module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.InnerProductSpace.Projection.Reflection | {
"line": 155,
"column": 4
} | {
"line": 155,
"column": 26
} | {
"line": 155,
"column": 27
} | [
{
"pp": "F : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nv w : F\nh : ‖v‖ = ‖w‖\nR : F ≃ₗᵢ[ℝ] F := ⋯\nthis : R v + R v = w + w\n⊢ (fun x ↦ 2 • x) (R v) = (fun x ↦ 2 • x) w",
"ppTerm": "?m.115",
"assigned": true,
"usedConstants": [
"LinearIsometryEquiv.instEquivLike"... | [
"F : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nv w : F\nh : ‖v‖ = ‖w‖\nR : F ≃ₗᵢ[ℝ] F := (ℝ ∙ (v - w))ᗮ.reflection\nthis : R v + R v = w + w\n⊢ R v + R v = w + w"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 336,
"column": 4
} | {
"line": 336,
"column": 50
} | {
"line": 336,
"column": 51
} | [
{
"pp": "case mp\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nx : E\n⊢ x ∈ T.rangeᗮ → x ∈ T.ker",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toN... | [
"case mp\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nx : E\n⊢ (∀ (a : E), ⟪a, T x⟫ = 0) → T x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.Submodule | {
"line": 63,
"column": 6
} | {
"line": 63,
"column": 40
} | {
"line": 63,
"column": 41
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\ninst✝ : K.HasOrthogonalProjection\ny : E\nhy : y ∈ K\nz : E\nhz : z ∈ Kᗮ\nhv : y + z ∈ Kᗮᗮ\nhyz : ⟪z, y⟫ = 0\n⊢ z = 0",
"ppTerm": "?m.96",
"assigned": false,
... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\ninst✝ : K.HasOrthogonalProjection\ny : E\nhy : y ∈ K\nz : E\nhz : z ∈ Kᗮ\nhv : y + z ∈ Kᗮᗮ\nhyz : ⟪z, y⟫ = 0\n⊢ z = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.Submodule | {
"line": 71,
"column": 14
} | {
"line": 71,
"column": 25
} | {
"line": 71,
"column": 26
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nK₀ K₁ : Submodule 𝕜 E\ninst✝¹ : K₀.HasOrthogonalProjection\ninst✝ : K₁.HasOrthogonalProjection\nh : K₀ᗮ ≤ K₁ᗮ\n⊢ K₁ ≤ K₀",
"ppTerm": "?m.41",
"assigned": false,
"usedConstants":... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nK₀ K₁ : Submodule 𝕜 E\ninst✝¹ : K₀.HasOrthogonalProjection\ninst✝ : K₁.HasOrthogonalProjection\nh : K₀ᗮ ≤ K₁ᗮ\n⊢ K₁ ≤ K₀"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.Submodule | {
"line": 103,
"column": 14
} | {
"line": 103,
"column": 25
} | {
"line": 103,
"column": 26
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nK L : Submodule 𝕜 E\ninst✝¹ : K.HasOrthogonalProjection\ninst✝ : L.HasOrthogonalProjection\nh : Kᗮ = Lᗮ\n⊢ K = L",
"ppTerm": "?m.36",
"assigned": false,
"usedConstants": [],
... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nK L : Submodule 𝕜 E\ninst✝¹ : K.HasOrthogonalProjection\ninst✝ : L.HasOrthogonalProjection\nh : Kᗮ = Lᗮ\n⊢ K = L"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.Submodule | {
"line": 123,
"column": 2
} | {
"line": 142,
"column": 71
} | {
"line": 144,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_4\ninst✝² : Preorder ι\nU : ι → Submodule 𝕜 E\ninst✝¹ : ∀ (i : ι), (U i).HasOrthogonalProjection\ninst✝ : (⨆ i, U i).topologicalClosure.HasOrthogonalProjection\nhU : Monotone U\n... | [] | refine .of_neBot_imp fun h ↦ ?_
cases atTop_neBot_iff.mp h
let y := (⨆ i, U i).topologicalClosure.starProjection x
have proj_x : ∀ i, (U i).orthogonalProjectionOnto x = (U i).orthogonalProjectionOnto y := fun i =>
