module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Dynamics.Ergodic.MeasurePreserving | {
"line": 101,
"column": 19
} | {
"line": 101,
"column": 58
} | {
"line": 101,
"column": 58
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμa : Measure α\nμb : Measure β\nμc : Measure γ\ng : β → γ\nf : α → β\nhg : MeasurePreserving g μb μc\nhf : MeasurePreserving f μa μb\n⊢ map (g ∘ f) μa = μc",
"ppTerm": "?m.49... | [] | by rw [← map_map hg.1 hf.1, hf.2, hg.2] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 524,
"column": 4
} | {
"line": 524,
"column": 40
} | {
"line": 525,
"column": 2
} | [
{
"pp": "case pos\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed[Pi.topologicalSpace] s\nhne : s.Nonempty\nx : (n : ℕ) → E n\nhx : x ∈ s\n⊢ (s ∩ cylinder x (longestPrefix x s)).Nonempty",
"ppTerm": "?pos✝",
... | [] | exact ⟨x, hx, self_mem_cylinder _ _⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 524,
"column": 4
} | {
"line": 524,
"column": 40
} | {
"line": 525,
"column": 2
} | [
{
"pp": "case pos\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed[Pi.topologicalSpace] s\nhne : s.Nonempty\nx : (n : ℕ) → E n\nhx : x ∈ s\n⊢ (s ∩ cylinder x (longestPrefix x s)).Nonempty",
"ppTerm": "?pos✝",
... | [] | exact ⟨x, hx, self_mem_cylinder _ _⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 524,
"column": 4
} | {
"line": 524,
"column": 40
} | {
"line": 525,
"column": 2
} | [
{
"pp": "case pos\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed[Pi.topologicalSpace] s\nhne : s.Nonempty\nx : (n : ℕ) → E n\nhx : x ∈ s\n⊢ (s ∩ cylinder x (longestPrefix x s)).Nonempty",
"ppTerm": "?pos✝",
... | [] | exact ⟨x, hx, self_mem_cylinder _ _⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 70
} | {
"line": 99,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : Measure α\nc : β\ns : Set β\nhs : MeasurableSet s\nhsc : c ∈ s\n⊢ μ (if c ∈ s then univ else ∅) = μ univ * s.indicator 1 c",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Iff.mp... | [
"case neg\nα : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : Measure α\nc : β\ns : Set β\nhs : MeasurableSet s\nhsc : c ∉ s\n⊢ μ (if c ∈ s then univ else ∅) = μ univ * s.indicator 1 c"
] | · rw [(Set.indicator_eq_one_iff_mem _).mpr hsc, mul_one, if_pos hsc] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.SetTheory.Cardinal.ENNReal | {
"line": 20,
"column": 2
} | {
"line": 20,
"column": 28
} | {
"line": 20,
"column": 29
} | [
{
"pp": "α : Type u_1\n⊢ (↑(ENat.card α)).toReal = ↑(Nat.card α)",
"ppTerm": "?m.2",
"assigned": true,
"usedConstants": [
"Real",
"Finite",
"finite_or_infinite",
"Nat.card",
"ENat.toENNReal",
"Nat.cast",
"Or.casesOn",
"ENNReal.toReal",
"Eq.refl",... | [
"case inl\nα : Type u_1\nh✝ : Finite α\n⊢ (↑(ENat.card α)).toReal = ↑(Nat.card α)",
"case inr\nα : Type u_1\nh✝ : Infinite α\n⊢ (↑(ENat.card α)).toReal = ↑(Nat.card α)"
] | cases finite_or_infinite α | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 268,
"column": 75
} | {
"line": 272,
"column": 69
} | {
"line": 274,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\ns : Set α\na : α\nhs : MeasurableSet s\ninst✝ : Decidable (a ∈ s)\n⊢ (dirac a).restrict s = if a ∈ s then dirac a else 0",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"MeasureTheory.Measure... | [] | by
split_ifs with has
· apply restrict_eq_self_of_ae_mem
rw [ae_dirac_iff] <;> assumption
· rw [restrict_eq_zero, dirac_apply' _ hs, indicator_of_notMem has] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 654,
"column": 14
} | {
"line": 654,
"column": 18
} | {
"line": 655,
"column": 14
} | [
{
"pp": "case inr\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed[Pi.topologicalSpace] s\nhne : s.Nonempty\nf : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x ↦ if x ∈ s then x else ⋯.some\nfs : ∀ x ∈ s, f x = x\nx y : (n ... | [
"case inr\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed[Pi.topologicalSpace] s\nhne : s.Nonempty\nf : ((n : ℕ) → E n) → (n : ℕ) → E n := ⋯\nfs : ∀ x ∈ s, f x = x\nx y : (n : ℕ) → E n\nhxy : x ≠ y\nhfxfy : f x ≠ f y\nI2... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.MeasureTheory.Measure.GiryMonad | {
"line": 297,
"column": 24
} | {
"line": 297,
"column": 42
} | {
"line": 297,
"column": 43
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_3\ninst✝ : MeasurableSpace γ\nm : Measure α\nf : α → Measure β\ng : β → Measure γ\nhf : AEMeasurable f m\nhg : AEMeasurable g (m.bind f)\ns : Set γ\nhs : MeasurableSet s\n⊢ ∫⁻ (a : β), (g a) s ∂m.bind f = (m.bind fun... | [
"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_3\ninst✝ : MeasurableSpace γ\nm : Measure α\nf : α → Measure β\ng : β → Measure γ\nhf : AEMeasurable f m\nhg : AEMeasurable g (m.bind f)\ns : Set γ\nhs : MeasurableSet s\n⊢ ∫⁻ (a : α), ∫⁻ (x : β), (g x) s ∂f a ∂m = (m.bind fun a... | lintegral_bind hf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.OpenPos | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 65
} | {
"line": 122,
"column": 4
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\nm : MeasurableSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\nμ : Measure X\ninst✝ : μ.IsOpenPosMeasure\nU : Set X\nf g : X → Y\nhU : IsOpen[inst✝³] U\nhf : ContinuousOn f U\nhg : ContinuousOn g U\nh : μ {a | a ∈ U ∧ ¬f a = g a} = 0\na ... | [
"X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\nm : MeasurableSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\nμ : Measure X\ninst✝ : μ.IsOpenPosMeasure\nU : Set X\nf g : X → Y\nhU : IsOpen[inst✝³] U\nhf : ContinuousOn f U\nhg : ContinuousOn g U\nh : μ {a | a ∈ U ∧ ¬f a = g a} = 0\na : X\nha : a ... | rcases ha with ⟨ha : a ∈ U, ha' : (f a, g a) ∈ (diagonal Y)ᶜ⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 245,
"column": 39
} | {
"line": 250,
"column": 45
} | {
"line": 252,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : Measure α\npa qa : Set α → Prop\nH : μ.InnerRegularWRT pa qa\nf : α → β\nhf : AEMeasurable f μ\npb qb : Set β → Prop\nhAB : ∀ (U : Set β), qb U → qa (f ⁻¹' U)\nhAB' : ∀ (K : Set α), pa K → pb (f '' K)\nhB₂ : ∀ (U : S... | [] | by
intro U hU r hr
rw [map_apply_of_aemeasurable hf (hB₂ _ hU)] at hr
rcases H (hAB U hU) r hr with ⟨K, hKU, hKc, hK⟩
refine ⟨f '' K, image_subset_iff.2 hKU, hAB' _ hKc, ?_⟩
exact hK.trans_le (le_map_apply_image hf _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 639,
"column": 2
} | {
"line": 639,
"column": 61
} | {
"line": 640,
"column": 2
