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.GroupTheory.SpecificGroups.Alternating.Simple | {
"line": 94,
"column": 2
} | {
"line": 96,
"column": 68
} | [
{
"pp": "α : Type u_2\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nhα : 5 ≤ Nat.card α\nN : Subgroup (Perm α)\ninst✝ : N.Normal\nntN : Nontrivial ↥N\nthis : IsPreprimitive (Perm α) ↑(Set.powersetCard α 2)\n⊢ _root_.commutator (Perm α) ≤ N",
"usedConstants": [
"MulAction.IwasawaStructure.commutator_le"... | classical
apply iwasawaStructure_two.commutator_le
exact fixedPoints_ne_univ_of_faithfulSMul (by norm_num) (by grind) | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.GroupTheory.SpecificGroups.ZGroup | {
"line": 123,
"column": 32
} | {
"line": 123,
"column": 68
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : Finite G\ninst✝ : IsZGroup G\np : ℕ\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nP : Sylow p G := default\n⊢ Nat.card ↥↑P ∣ Monoid.exponent G",
"usedConstants": [
"Sylow.toSubgroup",
"Eq.mpr",
"Dvd.dvd",
"congrArg",
"CancelMono... | ← (isZGroup p hp P).exponent_eq_card | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.PushoutI | {
"line": 638,
"column": 4
} | {
"line": 638,
"column": 26
} | [
{
"pp": "ι : Type u_1\nG : ι → Type u_2\nH : Type u_3\ninst✝¹ : (i : ι) → Group (G i)\ninst✝ : Group H\nφ : (i : ι) → H →* G i\nhφ : ∀ (i : ι), Injective ⇑(φ i)\nw : Word G\nhw : Reduced φ w\nd : Transversal φ\nw' : NormalWord d\nhw'prod : w'.prod = ofCoprodI w.prod\nhw'map : List.map Sigma.fst w'.toList = List... | simp [NormalWord.prod] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.SpecificGroups.ZGroup | {
"line": 205,
"column": 2
} | {
"line": 210,
"column": 74
} | [
{
"pp": "case neg.inr\nG : Type u_1\ninst✝⁴ : Group G\np : ℕ\ninst✝³ : Fact (Nat.Prime p)\ninst✝² : IsCyclic G\nK : Type u_4\ninst✝¹ : Group K\ninst✝ : MulDistribMulAction K G\nhGK : (Nat.card G).Coprime (Nat.card K)\nhc : ¬Nat.card G = 0\nthis : Finite G\nϕ : K →* ZMod (Nat.card G) := MulDistribMulAction.toMon... | · obtain ⟨⟨u, v, -, hvu⟩, hu : u = ϕ k - 1⟩ := (hG k).resolve_left hk
rw [← u.intCast_zmod_cast] at hu hvu
rw [← v.intCast_zmod_cast, ← Int.cast_mul, ← Int.cast_one, ZMod.intCast_eq_intCast_iff] at hvu
refine Or.inr fun p ↦ zpow_one p ▸ ⟨k, p ^ (v.cast : ℤ), ?_⟩
rw [h (p ^ v.cast) k u.cast hu.symm, ← zp... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.Basic | {
"line": 396,
"column": 2
} | {
"line": 396,
"column": 27
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ η : Kernel α β\ns : Set α\nhs : MeasurableSet s\ninst✝ : DecidablePred fun x ↦ x ∈ s\na : α\ng : β → ℝ≥0∞\n⊢ ∫⁻ (b : β), g b ∂(piecewise hs κ η) a = if a ∈ s then ∫⁻ (b : β), g b ∂κ a else ∫⁻ (b : β), g b ∂η a",
"usedCons... | simp_rw [piecewise_apply] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Probability.Kernel.Basic | {
"line": 401,
"column": 2
} | {
"line": 401,
"column": 27
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ η : Kernel α β\ns : Set α\nhs : MeasurableSet s\ninst✝ : DecidablePred fun x ↦ x ∈ s\na : α\ng : β → ℝ≥0∞\nt : Set β\n⊢ ∫⁻ (b : β) in t, g b ∂(piecewise hs κ η) a = if a ∈ s then ∫⁻ (b : β) in t, g b ∂κ a else ∫⁻ (b : β) in t... | simp_rw [piecewise_apply] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Probability.Kernel.MeasurableLIntegral | {
"line": 80,
"column": 6
} | {
"line": 80,
"column": 31
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\ninst✝ : IsSFiniteKernel κ\nt : Set (α × β)\nht : MeasurableSet t\n⊢ Measurable fun a ↦ (κ a) (Prod.mk a ⁻¹' t)",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"Mea... | ← Kernel.kernel_sum_seq κ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Composition.MapComap | {
"line": 192,
"column": 43
} | {
"line": 195,
"column": 53
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ✝ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ✝ : MeasurableSpace γ✝\nγ : Type u_4\nδ : Type u_5\nmγ : MeasurableSpace γ\nmδ : MeasurableSpace δ\nf : β → γ\ng : γ → α\nκ : Kernel α β\ninst✝ : IsZeroOrMarkovKernel κ\nhg : Measurable g\n⊢ IsZeroOrMarkovKerne... | by
rcases eq_zero_or_isMarkovKernel κ with rfl | h
· simp only [comap_zero]; infer_instance
· have := IsMarkovKernel.comap κ hg; infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.Composition.MapComap | {
"line": 281,
"column": 2
} | {
"line": 282,
"column": 51
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nι : Type u_5\ninst✝ : Countable ι\nκ : ι → Kernel α β\n⊢ (Kernel.sum fun i ↦ prodMkRight γ (κ i)) = prodMkRight γ (Kernel.sum κ)",
"usedConstants": [
"MeasureTheory.Measure",
... | ext
simp_rw [sum_apply, prodMkRight_apply, sum_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Composition.MapComap | {
"line": 281,
"column": 2
} | {
"line": 282,
"column": 51
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nι : Type u_5\ninst✝ : Countable ι\nκ : ι → Kernel α β\n⊢ (Kernel.sum fun i ↦ prodMkRight γ (κ i)) = prodMkRight γ (Kernel.sum κ)",
"usedConstants": [
"MeasureTheory.Measure",
... | ext
simp_rw [sum_apply, prodMkRight_apply, sum_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Composition.ParallelComp | {
"line": 160,
"column": 4
} | {
"line": 160,
"column": 43
} | [
{
"pp": "case h₂\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nmδ : MeasurableSpace δ\nκ : Kernel α β\nη : Kernel γ δ\nx : α × γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α × γ\n⊢ (η a.2) Set.univ ≤ η.bound",
... | · exact measure_le_bound η a.2 Set.univ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.Composition.KernelLemmas | {
"line": 92,
"column": 2
} | {
"line": 93,
"column": 59
} | [
{
"pp": "case pos.h.h.e_f.h.e_f.h\nX : Type u_1\nY : Type u_2\nZ : Type u_3\nT : Type u_4\nmX : MeasurableSpace X\nmY : MeasurableSpace Y\nmZ : MeasurableSpace Z\nmT : MeasurableSpace T\nκ : Kernel X Y\nX' : Type u_5\nmX' : MeasurableSpace X'\nη : Kernel X' Z\ninst✝¹ : IsSFiniteKernel η\nξ : Kernel Z T\ninst✝ :... | rw [parallelComp_apply' hs, Kernel.id_apply,
lintegral_dirac' _ (measurable_measure_prodMk_left hs)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.Composition.MeasureComp | {