(orthogonalProjectionOnto_starProjection_of_le
((le_iSup U i).trans (iSup U).le_topologi... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Projection.Submodule | {
"line": 123,
"column": 2
} | {
"line": 142,
"column": 71
} | {
"line": 144,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_4\ninst✝² : Preorder ι\nU : ι → Submodule 𝕜 E\ninst✝¹ : ∀ (i : ι), (U i).HasOrthogonalProjection\ninst✝ : (⨆ i, U i).topologicalClosure.HasOrthogonalProjection\nhU : Monotone U\n... | [] | refine .of_neBot_imp fun h ↦ ?_
cases atTop_neBot_iff.mp h
let y := (⨆ i, U i).topologicalClosure.starProjection x
have proj_x : ∀ i, (U i).orthogonalProjectionOnto x = (U i).orthogonalProjectionOnto y := fun i =>
(orthogonalProjectionOnto_starProjection_of_le
((le_iSup U i).trans (iSup U).le_topologi... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Projection.Submodule | {
"line": 225,
"column": 14
} | {
"line": 225,
"column": 25
} | {
"line": 225,
"column": 26
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nK₁ K₂ : ClosedSubmodule 𝕜 E\ninst✝¹ : (↑K₁).HasOrthogonalProjection\ninst✝ : (↑K₂).HasOrthogonalProjection\nh : K₁ᗮ = K₂ᗮ\n⊢ K₁ = K₂",
"ppTerm": "?m.41",
"assigned": false,
"use... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nK₁ K₂ : ClosedSubmodule 𝕜 E\ninst✝¹ : (↑K₁).HasOrthogonalProjection\ninst✝ : (↑K₂).HasOrthogonalProjection\nh : K₁ᗮ = K₂ᗮ\n⊢ K₁ = K₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.Submodule | {
"line": 235,
"column": 2
} | {
"line": 235,
"column": 13
} | {
"line": 235,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nK₁ K₂ : ClosedSubmodule 𝕜 E\n⊢ K₁ᗮ ⊔ K₂ᗮ = (K₁ ⊓ K₂)ᗮ",
"ppTerm": "?m.37",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals":... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nK₁ K₂ : ClosedSubmodule 𝕜 E\n⊢ K₁ᗮ ⊔ K₂ᗮ = (K₁ ⊓ K₂)ᗮ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 381,
"column": 2
} | {
"line": 382,
"column": 9
} | {
"line": 382,
"column": 10
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\ninst✝ : K.HasOrthogonalProjection\nhK : K ≠ ⊥\nx : E\nhxK : x ∈ K\nhx_ne_zero : x ≠ 0\n⊢ 1 ≤ ‖K.orthogonalProjectionOnto‖",
"ppTerm": "?m.60",
"assigned": false,
... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\ninst✝ : K.HasOrthogonalProjection\nhK : K ≠ ⊥\nx : E\nhxK : x ∈ K\nhx_ne_zero : x ≠ 0\n⊢ 1 ≤ ‖K.orthogonalProjectionOnto‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 396,
"column": 4
} | {
"line": 396,
"column": 15
} | {
"line": 396,
"column": 16
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nv w : E\nthis : (𝕜 ∙ v).starProjection (↑‖v‖ ^ 2 • w) = ⟪v, w⟫ • v\n⊢ ↑(‖v‖ ^ 2) • (𝕜 ∙ v).starProjection w = ⟪v, w⟫ • v",
"ppTerm": "?m.91",
"assigned": true,
"usedConstants": ... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nv w : E\nthis : (𝕜 ∙ v).starProjection (↑‖v‖ ^ 2 • w) = ⟪v, w⟫ • v\n⊢ (algebraMap ℝ 𝕜) ‖v‖ ^ 2 • (𝕜 ∙ v).starProjection w = ⟪v, w⟫ • v"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 477,
"column": 2
} | {
"line": 477,
"column": 13
} | {