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\nγ : Type u_6\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nμ ν : Measure (α × β × γ)\ninst✝ : IsFiniteMeasure μ\nh :\n ∀ {s : Set α} {t : Set β} {u : Set γ},\n MeasurableSet s → MeasurableSet t → MeasurableSet u → μ (s ×ˢ t ×ˢ u) = ν (s ×ˢ t ×ˢ... | [
"α : Type u_4\nβ : Type u_5\nγ : Type u_6\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nμ ν : Measure (α × β × γ)\ninst✝ : IsFiniteMeasure μ\nh :\n ∀ {s : Set α} {t : Set β} {u : Set γ},\n MeasurableSet s → MeasurableSet t → MeasurableSet u → μ (s ×ˢ t ×ˢ u) = ν (s ×ˢ t ×ˢ u)\ns : Set... | let C := image2 (· ×ˢ ·) { s : Set α | MeasurableSet s } C₂ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 836,
"column": 2
} | {
"line": 837,
"column": 76
} | {
"line": 838,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nδ : Type u_4\ninst✝² : MeasurableSpace δ\nf : α → β\ng : γ → δ\nμa : Measure α\nμc : Measure γ\ninst✝¹ : SFinite μa\ninst✝ : SFinite μc\nhf : Measurable f\nhg : Measurable g\n⊢ ... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nδ : Type u_4\ninst✝² : MeasurableSpace δ\nf : α → β\ng : γ → δ\nμa : Measure α\nμc : Measure γ\ninst✝¹ : SFinite μa\ninst✝ : SFinite μc\nhf : Measurable f\nhg : Measurable g\n⊢ (sum fun p ↦... | simp_rw [← sum_sfiniteSeq μa, ← sum_sfiniteSeq μc, map_sum hf.aemeasurable,
map_sum hg.aemeasurable, prod_sum, map_sum (hf.prodMap hg).aemeasurable] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.Group.Measure | {
"line": 443,
"column": 11
} | {
"line": 443,
"column": 30
} | {
"line": 444,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulLeftInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ (comap ... | [
"G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulLeftInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ μ (⇑f '' (fun x ↦ g... | rw [hf.comap_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Group.Measure | {
"line": 443,
"column": 11
} | {
"line": 443,
"column": 30
} | {
"line": 444,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulLeftInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ (comap ... | [
"G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulLeftInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ μ (⇑f '' (fun x ↦ g... | rw [hf.comap_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Group.Measure | {
"line": 443,
"column": 11
} | {
"line": 443,
"column": 30
} | {
"line": 444,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulLeftInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ (comap ... | [
"G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulLeftInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ μ (⇑f '' (fun x ↦ g... | rw [hf.comap_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Group.Measure | {
"line": 443,
"column": 11
} | {
"line": 443,
"column": 30
} | {
"line": 444,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulLeftInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ μ (⇑f '... | [
"G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulLeftInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ μ (⇑f '' (fun x ↦ g... | rw [hf.comap_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Group.Measure | {
"line": 443,
"column": 11
} | {
"line": 443,
"column": 30
} | {
"line": 444,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulLeftInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ μ (⇑f '... | [
"G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulLeftInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ μ (⇑f '' (fun x ↦ g... | rw [hf.comap_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Group.Measure | {
"line": 443,
"column": 11
} | {
"line": 443,
"column": 30
} | {
"line": 444,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulLeftInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ μ (⇑f '... | [
"G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulLeftInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ μ (⇑f '' (fun x ↦ g... | rw [hf.comap_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Group.Measure | {
"line": 443,
"column": 11
} | {
"line": 443,
"column": 30
} | {
"line": 444,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulLeftInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ μ (⇑f '... | [
"G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulLeftInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ μ (⇑f '' (fun x ↦ g... | rw [hf.comap_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Group.Measure | {
"line": 443,
"column": 11
} | {
"line": 443,
"column": 30
} | {
"line": 444,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulLeftInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ μ (⇑f '... | [
"G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulLeftInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ μ (⇑f '' (fun x ↦ g... | rw [hf.comap_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Group.Measure | {
"line": 443,
"column": 11
} | {
"line": 443,
"column": 30
} | {
"line": 444,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulLeftInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ μ (⇑f '... | [
"G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulLeftInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ μ (⇑f '' (fun x ↦ g... | rw [hf.comap_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Group.Measure | {
"line": 461,
"column": 11
} | {
"line": 461,
"column": 30
} | {
"line": 462,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulRightInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ (comap... | [
"G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulRightInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ μ (⇑f '' (fun x ↦ ... | rw [hf.comap_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Group.Measure | {
"line": 461,
"column": 11
} | {
"line": 461,
"column": 30
} | {
"line": 462,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulRightInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ (comap... | [
"G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulRightInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ μ (⇑f '' (fun x ↦ ... | rw [hf.comap_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Group.Measure | {
"line": 461,
"column": 11
} | {
"line": 461,
"column": 30
} | {
"line": 462,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulRightInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ (comap... | [
"G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulRightInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ μ (⇑f '' (fun x ↦ ... | rw [hf.comap_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Group.Measure | {
