ι\ninst✝ : FiniteDimensional 𝕜 E\nh : finrank 𝕜 E = Fintype.card ι\nf : ι → E\nhf : P... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.UnitPartition | {
"line": 287,
"column": 43
} | {
"line": 292,
"column": 28
} | {
"line": 294,
"column": 0
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\nx : ℝ\na : ℤ\n⊢ ↑a < x ↔ ↑a ≤ (↑⌈↑n * x⌉ - 1) / ↑n",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Int.lt_ceil",
"Real.instIsOrderedRing",
"Int.cast",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NonAss... | [] | by
have h : 0 < (n : ℝ) := Nat.cast_pos.mpr <| n.pos_of_neZero
rw [le_div_iff₀' h, le_sub_iff_add_le,
show (n : ℝ) * a + 1 = (n * a + 1 : ℤ) by norm_cast,
Int.cast_le, Int.add_one_le_iff, Int.lt_ceil, Int.cast_mul, Int.cast_natCast,
mul_lt_mul_iff_right₀ h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho | {
"line": 362,
"column": 2
} | {
"line": 362,
"column": 49
} | {
"line": 362,
"column": 50
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝⁴ : LinearOrder ι\ninst✝³ : LocallyFiniteOrderBot ι\ninst✝² : WellFoundedLT ι\ninst✝¹ : Fintype ι\ninst✝ : FiniteDimensional 𝕜 E\nh : finrank 𝕜 E = Fintype.card ι\nf : ι... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝⁴ : LinearOrder ι\ninst✝³ : LocallyFiniteOrderBot ι\ninst✝² : WellFoundedLT ι\ninst✝¹ : Fintype ι\ninst✝ : FiniteDimensional 𝕜 E\nh : finrank 𝕜 E = Fintype.card ι\nf : ι → E\ni j : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Integrability | {
"line": 135,
"column": 2
} | {
"line": 140,
"column": 30
} | {
"line": 141,
"column": 2
} | [
{
"pp": "ι : Type u\nE : Type v\ninst✝³ : Fintype ι\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nl : IntegrationParams\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhl : l.bRiemann = false\nε : ℝ≥0\nε0 : 0 < ε\nδ : ℕ → ℝ≥0\nδ0 : ∀ (i : ℕ), 0 < δ i\nc✝ : ℝ≥0\nhδ... | [
"ι : Type u\nE : Type v\ninst✝³ : Fintype ι\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nl : IntegrationParams\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhl : l.bRiemann = false\nε : ℝ≥0\nε0 : 0 < ε\nδ : ℕ → ℝ≥0\nδ0 : ∀ (i : ℕ), 0 < δ i\nc✝ : ℝ≥0\nhδc : HasSum δ... | have : ∀ J ∈ π.filter fun J => N (π.tag J) = n,
‖μ.real ↑J • f (π.tag J)‖ ≤ μ.real J * n := fun J hJ ↦ by
rw [TaggedPrepartition.mem_filter] at hJ
rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg measureReal_nonneg]
gcongr
exact hJ.2 ▸ Nat.le_ceil _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 97,
"column": 6
} | {
"line": 97,
"column": 32
} | {
"line": 97,
"column": 32
} | [
{
"pp": "ι : Type u\nE : Type v\nF : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nI : Box ι\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nπ : Prepartition I\nπi : (J : Box ι) → TaggedPrepartition J\nJ : Box ι\nhJ : J ∈ π.boxes\nJ' : Box ι... | [
"ι : Type u\nE : Type v\nF : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nI : Box ι\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nπ : Prepartition I\nπi : (J : Box ι) → TaggedPrepartition J\nJ : Box ι\nhJ : J ∈ π.boxes\nJ' : Box ι\nhJ' : J' ∈... | π.tag_biUnionTagged hJ hJ' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Orientation | {
"line": 220,
"column": 2
} | {
"line": 220,
"column": 13
} | {
"line": 220,
"column": 14
} | [
{
"pp": "case h\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_3\ninst✝ : IsEmpty ι\nx : M [⋀^ι]→ₗ[R] R\nhx : x ≠ 0\nh : LinearIndependent R ![x, AlternatingMap.constLinearEquivOfIsEmpty 1]\nf : M... | [
"case h\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_3\ninst✝ : IsEmpty ι\nx : M [⋀^ι]→ₗ[R] R\nhx : x ≠ 0\nh : LinearIndependent R ![x, AlternatingMap.constLinearEquivOfIsEmpty 1]\nf : M [⋀^ι]→ₗ[R] ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 260,
"column": 2
} | {
"line": 260,
"column": 51
} | {
"line": 260,
"column": 52
} | [
{
"pp": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI : Box ι\ninst✝ : Fintype ι\nl : IntegrationParams\nf g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\nh : HasIntegral I l f vol y\nh' : HasIntegral... | [
"ι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI : Box ι\ninst✝ : Fintype ι\nl : IntegrationParams\nf g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\nh : HasIntegral I l f vol y\nh' : HasIntegral I l g vol y... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 271,
"column": 2
} | {
"line": 271,
"column": 51
} | {
"line": 271,
"column": 52