"line": 477,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nU V : Submodule 𝕜 E\ninst✝¹ : U.HasOrthogonalProjection\ninst✝ : V.HasOrthogonalProjection\nh : ∀ (x : E), U.orthogonalProjectionOnto ↑(V.orthogonalProjectionOnto x) = 0\nx : ↥V\n⊢ U.orthog... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nU V : Submodule 𝕜 E\ninst✝¹ : U.HasOrthogonalProjection\ninst✝ : V.HasOrthogonalProjection\nh : ∀ (x : E), U.orthogonalProjectionOnto ↑(V.orthogonalProjectionOnto x) = 0\nx : ↥V\n⊢ ↑x ∈ Uᗮ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 498,
"column": 4
} | {
"line": 498,
"column": 43
} | {
"line": 499,
"column": 6
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nU V : Submodule 𝕜 E\ninst✝¹ : U.HasOrthogonalProjection\ninst✝ : V.HasOrthogonalProjection\nh : U ≤ V\nx : E\n⊢ U.orthogonalProjectionOnto x = U.orthogonalProjectionOnto (V.starProjection x... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nU V : Submodule 𝕜 E\ninst✝¹ : U.HasOrthogonalProjection\ninst✝ : V.HasOrthogonalProjection\nh : U ≤ V\nx : E\n⊢ U.orthogonalProjectionOnto x = U.orthogonalProjectionOnto (V.starProjection x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 497,
"column": 13
} | {
"line": 500,
"column": 73
} | {
"line": 502,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nU V : Submodule 𝕜 E\ninst✝¹ : U.HasOrthogonalProjection\ninst✝ : V.HasOrthogonalProjection\nh : U ≤ V\nx : E\n⊢ U.orthogonalProjectionOnto x = U.orthogonalProjectionOnto (V.starProjection x... | [] | by
simpa only [sub_eq_zero, map_sub] using
orthogonalProjectionOnto_apply_of_mem_orthogonal
(Submodule.orthogonal_le h (sub_starProjection_mem_orthogonal x)) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 566,
"column": 2
} | {
"line": 566,
"column": 76
} | {
"line": 567,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nU : Submodule 𝕜 E\ninst✝ : U.HasOrthogonalProjection\nv : E\nh : ‖U.starProjection v‖ = ‖v‖\n⊢ v ∈ U",
"ppTerm": "?m.44",
"assigned": false,
"usedConstants": [],
"usedFVars"... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nU : Submodule 𝕜 E\ninst✝ : U.HasOrthogonalProjection\nv : E\nh : ‖U.starProjection v‖ = ‖v‖\n⊢ v ∈ U"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 150,
"column": 12
} | {
"line": 150,
"column": 23
} | {
"line": 150,
"column": 24
} | [
{
"pp": "case h₁\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜' F\nf : E →SL[σ] F\nσ' : 𝕜' →+... | [
"case h₁\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜' F\nf : E →SL[σ] F\nσ' : 𝕜' →+* 𝕜\ninst✝³... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 150,
"column": 12
} | {
"line": 150,
"column": 28
} | {
"line": 151,
"column": 10
} | [
{
"pp": "case h₁\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜' F\nf : E →SL[σ] F\nσ' : 𝕜' →+... | [] | simpa using dinv | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 631,
"column": 2
} | {
"line": 631,
"column": 13
} | {
"line": 631,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\np : E →ₗ[𝕜] E\nhp : p.IsSymmetricProjection\nthis : p.range.HasOrthogonalProjection\nx : E\n⊢ x - p x ∈ p.ker",
"ppTerm": "?m.164",
"assigned": true,
"usedConstants": [
"Eq... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\np : E →ₗ[𝕜] E\nhp : p.IsSymmetricProjection\nthis : p.range.HasOrthogonalProjection\nx : E\n⊢ p x - p (p x) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 150,
"column": 12
} | {
"line": 150,
"column": 28
} | {
"line": 151,
"column": 10
} | [
{