"line": 461,
"column": 11
} | {
"line": 461,
"column": 30
} | {
"line": 462,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulRightInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ μ (⇑f ... | [
"G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulRightInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ μ (⇑f '' (fun x ↦ ... | rw [hf.comap_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Group.Measure | {
"line": 461,
"column": 11
} | {
"line": 461,
"column": 30
} | {
"line": 462,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulRightInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ μ (⇑f ... | [
"G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulRightInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ μ (⇑f '' (fun x ↦ ... | rw [hf.comap_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Group.Measure | {
"line": 461,
"column": 11
} | {
"line": 461,
"column": 30
} | {
"line": 462,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulRightInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ μ (⇑f ... | [
"G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulRightInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ μ (⇑f '' (fun x ↦ ... | rw [hf.comap_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Group.Measure | {
"line": 461,
"column": 11
} | {
"line": 461,
"column": 30
} | {
"line": 462,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulRightInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ μ (⇑f ... | [
"G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulRightInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ μ (⇑f '' (fun x ↦ ... | rw [hf.comap_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Group.Measure | {
"line": 461,
"column": 11
} | {
"line": 461,
"column": 30
} | {
"line": 462,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulRightInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ μ (⇑f ... | [
"G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulRightInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ μ (⇑f '' (fun x ↦ ... | rw [hf.comap_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Group.Measure | {
"line": 461,
"column": 11
} | {
"line": 461,
"column": 30
} | {
"line": 462,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulRightInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ μ (⇑f ... | [
"G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulRightInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet s\n⊢ μ (⇑f '' (fun x ↦ ... | rw [hf.comap_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 758,
"column": 6
} | {
"line": 758,
"column": 19
} | {
"line": 760,
"column": 4
} | [
{
"pp": "γ : Type u_3\nβ : Type u_4\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : PolishSpace γ\ninst✝³ : TopologicalSpace β\ninst✝² : T2Space β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\nf : γ → β\nf_cont : Continuous[inst✝⁵, inst✝³] f\nf_inj : Injective f\nthis✝ : UpgradedIsCompletelyMetrizableSpac... | [] | exact A.2 B.1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 1047,
"column": 2
} | {
"line": 1052,
"column": 74
} | {
"line": 1054,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝¹ : SFinite ν\ninst✝ : SFinite μ\ns : Set α\nt : Set β\nf : α × β → ℝ≥0∞\nhf : AEMeasurable f ((μ.prod ν).restrict (s ×ˢ t))\n⊢ ∫⁻ (z : α × β) in s ×ˢ t, f z ∂μ.prod ν = ∫⁻ (y : β) in ... | [] | rw [← Measure.prod_restrict, ← lintegral_prod_swap, Measure.prod_restrict,
setLIntegral_prod]
· rfl
· refine AEMeasurable.comp_measurable ?_ measurable_swap
convert! hf
rw [← Measure.prod_restrict, Measure.prod_swap, Measure.prod_restrict] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 1047,
"column": 2
} | {
"line": 1052,
"column": 74
} | {
"line": 1054,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝¹ : SFinite ν\ninst✝ : SFinite μ\ns : Set α\nt : Set β\nf : α × β → ℝ≥0∞\nhf : AEMeasurable f ((μ.prod ν).restrict (s ×ˢ t))\n⊢ ∫⁻ (z : α × β) in s ×ˢ t, f z ∂μ.prod ν = ∫⁻ (y : β) in ... | [] | rw [← Measure.prod_restrict, ← lintegral_prod_swap, Measure.prod_restrict,
setLIntegral_prod]
· rfl
· refine AEMeasurable.comp_measurable ?_ measurable_swap
convert! hf
rw [← Measure.prod_restrict, Measure.prod_swap, Measure.prod_restrict] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 955,
"column": 8
} | {
"line": 955,
"column": 49
} | {
"line": 956,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : MeasurableSpace α\nμ : Measure α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : μ.InnerRegularCompactLTTop\ninst✝² : IsLocallyFiniteMeasure μ\ninst✝¹ : R1Space α\ninst✝ : BorelSpace α\ns : Set α\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nε : ℝ≥0∞\nhε : ε ≠ 0\nthis : ε / 2 ≠ 0\nK : Set α\nhKs ... | [] | exact ne_top_of_le_ne_top hμs (by gcongr) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Group.Prod | {
"line": 192,
"column": 2
} | {
"line": 192,
"column": 6
} | {
"line": 193,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SFinite ν\ninst✝³ : SFinite μ\ninst✝² : MeasurableInv G\ninst✝¹ : μ.IsMulLeftInvariant\ninst✝ : ν.IsMulLeftInvariant\nf : G → G → ℝ≥0∞\nhf : AEMeasurable (uncurry f) (μ.prod ν)\nh : Measurab... | [
"G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SFinite ν\ninst✝³ : SFinite μ\ninst✝² : MeasurableInv G\ninst✝¹ : μ.IsMulLeftInvariant\ninst✝ : ν.IsMulLeftInvariant\nf : G → G → ℝ≥0∞\nhf : AEMeasurable (uncurry f) (μ.prod ν)\nh : Measurable fun z ↦ (... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.MeasureTheory.Group.Prod | {
"line": 237,
"column": 15
} | {
"line": 237,
"column": 19
} | {
"line": 237,
"column": 19
} | [
{
"pp": "G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SFinite ν\ninst✝³ : SFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : μ.IsMulLeftInvariant\ninst✝ : ν.IsMulLeftInvariant\nsm : MeasurableSet s\nf : G → ℝ≥0∞\nhf : Measurable f\nms : ∀... | [
"G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SFinite ν\ninst✝³ : SFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : μ.IsMulLeftInvariant\ninst✝ : ν.IsMulLeftInvariant\nsm : MeasurableSet s\nf : G → ℝ≥0∞\nhf : Measurable f\nms : ∀ (x : G), Me... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 943,
"column": 4
} | {
"line": 950,
"column": 55
} | {
"line": 951,
"column": 2
} | [
{