} | [
{
"pp": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI : Box ι\ninst✝ : Fintype ι\nl : IntegrationParams\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny : F\nhf : HasIntegral I l f vol y\n⊢ HasIntegral I l (-... | [
"ι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI : Box ι\ninst✝ : Fintype ι\nl : IntegrationParams\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny : F\nhf : HasIntegral I l f vol y\n⊢ Tendsto (integralSum (-f) vol)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 290,
"column": 47
} | {
"line": 290,
"column": 80
} | {
"line": 290,
"column": 81
} | [
{
"pp": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI : Box ι\ninst✝ : Fintype ι\nl : IntegrationParams\nf g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\nh : HasIntegral I l f vol y\nh' : HasIntegral... | [
"ι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI : Box ι\ninst✝ : Fintype ι\nl : IntegrationParams\nf g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\nh : HasIntegral I l f vol y\nh' : HasIntegral I l g vol y... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 312,
"column": 2
} | {
"line": 312,
"column": 39
} | {
"line": 312,
"column": 40
} | [
{
"pp": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI : Box ι\ninst✝ : Fintype ι\nl : IntegrationParams\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\n⊢ HasIntegral I l (fun x ↦ 0) vol 0",
"ppTerm": "?m.33",
"assigned... | [
"ι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI : Box ι\ninst✝ : Fintype ι\nl : IntegrationParams\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\n⊢ HasIntegral I l (fun x ↦ 0) vol ((vol I) 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 332,
"column": 2
} | {
"line": 332,
"column": 52
} | {
"line": 333,
"column": 4
} | [
{
"pp": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI : Box ι\ninst✝ : Fintype ι\nl : IntegrationParams\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny : F\nhf : HasIntegral I l f vol y\nc : ℝ\n⊢ HasIntegral... | [
"ι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI : Box ι\ninst✝ : Fintype ι\nl : IntegrationParams\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny : F\nhf : HasIntegral I l f vol y\nc : ℝ\n⊢ Tendsto (integralSum (c... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Integrability | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 13
} | {
"line": 163,
"column": 14
} | [
{
"pp": "ι : Type u\nE : Type v\ninst✝³ : Fintype ι\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nl : IntegrationParams\nI : Box ι\ny : E\nf g : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhf : HasIntegral I l f μ.toBoxAdditive.toSMul y\nhfg : f =ᵐ[μ.restrict ↑I] g\nhl : l.bR... | [
"ι : Type u\nE : Type v\ninst✝³ : Fintype ι\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nl : IntegrationParams\nI : Box ι\ny : E\nf g : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhf : HasIntegral I l f μ.toBoxAdditive.toSMul y\nhfg : f =ᵐ[μ.restrict ↑I] g\nhl : l.bRiemann = fal... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 340,
"column": 2
} | {
"line": 340,
"column": 33
} | {
"line": 340,
"column": 34
} | [
{
"pp": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI : Box ι\ninst✝ : Fintype ι\nl : IntegrationParams\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nc : ℝ\nhf : Integrable I l (c • f) vol\nhc : c ≠ 0\n⊢ Int... | [
"ι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI : Box ι\ninst✝ : Fintype ι\nl : IntegrationParams\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nc : ℝ\nhf : Integrable I l (c • f) vol\nhc : c ≠ 0\n⊢ Integrable I l ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 468,
"column": 2
} | {
"line": 468,
"column": 39
} | {
"line": 468,
"column": 40
} | [
{
"pp": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI : Box ι\ninst✝ : Fintype ι\nl : IntegrationParams\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nc₁ c₂ : ℝ≥0\nε₁ ε₂ : ℝ\nπ₁ π₂ : TaggedPrepartition I\nh :... | [
"ι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI : Box ι\ninst✝ : Fintype ι\nl : IntegrationParams\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nc₁ c₂ : ℝ≥0\nε₁ ε₂ : ℝ\nπ₁ π₂ : TaggedPrepartition I\nh : Integrable ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orientation | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 15
} | {
"line": 61,
"column": 16