"line": 48,
"column": 36
} | {
"line": 48,
"column": 49
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure α\ninst✝¹ : SFinite μ\nκ : Kernel α β\ninst✝ : IsSFiniteKernel κ\ns : Set β\nhs : MeasurableSet s\n⊢ (μ ⊗ₘ κ).snd s = ∫⁻ (a : α), (κ a) s ∂μ",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Meas... | snd_apply hs, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Composition.MeasureComp | {
"line": 97,
"column": 63
} | {
"line": 99,
"column": 5
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure α\nf : α → β\nhf : AEMeasurable f μ\n⊢ ⇑(Kernel.copy β) ∘ₘ map f μ = map (fun a ↦ (f a, f a)) μ",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"ProbabilityTheory.Kernel.copy.eq_1",
... | by
rw [Kernel.copy, deterministic_comp_eq_map, AEMeasurable.map_map_of_aemeasurable (by fun_prop) hf]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.MeasurableIntegral | {
"line": 104,
"column": 6
} | {
"line": 107,
"column": 94
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nE : Type u_4\ninst✝² : NormedAddCommGroup E\ninst✝¹ : IsSFiniteKernel κ\ninst✝ : NormedSpace ℝ E\nf : α → β → E\nhf : StronglyMeasurable (uncurry f)\nhE : CompleteSpace E\nthis✝¹ : MeasurableSpace E :=... | have (n : _) : Integrable (s' n x) (κ x) := by
apply (hfx.norm.add hfx.norm).mono' (s' n x).aestronglyMeasurable
filter_upwards with y
simp_rw [s', SimpleFunc.coe_comp]; exact SimpleFunc.norm_approxOn_zero_le _ _ (x, y) n | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Kernel.Composition.CompProd | {
"line": 543,
"column": 8
} | {
"line": 543,
"column": 25
} | [
{
"pp": "case h.h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nδ : Type u_4\nmδ : MeasurableSpace δ\nκ : Kernel α β\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (α × β) γ\ninst✝ : IsSFiniteKernel η\nf : δ → γ\nhf : MeasurableEmbedding f\na : α\nt ... | comapRight_apply' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Composition.IntegralCompProd | {
"line": 253,
"column": 4
} | {
"line": 253,
"column": 29
} | [
{
"pp": "case pos.h_add\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nE : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝³ : NormedAddCommGroup E\na : α\nκ : Kernel α β\ninst✝² : IsSFiniteKernel κ\nη : Kernel (α × β) γ\ninst✝¹ : IsSFiniteKernel η\ninst✝ : NormedSpace ℝ E\nf✝... | intro f g _ i_f i_g hf hg | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.MeasureTheory.Measure.LogLikelihoodRatio | {
"line": 212,
"column": 8
} | {
"line": 212,
"column": 49
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : SigmaFinite ν\nhμν : μ ≪ ν\nhf : Integrable f μ\nh_int : Integrable (llr μ ν) μ\nhfμ : Integrable (fun x ↦ rexp (f x)) μ\nhfν : AEMeasurable f ν\n⊢ ∫ (a : α), f a - log (∫ (x : α), rexp (f x) ∂μ) ... | rw [integral_sub hf (integrable_const _)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.InformationTheory.KullbackLeibler.Basic | {
"line": 267,
"column": 8
} | {
"line": 267,
"column": 43
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nc : ℝ≥0\nhc : c ≠ 0\nhc' : ↑c ≠ 0\nhμ_smul : μ = c • c⁻¹ • μ\nhν_smul : ν = c⁻¹ • c • ν\nhμν : μ ≪ ν\nhμν_right : μ ≪ c • ν\nh_int : ¬Integrable (llr μ ν) μ\nh_contra : Integrable (llr μ (c • ν... | simp only [Measure.coe_nnreal_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable | {
"line": 382,
"column": 8
} | {
"line": 382,
"column": 35
} | [
{
"pp": "case refine_2\nα : Type u_1\nF : Type u_2\np : ℝ≥0∞\ninst✝² : NormedAddCommGroup F\nm m0 : MeasurableSpace α\nμ : Measure α\ninst✝¹ : Fact (1 ≤ p)\ninst✝ : NormedSpace ℝ F\nhm : m ≤ m0\nhp_ne_top : p ≠ ∞\nP : ↥(Lp F p μ) → Prop\nh_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), P ↑(... | LinearIsometryEquiv.map_add | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2 | {
"line": 158,
"column": 6
} | {
"line": 158,
"column": 42
} | [
{
"pp": "α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhm : m ≤ m0\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nf : ↥(Lp ℝ 2 μ)\nh_meas : AEStronglyMeasurable (↑↑↑((condExpL2 ℝ ℝ hm) f)) μ := lpMeas.aestronglyMeasurable ((condExpL2 ℝ ℝ hm) f)\ng : α → ℝ := Exists.choose h_meas\nhg_meas : StronglyMe... | lintegral_congr_ae hg_nnnorm_eq.symm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2 | {
"line": 367,
"column": 53
} | {
"line": 367,
"column": 65
} | [
{
"pp": "α : Type u_1\nG : Type u_5\ninst✝¹ : NormedAddCommGroup G\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\ninst✝ : NormedSpace ℝ G\nhm : m ≤ m0\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nx y : G\n⊢ (compLpL 2 μ (toSpanSingleton ℝ x + toSpanSingleton ℝ y)) ↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs ... | add_compLpL, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1 | {
"line": 214,
"column": 83
} | {
"line": 220,
"column": 58
} | [
{
"pp": "α : Type u_1\nG : Type u_4\ninst✝² : NormedAddCommGroup G\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝¹ : NormedSpace ℝ G\nhm : m ≤ m0\ninst✝ : SigmaFinite (μ.trim hm)\nhs : MeasurableSet s\nht : MeasurableSet t\nhμs : μ s ≠ ∞\nhμt : μ t ≠ ∞\nhst : Disjoint s t\nx : G\n⊢ condExpIndL1 hm... | by
have hμst : μ (s ∪ t) ≠ ∞ :=
((measure_union_le s t).trans_lt (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne
rw [condExpIndL1_of_measurableSet_of_measure_ne_top hs hμs x,
condExpIndL1_of_measurableSet_of_measure_ne_top ht hμt x,
condExpIndL1_of_measurableSet_of_measure_ne_top (hs.uni... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1 | {
"line": 376,
"column": 4
} | {
"line": 376,
"column": 76
} | [
{
"pp": "α : Type u_1\nF' : Type u_3\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\ninst✝¹ : SigmaFinite (μ.trim hm)\ns : Set α\ninst✝ : CompleteSpace F'\nf : ↥(Lp F' 1 μ)\nhs : MeasurableSet s\nS : ℕ → Set α := spanningSets (μ.trim hm)\nhS_meas... | rw [← Set.inter_iUnion, iUnion_spanningSets (μ.trim hm), Set.inter_univ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1 | {
"line": 433,
"column": 8
} | {
"line": 433,
"column": 35
} | [
{