"pp": "case h₁\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜' F\nf : E →SL[σ] F\nσ' : 𝕜' →+... | [] | simpa using dinv | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 150,
"column": 12
} | {
"line": 150,
"column": 28
} | {
"line": 151,
"column": 10
} | [
{
"pp": "case h₁\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜' F\nf : E →SL[σ] F\nσ' : 𝕜' →+... | [] | simpa using dinv | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 654,
"column": 2
} | {
"line": 654,
"column": 28
} | {
"line": 654,
"column": 29
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\ninst✝ : K.HasOrthogonalProjection\nv : E\n⊢ ‖v‖ * ‖v‖ =\n ‖v - ↑(K.orthogonalProjectionOnto v)‖ * ‖v - ↑(K.orthogonalProjectionOnto v)‖ +\n ‖↑(K.orthogonalProject... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\ninst✝ : K.HasOrthogonalProjection\nv : E\n⊢ ‖v‖ * ‖v‖ = ‖K.starProjection v‖ * ‖K.starProjection v‖ + ‖v - K.starProjection v‖ * ‖v - K.starProjection v‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 649,
"column": 71
} | {
"line": 654,
"column": 70
} | {
"line": 656,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\ninst✝ : K.HasOrthogonalProjection\nv : E\n⊢ re ⟪K.starProjection v, v⟫ = ‖K.orthogonalProjectionOnto v‖ ^ 2",
"ppTerm": "?m.46",
"assigned": true,
"usedConsta... | [] | by
rw [starProjection_apply,
re_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two,
div_eq_iff (NeZero.ne' 2).symm, pow_two, add_sub_assoc, ← eq_sub_iff_add_eq', coe_norm,
← mul_sub_one, show (2 : ℝ) - 1 = 1 by norm_num, mul_one, sub_eq_iff_eq_add', norm_sub_rev]
simpa [sq, add_comm]... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 660,
"column": 2
} | {
"line": 660,
"column": 28
} | {
"line": 660,
"column": 29
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\ninst✝ : K.HasOrthogonalProjection\nv : E\n⊢ ‖v‖ * ‖v‖ =\n ‖v - orthogonalProjectionFn v‖ * ‖v - orthogonalProjectionFn v‖ +\n ‖orthogonalProjectionFn v‖ * ‖orthog... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\ninst✝ : K.HasOrthogonalProjection\nv : E\n⊢ ‖v‖ * ‖v‖ =\n ‖orthogonalProjectionFn v‖ * ‖orthogonalProjectionFn v‖ +\n ‖v - orthogonalProjectionFn v‖ * ‖v - orthogonalProjecti... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional | {
"line": 327,
"column": 36
} | {
"line": 327,
"column": 68
} | {
"line": 328,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nι : Type u_4\ninst✝² : DecidableEq ι\ninst✝¹ : Fintype ι\nV : ι → Submodule 𝕜 E\ninst✝ : ∀ (... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nι : Type u_4\ninst✝² : DecidableEq ι\ninst✝¹ : Fintype ι\nV : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), Comp... | rw [DFinsupp.sum_eq_sum_fintype] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional | {
"line": 361,
"column": 8
} | {
"line": 361,
"column": 54
} | {
"line": 361,
"column": 55
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhev : e ∈ v\nthis : e ∈ span 𝕜 v ⊓ (span �... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhev : e ∈ v\nthis : e ∈ span 𝕜 v ⊓ (span 𝕜 v)ᗮ\n⊢ e =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.UnitaryGroup | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 36
} | {
"line": 74,
"column": 37
} | [
{
"pp": "n : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα : Type v\ninst✝¹ : CommRing α\ninst✝ : StarRing α\nA : Matrix n n α\nhA : A * star A = 1\n⊢ star A * A = 1",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Matrix.instMulOneOfFint... | [