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\ne f : OrthonormalBasis ι ℝ E\nh : 0 < e.toBasis.det ⇑f.toBasis\n⊢ 0 < e.toBasis.det ⇑f",
"ppTerm": "?m.126",
"assigned": false,
"usedConstants": [],
"use... | [
"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\ne f : OrthonormalBasis ι ℝ E\nh : 0 < e.toBasis.det ⇑f.toBasis\n⊢ 0 < e.toBasis.det ⇑f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orientation | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 13
} | {
"line": 103,
"column": 14
} | [
{
"pp": "E : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace ℝ E\nι : Type u_2\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ne : OrthonormalBasis ι ℝ E\nx : Orientation ℝ E ι\ninst✝ : Nonempty ι\n⊢ ∀ (i : ι), (e.toBasis.adjustToOrientation x) i = e i ∨ (e.toBasis.adjustToOrientation x) i = -e... | [
"E : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace ℝ E\nι : Type u_2\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ne : OrthonormalBasis ι ℝ E\nx : Orientation ℝ E ι\ninst✝ : Nonempty ι\n⊢ ∀ (i : ι), (e.toBasis.adjustToOrientation x) i = e i ∨ (e.toBasis.adjustToOrientation x) i = -e i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orientation | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 47
} | {
"line": 126,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace ℝ E\nι : Type u_2\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ne : OrthonormalBasis ι ℝ E\nx : Orientation ℝ E ι\ninst✝ : Nonempty ι\ni : ι\n⊢ (e.adjustToOrientation x) i = e i ∨ (e.adjustToOrientation x) i = -e i",
"ppTerm": "... | [
"E : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace ℝ E\nι : Type u_2\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ne : OrthonormalBasis ι ℝ E\nx : Orientation ℝ E ι\ninst✝ : Nonempty ι\ni : ι\n⊢ (e.adjustToOrientation x) i = e i ∨ (e.adjustToOrientation x) i = -e i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orientation | {
"line": 192,
"column": 2
} | {
"line": 192,
"column": 13
} | {
"line": 193,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\n_i : Fact (finrank ℝ E = 0)\nh : SameRay ℝ (AlternatingMap.constLinearEquivOfIsEmpty 1) (-AlternatingMap.constLinearEquivOfIsEmpty 1)\n⊢ False",
"ppTerm": "?m.90",
"assigned": false,
"usedConstants": [],
"usedFV... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\n_i : Fact (finrank ℝ E = 0)\nh : SameRay ℝ (AlternatingMap.constLinearEquivOfIsEmpty 1) (-AlternatingMap.constLinearEquivOfIsEmpty 1)\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orientation | {
"line": 248,
"column": 55
} | {
"line": 248,
"column": 66
} | {
"line": 248,
"column": 67
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nn : ℕ\n_i : Fact (finrank ℝ E = n + 1)\no : Orientation ℝ E (Fin (n + 1))\nv : Fin (n + 1) → E\nthis : FiniteDimensional ℝ E\n⊢ finrank ℝ E = Fintype.card (Fin n.succ)",
"ppTerm": "?m.93",
"assigned": true,
"usedCon... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nn : ℕ\n_i : Fact (finrank ℝ E = n + 1)\no : Orientation ℝ E (Fin (n + 1))\nv : Fin (n + 1) → E\nthis : FiniteDimensional ℝ E\n⊢ finrank ℝ E = n + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orientation | {
"line": 270,
"column": 60
} | {
"line": 270,
"column": 71
} | {
"line": 270,
"column": 72
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nn : ℕ\n_i : Fact (finrank ℝ E = n + 1)\no : Orientation ℝ E (Fin (n + 1))\nv : Fin (n + 1) → E\nhv : Pairwise fun i j ↦ ⟪v i, v j⟫ = 0\nthis : FiniteDimensional ℝ E\n⊢ finrank ℝ E = Fintype.card (Fin n.succ)",
"ppTerm": "?m... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nn : ℕ\n_i : Fact (finrank ℝ E = n + 1)\no : Orientation ℝ E (Fin (n + 1))\nv : Fin (n + 1) → E\nhv : Pairwise fun i j ↦ ⟪v i, v j⟫ = 0\nthis : FiniteDimensional ℝ E\n⊢ finrank ℝ E = n + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orientation | {
"line": 290,
"column": 2
} | {
"line": 290,
"column": 40
} | {
"line": 290,
"column": 41
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nn : ℕ\n_i : Fact (finrank ℝ E = n)\no : Orientation ℝ E (Fin n)\nv : OrthonormalBasis (Fin n) ℝ E\n⊢ |o.volumeForm ⇑v| = 1",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"AlternatingMap",
"Eq.... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nn : ℕ\n_i : Fact (finrank ℝ E = n)\no : Orientation ℝ E (Fin n)\nv : OrthonormalBasis (Fin n) ℝ E\n⊢ |v.toBasis.det ⇑v| = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 679,