"pp": "case refine_2\nα : Type u_1\nF' : Type u_3\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\ninst✝¹ : SigmaFinite (μ.trim hm)\ninst✝ : CompleteSpace F'\nf✝ : ↥(lpMeas F' ℝ m 1 μ)\ng✝ : ↥(Lp F' 1 (μ.trim hm)) := (lpMeasToLpTrimLie F' ℝ 1 μ ... | LinearIsometryEquiv.map_add | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.FixedSubmodule | {
"line": 82,
"column": 24
} | {
"line": 82,
"column": 45
} | [
{
"pp": "R : Type u_1\ninst✝² : Semiring R\nV : Type u_3\ninst✝¹ : AddCommMonoid V\ninst✝ : Module R V\ne : V ≃ₗ[R] V\nW : Submodule R V\nhW : W ≤ (↑e).fixedSubmodule\nx : V\nhx : x ∈ W\nthis : e x = x\n⊢ (DistribSMul.toLinearMap R V e) x ∈ W",
"usedConstants": [
"Eq.mpr",
"Submodule",
"in... | simpa [this, coe_coe] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.FixedSubmodule | {
"line": 82,
"column": 24
} | {
"line": 82,
"column": 45
} | [
{
"pp": "R : Type u_1\ninst✝² : Semiring R\nV : Type u_3\ninst✝¹ : AddCommMonoid V\ninst✝ : Module R V\ne : V ≃ₗ[R] V\nW : Submodule R V\nhW : W ≤ (↑e).fixedSubmodule\nx : V\nhx : x ∈ W\nthis : e x = x\n⊢ (DistribSMul.toLinearMap R V e) x ∈ W",
"usedConstants": [
"Eq.mpr",
"Submodule",
"in... | simpa [this, coe_coe] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.FixedSubmodule | {
"line": 82,
"column": 24
} | {
"line": 82,
"column": 45
} | [
{
"pp": "R : Type u_1\ninst✝² : Semiring R\nV : Type u_3\ninst✝¹ : AddCommMonoid V\ninst✝ : Module R V\ne : V ≃ₗ[R] V\nW : Submodule R V\nhW : W ≤ (↑e).fixedSubmodule\nx : V\nhx : x ∈ W\nthis : e x = x\n⊢ (DistribSMul.toLinearMap R V e) x ∈ W",
"usedConstants": [
"Eq.mpr",
"Submodule",
"in... | simpa [this, coe_coe] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Center | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 59
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nf : V →ₗ[R] V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nh : ∀ (v : V), ¬LinearIndependent R ![v, f v]\ni j : ι\n⊢ (b.repr (f (b i))) j = (b.repr ((b.coord i) (f (b i)) • b i)) j",... | simp only [LinearIndependent.pair_iff, not_forall] at h | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 402,
"column": 6
} | {
"line": 402,
"column": 73
} | [
{
"pp": "V : Type u_2\ninst✝² : AddCommGroup V\nK : Type u_3\ninst✝¹ : DivisionRing K\ninst✝ : Module K V\ne : V ≃ₗ[K] V\nu : V →ₗ[K] V := ↑e - LinearMap.id\nhu : u + LinearMap.id = ↑e\nhr : 1 ≤ Module.rank K ↥u.range\nb : Basis Unit K ↥u.range\nx : V\nthis : ↑(u.rangeRestrict x) = u x\n⊢ (Finsupp.linearCombina... | rw [Finsupp.linearCombination_apply, Finsupp.sum_eq_single default] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 654,
"column": 8
} | {
"line": 654,
"column": 47
} | [
{
"pp": "R : Type u_3\nV : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : Free R V\ninst✝ : Module.Finite R V\nf : Dual R V\nv : V\nhR : Nontrivial R\nn : ℕ := finrank R V\nb : Basis (Fin n) R V\nS : Type := MvPolynomial (Fin n ⊕ Fin n) ℤ\nγ : S →+* R := ↑(MvPolynomial.aev... | Fintype.linearCombination_apply_single, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 661,
"column": 33
} | {
"line": 661,
"column": 72
} | [
{
"pp": "R : Type u_3\nV : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : Free R V\ninst✝ : Module.Finite R V\nf : Dual R V\nv : V\nhR : Nontrivial R\nn : ℕ := finrank R V\nb : Basis (Fin n) R V\nS : Type := MvPolynomial (Fin n ⊕ Fin n) ℤ\nγ : S →+* R := ↑(MvPolynomial.aev... | Fintype.linearCombination_apply_single, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.QuadraticForm.TensorProduct | {
"line": 143,
"column": 16
} | {
"line": 143,
"column": 27
} | [
{
"pp": "R : Type uR\nA : Type uA\nM₂ : Type uM₂\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : Algebra R A\ninst✝² : Module R M₂\ninst✝¹ : Invertible 2\ninst✝ : Invertible 2\nQ : QuadraticForm R M₂\n⊢ (BilinForm.tensorDistrib R A) (⅟2 • QuadraticMap.sq.polarBilin ⊗ₜ[R] polarBilin... | smul_tmul', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 76
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nm : M\n⊢ e0 Q * (v Q) m * e0 Q = (v Q) m",
"usedConstants": [
"CliffordAlgebra.EquivEven.Q'",
"Eq.mpr",
"NegZeroClass.toNeg",
"MulOne.toOne",
"Semigroup... | rw [← neg_v_mul_e0, ← neg_mul, mul_assoc, e0_mul_e0, mul_neg_one, neg_neg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 76
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nm : M\n⊢ e0 Q * (v Q) m * e0 Q = (v Q) m",
"usedConstants": [
"CliffordAlgebra.EquivEven.Q'",
"Eq.mpr",
"NegZeroClass.toNeg",
"MulOne.toOne",
"Semigroup... | rw [← neg_v_mul_e0, ← neg_mul, mul_assoc, e0_mul_e0, mul_neg_one, neg_neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 76
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nm : M\n⊢ e0 Q * (v Q) m * e0 Q = (v Q) m",
"usedConstants": [
"CliffordAlgebra.EquivEven.Q'",
"Eq.mpr",
"NegZeroClass.toNeg",
"MulOne.toOne",
"Semigroup... | rw [← neg_v_mul_e0, ← neg_mul, mul_assoc, e0_mul_e0, mul_neg_one, neg_neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.CliffordAlgebra.Prod | {
"line": 152,
"column": 2
} | {
"line": 153,
"column": 36
} | [
{
"pp": "R : Type u_1\nM₁ : Type u_2\nM₂ : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M₁\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R M₁\ninst✝ : Module R M₂\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nm₂ : M₂\n⊢ (toProd Q₁ Q₂) (1 ᵍ⊗ₜ[R] (ι Q₂) m₂) = (ι (QuadraticMap.prod Q₁ Q₂)) (0, m₂)",
... | rw [toProd, GradedTensorProduct.lift_tmul, map_one, one_mul, map_apply_ι,
QuadraticMap.Isometry.inr_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.CliffordAlgebra.Prod | {
"line": 152,
"column": 2
} | {
"line": 153,
"column": 36
} | [
{
"pp": "R : Type u_1\nM₁ : Type u_2\nM₂ : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M₁\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R M₁\ninst✝ : Module R M₂\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nm₂ : M₂\n⊢ (toProd Q₁ Q₂) (1 ᵍ⊗ₜ[R] (ι Q₂) m₂) = (ι (QuadraticMap.prod Q₁ Q₂)) (0, m₂)",
... | rw [toProd, GradedTensorProduct.lift_tmul, map_one, one_mul, map_apply_ι,