"n : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα : Type v\ninst✝¹ : CommRing α\ninst✝ : StarRing α\nA : Matrix n n α\nhA : A * star A = 1\n⊢ A * star A = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.UnitaryGroup | {
"line": 83,
"column": 4
} | {
"line": 83,
"column": 37
} | {
"line": 83,
"column": 38
} | [
{
"pp": "case left\nn : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα : Type v\ninst✝¹ : CommRing α\ninst✝ : StarRing α\nA : Matrix n n α\nhA : A ∈ unitaryGroup n α\n⊢ star A.det * A.det = 1",
"ppTerm": "?left",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []... | [
"case left\nn : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα : Type v\ninst✝¹ : CommRing α\ninst✝ : StarRing α\nA : Matrix n n α\nhA : A ∈ unitaryGroup n α\n⊢ star A.det * A.det = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.UnitaryGroup | {
"line": 84,
"column": 4
} | {
"line": 84,
"column": 37
} | {
"line": 84,
"column": 38
} | [
{
"pp": "case right\nn : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα : Type v\ninst✝¹ : CommRing α\ninst✝ : StarRing α\nA : Matrix n n α\nhA : A ∈ unitaryGroup n α\n⊢ A.det * star A.det = 1",
"ppTerm": "?right",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": ... | [
"case right\nn : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα : Type v\ninst✝¹ : CommRing α\ninst✝ : StarRing α\nA : Matrix n n α\nhA : A ∈ unitaryGroup n α\n⊢ A.det * star A.det = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional | {
"line": 380,
"column": 8
} | {
"line": 380,
"column": 24
} | {
"line": 380,
"column": 25
} | [
{
"pp": "case mp.refine_2.inl\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhe'' : e ∉ v\nh_end :... | [
"case mp.refine_2.inl\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhe'' : e ∉ v\nh_end : ∀ a ∈ v, ⟪a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional | {
"line": 383,
"column": 8
} | {
"line": 383,
"column": 24
} | {
"line": 383,
"column": 25
} | [
{
"pp": "case mp.refine_2.inr.inl\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhe'' : e ∉ v\nh_e... | [
"case mp.refine_2.inr.inl\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhe'' : e ∉ v\nh_end : ∀ a ∈ v... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 206,
"column": 2
} | {
"line": 208,
"column": 9
} | {
"line": 209,
"column": 2
} | [
{
"pp": "p : ℝ≥0∞\n𝕜 : Type u_1\ninst✝³ : Semiring 𝕜\nη : Type u_5\nιs : η → Type u_6\nMs : η → Type u_7\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module 𝕜 (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent 𝕜 (v i)\n⊢ LinearIndependent 𝕜 fun ji ↦ s... | [
"p : ℝ≥0∞\n𝕜 : Type u_1\ninst✝³ : Semiring 𝕜\nη : Type u_5\nιs : η → Type u_6\nMs : η → Type u_7\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module 𝕜 (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent 𝕜 (v i)\n⊢ LinearIndependent 𝕜 (⇑↑(WithLp.linearEquiv... | suffices LinearIndependent 𝕜 ((WithLp.linearEquiv p 𝕜 _).symm.toLinearMap ∘
fun ji : Σ j, ιs j ↦ Pi.single ji.1 (v ji.1 ji.2)) by
simpa | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional | {
"line": 387,
"column": 8
} | {
"line": 387,
"column": 19
} | {
"line": 387,
"column": 20