"column": 4
} | {
"line": 679,
"column": 75
} | {
"line": 679,
"column": 76
} | [
{
"pp": "ι : Type u\nE : Type v\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : Fintype ι\nl : IntegrationParams\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box... | [
"ι : Type u\nE : Type v\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : Fintype ι\nl : IntegrationParams\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace | {
"line": 231,
"column": 31
} | {
"line": 231,
"column": 42
} | {
"line": 231,
"column": 43
} | [
{
"pp": "E : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\ninst✝ : FiniteDimensional ℝ E\nh : finrank ℝ E = 1\nv : E\nhv : v ≠ 0\n⊢ ‖v‖⁻¹ ≠ 0",
"ppTerm": "?m.478",
"assigned": true,
"usedConstants": [
"AddGroup.toSub... | [
"E : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\ninst✝ : FiniteDimensional ℝ E\nh : finrank ℝ E = 1\nv : E\nhv : v ≠ 0\n⊢ ¬v = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Integrability | {
"line": 297,
"column": 6
} | {
"line": 297,
"column": 98
} | {
"line": 297,
"column": 98
} | [
{
"pp": "case refine_3\nι : Type u\nE : Type v\ninst✝⁴ : Fintype ι\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nI : Box ι\nl : IntegrationParams\nhl : l.bRiemann = false\nthis✝¹ : MeasurableSpace E := borel E\nthis✝ : ... | [
"case refine_3\nι : Type u\nE : Type v\ninst✝⁴ : Fintype ι\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nI : Box ι\nl : IntegrationParams\nhl : l.bRiemann = false\nthis✝¹ : MeasurableSpace E := borel E\nthis✝ : BorelSpace E... | integral_biUnion_finset π.boxes (fun J _ => J.measurableSet_coe) π.pairwiseDisjoint (hfgi _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SumOverResidueClass | {
"line": 42,
"column": 55
} | {
"line": 42,
"column": 77
} | {
"line": 42,
"column": 78
} | [
{
"pp": "R : Type u_1\ninst✝² : AddCommGroup R\ninst✝¹ : TopologicalSpace R\ninst✝ : IsTopologicalAddGroup R\nm✝ : ℕ\nhm : NeZero m✝\nk : ℕ\nf : ℕ → R\ng : ℕ → ℕ := fun n ↦ m✝ * n + k\nm n : ℕ\nhmn : g m = g n\n⊢ m = n",
"ppTerm": "?m.69",
"assigned": false,
"usedConstants": [],
"usedFVars": [],... | [
"R : Type u_1\ninst✝² : AddCommGroup R\ninst✝¹ : TopologicalSpace R\ninst✝ : IsTopologicalAddGroup R\nm✝ : ℕ\nhm : NeZero m✝\nk : ℕ\nf : ℕ → R\ng : ℕ → ℕ := fun n ↦ m✝ * n + k\nm n : ℕ\nhmn : g m = g n\n⊢ m = n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.SpecificAsymptotics | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 23
} | {
"line": 45,
"column": 24
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : NormMulClass R\np q : ℕ\nhpq : p < q\n⊢ (fun x ↦ x ^ p) =o[cobounded R] fun x ↦ x ^ (p + (q - p))",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedRing.toRing",
"HMul.hMul",
"PseudoMetricSpace.toB... | [
"R : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : NormMulClass R\np q : ℕ\nhpq : p < q\n⊢ (fun x ↦ x ^ p) =o[cobounded R] fun x ↦ x ^ p * x ^ (q - p)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.SpecificAsymptotics | {
"line": 145,
"column": 4
} | {
"line": 145,
"column": 42
} | {
"line": 145,
"column": 43
} | [
{
"pp": "α : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n ↦ ∑ i ∈ Finset.range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i ∈ Finset.range n, g i‖ = ∑ i ∈ Finset.range n, g i\nε : ℝ\nεpos : 0 < ε\n⊢ ∃ N, ∀ (b : ℕ), N ≤ ... | [
"α : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n ↦ ∑ i ∈ Finset.range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i ∈ Finset.range n, g i‖ = ∑ i ∈ Finset.range n, g i\nε : ℝ\nεpos : 0 < ε\n⊢ ∃ N, ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.PSeries | {
"line": 90,
"column": 95
} | {
"line": 103,
"column": 8
} | {
"line": 105,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝² : AddCommMonoid M\ninst✝¹ : PartialOrder M\ninst✝ : IsOrderedAddMonoid M\nf : ℕ → M\nu : ℕ → ℕ\nhf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m\nh_pos : ∀ (n : ℕ), 0 < u n\nhu : Monotone u\nn : ℕ\n⊢ ∑ k ∈ range n, (u (k + 1) - u k) • f (u (k + 1)) ≤ ∑ k ∈ Ico (u 0 + 1) (u n + 1), f k",... | [] | by
induction n with
| zero => simp
| succ n ihn =>
suffices (u (n + 1) - u n) • f (u (n + 1)) ≤ ∑ k ∈ Ico (u n + 1) (u (n + 1) + 1), f k by
rw [sum_range_succ, ← sum_Ico_consecutive]
exacts [add_le_add ihn this,
(add_le_add_left (hu n.zero_le) _ : u 0 + 1 ≤ u n + 1),
add_le_add_lef... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Asymptotics.SpecificAsymptotics | {