"pp": "case h₁\nα : Type u_6\nβ : Type u_7\nt : Set α\ng : α → β\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : MeasurableSpace α\ninst✝⁸ : BorelSpace α\ninst✝⁷ : LinearOrder α\ninst✝⁶ : OrderTopology α\ninst✝⁵ : PolishSpace α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : MeasurableSpace β\ninst✝² : BorelSpace β\ninst✝¹ : ... | [] | apply (ht.diff ht').image_of_continuousOn_injOn (h'g.mono sdiff_subset)
intro x hx y hy hxy
contrapose! hxy
wlog! H : x < y generalizing x y with h
· have : y < x := lt_of_le_of_ne H hxy.symm
exact (h hy hx hxy.symm this).symm
intro h
exact hx.2 ⟨hx.1, x, y, hx.1, hy.1, H, rfl, h.symm⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 943,
"column": 4
} | {
"line": 950,
"column": 55
} | {
"line": 951,
"column": 2
} | [
{
"pp": "case h₁\nα : Type u_6\nβ : Type u_7\nt : Set α\ng : α → β\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : MeasurableSpace α\ninst✝⁸ : BorelSpace α\ninst✝⁷ : LinearOrder α\ninst✝⁶ : OrderTopology α\ninst✝⁵ : PolishSpace α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : MeasurableSpace β\ninst✝² : BorelSpace β\ninst✝¹ : ... | [] | apply (ht.diff ht').image_of_continuousOn_injOn (h'g.mono sdiff_subset)
intro x hx y hy hxy
contrapose! hxy
wlog! H : x < y generalizing x y with h
· have : y < x := lt_of_le_of_ne H hxy.symm
exact (h hy hx hxy.symm this).symm
intro h
exact hx.2 ⟨hx.1, x, y, hx.1, hy.1, H, rfl, h.symm⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Group.Prod | {
"line": 320,
"column": 4
} | {
"line": 320,
"column": 30
} | {
"line": 321,
"column": 2
} | [
{
"pp": "case inr\nG : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\ninst✝⁴ : MeasurableInv G\nμ' ν' : Measure G\ninst✝³ : SigmaFinite μ'\ninst✝² : SigmaFinite ν'\ninst✝¹ : μ'.IsMulLeftInvariant\ninst✝ : ν'.IsMulLeftInvariant\nt s : Set G\nh2s : ν' s ≠ 0\nh3s : ν' s ≠ ∞\nhs ... | [] | rw [← hμ, ← hν, this _ hm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral | {
"line": 92,
"column": 67
} | {
"line": 93,
"column": 54
} | {
"line": 95,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf : α → β\n⊢ HasFiniteIntegral f μ ↔ ∫⁻ (a : α), ENNReal.ofReal ‖f a‖ ∂μ < ∞",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
... | [] | by
simp only [hasFiniteIntegral_iff_enorm, ofReal_norm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.Egorov | {
"line": 84,
"column": 4
} | {
"line": 84,
"column": 28
} | {
"line": 85,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\ninst✝² : PseudoEMetricSpace β\nμ : Measure α\ns : Set α\nf : ι → α → β\ng : α → β\ninst✝¹ : SemilatticeSup ι\ninst✝ : Countable ι\nhf : ∀ (n : ι), Measurable fun a ↦ edist (f n a) (g a)\nhsm : MeasurableSet s\nhs : μ s ≠ ∞\nhfg ... | [] | exact tendsto_const_nhds | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 495,
"column": 2
} | {
"line": 497,
"column": 15
} | {
"line": 498,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nf_meas : Measurable f\nhf : ∀ᵐ (x : α) ∂μ, f x < ∞\ng i : α → ℝ≥0∞\ni_meas : Measurable i\nhi : i ≤ f * g\nA : (fun x ↦ (f x)⁻¹ * i x) ≤ g\nx : α\nh'x : f x < ∞\nhx : f x = 0\n⊢ i x ≤ (f * fun i_1 ↦ (f i_1)⁻¹ * i i_1) x",
... | [
"case inr\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nf_meas : Measurable f\nhf : ∀ᵐ (x : α) ∂μ, f x < ∞\ng i : α → ℝ≥0∞\ni_meas : Measurable i\nhi : i ≤ f * g\nA : (fun x ↦ (f x)⁻¹ * i x) ≤ g\nx : α\nh'x : f x < ∞\nhx : f x ≠ 0\n⊢ i x ≤ (f * fun i_1 ↦ (f i_1)⁻¹ * i i_1) x"
] | · have := hi x
simp only [hx, zero_mul, Pi.mul_apply, nonpos_iff_eq_zero] at this
simp [this] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 515,
"column": 31
} | {
"line": 519,
"column": 17
} | {
"line": 520,
"column": 4
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f μ\nh'f : ∀ᵐ (x : α) ∂μ, f x < ∞\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\n⊢ ∫⁻ (a : α), g a ∂μ.withDensity f' = ∫⁻ (a : α), (f' * g) a ∂μ",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants"... | [] | by
apply lintegral_withDensity_eq_lintegral_mul_non_measurable _ hf.measurable_mk
filter_upwards [h'f, hf.ae_eq_mk]
intro x hx h'x
rwa [← h'x] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Complex.Basic | {
"line": 328,
"column": 19
} | {
"line": 328,
"column": 67
} | {
"line": 329,
"column": 2
} | [
{
"pp": "⊢ ∀ (n : ℕ) (x : ℂ), (n + 1) • x = n • x + x",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"instHSMul",
"NSMul.ofSMul",
"Mathlib.Tacti... | [] | intros; ext <;> simp [smul_re, smul_im] <;> ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Complex.Basic | {
"line": 328,
"column": 19
} | {
"line": 328,
"column": 67
} | {
"line": 329,
"column": 2
} | [
{
"pp": "⊢ ∀ (n : ℕ) (x : ℂ), (n + 1) • x = n • x + x",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"instHSMul",
"NSMul.ofSMul",
"Mathlib.Tacti... | [] | intros; ext <;> simp [smul_re, smul_im] <;> ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Complex.Basic | {
"line": 329,
"column": 20
} | {
"line": 329,
"column": 68
} | {
"line": 330,
"column": 2
} | [
{
"pp": "⊢ ∀ (n : ℕ) (a : ℂ), ↑n.succ • a = ↑n • a + a",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Int.cast",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Int.cast_natCast",
"zsmul_eq_mul",
"R... | [] | intros; ext <;> simp [smul_re, smul_im] <;> ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Complex.Basic | {
"line": 329,
"column": 20
} | {
"line": 329,
"column": 68
} | {
"line": 330,
"column": 2
} | [
{
"pp": "⊢ ∀ (n : ℕ) (a : ℂ), ↑n.succ • a = ↑n • a + a",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Int.cast",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Int.cast_natCast",
"zsmul_eq_mul",
"R... | [] | intros; ext <;> simp [smul_re, smul_im] <;> ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Complex.Basic | {
"line": 330,
"column": 19
} | {
"line": 330,
"column": 67
} | {
"line": 331,
"column": 2
} | [
{
"pp": "⊢ ∀ (n : ℕ) (a : ℂ), Int.negSucc n • a = -(↑n.succ • a)",
"ppTerm": "?m.71",
"assigned": true,
"usedConstants": [
"neg_add_rev",
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Mathlib.Tactic.Ring.Common.neg_zero",
"... | [] | intros; ext <;> simp [smul_re, smul_im] <;> ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Complex.Basic | {
"line": 330,
"column": 19
} | {
"line": 330,
"column": 67
} | {
"line": 331,
"column": 2
} | [
{
"pp": "⊢ ∀ (n : ℕ) (a : ℂ), Int.negSucc n • a = -(↑n.succ • a)",
"ppTerm": "?m.71",
"assigned": true,
"usedConstants": [
"neg_add_rev",
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Mathlib.Tactic.Ring.Common.neg_zero",
"... | [] | intros; ext <;> simp [smul_re, smul_im] <;> ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Norm | {
"line": 39,
"column": 6