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := ⋯\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhe'' : e ∉ v\nh_end : ∀ a ∈ v, ⟪a, e⟫_𝕜 = 0\na : E\nh... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhe'' : e ∉ v\nh_end : ∀ a ∈ v, ⟪a, e⟫_𝕜 = 0\na : E\nha... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 408,
"column": 6
} | {
"line": 416,
"column": 36
} | {
"line": 418,
"column": 0
} | [
{
"pp": "case inr\np : ℝ≥0∞\n𝕜 : Type u_1\nι : Type u_2\nα : ι → Type u_3\nβ : ι → Type u_4\ninst✝³ : Fact (1 ≤ p)\ninst✝² : (i : ι) → PseudoMetricSpace (α i)\ninst✝¹ : (i : ι) → PseudoEMetricSpace (β i)\ninst✝ : Fintype ι\nf g h : PiLp p β\nhp : 1 ≤ p.toReal\n⊢ (∑ i, edist (f.ofLp i) (h.ofLp i) ^ p.toReal) ^ ... | [] | calc
(∑ i, edist (f i) (h i) ^ p.toReal) ^ (1 / p.toReal) ≤
(∑ i, (edist (f i) (g i) + edist (g i) (h i)) ^ p.toReal) ^ (1 / p.toReal) := by
gcongr
apply edist_triangle
_ ≤
(∑ i, edist (f i) (g i) ^ p.toReal) ^ (1 / p.toReal) +
(∑ i, edist (g i) ... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 434,
"column": 4
} | {
"line": 434,
"column": 83
} | {
"line": 435,
"column": 4
} | [
{
"pp": "case inr\np : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : WithLp p (α × β)\nh : 1 ≤ p.toReal\npos : 0 < p.toReal\nnonneg : 0 ≤ 1 / p.toReal\n⊢ edist x y ≤ ↑(2 ^ (1 / p).toReal) * edist x.ofLp y.ofLp",
"ppTerm": "?inr",
... | [
"case inr\np : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : WithLp p (α × β)\nh : 1 ≤ p.toReal\npos : 0 < p.toReal\nnonneg : 0 ≤ 1 / p.toReal\ncancel : p.toReal * (1 / p.toReal) = 1\n⊢ edist x y ≤ ↑(2 ^ (1 / p).toReal) * edist x.ofLp y.ofLp"... | have cancel : p.toReal * (1 / p.toReal) = 1 := mul_div_cancel₀ 1 (ne_of_gt pos) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.UnitaryGroup | {
"line": 267,
"column": 24
} | {
"line": 267,
"column": 35
} | {
"line": 267,
"column": 36
} | [
{
"pp": "n : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα : Type v\ninst✝¹ : CommRing α\ninst✝ : StarRing α\nA✝ : Matrix n n α\nA : ↥(specialUnitaryGroup n α)\n⊢ star ↑A ∈ ↑(unitaryGroup n α)",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matrix.instStar",
... | [
"n : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα : Type v\ninst✝¹ : CommRing α\ninst✝ : StarRing α\nA✝ : Matrix n n α\nA : ↥(specialUnitaryGroup n α)\n⊢ ↑A ∈ unitary (Matrix n n α)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 632,
"column": 2
} | {
"line": 632,
"column": 43
} | {
"line": 632,
"column": 44
} | [
{
"pp": "p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nx y : WithLp p (α × β)\n⊢ nndist x.fst y.fst ≤ nndist x y",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nx y : WithLp p (α × β)\n⊢ nndist x.fst y.fst ≤ nndist x y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 636,
"column": 2
} | {
"line": 636,
"column": 43
} | {
"line": 636,
"column": 44
} | [
{
"pp": "p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nx y : WithLp p (α × β)\n⊢ nndist x.snd y.snd ≤ nndist x y",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nx y : WithLp p (α × β)\n⊢ nndist x.snd y.snd ≤ nndist x y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 667,
"column": 18
} | {
"line": 667,
"column": 61
} | {
"line": 667,
"column": 62