"line": 242,
"column": 4
} | {
"line": 242,
"column": 15
} | {
"line": 242,
"column": 16
} | [
{
"pp": "case h\nE : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\nD : Type u_2\ninst✝ : TopologicalSpace D\nf : D → E\nhf : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\nc : ℝ\nhc : ∀ (a : D), ‖f a‖ ≤ c\n⊢ ∀ (x : D), ‖f x‖ ≤ c * ‖1 x‖",
"ppTerm": "?h",
"assigned": true,
"... | [
"case h\nE : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\nD : Type u_2\ninst✝ : TopologicalSpace D\nf : D → E\nhf : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\nc : ℝ\nhc : ∀ (a : D), ‖f a‖ ≤ c\n⊢ ∀ (x : D), ‖f x‖ ≤ c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.SpecificAsymptotics | {
"line": 253,
"column": 6
} | {
"line": 253,
"column": 17
} | {
"line": 253,
"column": 18
} | [
{
"pp": "case mpr.ht\nE : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\nD : Type u_2\ninst✝ : TopologicalSpace D\nf : D → E\nhf : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\nc : ℝ\nt : Set D\nhcompact : IsCompact t\nh : tᶜ ⊆ {x | ‖f x‖ ≤ c}\ny : D\nhy : y ∈ tᶜ\n⊢ ‖f y‖ ≤ c",
"pp... | [
"case mpr.ht\nE : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\nD : Type u_2\ninst✝ : TopologicalSpace D\nf : D → E\nhf : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\nc : ℝ\nt : Set D\nhcompact : IsCompact t\nh : tᶜ ⊆ {x | ‖f x‖ ≤ c}\ny : D\nhy : y ∈ tᶜ\n⊢ ‖f y‖ ≤ c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.PSeries | {
"line": 170,
"column": 2
} | {
"line": 170,
"column": 13
} | {
"line": 170,
"column": 14
} | [
{
"pp": "u : ℕ → ℕ\nf : ℕ → ℝ≥0∞\nC : ℕ\nhf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m\nh_pos : ∀ (n : ℕ), 0 < u n\nh_nonneg : ∀ (n : ℕ), 0 ≤ f n\nhu : Monotone u\nh_succ_diff : SuccDiffBounded C u\nn : ℕ\n⊢ ∑ a ∈ range n.succ, (↑(u (a + 1)) - ↑(u a)) * f (u a) ≤\n (↑(u 1) - ↑(u 0)) * f (u 0) + ↑C * ∑ x ∈ Ico ... | [
"u : ℕ → ℕ\nf : ℕ → ℝ≥0∞\nC : ℕ\nhf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m\nh_pos : ∀ (n : ℕ), 0 < u n\nh_nonneg : ∀ (n : ℕ), 0 ≤ f n\nhu : Monotone u\nh_succ_diff : SuccDiffBounded C u\nn : ℕ\n⊢ ∑ a ∈ range (n + 1), (↑(u (a + 1)) - ↑(u a)) * f (u a) ≤\n (↑(u 1) - ↑(u 0)) * f (u 0) + ↑C * ∑ x ∈ Ico (u 0 + 1) (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.PSeries | {
"line": 177,
"column": 2
} | {
"line": 177,
"column": 13
} | {
"line": 177,
"column": 14
} | [
{
"pp": "f : ℕ → ℝ≥0∞\nhf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m\nn : ℕ\n⊢ ∑ a ∈ range n.succ, 2 ^ a * f (2 ^ a) ≤ f 1 + 2 • ∑ x ∈ Ico 2 (2 ^ n + 1), f x",
"ppTerm": "?m.160",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"ENNReal.instA... | [
"f : ℕ → ℝ≥0∞\nhf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m\nn : ℕ\n⊢ ∑ a ∈ range (n + 1), 2 ^ a * f (2 ^ a) ≤ f 1 + 2 * ∑ x ∈ Ico 2 (2 ^ n + 1), f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 716,
"column": 6
} | {
"line": 716,
"column": 75
} | {
"line": 716,
"column": 76
} | [
{
"pp": "case hbc\nι : Type u\nE : Type v\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : Fintype ι\nl : IntegrationParams\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithi... | [
"case hbc\nι : Type u\nE : Type v\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : Fintype ι\nl : IntegrationParams\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.PSeries | {
"line": 199,
"column": 4
} | {
"line": 199,
"column": 64
} | {
"line": 199,
"column": 65
} | [
{
"pp": "case mp\nC : ℕ\nu : ℕ → ℕ\nf : ℕ → ℝ≥0\nh_pos : ∀ (n : ℕ), 0 < u n\nhu_strict : StrictMono u\nhC_nonzero : C ≠ 0\nh_succ_diff : SuccDiffBounded C u\nh : ∑' (b : ℕ), ↑(↑(u (b + 1)) - ↑(u b)) * ↑(f (u b)) = ∞\nhf : ∀ (m n : ℕ), 1 < m → m ≤ n → ↑(f n) ≤ ↑(f m)\nh_nonneg : ∀ (n : ℕ), 0 ≤ ↑(f n)\nhC : ∑' (k... | [
"case mp\nC : ℕ\nu : ℕ → ℕ\nf : ℕ → ℝ≥0\nh_pos : ∀ (n : ℕ), 0 < u n\nhu_strict : StrictMono u\nhC_nonzero : C ≠ 0\nh_succ_diff : SuccDiffBounded C u\nh : ∑' (b : ℕ), ↑(↑(u (b + 1)) - ↑(u b)) * ↑(f (u b)) = ∞\nhf : ∀ (m n : ℕ), 1 < m → m ≤ n → ↑(f n) ≤ ↑(f m)\nh_nonneg : ∀ (n : ℕ), 0 ≤ ↑(f n)\nhC : ∑' (k : ℕ), (↑(u ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.PSeries | {
"line": 200,
"column": 4
} | {
"line": 201,
"column": 38
} | {
"line": 202,
"column": 4
} | [
{