} | {
"line": 39,
"column": 73
} | {
"line": 39,
"column": 74
} | [
{
"pp": "z : ℂ\n⊢ |z.re| ≤ ‖z‖",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"Eq.mpr",
"Complex.norm_nonneg",
"Real.partialOrder",
"Real.instLE",
"Real",
"HMul.hMul",
"Real.lattice",
"abs",
... | [
"z : ℂ\n⊢ |z.re| * |z.re| ≤ ‖z‖ * ‖z‖"
] | mul_self_le_mul_self_iff (abs_nonneg z.re) (Complex.norm_nonneg _), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.BigOperators.Expect | {
"line": 42,
"column": 2
} | {
"line": 42,
"column": 57
} | {
"line": 44,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝³ : AddCommMonoid α\ninst✝² : PartialOrder α\ninst✝¹ : IsOrderedAddMonoid α\ninst✝ : Module ℚ≥0 α\ns : Finset ι\nf : ι → α\nhf : ∀ i ∈ s, f i ≤ 0\n⊢ 𝔼 i ∈ s, f i = 0 ↔ ∀ i ∈ s, f i = 0",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"inv_eq_... | [] | simp +contextual [expect, sum_eq_zero_iff_of_nonpos hf] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Order.BigOperators.Expect | {
"line": 42,
"column": 2
} | {
"line": 42,
"column": 57
} | {
"line": 44,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝³ : AddCommMonoid α\ninst✝² : PartialOrder α\ninst✝¹ : IsOrderedAddMonoid α\ninst✝ : Module ℚ≥0 α\ns : Finset ι\nf : ι → α\nhf : ∀ i ∈ s, f i ≤ 0\n⊢ 𝔼 i ∈ s, f i = 0 ↔ ∀ i ∈ s, f i = 0",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"inv_eq_... | [] | simp +contextual [expect, sum_eq_zero_iff_of_nonpos hf] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.BigOperators.Expect | {
"line": 42,
"column": 2
} | {
"line": 42,
"column": 57
} | {
"line": 44,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝³ : AddCommMonoid α\ninst✝² : PartialOrder α\ninst✝¹ : IsOrderedAddMonoid α\ninst✝ : Module ℚ≥0 α\ns : Finset ι\nf : ι → α\nhf : ∀ i ∈ s, f i ≤ 0\n⊢ 𝔼 i ∈ s, f i = 0 ↔ ∀ i ∈ s, f i = 0",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"inv_eq_... | [] | simp +contextual [expect, sum_eq_zero_iff_of_nonpos hf] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.Basic | {
"line": 267,
"column": 2
} | {
"line": 267,
"column": 98
} | {
"line": 269,
"column": 0
} | [
{
"pp": "E : Type u_2\ninst✝² : NonUnitalNormedRing E\ninst✝¹ : StarRing E\ninst✝ : CStarRing E\ne : E\nhe : IsStarProjection e\n⊢ ‖e‖ * (‖e‖ - 1) = 0",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Norm.norm",
"_private.Mathlib.Analysis.CStarAlgebra.Basic.0.IsStarProjection.n... | [] | simp [mul_sub, ← CStarRing.norm_star_mul_self, he.isSelfAdjoint.star_eq, he.isIdempotentElem.eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Star.Unitary | {
"line": 139,
"column": 38
} | {
"line": 140,
"column": 71
} | {
"line": 142,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Monoid R\ninst✝ : StarMul R\nu : R\nhu : IsUnit u\n⊢ u ∈ unitary R ↔ u * star u = 1",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"IsUnit.star",
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"star_star",
"Monoid.toMulOneClass",... | [] | by
rw [← star_mem_iff, hu.star.mem_unitary_iff_star_mul_self, star_star] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Complex.Module | {
"line": 216,
"column": 14
} | {
"line": 216,
"column": 90
} | {
"line": 216,
"column": 90
} | [
{
"pp": "E : Type u_1\ninst✝³ : AddCommGroup E\ninst✝² : Star E\ninst✝¹ : Module ℂ E\ninst✝ : StarModule ℂ E\nr : ℝ\na : E\n⊢ star (r • a) = star r • star a",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Real",
"instHSMul",
"NonUnit... | [] | by rw [← smul_one_smul ℂ r a, star_smul, star_smul, star_one, smul_one_smul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Basic | {
"line": 727,
"column": 7
} | {
"line": 730,
"column": 43
} | {
"line": 732,
"column": 0
} | [] | [] | ‖imaginaryPart x‖ = ‖realPart (Complex.I • (-x))‖ := by simp
_ ≤ ‖x‖ := by simpa only [smul_neg, map_neg, realPart_I_smul, neg_neg,
AddSubgroupClass.coe_norm, norm_neg, norm_smul, Complex.norm_I, one_mul] using
realPart.norm_le (Complex.I • (-x)) | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Analysis.RCLike.Basic | {
"line": 474,
"column": 2
} | {
"line": 474,
"column": 79
} | {
"line": 476,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz w : K\n⊢ normSq (z - w) = normSq z + normSq w - 2 * re (z * (starRingEnd K) w)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"RingHom.instRingHomClass",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"AddMonoidH... | [] | simp only [normSq_add, sub_eq_add_neg, map_neg, mul_neg, normSq_neg, map_neg] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.RCLike.Basic | {
"line": 474,
"column": 2
} | {
"line": 474,
"column": 79
} | {
"line": 476,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz w : K\n⊢ normSq (z - w) = normSq z + normSq w - 2 * re (z * (starRingEnd K) w)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"RingHom.instRingHomClass",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"AddMonoidH... | [] | simp only [normSq_add, sub_eq_add_neg, map_neg, mul_neg, normSq_neg, map_neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.RCLike.Basic | {
"line": 474,
"column": 2
} | {
"line": 474,
"column": 79
} | {
"line": 476,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz w : K\n⊢ normSq (z - w) = normSq z + normSq w - 2 * re (z * (starRingEnd K) w)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"RingHom.instRingHomClass",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"AddMonoidH... | [] | simp only [normSq_add, sub_eq_add_neg, map_neg, mul_neg, normSq_neg, map_neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.RCLike.Basic | {
"line": 493,
"column": 53
} | {
"line": 494,
"column": 80
} | {
"line": 496,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ re z⁻¹ = re z / normSq z",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"RCLike.conj_re",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Norm.norm",
"Eq.mpr",
"Real",
"instHDiv",
"No... | [] | by
rw [inv_def, normSq_eq_def', mul_comm, re_ofReal_mul, conj_re, div_eq_inv_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.RCLike.Basic | {
"line": 906,
"column": 2
} | {
"line": 907,
"column": 25
} | {
"line": 909,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nx : K\nhx : 0 ≤ x\n⊢ ↑‖x‖ = x",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"Real.instZero",
"Real.instAddMonoid"... | [] | rw [eq_comm, ← norm_le_re_iff_eq_norm, ← sqrt_normSq_eq_norm, normSq_apply]
simp [nonneg_iff.mp hx] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.RCLike.Basic | {
"line": 906,
"column": 2
} | {
"line": 907,
"column": 25
} | {