QuadraticMap.Isometry.inr_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.CliffordAlgebra.Prod | {
"line": 152,
"column": 2
} | {
"line": 153,
"column": 36
} | [
{
"pp": "R : Type u_1\nM₁ : Type u_2\nM₂ : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M₁\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R M₁\ninst✝ : Module R M₂\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nm₂ : M₂\n⊢ (toProd Q₁ Q₂) (1 ᵍ⊗ₜ[R] (ι Q₂) m₂) = (ι (QuadraticMap.prod Q₁ Q₂)) (0, m₂)",
... | rw [toProd, GradedTensorProduct.lift_tmul, map_one, one_mul, map_apply_ι,
QuadraticMap.Isometry.inr_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.TensorProduct.Graded.External | {
"line": 227,
"column": 38
} | {
"line": 227,
"column": 49
} | [
{
"pp": "case a.H.h.H.h\nR : Type u_1\nι : Type u_2\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : Module ι (Additive ℤˣ)\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → Type u_3\nℬ : ι → Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝⁶ : (i : ι) → AddCommGroup (ℬ i)\ninst✝⁵ : (i : ι) → Module R (𝒜 i)\nin... | smul_tmul', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.TensorProduct.Graded.Internal | {
"line": 343,
"column": 17
} | {
"line": 343,
"column": 74
} | [
{
"pp": "R : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : DecidableEq ι\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : Ring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\n𝒜 : ι → Submodule R A\nℬ : ι → Submodule R B\ninst✝⁴ : GradedAlgebra 𝒜\ninst✝³ : GradedAlgebra ... | by ext <;> (dsimp; simp only [map_one, mul_one, one_mul]) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.CharP | {
"line": 28,
"column": 4
} | {
"line": 28,
"column": 33
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝³ : AddMonoidWithOne R\ninst✝² : DecidableEq n\ninst✝¹ : Nonempty n\np : ℕ\ninst✝ : CharP R p\nk : ℕ\n⊢ (∀ (i : n), ↑k = 0) ↔ p ∣ k",
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"outParam",
"congrArg",
"AddMonoid.toAddZeroClass",
"Add... | CharP.cast_eq_zero_iff R p k, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Matrix.Charpoly.FiniteField | {
"line": 33,
"column": 4
} | {
"line": 33,
"column": 32
} | [
{
"pp": "case inl\nn : Type u_1\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Fintype K\nM : Matrix n n K\nh✝ : Nonempty n\np : ℕ\nhp✝ : CharP K p\nk : ℕ\nkpos : 0 < k\nhp : Nat.Prime p\nhk : Fintype.card K = p ^ ↑⟨k, kpos⟩\n⊢ (M ^ Fintype.card K).charpoly = M.charpoly",
... | haveI : Fact p.Prime := ⟨hp⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.LinearAlgebra.Matrix.Determinant.TotallyUnimodular | {
"line": 174,
"column": 2
} | {
"line": 178,
"column": 76
} | [
{
"pp": "m : Type u_1\nn : Type u_3\nR : Type u_5\ninst✝¹ : CommRing R\ninst✝ : DecidableEq n\nA : Matrix m n R\n⊢ (fromRows 1 A).IsTotallyUnimodular ↔ A.IsTotallyUnimodular",
"usedConstants": [
"Eq.mpr",
"Sum.swap",
"Matrix.submatrix",
"Equiv.instEquivLike",
"congrArg",
... | have hA :
fromRows (1 : Matrix n n R) A =
(fromRows A (1 : Matrix n n R)).reindex (Equiv.sumComm m n) (Equiv.refl n) := by
aesop
rw [hA, reindex_isTotallyUnimodular, fromRows_one_isTotallyUnimodular_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.Determinant.TotallyUnimodular | {
"line": 174,
"column": 2
} | {
"line": 178,
"column": 76
} | [
{
"pp": "m : Type u_1\nn : Type u_3\nR : Type u_5\ninst✝¹ : CommRing R\ninst✝ : DecidableEq n\nA : Matrix m n R\n⊢ (fromRows 1 A).IsTotallyUnimodular ↔ A.IsTotallyUnimodular",
"usedConstants": [
"Eq.mpr",
"Sum.swap",
"Matrix.submatrix",
"Equiv.instEquivLike",
"congrArg",
... | have hA :
fromRows (1 : Matrix n n R) A =
(fromRows A (1 : Matrix n n R)).reindex (Equiv.sumComm m n) (Equiv.refl n) := by
aesop
rw [hA, reindex_isTotallyUnimodular, fromRows_one_isTotallyUnimodular_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Irreducible.Defs | {
"line": 129,
"column": 4
} | {
"line": 154,
"column": 42
} | [
{
"pp": "case succ\nn : Type u_1\nR : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrder R\nA : Matrix n n R\ninst✝⁴ : Fintype n\ninst✝³ : IsOrderedRing R\ninst✝² : PosMulStrictMono R\ninst✝¹ : Nontrivial R\ninst✝ : DecidableEq n\nhA : ∀ (i j : n), 0 ≤ A i j\nthis : Quiver n := A.toQuiver\nm : ℕ\nih : ∀ (i j : n),... | rw [pow_succ, mul_apply]
constructor
· intro h_pos
obtain ⟨l, hl_mem, hl_pos⟩ :
∃ l ∈ (Finset.univ : Finset n), 0 < (A ^ m) i l * A l j := by
simpa [Finset.sum_pos_iff_of_nonneg
(fun x _ => mul_nonneg (pow_apply_nonneg hA m i x) (hA x j))]
using h_pos
hav... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.Irreducible.Defs | {
"line": 129,
"column": 4
} | {
"line": 154,
"column": 42
} | [
{
"pp": "case succ\nn : Type u_1\nR : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrder R\nA : Matrix n n R\ninst✝⁴ : Fintype n\ninst✝³ : IsOrderedRing R\ninst✝² : PosMulStrictMono R\ninst✝¹ : Nontrivial R\ninst✝ : DecidableEq n\nhA : ∀ (i j : n), 0 ≤ A i j\nthis : Quiver n := A.toQuiver\nm : ℕ\nih : ∀ (i j : n),... | rw [pow_succ, mul_apply]
constructor
· intro h_pos
obtain ⟨l, hl_mem, hl_pos⟩ :
∃ l ∈ (Finset.univ : Finset n), 0 < (A ^ m) i l * A l j := by
simpa [Finset.sum_pos_iff_of_nonneg
(fun x _ => mul_nonneg (pow_apply_nonneg hA m i x) (hA x j))]
using h_pos
hav... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.IsDiag | {
"line": 142,
"column": 2
} | {
"line": 146,
"column": 35
} | [
{
"pp": "α : Type u_1\nn : Type u_4\nm : Type u_5\ninst✝ : Zero α\nA : Matrix m m α\nD : Matrix n n α\nha : A.IsDiag\nhd : D.IsDiag\n⊢ (Matrix.fromBlocks A 0 0 D).IsDiag",
"usedConstants": [
"Matrix.fromBlocks",
"Matrix",
"Sum.casesOn",
"Sum",
"Ne",
"Sum.inl",
"ne_o... | rintro (i | i) (j | j) hij
· exact ha (ne_of_apply_ne _ hij)
· rfl
· rfl
· exact hd (ne_of_apply_ne _ hij) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.IsDiag | {
"line": 142,
"column": 2
} | {
"line": 146,
"column": 35
} | [
{
"pp": "α : Type u_1\nn : Type u_4\nm : Type u_5\ninst✝ : Zero α\nA : Matrix m m α\nD : Matrix n n α\nha : A.IsDiag\nhd : D.IsDiag\n⊢ (Matrix.fromBlocks A 0 0 D).IsDiag",
"usedConstants": [
"Matrix.fromBlocks",