"pp": "case mpr\nC : ℕ\nu : ℕ → ℕ\nf : ℕ → ℝ≥0\nhf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m\nh_pos : ∀ (n : ℕ), 0 < u n\nhu_strict : StrictMono u\nhC_nonzero : C ≠ 0\nh_succ_diff : SuccDiffBounded C u\nh : ∑' (b : ℕ), ↑(f b) = ∞\n⊢ ∑' (b : ℕ), ↑(↑(u (b + 1)) - ↑(u b)) * ↑(f (u b)) = ∞",
"ppTerm": "?mpr",
... | [
"case mpr\nC : ℕ\nu : ℕ → ℕ\nf : ℕ → ℝ≥0\nh_pos : ∀ (n : ℕ), 0 < u n\nhu_strict : StrictMono u\nhC_nonzero : C ≠ 0\nh_succ_diff : SuccDiffBounded C u\nh : ∑' (b : ℕ), ↑(f b) = ∞\nhf : ∀ (m n : ℕ), 0 < m → m ≤ n → ↑(f n) ≤ ↑(f m)\n⊢ ∑' (b : ℕ), ↑(↑(u (b + 1)) - ↑(u b)) * ↑(f (u b)) = ∞"
] | replace hf : ∀ m n, 0 < m → m ≤ n → (f n : ℝ≥0∞) ≤ f m := fun m n hm hmn =>
ENNReal.coe_le_coe.2 (hf hm hmn) | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.Analysis.PSeries | {
"line": 203,
"column": 4
} | {
"line": 203,
"column": 37
} | {
"line": 203,
"column": 38
} | [
{
"pp": "case mpr\nC : ℕ\nu : ℕ → ℕ\nf : ℕ → ℝ≥0\nh_pos : ∀ (n : ℕ), 0 < u n\nhu_strict : StrictMono u\nhC_nonzero : C ≠ 0\nh_succ_diff : SuccDiffBounded C u\nh : ∑' (b : ℕ), ↑(f b) = ∞\nhf : ∀ (m n : ℕ), 0 < m → m ≤ n → ↑(f n) ≤ ↑(f m)\nthis : ∑ k ∈ range (u 0), ↑(f k) ≠ ∞\n⊢ ∑' (b : ℕ), ↑(↑(u (b + 1)) - ↑(u b... | [
"case mpr\nC : ℕ\nu : ℕ → ℕ\nf : ℕ → ℝ≥0\nh_pos : ∀ (n : ℕ), 0 < u n\nhu_strict : StrictMono u\nhC_nonzero : C ≠ 0\nh_succ_diff : SuccDiffBounded C u\nh : ∑' (b : ℕ), ↑(f b) = ∞\nhf : ∀ (m n : ℕ), 0 < m → m ≤ n → ↑(f n) ≤ ↑(f m)\nthis : ∑ k ∈ range (u 0), ↑(f k) ≠ ∞\n⊢ ∑' (b : ℕ), (↑(u (b + 1)) - ↑(u b)) * ↑(f (u b... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.PSeries | {
"line": 307,
"column": 4
} | {
"line": 308,
"column": 27
} | {
"line": 308,
"column": 28
} | [
{
"pp": "case inr\np : ℝ\nhp : p < 0\nh : Summable fun n ↦ (↑n ^ p)⁻¹\nk : ℕ\nhk₁ : (↑k ^ p)⁻¹ < 1\nhk₀ : 0 < ↑k\n⊢ k = 0",
"ppTerm": "?inr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case inr\np : ℝ\nhp : p < 0\nh : Summable fun n ↦ (↑n ^ p)⁻¹\nk : ℕ\nhk₁ : (↑k ^ p)⁻¹ < 1\nhk₀ : 0 < ↑k\n⊢ k = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.PSeries | {
"line": 358,
"column": 2
} | {
"line": 358,
"column": 35
} | {
"line": 358,
"column": 36
} | [
{
"pp": "⊢ ¬Summable fun n ↦ 1 / ↑n",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ ¬Summable fun n ↦ 1 / ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.PSeries | {
"line": 402,
"column": 29
} | {
"line": 402,
"column": 40
} | {
"line": 402,
"column": 41
} | [
{
"pp": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\nk n✝ : ℕ\nhk : k ≠ 0\nh : k ≤ n✝\nn : ℕ\nhn : k ≤ n\nIH : ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹\n⊢ 0 < ↑n",
"ppTerm": "?m.108",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.t... | [
"α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\nk n✝ : ℕ\nhk : k ≠ 0\nh : k ≤ n✝\nn : ℕ\nhn : k ≤ n\nIH : ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹\n⊢ 0 < n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.PSeries | {
"line": 428,
"column": 6
} | {
"line": 428,
"column": 17
} | {
"line": 428,
"column": 18
} | [
{
"pp": "case hdb\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\nk n : ℕ\nA : 1 ≤ ↑k + 1\n⊢ ↑k + 1 ≤ (↑k + 1) ^ 2",
"ppTerm": "?hdb",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case hdb\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\nk n : ℕ\nA : 1 ≤ ↑k + 1\n⊢ ↑k + 1 ≤ (↑k + 1) ^ 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.PSeries | {
"line": 474,
"column": 2
} | {
"line": 476,
"column": 11
} | {
"line": 476,
"column": 12
} | [
{
"pp": "α : Type u_1\nx : α\ninst✝ : RCLike α\nq k : ℕ\nhq : 1 < q\n⊢ Summable fun x ↦ 1 / ‖(↑x + ↑k) ^ q‖",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [
"α : Type u_1\nx : α\ninst✝ : RCLike α\nq k : ℕ\nhq : 1 < q\n⊢ Summable fun x ↦ (‖↑x + ↑k‖ ^ q)⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.Grading | {
"line": 116,
"column": 6
} | {
"line": 116,
"column": 91
} | {
"line": 117,
"column": 8
} | [
{
"pp": "M : Type u_1\nι : Type u_2\nR : Type u_3\ninst✝³ : AddMonoid M\ninst✝² : DecidableEq ι\ninst✝¹ : AddMonoid ι\ninst✝ : CommSemiring R\nf : M →+ ι\ni j : Multiplicative M\n⊢ (DirectSum.of (fun i ↦ ↥(gradeBy R (⇑f) i)) (f (Multiplicative.toAdd (i * j))))\n ⟨single (Multiplicative.toAdd (i * j)) 1, ⋯⟩... | [
"M : Type u_1\nι : Type u_2\nR : Type u_3\ninst✝³ : AddMonoid M\ninst✝² : DecidableEq ι\ninst✝¹ : AddMonoid ι\ninst✝ : CommSemiring R\nf : M →+ ι\ni j : Multiplicative M\n⊢ (DirectSum.of (fun i ↦ ↥(gradeBy R (⇑f) i)) (f (Multiplicative.toAdd i + Multiplicative.toAdd j)))\n ⟨single (Multiplicative.toAdd i + Mul... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.Ideal | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 15