"line": 909,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nx : K\nhx : 0 ≤ x\n⊢ ↑‖x‖ = x",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"Real.instZero",
"Real.instAddMonoid"... | [] | rw [eq_comm, ← norm_le_re_iff_eq_norm, ← sqrt_normSq_eq_norm, normSq_apply]
simp [nonneg_iff.mp hx] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.RCLike.Basic | {
"line": 1002,
"column": 53
} | {
"line": 1003,
"column": 80
} | {
"line": 1005,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nx : ℝ\nz : K\n⊢ ↑x * z < 0 ↔ x < 0 ∧ 0 < z ∨ 0 < x ∧ z < 0",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"Real",
"Preorder.toLT",
"NonUnitalCommRing.toNonUnitalNon... | [] | by
simpa only [mul_neg, neg_pos, neg_neg_iff_pos] using ofReal_mul_pos_iff x (-z) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.RCLike.Basic | {
"line": 1321,
"column": 30
} | {
"line": 1321,
"column": 57
} | {
"line": 1322,
"column": 4
} | [
{
"pp": "K : Type u_1\nE : Type u_2\ninst✝ : RCLike K\n𝕜 : Type u_3\nh : RCLike 𝕜\n__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField\n⊢ (algebraMap ℝ 𝕜) 1 = 1",
"ppTerm": "?m.186",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"MulOne.toOne",
"Real",
... | [] | exact h.algebraMap.map_one' | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Order.CauSeq.BigOperators | {
"line": 41,
"column": 2
} | {
"line": 41,
"column": 33
} | {
"line": 42,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → β\na : ℕ → α\nn : ℕ\nhm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ a m\nhg : IsCauSeq abs fun n ↦ ∑ i ∈ range n, a i\nε : α\nε0 : ε > 0\ni : ℕ\n... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → β\na : ℕ → α\nn : ℕ\nhm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ a m\nhg : IsCauSeq abs fun n ↦ ∑ i ∈ range n, a i\nε : α\nε0 : ε > 0\ni : ℕ\nhi : ∀ j ≥ i... | generalize hk : j - max n i = k | Lean.Elab.Tactic.evalGeneralize | Lean.Parser.Tactic.generalize |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1189,
"column": 4
} | {
"line": 1189,
"column": 23
} | {
"line": 1190,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\nE'' : Type u_9\nF'' : Type u_10\ninst✝² : NormedAddCommGroup E''\ninst✝¹ : NormedAddCommGroup F''\nc : E''\nl : Filter α\ninst✝ : l.NeBot\n⊢ ((fun _x ↦ c) =O[l] fun _x ↦ 0) ↔ 0 = 0 → c = 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"congrArg",
... | [] | simp [EventuallyEq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1189,
"column": 4
} | {
"line": 1189,
"column": 23
} | {
"line": 1190,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\nE'' : Type u_9\nF'' : Type u_10\ninst✝² : NormedAddCommGroup E''\ninst✝¹ : NormedAddCommGroup F''\nc : E''\nl : Filter α\ninst✝ : l.NeBot\n⊢ ((fun _x ↦ c) =O[l] fun _x ↦ 0) ↔ 0 = 0 → c = 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"congrArg",
... | [] | simp [EventuallyEq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1189,
"column": 4
} | {
"line": 1189,
"column": 23
} | {
"line": 1190,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\nE'' : Type u_9\nF'' : Type u_10\ninst✝² : NormedAddCommGroup E''\ninst✝¹ : NormedAddCommGroup F''\nc : E''\nl : Filter α\ninst✝ : l.NeBot\n⊢ ((fun _x ↦ c) =O[l] fun _x ↦ 0) ↔ 0 = 0 → c = 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"congrArg",
... | [] | simp [EventuallyEq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 354,
"column": 4
} | {
"line": 354,
"column": 15
} | {
"line": 354,
"column": 16
} | [
{
"pp": "case cons.inr\nα : Type u_1\nR : Type u_13\n𝕜 : Type u_15\ninst✝¹ : SeminormedRing R\ninst✝ : NormedDivisionRing 𝕜\nl : Filter α\nι : Type u_17\nf : ι → α → R\ng : ι → α → 𝕜\ni : ι\nL : List ι\nihL :\n (∀ i ∈ L, f i =O[l] g i) →\n (∃ i ∈ L, f i =o[l] g i) →\n (fun x ↦ (List.map (fun x_1 ↦ f... | [] | | inr hL => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 546,
"column": 2
} | {
"line": 547,
"column": 32
} | {
"line": 548,
"column": 2
} | [
{
"pp": "α : Type u_1\nE' : Type u_6\n𝕜 : Type u_15\ninst✝¹ : SeminormedAddCommGroup E'\ninst✝ : NormedDivisionRing 𝕜\nu : α → E'\nv : α → 𝕜\nl : Filter α\ny : 𝕜\nhuv : u =o[l] v\nhv : Tendsto v l (𝓝 y)\n⊢ Tendsto u l (𝓝 0)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"AddGr... | [
"α : Type u_1\nE' : Type u_6\n𝕜 : Type u_15\ninst✝¹ : SeminormedAddCommGroup E'\ninst✝ : NormedDivisionRing 𝕜\nu : α → E'\nv : α → 𝕜\nl : Filter α\ny : 𝕜\nhuv : u =o[l] v\nhv : Tendsto v l (𝓝 y)\n⊢ u =o[l] fun _x ↦ 1"
] | suffices h : u =o[l] fun _x => (1 : 𝕜) by
rwa [isLittleO_one_iff] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Tactic.NormNum.NatFactorial | {
"line": 29,
"column": 2
} | {
"line": 29,
"column": 6
} | {
"line": 30,
"column": 2
} | [
{
"pp": "n l m a b : ℕ\nh₁ : n.ascFactorial l = a\nh₂ : (n + l).ascFactorial m = b\n⊢ n.ascFactorial (l + m) = n.ascFactorial l * (n + l).ascFactorial m",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Nat.ascFactorial",
"instMulNat",
"instHAdd",
... | [
"n l m a b : ℕ\nh₁ : n.ascFactorial l = a\nh₂ : (n + l).ascFactorial m = b\n⊢ n.ascFactorial l * (n + l).ascFactorial m = n.ascFactorial (l + m)"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 748,
"column": 4
} | {
"line": 748,
"column": 14
} | {
"line": 749,
"column": 4
} | [
{
"pp": "E'' : Type u_9\nF'' : Type u_10\ninst✝¹ : NormedAddCommGroup E''\ninst✝ : NormedAddCommGroup F''\nf : ℕ → E''\ng : ℕ → F''\nh✝ : ∀ᶠ (x : ℕ) in atTop, ∀ (y : ℕ), x ≤ y → g y = 0 → f y = 0\nhrec✝ :\n ∀ᶠ (n₀ : ℕ) in atTop,\n ∃ C₀, ∀ᶠ (n : ℕ) in atTop, ∀ C ≥ C₀, (∀ m ∈ Finset.Ico n₀ n, ‖f m‖ ≤ C * ‖g m... | [
"E'' : Type u_9\nF'' : Type u_10\ninst✝¹ : NormedAddCommGroup E''\ninst✝ : NormedAddCommGroup F''\nf : ℕ → E''\ng : ℕ → F''\nh✝ : ∀ᶠ (x : ℕ) in atTop, ∀ (y : ℕ), x ≤ y → g y = 0 → f y = 0\nhrec✝ :\n ∀ᶠ (n₀ : ℕ) in atTop,\n ∃ C₀, ∀ᶠ (n : ℕ) in atTop, ∀ C ≥ C₀, (∀ m ∈ Finset.Ico n₀ n, ‖f m‖ ≤ C * ‖g m‖) → ‖f n‖ ≤... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 749,
"column": 47
} | {
"line": 749,
"column": 69
} | {
"line": 749,
"column": 70
} | [
{
"pp": "E'' : Type u_9\nF'' : Type u_10\ninst✝¹ : NormedAddCommGroup E''\ninst✝ : NormedAddCommGroup F''\nf : ℕ → E''\ng : ℕ → F''\nh✝ : ∀ᶠ (x : ℕ) in atTop, ∀ (y : ℕ), x ≤ y → g y = 0 → f y = 0\nhrec✝ :\n ∀ᶠ (n₀ : ℕ) in atTop,\n ∃ C₀, ∀ᶠ (n : ℕ) in atTop, ∀ C ≥ C₀, (∀ m ∈ Finset.Ico n₀ n, ‖f m‖ ≤ C * ‖g m... | [
"case pos\nE'' : Type u_9\nF'' : Type u_10\ninst✝¹ : NormedAddCommGroup E''\ninst✝ : NormedAddCommGroup F''\nf : ℕ → E''\ng : ℕ → F''\nh✝ : ∀ᶠ (x : ℕ) in atTop, ∀ (y : ℕ), x ≤ y → g y = 0 → f y = 0\nhrec✝ :\n ∀ᶠ (n₀ : ℕ) in atTop,\n ∃ C₀, ∀ᶠ (n : ℕ) in atTop, ∀ C ≥ C₀, (∀ m ∈ Finset.Ico n₀ n, ‖f m‖ ≤ C * ‖g m‖)... | by_cases hm' : g m = 0 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Analysis.Complex.Exponential | {