"Matrix",
"Sum.casesOn",
"Sum",
"Ne",
"Sum.inl",
"ne_o... | rintro (i | i) (j | j) hij
· exact ha (ne_of_apply_ne _ hij)
· rfl
· rfl
· exact hd (ne_of_apply_ne _ hij) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Multilinear.Pi | {
"line": 98,
"column": 4
} | {
"line": 102,
"column": 38
} | [
{
"pp": "case h.inr\nι : Type uι\nκ : ι → Type uκ\nR : Type uR\nM : (i : ι) → κ i → Type uM\nN : ((i : ι) → κ i) → Type uN\ninst✝⁶ : Semiring R\ninst✝⁵ : (i : ι) → (k : κ i) → AddCommMonoid (M i k)\ninst✝⁴ : (p : (i : ι) → κ i) → AddCommMonoid (N p)\ninst✝³ : (i : ι) → (k : κ i) → Module R (M i k)\ninst✝² : (p ... | rw [Pi.single_eq_of_ne' hpq]
rw [Function.ne_iff] at hpq
obtain ⟨i, hpqi⟩ := hpq
apply (f q).map_coord_zero i
simp_rw [Pi.single_eq_of_ne' hpqi] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Multilinear.Pi | {
"line": 98,
"column": 4
} | {
"line": 102,
"column": 38
} | [
{
"pp": "case h.inr\nι : Type uι\nκ : ι → Type uκ\nR : Type uR\nM : (i : ι) → κ i → Type uM\nN : ((i : ι) → κ i) → Type uN\ninst✝⁶ : Semiring R\ninst✝⁵ : (i : ι) → (k : κ i) → AddCommMonoid (M i k)\ninst✝⁴ : (p : (i : ι) → κ i) → AddCommMonoid (N p)\ninst✝³ : (i : ι) → (k : κ i) → Module R (M i k)\ninst✝² : (p ... | rw [Pi.single_eq_of_ne' hpq]
rw [Function.ne_iff] at hpq
obtain ⟨i, hpqi⟩ := hpq
apply (f q).map_coord_zero i
simp_rw [Pi.single_eq_of_ne' hpqi] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.WithConv | {
"line": 140,
"column": 6
} | {
"line": 140,
"column": 27
} | [
{
"pp": "n : Type u_2\nα : Type u_3\ninst✝³ : CommSemiring α\ninst✝² : StarRing α\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix n n α\n⊢ A.IsSymm ↔ star (toConv (toLin' A)) = toConv (toLin' (star A))",
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"Matrix.instStar",
"Wi... | intrinsicStar_toLin', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Projectivization.Action | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 41
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Field K\ninst✝ : Module K V\nthis : IsMultiplyPretransitive (V ≃ₗ[K] V) (ℙ K V) 2\n⊢ IsMultiplyPretransitive (SpecialLinearGroup K V) (ℙ K V) 2",
"usedConstants": [
"instHSMul",
"instSMulOfMul",
"congrArg",
"Distr... | rw [is_two_pretransitive_iff] at this ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Projectivization.Action | {
"line": 134,
"column": 4
} | {
"line": 134,
"column": 72
} | [
{
"pp": "case pos\nK : Type u_1\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Field K\ninst✝ : Module K V\nthis✝ : ∀ {a b c d : ℙ K V}, a ≠ b → c ≠ d → ∃ g, g • a = c ∧ g • b = d\nD D' E E' : ℙ K V\nhD : LinearIndependent K ![D.rep, D'.rep]\nhE : E ≠ E'\ng : V ≃ₗ[K] V\ngD : g • D = E\ngE : g • D' = E'\nhV : ... | have hD_mem : D.rep ∈ s := LinearIndepOn.subset_extend _ _ (by simp) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.QuadraticForm.Signature | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 19
} | [
{
"pp": "𝕜 : Type u_4\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\nι : Type u_5\ninst✝¹ : Fintype ι\nw : ι → 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns : Set ι\nhs : ∀ i ∈ s, 0 < w i\n⊢ ((weightedSumSquares 𝕜 w).restrict (Pi.spanSubset 𝕜 s)).PosDef",
"usedConstants": [
"Pi.Function.module",
"Submo... | intro ⟨v, hv⟩ hv' | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 142,
"column": 2
} | {
"line": 144,
"column": 48
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.Base\ni j ... | have hki : P.root k ≠ P.root i := fun contra ↦ by
replace h₁ : 2 • P.root i = P.root l := by rwa [contra, ← two_nsmul] at h₁
exact P.nsmul_notMem_range_root ⟨_, h₁.symm⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 88
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Finite ι\ninst✝⁹ : CommRing R\ninst✝⁸ : IsDomain R\ninst✝⁷ : CharZero R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsCrystallographic\nb : P.Base\ninst... | rcases exists_or_forall_not (fun x ↦ P.root k = P.root i + P.root x) with ⟨x, hx⟩ | h₁ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 158,
"column": 4
} | {
"line": 185,
"column": 24
} | [
{
"pp": "case a.inr.inr\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Finite ι\ninst✝⁹ : CommRing R\ninst✝⁸ : IsDomain R\ninst✝⁷ : CharZero R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsCrystallographic\n... | rcases eq_or_ne k i with rfl | hki
· have hx (x : ι) : ¬ (P.root x = P.root i + P.root l ∧ P.root i = P.root x - P.root i) := by
rintro ⟨-, contra⟩
refine P.nsmul_notMem_range_root (n := 2) (i := i) ⟨x, ?_⟩
rwa [eq_sub_iff_add_eq, ← two_smul ℕ, eq_comm] at contra
simp only [e, f, h, P.... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 158,
"column": 4
} | {
"line": 185,
"column": 24
} | [
{
"pp": "case a.inr.inr\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Finite ι\ninst✝⁹ : CommRing R\ninst✝⁸ : IsDomain R\ninst✝⁷ : CharZero R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsCrystallographic\n... | rcases eq_or_ne k i with rfl | hki
· have hx (x : ι) : ¬ (P.root x = P.root i + P.root l ∧ P.root i = P.root x - P.root i) := by
rintro ⟨-, contra⟩
refine P.nsmul_notMem_range_root (n := 2) (i := i) ⟨x, ?_⟩
rwa [eq_sub_iff_add_eq, ← two_smul ℕ, eq_comm] at contra
simp only [e, f, h, P.... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 327,
"column": 54
} | {
"line": 327,
"column": 72
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹¹ : Finite ι\ninst✝¹⁰ : CommRing R\ninst✝⁹ : IsDomain R\ninst✝⁸ : CharZero R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : P.IsCrystallographic\nb : P.Base\nins... | rw [hl', add_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 327,
"column": 54
} | {
"line": 327,
"column": 72
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹¹ : Finite ι\ninst✝¹⁰ : CommRing R\ninst✝⁹ : IsDomain R\ninst✝⁸ : CharZero R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : P.IsCrystallographic\nb : P.Base\nins... | rw [hl', add_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 327,
"column": 54
} | {
"line": 327,
"column": 72