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : WithLp ∞ (α × β)\n⊢ edist x.ofLp y.ofLp ≤ edist x y",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\nβ : Type u_3\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : WithLp ∞ (α × β)\n⊢ edist x.ofLp y.ofLp ≤ edist x y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 669,
"column": 6
} | {
"line": 670,
"column": 22
} | {
"line": 670,
"column": 23
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : WithLp ∞ (α × β)\n⊢ edist x y ≤ edist x.ofLp y.ofLp",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\nβ : Type u_3\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : WithLp ∞ (α × β)\n⊢ edist x y ≤ edist x.ofLp y.ofLp"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 695,
"column": 2
} | {
"line": 695,
"column": 13
} | {
"line": 695,
"column": 14
} | [
{
"pp": "p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nx : WithLp p (α × β)\n⊢ ‖x.fst‖ₑ ≤ ‖x‖ₑ",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nx : WithLp p (α × β)\n⊢ ‖x.fst‖ₑ ≤ ‖x‖ₑ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 699,
"column": 2
} | {
"line": 699,
"column": 13
} | {
"line": 699,
"column": 14
} | [
{
"pp": "p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nx : WithLp p (α × β)\n⊢ ‖x.snd‖ₑ ≤ ‖x‖ₑ",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nx : WithLp p (α × β)\n⊢ ‖x.snd‖ₑ ≤ ‖x‖ₑ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 703,
"column": 2
} | {
"line": 703,
"column": 13
} | {
"line": 703,
"column": 14
} | [
{
"pp": "p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nx : WithLp p (α × β)\n⊢ ‖x.fst‖₊ ≤ ‖x‖₊",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nx : WithLp p (α × β)\n⊢ ‖x.fst‖₊ ≤ ‖x‖₊"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 707,
"column": 2
} | {
"line": 707,
"column": 13
} | {
"line": 707,
"column": 14
} | [
{
"pp": "p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nx : WithLp p (α × β)\n⊢ ‖x.snd‖₊ ≤ ‖x‖₊",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nx : WithLp p (α × β)\n⊢ ‖x.snd‖₊ ≤ ‖x‖₊"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 711,
"column": 2
} | {
"line": 711,
"column": 13
} | {
"line": 711,
"column": 14
} | [
{
"pp": "p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nx : WithLp p (α × β)\n⊢ ‖x.fst‖ ≤ ‖x‖",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nx : WithLp p (α × β)\n⊢ ‖x.fst‖ ≤ ‖x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 715,
"column": 2
} | {
"line": 715,
"column": 13
} | {
"line": 715,
"column": 14
} | [
{
"pp": "p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nx : WithLp p (α × β)\n⊢ ‖x.snd‖ ≤ ‖x‖",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nx : WithLp p (α × β)\n⊢ ‖x.snd‖ ≤ ‖x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 469,
"column": 4
} | {
"line": 469,
"column": 36
} | {
"line": 469,
"column": 37
} | [
{
"pp": "case inl\nι : Type u_2\nβ : ι → Type u_4\ninst✝² : (i : ι) → PseudoEMetricSpace (β i)\ninst✝¹ : Fintype ι\ni : ι\ninst✝ : Fact (1 ≤ ∞)\nx y : PiLp ∞ β\n⊢ edist (x.ofLp i) (y.ofLp i) ≤ edist x y",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"PseudoEMetricSpace.toWeakPseudoEM... | [
"case inl\nι : Type u_2\nβ : ι → Type u_4\ninst✝² : (i : ι) → PseudoEMetricSpace (β i)\ninst✝¹ : Fintype ι\ni : ι\ninst✝ : Fact (1 ≤ ∞)\nx y : PiLp ∞ β\n⊢ edist (x.ofLp i) (y.ofLp i) ≤ ⨆ i, edist (x.ofLp i) (y.ofLp i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 840,