"line": 378,
"column": 2
} | {
"line": 403,
"column": 39
} | {
"line": 405,
"column": 0
} | [
{
"pp": "x : ℂ\nhx : ‖x‖ ≤ 1\nn : ℕ\nhn : 0 < n\n⊢ ‖cexp x - ∑ m ∈ range n, x ^ m / ↑m.factorial‖ ≤ ‖x‖ ^ n * (↑n.succ * (↑n.factorial * ↑n)⁻¹)",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"CauSeq.addGroup",
"Real.instIsOrderedRing",
"Norm.norm",
... | [] | rw [← lim_const (abv := norm) (∑ m ∈ range n, _), exp, sub_eq_add_neg,
← lim_neg, lim_add, ← lim_norm]
refine lim_le (CauSeq.le_of_exists ⟨n, fun j hj => ?_⟩)
simp_rw [← sub_eq_add_neg]
change
‖(∑ m ∈ range j, x ^ m / m.factorial) - ∑ m ∈ range n, x ^ m / m.factorial‖ ≤
‖x‖ ^ n * ((n.succ : ℝ) * (n.... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Exponential | {
"line": 378,
"column": 2
} | {
"line": 403,
"column": 39
} | {
"line": 405,
"column": 0
} | [
{
"pp": "x : ℂ\nhx : ‖x‖ ≤ 1\nn : ℕ\nhn : 0 < n\n⊢ ‖cexp x - ∑ m ∈ range n, x ^ m / ↑m.factorial‖ ≤ ‖x‖ ^ n * (↑n.succ * (↑n.factorial * ↑n)⁻¹)",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"CauSeq.addGroup",
"Real.instIsOrderedRing",
"Norm.norm",
... | [] | rw [← lim_const (abv := norm) (∑ m ∈ range n, _), exp, sub_eq_add_neg,
← lim_neg, lim_add, ← lim_norm]
refine lim_le (CauSeq.le_of_exists ⟨n, fun j hj => ?_⟩)
simp_rw [← sub_eq_add_neg]
change
‖(∑ m ∈ range j, x ^ m / m.factorial) - ∑ m ∈ range n, x ^ m / m.factorial‖ ≤
‖x‖ ^ n * ((n.succ : ℝ) * (n.... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Exponential | {
"line": 434,
"column": 8
} | {
"line": 434,
"column": 89
} | {
"line": 435,
"column": 8
} | [
{
"pp": "x : ℂ\nn : ℕ\nhx : ‖x‖ / ↑n.succ ≤ 1 / 2\nj : ℕ\nhj✝ : j ≥ n\nk : ℕ := ⋯\nhj : j = n + k\n⊢ -1 ≤ (‖x‖ / ↑n.succ) ^ k - 1",
"ppTerm": "?m.460",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Nat.cast_succ",
"Real.partialOrder",
"Real.instLE",
... | [
"x : ℂ\nn : ℕ\nhx : ‖x‖ / ↑n.succ ≤ 1 / 2\nj : ℕ\nhj✝ : j ≥ n\nk : ℕ := j - n\nhj : j = n + k\n⊢ 0 ≤ ‖x‖ ^ k / (↑n + 1) ^ k"
] | simp only [neg_le_sub_iff_le_add, div_pow, Nat.cast_succ, le_add_iff_nonneg_left] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.Exponential | {
"line": 443,
"column": 24
} | {
"line": 443,
"column": 30
} | {
"line": 443,
"column": 30
} | [
{
"pp": "x : ℂ\nhx : ‖x‖ ≤ 1\n⊢ 0 < 1",
"ppTerm": "?m.90",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
"Bool",
"Nat.decLt",
"Eq.refl",
"instLTNat",
"OfNat.ofNat",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Analysis.Complex.Exponential | {
"line": 443,
"column": 24
} | {
"line": 443,
"column": 30
} | {
"line": 443,
"column": 30
} | [
{
"pp": "x : ℂ\nhx : ‖x‖ ≤ 1\n⊢ 0 < 1",
"ppTerm": "?m.90",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
"Bool",
"Nat.decLt",
"Eq.refl",
"instLTNat",
"OfNat.ofNat",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Exponential | {
"line": 443,
"column": 24
} | {
"line": 443,
"column": 30
} | {
"line": 443,
"column": 30
} | [
{
"pp": "x : ℂ\nhx : ‖x‖ ≤ 1\n⊢ 0 < 1",
"ppTerm": "?m.90",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
"Bool",
"Nat.decLt",
"Eq.refl",
"instLTNat",
"OfNat.ofNat",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Exponential | {
"line": 451,
"column": 24
} | {
"line": 451,
"column": 30
} | {
"line": 451,
"column": 30
} | [
{
"pp": "x : ℂ\nhx : ‖x‖ ≤ 1\n⊢ 0 < 2",
"ppTerm": "?m.103",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
"Bool",
"Nat.decLt",
"Eq.refl",
"instLTNat",
"OfNat.ofNat",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Analysis.Complex.Exponential | {
"line": 451,
"column": 24
} | {
"line": 451,
"column": 30
} | {
"line": 451,
"column": 30
} | [
{
"pp": "x : ℂ\nhx : ‖x‖ ≤ 1\n⊢ 0 < 2",
"ppTerm": "?m.103",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
"Bool",
"Nat.decLt",
"Eq.refl",
"instLTNat",
"OfNat.ofNat",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Exponential | {
"line": 451,
"column": 24
} | {
"line": 451,
"column": 30
} | {
"line": 451,
"column": 30
} | [
{
"pp": "x : ℂ\nhx : ‖x‖ ≤ 1\n⊢ 0 < 2",
"ppTerm": "?m.103",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
"Bool",
"Nat.decLt",
"Eq.refl",
"instLTNat",
"OfNat.ofNat",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 790,
"column": 38
} | {
"line": 790,
"column": 46
} | {
"line": 790,
"column": 47
} | [
{
"pp": "x : ℝ\n⊢ (rexp x - rexp (-x)) / 2 / cosh x = (rexp x - rexp (-x)) / (rexp x + rexp (-x))",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"congrArg",
"Real.instDivInvMonoid",
"Real.instSub",
"Nat.instAtLeastTwo... | [
"x : ℝ\n⊢ (rexp x - rexp (-x)) / 2 / ((rexp x + rexp (-x)) / 2) = (rexp x - rexp (-x)) / (rexp x + rexp (-x))"
] | cosh_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 43
} | {
"line": 59,
"column": 2
} | [
{
"pp": "r : ℝ\nhr_nonneg : 0 ≤ r\nhr_le : r ≤ 1\nx y : ℂ\nhyx : ‖y - x‖ < r\nhy_eq : y = x + (y - x)\nhyx_sq_le : ‖y - x‖ ^ 2 ≤ r * ‖y - x‖\nz : ℂ\nhz : ‖z‖ ≤ 1\nthis : ‖cexp (x + z) - cexp x - z • cexp x‖ ≤ ‖cexp x‖ * ‖z‖ ^ 2\n⊢ ‖cexp (x + z) - cexp x‖ - ‖z • cexp x‖ ≤ ‖cexp x‖ * ‖z‖ ^ 2",
"ppTerm": "?m.2... | [] | exact (norm_sub_norm_le _ _).trans this | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 433,
"column": 2
} | {
"line": 434,
"column": 54
} | {
"line": 436,
"column": 0
} | [
{
"pp": "α : Type u_1\nl : Filter α\nf g : α → ℝ\n⊢ ((fun x ↦ rexp (f x)) =o[l] fun x ↦ rexp (g x)) ↔ Tendsto (fun x ↦ g x - f x) l atTop",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"False",
"Real.partialOrder",
"Real",
"_private.Math... | [] | simp only [isLittleO_iff_tendsto, exp_ne_zero, ← exp_sub, ← tendsto_neg_atTop_iff, false_imp_iff,
imp_true_iff, tendsto_exp_comp_nhds_zero, neg_sub] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 433,
"column": 2
} | {
"line": 434,
"column": 54
} | {
"line": 436,
"column": 0
} | [
{
"pp": "α : Type u_1\nl : Filter α\nf g : α → ℝ\n⊢ ((fun x ↦ rexp (f x)) =o[l] fun x ↦ rexp (g x)) ↔ Tendsto (fun x ↦ g x - f x) l atTop",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"False",
"Real.partialOrder",
"Real",
"_private.Math... | [] | simp only [isLittleO_iff_tendsto, exp_ne_zero, ← exp_sub, ← tendsto_neg_atTop_iff, false_imp_iff,
imp_true_iff, tendsto_exp_comp_nhds_zero, neg_sub] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 433,
"column": 2
} | {
"line": 434,
"column": 54
} | {
"line": 436,
"column": 0
} | [
{