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹¹ : Finite ι\ninst✝¹⁰ : CommRing R\ninst✝⁹ : IsDomain R\ninst✝⁸ : CharZero R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : P.IsCrystallographic\nb : P.Base\nins... | rw [hl', add_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 144,
"column": 52
} | {
"line": 144,
"column": 67
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : IsDomain R\ninst✝⁸ : CharZero R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : P.IsCrystallographic\ninst✝² : P.IsReduced\nb : P.Base\ni... | simp [this, hn] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 144,
"column": 52
} | {
"line": 144,
"column": 67
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : IsDomain R\ninst✝⁸ : CharZero R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : P.IsCrystallographic\ninst✝² : P.IsReduced\nb : P.Base\ni... | simp [this, hn] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 144,
"column": 52
} | {
"line": 144,
"column": 67
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : IsDomain R\ninst✝⁸ : CharZero R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : P.IsCrystallographic\ninst✝² : P.IsReduced\nb : P.Base\ni... | simp [this, hn] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Logic.Hydra | {
"line": 127,
"column": 4
} | {
"line": 127,
"column": 67
} | [
{
"pp": "case inr\nα : Type u_1\nr : α → α → Prop\ns₁ s₂ t : Multiset α\na : α\nhr : ∀ a' ∈ t, r a' a\nhe : (s₁ + s₂ + t).erase a + {a} = s₁ + s₂ + t\nh : a ∈ s₂ + t\n⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.1 + a'.2 = (s₁ + s₂ + t).erase a",
"usedConstants": [
"Prod.GameAdd",
... | refine ⟨(s₁, (s₂ + t).erase a), GameAdd.snd ⟨t, a, hr, ?_⟩, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Logic.Hydra | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 43
} | [
{
"pp": "α : Type u_1\nr : α → α → Prop\ninst✝ : Std.Irrefl r\ns : Multiset α\nhs : ∀ a ∈ s, Acc (CutExpand r) {a}\n⊢ Acc (CutExpand r) s",
"usedConstants": [
"Eq.mpr",
"Multiset.singleton_add",
"congrArg",
"Prod.GameAdd",
"False.elim",
"Membership.mem",
"Relation.C... | induction s using Multiset.induction with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 251,
"column": 56
} | {
"line": 251,
"column": 84
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Field K\ninst✝⁸ : CharZero K\ninst✝⁷ : DecidableEq ι\ninst✝⁶ : Fintype ι\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module K M\ninst✝³ : AddCommGroup N\ninst✝² : Module K N\nP : RootPairing ι K M N\ninst✝¹ : P.IsRootSystem\ninst✝ : P.IsCrystallog... | diagonal_elim_mem_span_h_iff | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 265,
"column": 65
} | {
"line": 265,
"column": 80
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Field K\ninst✝⁸ : CharZero K\ninst✝⁷ : DecidableEq ι\ninst✝⁶ : Fintype ι\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module K M\ninst✝³ : AddCommGroup N\ninst✝² : Module K N\nP : RootPairing ι K M N\ninst✝¹ : P.IsRootSystem\ninst✝ : P.IsCrystallog... | biSup_congr hU, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 272,
"column": 2
} | {
"line": 273,
"column": 59
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Field K\ninst✝⁹ : CharZero K\ninst✝⁸ : DecidableEq ι\ninst✝⁷ : Fintype ι\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module K M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module K N\nP : RootPairing ι K M N\ninst✝² : P.IsCrystallographic\nb : P.Base\nins... | have hωu (i : b.support) : ω b *ᵥ (u i) = u i := by
ext (j | j) <;> simp [ω, u, Pi.single_apply, one_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Constructions.Cylinders | {
"line": 351,
"column": 2
} | {
"line": 352,
"column": 75
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\ninst✝ : (i : ι) → MeasurableSpace (α i)\ns t : Set ((i : ι) → α i)\nhs : s ∈ measurableCylinders α\nht : t ∈ measurableCylinders α\n⊢ s \\ t ∈ measurableCylinders α",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"MeasureTheory.measu... | rw [diff_eq_compl_inter]
exact inter_mem_measurableCylinders (compl_mem_measurableCylinders ht) hs | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Constructions.Cylinders | {
"line": 351,
"column": 2
} | {
"line": 352,
"column": 75
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\ninst✝ : (i : ι) → MeasurableSpace (α i)\ns t : Set ((i : ι) → α i)\nhs : s ∈ measurableCylinders α\nht : t ∈ measurableCylinders α\n⊢ s \\ t ∈ measurableCylinders α",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"MeasureTheory.measu... | rw [diff_eq_compl_inter]
exact inter_mem_measurableCylinders (compl_mem_measurableCylinders ht) hs | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Constructions.Cylinders | {
"line": 360,
"column": 4
} | {
"line": 360,
"column": 22
} | [
{
"pp": "case a\nι : Type u_1\nα : ι → Type u_2\ninst✝ : (i : ι) → MeasurableSpace (α i)\ns : Finset ι\nS : Set ((i : ↥s) → α ↑i)\nhSm : MeasurableSet S\nhS : cylinder s S ∈ measurableCylinders α\n⊢ MeasurableSet (cylinder s S)",
"usedConstants": [
"MeasurableSet.cylinder"
]
}
] | exact hSm.cylinder | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.SetSemiring | {
"line": 113,
"column": 68
} | {
"line": 113,
"column": 92
} | [
{
"pp": "case insert.inr\nα : Type u_1\nC : Set (Set α)\nhC : IsSetSemiring C\ns : Set α\nS : Finset (Set α)\na✝ : s ∉ S\nih : ↑S ⊆ C → ∃ P, ↑P.parts ⊆ C\nhsC : s ∈ C\nhSC : ↑S ⊆ C\nP : Finpartition (S.sup id)\nhP : ↑P.parts ⊆ C\nhs : s ≠ ⊥\nQ : (t : Set α) → t ∈ (P.avoid s).parts → Finpartition t\nhQ : ∀ (t : ... | Finpartition.bind_parts, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.SetAlgebra | {
"line": 211,
"column": 57
} | {
"line": 211,
"column": 62
} | [
{
"pp": "α : Type u_1\n𝒜 : Set (Set α)\ns u v : Set α\nhs✝ : generateSetAlgebra 𝒜 u\nht✝ : generateSetAlgebra 𝒜 v\nAu : Set (Set (Set α))\nAu_fin : Au.Finite\nmem_Au : ∀ a ∈ Au, a.Finite\nhAu : ∀ a ∈ Au, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜\nu_eq : u = ⋃ a ∈ Au, ⋂ t ∈ a, t\nAv : Set (Set (Set α))\nAv_fin : Av.Finite\nm... | u_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 270,