} | {
"line": 49,
"column": 16
} | [
{
"pp": "case mp\nk : Type u_1\nG : Type u_3\ninst✝¹ : Monoid G\ninst✝ : Semiring k\ns : Set G\nx : k[G]\nRHS : Ideal k[G] :=\n { carrier := {p | ∀ m ∈ p.coeff.support, ∃ m' ∈ s, ∃ d, m = d * m'}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }\ni : G\nhi : i ∈ s\nm : G\nhm : m ∈ ((of k G) i).coeff.support\n⊢ ... | [
"case mp\nk : Type u_1\nG : Type u_3\ninst✝¹ : Monoid G\ninst✝ : Semiring k\ns : Set G\nx : k[G]\nRHS : Ideal k[G] :=\n { carrier := {p | ∀ m ∈ p.coeff.support, ∃ m' ∈ s, ∃ d, m = d * m'}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }\ni : G\nhi : i ∈ s\nm : G\nhm : m ∈ ((of k G) i).coeff.support\n⊢ m = i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.Ideal | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 15
} | {
"line": 54,
"column": 16
} | [
{
"pp": "k : Type u_1\nG : Type u_3\ninst✝¹ : Monoid G\ninst✝ : Semiring k\ns : Set G\nx : k[G]\nRHS : Ideal k[G] :=\n { carrier := {p | ∀ m ∈ p.coeff.support, ∃ m' ∈ s, ∃ d, m = d * m'}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }\nhx : x ∈ RHS\nd : G\nhd : d ∈ s\nd2 : G\nhi : d2 * d ∈ x.coeff.support\n⊢ ... | [
"k : Type u_1\nG : Type u_3\ninst✝¹ : Monoid G\ninst✝ : Semiring k\ns : Set G\nx : k[G]\nRHS : Ideal k[G] :=\n { carrier := {p | ∀ m ∈ p.coeff.support, ∃ m' ∈ s, ∃ d, m = d * m'}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }\nhx : x ∈ RHS\nd : G\nhd : d ∈ s\nd2 : G\nhi : d2 * d ∈ x.coeff.support\n⊢ single (d2 *... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.Ideal | {
"line": 31,
"column": 2
} | {
"line": 55,
"column": 54
} | {
"line": 57,
"column": 0
} | [
{
"pp": "k : Type u_1\nG : Type u_3\ninst✝¹ : Monoid G\ninst✝ : Semiring k\ns : Set G\nx : k[G]\n⊢ x ∈ Ideal.span (⇑(of k G) '' s) ↔ ∀ m ∈ x.coeff.support, ∃ m' ∈ s, ∃ d, m = d * m'",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"MonoidAlgebra.semiring",
"Ideal.span_le",
... | [] | let RHS : Ideal (MonoidAlgebra k G) :=
{ carrier := { p | ∀ m : G, m ∈ p.coeff.support → ∃ m' ∈ s, ∃ d, m = d * m' }
add_mem' {x y} hx hy m hm := (Finset.mem_union.1 <| Finsupp.support_add hm).elim (hx m) (hy m)
zero_mem' := by simp
smul_mem' x y hy m hm := by
simp only [smul_eq_mul, mul_d... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MonoidAlgebra.Ideal | {
"line": 31,
"column": 2
} | {
"line": 55,
"column": 54
} | {
"line": 57,
"column": 0
} | [
{
"pp": "k : Type u_1\nG : Type u_3\ninst✝¹ : Monoid G\ninst✝ : Semiring k\ns : Set G\nx : k[G]\n⊢ x ∈ Ideal.span (⇑(of k G) '' s) ↔ ∀ m ∈ x.coeff.support, ∃ m' ∈ s, ∃ d, m = d * m'",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"MonoidAlgebra.semiring",
"Ideal.span_le",
... | [] | let RHS : Ideal (MonoidAlgebra k G) :=
{ carrier := { p | ∀ m : G, m ∈ p.coeff.support → ∃ m' ∈ s, ∃ d, m = d * m' }
add_mem' {x y} hx hy m hm := (Finset.mem_union.1 <| Finsupp.support_add hm).elim (hx m) (hy m)
zero_mem' := by simp
smul_mem' x y hy m hm := by
simp only [smul_eq_mul, mul_d... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MonoidAlgebra.Ideal | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 15
} | {
"line": 87,
"column": 16
} | [
{
"pp": "case mp\nk : Type u_1\nA : Type u_2\ninst✝¹ : AddMonoid A\ninst✝ : Semiring k\ns : Set A\nx : k[A]\nRHS : Ideal k[A] :=\n { carrier := {p | ∀ m ∈ p.coeff.support, ∃ m' ∈ s, ∃ d, m = d + m'}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }\ni : A\nhi : i ∈ s\nm : A\nhm : m ∈ (of' k A i).coeff.support\n... | [
"case mp\nk : Type u_1\nA : Type u_2\ninst✝¹ : AddMonoid A\ninst✝ : Semiring k\ns : Set A\nx : k[A]\nRHS : Ideal k[A] :=\n { carrier := {p | ∀ m ∈ p.coeff.support, ∃ m' ∈ s, ∃ d, m = d + m'}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }\ni : A\nhi : i ∈ s\nm : A\nhm : m ∈ (of' k A i).coeff.support\n⊢ m = i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.Ideal | {
"line": 92,
"column": 4
} | {
"line": 92,
"column": 15
} | {
"line": 92,
"column": 16
} | [
{
"pp": "k : Type u_1\nA : Type u_2\ninst✝¹ : AddMonoid A\ninst✝ : Semiring k\ns : Set A\nx : k[A]\nRHS : Ideal k[A] :=\n { carrier := {p | ∀ m ∈ p.coeff.support, ∃ m' ∈ s, ∃ d, m = d + m'}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }\nhx : x ∈ RHS\nd : A\nhd : d ∈ s\nd2 : A\nhi : d2 + d ∈ x.coeff.support\... | [