"pp": "α : Type u_1\nl : Filter α\nf g : α → ℝ\n⊢ ((fun x ↦ rexp (f x)) =o[l] fun x ↦ rexp (g x)) ↔ Tendsto (fun x ↦ g x - f x) l atTop",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"False",
"Real.partialOrder",
"Real",
"_private.Math... | [] | simp only [isLittleO_iff_tendsto, exp_ne_zero, ← exp_sub, ← tendsto_neg_atTop_iff, false_imp_iff,
imp_true_iff, tendsto_exp_comp_nhds_zero, neg_sub] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Group.Pointwise | {
"line": 41,
"column": 2
} | {
"line": 41,
"column": 6
} | {
"line": 42,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝ : SeminormedGroup E\ns t : Set E\nhst : IsBounded (s * t)\n⊢ IsBounded s ∨ IsBounded t",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"PseudoMetricSpace.toBornology",
"SeminormedGroup.toPseudoMetricSpace",
"Bornology.IsBounded",
"Or.sym... | [
"E : Type u_1\ninst✝ : SeminormedGroup E\ns t : Set E\nhst : IsBounded (s * t)\n⊢ IsBounded t ∨ IsBounded s"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Analysis.Normed.Group.Pointwise | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 69
} | {
"line": 151,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝ : SeminormedCommGroup E\nδ : ℝ\nx y : E\n⊢ {x} * closedBall y δ = closedBall (x * y) δ",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"instSMulOfMul",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
... | [] | simp_rw [singleton_mul, ← smul_eq_mul, image_smul, smul_closedBall] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Normed.Group.Pointwise | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 69
} | {
"line": 151,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝ : SeminormedCommGroup E\nδ : ℝ\nx y : E\n⊢ {x} * closedBall y δ = closedBall (x * y) δ",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"instSMulOfMul",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
... | [] | simp_rw [singleton_mul, ← smul_eq_mul, image_smul, smul_closedBall] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Group.Pointwise | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 69
} | {
"line": 151,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝ : SeminormedCommGroup E\nδ : ℝ\nx y : E\n⊢ {x} * closedBall y δ = closedBall (x * y) δ",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"instSMulOfMul",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
... | [] | simp_rw [singleton_mul, ← smul_eq_mul, image_smul, smul_closedBall] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 554,
"column": 17
} | {
"line": 554,
"column": 28
} | {
"line": 554,
"column": 28
} | [
{
"pp": "x y : ℝ\nhx₁ : -(π / 2) ≤ x\nhy₂ : y ≤ π / 2\nhxy : x < y\n⊢ 0 < sin y - sin x",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Real",
"instHDiv",
"HMul.hMul",
"Real.cos",
"congrArg",
"Real.i... | [
"x y : ℝ\nhx₁ : -(π / 2) ≤ x\nhy₂ : y ≤ π / 2\nhxy : x < y\n⊢ 0 < 2 * sin ((y - x) / 2) * cos ((y + x) / 2)"
] | sin_sub_sin | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 789,
"column": 88
} | {
"line": 789,
"column": 100
} | {
"line": 790,
"column": 4
} | [
{
"pp": "⊢ √((1 + 1 / 2) / 2) = √3 / 2",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"instHDiv",
"GroupWithZero.toDivInvMonoid",
"congrArg",
"Real.instDivInvMonoid",
"Nat.instAtLe... | [
"⊢ √((2 + 1) / 2 / 2) = √3 / 2",
"⊢ 2 ≠ 0",
"case hl\n⊢ -π ≤ π / 3",
"case hr\n⊢ π / 3 ≤ π"
] | one_add_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 912,
"column": 2
} | {
"line": 913,
"column": 100
} | {
"line": 914,
"column": 2
} | [
{
"pp": "x : ℝ\nhx : x ∈ Ioo (-(π / 2)) (π / 2)\ny : ℝ\nhy : y ∈ Ioo (-(π / 2)) (π / 2)\nhlt : x < y\n⊢ tan x < tan y",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"sub_pos",
"AddGroup.toSubtractionMonoid",
"N... | [
"x : ℝ\nhx : x ∈ Ioo (-(π / 2)) (π / 2)\ny : ℝ\nhy : y ∈ Ioo (-(π / 2)) (π / 2)\nhlt : x < y\n⊢ 0 < sin (y - x)"
] | rw [tan_eq_sin_div_cos, tan_eq_sin_div_cos,
div_lt_div_iff₀ (cos_pos_of_mem_Ioo hx) (cos_pos_of_mem_Ioo hy), mul_comm, ← sub_pos, ← sin_sub] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Module.Ball.Pointwise | {
"line": 188,
"column": 12
} | {
"line": 188,
"column": 53
} | {
"line": 188,
"column": 53
} | [
{
"pp": "E : Type u_2\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nx z : E\nδ ε : ℝ\nhδ : 0 < δ\nhε : 0 ≤ ε\nh : dist x z < ε + δ\ny : E\nyz : dist z y ≤ ε\nxy : dist y x < δ\n⊢ dist x y < δ ∧ dist y z ≤ ε",
"ppTerm": "?m.78",
"assigned": true,
"usedConstants": [
"Real.instLE",... | [] | by simp [dist_comm x y, dist_comm y z, *] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.CompactOpen | {
"line": 97,
"column": 11
} | {
"line": 97,
"column": 28
} | {
"line": 97,
"column": 29
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : C(X, Y)\ns : Set C(X, Y)\n⊢ s ∈ 𝓝 f ↔\n ∃ i,\n (i.Finite ∧ ∀ (K : Set X) (U : Set Y), (K, U) ∈ i → IsCompact K ∧ IsOpen[inst✝] U ∧ MapsTo (⇑f) K U) ∧\n ⋂ KU ∈ i, {g | MapsTo (⇑g) KU.1 KU.2} ⊆ s",
... | [
"X : Type u_2\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : C(X, Y)\ns : Set C(X, Y)\n⊢ s ∈ ⨅ K, ⨅ (_ : IsCompact K), ⨅ U, ⨅ (_ : IsOpen[inst✝] U), ⨅ (_ : MapsTo (⇑f) K U), 𝓟 {g | MapsTo (⇑g) K U} ↔\n ∃ i,\n (i.Finite ∧ ∀ (K : Set X) (U : Set Y), (K, U) ∈ i → IsCompact K ∧ IsO... | nhds_compactOpen, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Normed.Module.Ball.Pointwise | {
"line": 311,
"column": 48
} | {
"line": 312,
"column": 80
} | {
"line": 314,
"column": 0
} | [
{
"pp": "E : Type u_2\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nδ ε : ℝ\nhε : 0 < ε\nhδ : 0 < δ\nx : E\n⊢ thickening ε (Metric.ball x δ) = Metric.ball x (ε + δ)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Set.ins... | [] | by
rw [← thickening_singleton, thickening_thickening hε hδ, thickening_singleton] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 527,
"column": 4
} | {
"line": 527,
"column": 8
} | {
"line": 528,
"column": 4
} | [
{
"pp": "case e'_6\nR : Type u_4\ninst✝¹ : NormedRing R\ninst✝ : HasSummableGeomSeries R\nx : R\nh : ‖x‖ < 1\nA : HasSum (fun n ↦ (↑n + 1) * x ^ n) ((1 - x)⁻¹ʳ ^ 2)\nB : HasSum (fun n ↦ x ^ n) (1 - x)⁻¹ʳ\n⊢ x * (1 - x)⁻¹ʳ ^ 2 = (1 - x)⁻¹ʳ ^ 2 - (1 - x)⁻¹ʳ",
"ppTerm": "?e'_6",
"assigned": true,
"used... | [
"case e'_6\nR : Type u_4\ninst✝¹ : NormedRing R\ninst✝ : HasSummableGeomSeries R\nx : R\nh : ‖x‖ < 1\nA : HasSum (fun n ↦ (↑n + 1) * x ^ n) ((1 - x)⁻¹ʳ ^ 2)\nB : HasSum (fun n ↦ x ^ n) (1 - x)⁻¹ʳ\n⊢ (1 - x)⁻¹ʳ ^ 2 - (1 - x)⁻¹ʳ = x * (1 - x)⁻¹ʳ ^ 2"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
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