"column": 6
} | {
"line": 270,
"column": 44
} | [
{
"pp": "α : Type u_1\nC : Set (Set α)\ns t : Set α\nI✝ : Finset (Set α)\nG : Type u_2\ninst✝ : AddCommMonoid G\nm✝ m' m : AddContent G C\nhC : IsSetSemiring C\nI : Finset (Set α)\nhI : ↑I ⊆ _root_.supClosure C\nh'I : (↑I).PairwiseDisjoint id\nhh'I : ⋃₀ ↑I ∈ _root_.supClosure C\nJ : Set α → Finset (Set α)\nhJC ... | refine (h'I.mono_on ?_).biUnion hJdisj | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.SetSemiring | {
"line": 470,
"column": 64
} | {
"line": 470,
"column": 67
} | [
{
"pp": "case h.refine_6\nα : Type u_1\nC : Set (Set α)\nJ✝ : Finset (Set α)\nhC : IsSetSemiring C\ns : Set α\nJ : Finset (Set α)\nhJ : s ∉ J\nhind :\n ↑J ⊆ C →\n ∃ K,\n (↑J).PairwiseDisjoint K ∧\n (∀ i ∈ J, ↑(K i) ⊆ C) ∧\n (⋃ x ∈ J, ↑(K x)).PairwiseDisjoint id ∧\n (∀ j ∈ J, ... | hK5 | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Covering.LiminfLimsup | {
"line": 171,
"column": 4
} | {
"line": 174,
"column": 54
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝⁵ : PseudoMetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr₁ r₂ : ℕ → ℝ\nhr : Tendsto ... | apply HasSubset.Subset.eventuallyLE
change _ ≤ _
refine mono_blimsup' (hMr.mono fun i hi _ => cthickening_mono ?_ (s i))
exact (le_mul_of_one_le_left (hRp i) hM').trans hi | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Covering.LiminfLimsup | {
"line": 171,
"column": 4
} | {
"line": 174,
"column": 54
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝⁵ : PseudoMetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr₁ r₂ : ℕ → ℝ\nhr : Tendsto ... | apply HasSubset.Subset.eventuallyLE
change _ ≤ _
refine mono_blimsup' (hMr.mono fun i hi _ => cthickening_mono ?_ (s i))
exact (le_mul_of_one_le_left (hRp i) hM').trans hi | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 164,
"column": 4
} | {
"line": 164,
"column": 68
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\nf : α → E\ns : Set α\nm m₂ m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\nhm₂ : m₂ ≤ m0\ninst✝¹ : SigmaFinite (μ.trim hm)\ninst✝ : SigmaFinite (μ.trim hm₂)\nhs_m : MeasurableSet s\nhs : ∀... | rw [Measure.restrict_restrict (MeasurableSet.compl (hm _ hs_m))] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 546,
"column": 2
} | {
"line": 557,
"column": 68
} | [
{
"pp": "α : Type u_1\nC : Set (Set α)\nG : Type u_2\ninst✝ : AddCommMonoid G\nm : AddContent G C\nι : Type u_3\nhC : IsSetRing C\ns : ι → Set α\nS : Finset ι\nhs : ∀ n ∈ S, s n ∈ C\nhS : (↑S).PairwiseDisjoint s\n⊢ m (⋃ i ∈ S, s i) = ∑ i ∈ S, m (s i)",
"usedConstants": [
"Eq.mpr",
"False",
... | classical
induction S using Finset.induction with
| empty => simp
| insert i S hiS h =>
rw [Finset.sum_insert hiS]
simp_rw [← Finset.mem_coe, Finset.coe_insert, Set.biUnion_insert]
simp only [Finset.mem_insert, forall_eq_or_imp] at hs
simp only [Finset.coe_insert, Set.pairwiseDisjoint_insert] at h... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 546,
"column": 2
} | {
"line": 557,
"column": 68
} | [
{
"pp": "α : Type u_1\nC : Set (Set α)\nG : Type u_2\ninst✝ : AddCommMonoid G\nm : AddContent G C\nι : Type u_3\nhC : IsSetRing C\ns : ι → Set α\nS : Finset ι\nhs : ∀ n ∈ S, s n ∈ C\nhS : (↑S).PairwiseDisjoint s\n⊢ m (⋃ i ∈ S, s i) = ∑ i ∈ S, m (s i)",
"usedConstants": [
"Eq.mpr",
"False",
... | classical
induction S using Finset.induction with
| empty => simp
| insert i S hiS h =>
rw [Finset.sum_insert hiS]
simp_rw [← Finset.mem_coe, Finset.coe_insert, Set.biUnion_insert]
simp only [Finset.mem_insert, forall_eq_or_imp] at hs
simp only [Finset.coe_insert, Set.pairwiseDisjoint_insert] at h... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 546,
"column": 2
} | {
"line": 557,
"column": 68
} | [
{
"pp": "α : Type u_1\nC : Set (Set α)\nG : Type u_2\ninst✝ : AddCommMonoid G\nm : AddContent G C\nι : Type u_3\nhC : IsSetRing C\ns : ι → Set α\nS : Finset ι\nhs : ∀ n ∈ S, s n ∈ C\nhS : (↑S).PairwiseDisjoint s\n⊢ m (⋃ i ∈ S, s i) = ∑ i ∈ S, m (s i)",
"usedConstants": [
"Eq.mpr",
"False",
... | classical
induction S using Finset.induction with
| empty => simp
| insert i S hiS h =>
rw [Finset.sum_insert hiS]
simp_rw [← Finset.mem_coe, Finset.coe_insert, Set.biUnion_insert]
simp only [Finset.mem_insert, forall_eq_or_imp] at hs
simp only [Finset.coe_insert, Set.pairwiseDisjoint_insert] at h... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 219,
"column": 4
} | {
"line": 219,
"column": 24
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nα : Type u_2\nf : α → E\nm mα : MeasurableSpace α\nμ : Measure α\nhm : m ≤ mα\nhμm : SigmaFinite (μ.trim hm)\nhf_int : ¬Integrable f μ\n⊢ (fun x ↦ 0) ≤ᶠ[ae μ] μ[fun x ↦ ‖f x‖ | m]",
"usedConsta... | apply condExp_nonneg | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 694,
"column": 4
} | {
"line": 694,
"column": 76
} | [
{
"pp": "α : Type u_1\nC : Set (Set α)\nm : AddContent ℝ≥0∞ C\nhC : IsSetRing C\nm_iUnion :\n ∀ (f : ℕ → Set α), (∀ (i : ℕ), f i ∈ C) → ⋃ i, f i ∈ C → Pairwise (Disjoint on f) → m (⋃ i, f i) = ∑' (i : ℕ), m (f i)\nf : ℕ → Set α\nhf : ∀ (i : ℕ), f i ∈ C\nhf_Union : ⋃ i, f i ∈ C\nh_tendsto : Tendsto (fun n ↦ m (... | rw [tendsto_add_atTop_iff_nat (f := (fun k ↦ ∑ i ∈ range k, m (f i))) 1] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan | {
"line": 345,
"column": 11
} | {
"line": 345,
"column": 13
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nj₁ : JordanDecomposition α\n⊢ ∀ ⦃a₂ : JordanDecomposition α⦄, j₁.toSignedMeasure = a₂.toSignedMeasure → j₁ = a₂",
"usedConstants": [
"MeasureTheory.JordanDecomposition"
]
}
] | j₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 489,
"column": 4
} | {
"line": 489,
"column": 25
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ∞\nf : ℕ → α → β\ng : α → β\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : MemLp g p μ\nhui : UnifIntegrable f p μ\nhf... | exact min_le_left _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.VectorMeasure.WithDensity | {
"line": 77,
"column": 9
} | {
"line": 77,
"column": 27
} | [
{