"k : Type u_1\nA : Type u_2\ninst✝¹ : AddMonoid A\ninst✝ : Semiring k\ns : Set A\nx : k[A]\nRHS : Ideal k[A] :=\n { carrier := {p | ∀ m ∈ p.coeff.support, ∃ m' ∈ s, ∃ d, m = d + m'}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }\nhx : x ∈ RHS\nd : A\nhd : d ∈ s\nd2 : A\nhi : d2 + d ∈ x.coeff.support\n⊢ single (d... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.ZLattice.Summable | {
"line": 127,
"column": 57
} | {
"line": 127,
"column": 72
} | {
"line": 127,
"column": 73
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝² : DiscreteTopology ↥L\nι : Type u_3\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nb : Basis ι ℤ ↥L\nn : ℕ\nr : ℝ\ns : ℕ → Finset (ι → ℤ) := fun n ↦ Fintype.piFinset fun i ↦ Icc... | [
"E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝² : DiscreteTopology ↥L\nι : Type u_3\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nb : Basis ι ℤ ↥L\nn : ℕ\nr : ℝ\ns : ℕ → Finset (ι → ℤ) := fun n ↦ Fintype.piFinset fun i ↦ Icc (-↑n) ↑n\nd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.ZLattice.Summable | {
"line": 129,
"column": 35
} | {
"line": 129,
"column": 69
} | {
"line": 129,
"column": 70
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝² : DiscreteTopology ↥L\nι : Type u_3\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nb : Basis ι ℤ ↥L\nn : ℕ\nr : ℝ\ns : ℕ → Finset (ι → ℤ) := fun n ↦ Fintype.piFinset fun i ↦ Icc... | [
"E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝² : DiscreteTopology ↥L\nι : Type u_3\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nb : Basis ι ℤ ↥L\nn : ℕ\nr : ℝ\ns : ℕ → Finset (ι → ℤ) := fun n ↦ Fintype.piFinset fun i ↦ Icc (-↑n) ↑n\nd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.ZLattice.Summable | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 64
} | {
"line": 165,
"column": 65
} | [
{
"pp": "case inl\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝ : DiscreteTopology ↥L\nh✝ : Subsingleton ↥L\nr : ℝ\nhr✝ : r < -↑(finrank ℤ ↥L)\ns : Finset ↥L\nhr : r ≠ 0\n⊢ ∑ z ∈ s, ‖z‖ ^ r ≤ 1 ^ r * ∑' (k : ℕ), ↑k ^ (↑(finrank ℤ... | [
"case inl\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝ : DiscreteTopology ↥L\nh✝ : Subsingleton ↥L\nr : ℝ\nhr✝ : r < -↑(finrank ℤ ↥L)\ns : Finset ↥L\nhr : r ≠ 0\n⊢ 0 ≤ ∑' (k : ℕ), ↑k ^ (↑(finrank ℤ ↥L) - 1 + r)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.ZLattice.Summable | {
"line": 180,
"column": 12
} | {
"line": 180,
"column": 23
} | {
"line": 180,
"column": 24
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝ : DiscreteTopology ↥L\nh✝ : Nontrivial ↥L\nI : Type u_1 := Free.ChooseBasisIndex ℤ ↥L\nthis : Fintype I\nb : Basis I ℤ ↥L := Free.chooseBasis ℤ ↥L\nd : ℕ := Fintype.card I\nhd... | [
"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝ : DiscreteTopology ↥L\nh✝ : Nontrivial ↥L\nI : Type u_1 := Free.ChooseBasisIndex ℤ ↥L\nthis : Fintype I\nb : Basis I ℤ ↥L := Free.chooseBasis ℤ ↥L\nd : ℕ := Fintype.card I\nhd : d ≠ 0\nε ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.ZLattice.Summable | {
"line": 186,
"column": 6
} | {
"line": 186,
"column": 39
} | {
"line": 187,
"column": 6
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝ : DiscreteTopology ↥L\nh✝ : Nontrivial ↥L\nI : Type u_1 := Free.ChooseBasisIndex ℤ ↥L\nthis : Fintype I\nb : Basis I ℤ ↥L := Free.chooseBasis ℤ ↥L\nd : ℕ := Fintype.card I\nhd... | [
"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝ : DiscreteTopology ↥L\nh✝ : Nontrivial ↥L\nI : Type u_1 := Free.ChooseBasisIndex ℤ ↥L\nthis : Fintype I\nb : Basis I ℤ ↥L := Free.chooseBasis ℤ ↥L\nd : ℕ := Fintype.card I\nhd : d ≠ 0\nε ... | refine ⟨⌊r⌋.toNat, fun x hx ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Module.ZLattice.Summable | {
"line": 250,
"column": 24
} | {
"line": 250,
"column": 35
} | {
"line": 250,
"column": 36
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝ : DiscreteTopology ↥L\nr : ℝ\nhr : r < -↑(finrank ℤ ↥L)\nx : E\nh✝ : Nontrivial ↥L\nH : IsClosed ↑L\nt : ↥L\nht : t ∈ {x_1 | (fun i ↦ ‖‖↑i - x‖ ^ r‖ ≤ (1 / 2) ^ r * ‖i‖ ^ r) x... | [
"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝ : DiscreteTopology ↥L\nr : ℝ\nhr : r < -↑(finrank ℤ ↥L)\nx : E\nh✝ : Nontrivial ↥L\nH : IsClosed ↑L\nt : ↥L\nht : t ∈ {x_1 | (fun i ↦ ‖‖↑i - x‖ ^ r‖ ≤ (1 / 2) ^ r * ‖i‖ ^ r) x_1}ᶜ\nht₁ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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