"pp": "case neg.hnc\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → E\nhf : ¬Integrable f μ\n⊢ ¬Integrable (-f) μ",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Pi.instNeg",
"congrArg",
... | integrable_neg_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 263,
"column": 25
} | {
"line": 263,
"column": 74
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nM : Type u_3\ninst✝⁴ : AddCommMonoid M\ninst✝³ : TopologicalSpace M\nR : Type u_4\ninst✝² : Semiring R\ninst✝¹ : DistribMulAction R M\ninst✝ : ContinuousConstSMul R M\nr : R\nv : VectorMeasure α M\nx✝ : ℕ → Set α\nhf₁ : ∀ (i : ℕ), MeasurableSet (x✝ i)\... | by exact HasSum.const_smul _ (v.m_iUnion hf₁ hf₂) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 493,
"column": 8
} | {
"line": 493,
"column": 45
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\nx✝ : ℕ → Set α\nhf₁ : ∀ (i : ℕ), MeasurableSet (x✝ i)\nhf₂ : Pairwise (Disjoint on x✝)\n⊢ HasSum (fun i ↦ if MeasurableSet (x✝ i) then μ (x✝ i) else 0) (if MeasurableSet (⋃ i, x✝ i) then μ (⋃ i, x✝ i) else 0)",
"usedConstants": [
... | Summable.hasSum_iff ENNReal.summable, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 510,
"column": 2
} | {
"line": 511,
"column": 91
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ni : Set α\nhi : MeasurableSet i\n⊢ ↑(μ + ν).toENNRealVectorMeasure i = ↑(μ.toENNRealVectorMeasure + ν.toENNRealVectorMeasure) i",
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"MeasureTheory.Measure",
"ENNReal.instAddCo... | rw [toENNRealVectorMeasure_apply_measurable hi, add_apply, VectorMeasure.add_apply,
toENNRealVectorMeasure_apply_measurable hi, toENNRealVectorMeasure_apply_measurable hi] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 673,
"column": 19
} | {
"line": 679,
"column": 66
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm inst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : TopologicalSpace M\nv✝ v : VectorMeasure α M\ni : Set α\nhi : MeasurableSet i\n⊢ ∀ ⦃f : ℕ → Set α⦄,\n (∀ (i : ℕ), MeasurableSet (f i)) →\n Pairwise (Disjoint on ... | by
intro f hf₁ hf₂
convert!
v.m_iUnion (fun n => (hf₁ n).inter hi)
(hf₂.mono fun i j => Disjoint.mono inf_le_left inf_le_left)
· rw [if_pos (hf₁ _)]
· rw [Set.iUnion_inter, if_pos (MeasurableSet.iUnion hf₁)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 733,
"column": 4
} | {
"line": 733,
"column": 39
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝³ : MeasurableSpace α\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousAdd M\nv w : VectorMeasure α M\ni : Set α\nhi : ¬MeasurableSet i\n⊢ (v + w).restrict i = v.restrict i + w.restrict i",
"usedConstants": [
"congrArg",
... | simp [restrict_not_measurable _ hi] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 733,
"column": 4
} | {
"line": 733,
"column": 39
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝³ : MeasurableSpace α\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousAdd M\nv w : VectorMeasure α M\ni : Set α\nhi : ¬MeasurableSet i\n⊢ (v + w).restrict i = v.restrict i + w.restrict i",
"usedConstants": [
"congrArg",
... | simp [restrict_not_measurable _ hi] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 733,
"column": 4
} | {
"line": 733,
"column": 39
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝³ : MeasurableSpace α\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousAdd M\nv w : VectorMeasure α M\ni : Set α\nhi : ¬MeasurableSet i\n⊢ (v + w).restrict i = v.restrict i + w.restrict i",
"usedConstants": [
"congrArg",
... | simp [restrict_not_measurable _ hi] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 769,
"column": 4
} | {
"line": 769,
"column": 39
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝³ : MeasurableSpace α\nM : Type u_4\ninst✝² : AddCommGroup M\ninst✝¹ : TopologicalSpace M\ninst✝ : IsTopologicalAddGroup M\nv : VectorMeasure α M\ni : Set α\nhi : ¬MeasurableSet i\n⊢ (-v).restrict i = -v.restrict i",
"usedConstants": [
"congrArg",
"AddCommGr... | simp [restrict_not_measurable _ hi] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 769,
"column": 4
} | {
"line": 769,
"column": 39
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝³ : MeasurableSpace α\nM : Type u_4\ninst✝² : AddCommGroup M\ninst✝¹ : TopologicalSpace M\ninst✝ : IsTopologicalAddGroup M\nv : VectorMeasure α M\ni : Set α\nhi : ¬MeasurableSet i\n⊢ (-v).restrict i = -v.restrict i",
"usedConstants": [
"congrArg",
"AddCommGr... | simp [restrict_not_measurable _ hi] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 769,
"column": 4
} | {
"line": 769,
"column": 39
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝³ : MeasurableSpace α\nM : Type u_4\ninst✝² : AddCommGroup M\ninst✝¹ : TopologicalSpace M\ninst✝ : IsTopologicalAddGroup M\nv : VectorMeasure α M\ni : Set α\nhi : ¬MeasurableSet i\n⊢ (-v).restrict i = -v.restrict i",
"usedConstants": [
"congrArg",
"AddCommGr... | simp [restrict_not_measurable _ hi] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 148,
"column": 2
} | {
"line": 151,
"column": 33
} | [
{
"pp": "case neg\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nR : ℝ≥0\nf : α → ℝ\nhbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R\nhnm : m ≤ m0\nhfint : ¬Integrable f μ\n⊢ ∀ᵐ (x : α) ∂μ, |μ[f | m] x| ≤ ↑R",
"usedConstants": [
"MeasureTheory.ae",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
... | · simp_rw [condExp_of_not_integrable hfint]
filter_upwards [hbdd] with x hx
rw [Pi.zero_apply, abs_zero]
exact (abs_nonneg _).trans hx | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 357,
"column": 2
} | {
"line": 357,
"column": 98
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nHeval : ∀ {u : Set α} {v : Set β}, MeasurableSet u → MeasurableSet v → Measurable fun a ↦ ↑a.1 u * ↑a.2 v\n⊢ Measurable fun a ↦ ((↑a.1).prod ↑a.2) univ",
"usedConstants": [
"Set.instSProd",
"MeasureTheory... | · simp_rw [← Set.univ_prod_univ, Measure.prod_prod, Heval MeasurableSet.univ MeasurableSet.univ] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
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