"column": 2
} | {
"line": 845,
"column": 91
} | {
"line": 847,
"column": 0
} | [
{
"pp": "p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nx : α\n⊢ ‖toLp p (x, 0)‖₊ = ‖x‖₊",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"ENNReal.coe_ne_top._simp_1",
"WithLp",
"GroupWithZero.t... | [] | induction p generalizing hp with
| top =>
simp [prod_nnnorm_eq_sup]
| coe p =>
have hp0 : (p : ℝ) ≠ 0 := mod_cast (zero_lt_one.trans_le <| Fact.out (p := 1 ≤ (p : ℝ≥0∞))).ne'
simp [prod_nnnorm_eq_add, NNReal.zero_rpow hp0, ← NNReal.rpow_mul, mul_inv_cancel₀ hp0] | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 840,
"column": 2
} | {
"line": 845,
"column": 91
} | {
"line": 847,
"column": 0
} | [
{
"pp": "p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nx : α\n⊢ ‖toLp p (x, 0)‖₊ = ‖x‖₊",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"ENNReal.coe_ne_top._simp_1",
"WithLp",
"GroupWithZero.t... | [] | induction p generalizing hp with
| top =>
simp [prod_nnnorm_eq_sup]
| coe p =>
have hp0 : (p : ℝ) ≠ 0 := mod_cast (zero_lt_one.trans_le <| Fact.out (p := 1 ≤ (p : ℝ≥0∞))).ne'
simp [prod_nnnorm_eq_add, NNReal.zero_rpow hp0, ← NNReal.rpow_mul, mul_inv_cancel₀ hp0] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 840,
"column": 2
} | {
"line": 845,
"column": 91
} | {
"line": 847,
"column": 0
} | [
{
"pp": "p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nx : α\n⊢ ‖toLp p (x, 0)‖₊ = ‖x‖₊",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"ENNReal.coe_ne_top._simp_1",
"WithLp",
"GroupWithZero.t... | [] | induction p generalizing hp with
| top =>
simp [prod_nnnorm_eq_sup]
| coe p =>
have hp0 : (p : ℝ) ≠ 0 := mod_cast (zero_lt_one.trans_le <| Fact.out (p := 1 ≤ (p : ℝ≥0∞))).ne'
simp [prod_nnnorm_eq_add, NNReal.zero_rpow hp0, ← NNReal.rpow_mul, mul_inv_cancel₀ hp0] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 848,
"column": 2
} | {
"line": 853,
"column": 91
} | {
"line": 855,
"column": 0
} | [
{
"pp": "p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\ny : β\n⊢ ‖toLp p (0, y)‖₊ = ‖y‖₊",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"ENNReal.coe_ne_top._simp_1",
"WithLp",
"GroupWithZero.t... | [] | induction p generalizing hp with
| top =>
simp [prod_nnnorm_eq_sup]
| coe p =>
have hp0 : (p : ℝ) ≠ 0 := mod_cast (zero_lt_one.trans_le <| Fact.out (p := 1 ≤ (p : ℝ≥0∞))).ne'
simp [prod_nnnorm_eq_add, NNReal.zero_rpow hp0, ← NNReal.rpow_mul, mul_inv_cancel₀ hp0] | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 848,
"column": 2
} | {
"line": 853,
"column": 91
} | {
"line": 855,
"column": 0
} | [
{
"pp": "p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\ny : β\n⊢ ‖toLp p (0, y)‖₊ = ‖y‖₊",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"ENNReal.coe_ne_top._simp_1",
"WithLp",
"GroupWithZero.t... | [] | induction p generalizing hp with
| top =>
simp [prod_nnnorm_eq_sup]
| coe p =>
have hp0 : (p : ℝ) ≠ 0 := mod_cast (zero_lt_one.trans_le <| Fact.out (p := 1 ≤ (p : ℝ≥0∞))).ne'
simp [prod_nnnorm_eq_add, NNReal.zero_rpow hp0, ← NNReal.rpow_mul, mul_inv_cancel₀ hp0] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 848,
"column": 2
} | {
"line": 853,
"column": 91
} | {
"line": 855,
"column": 0
} | [
{
"pp": "p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\ny : β\n⊢ ‖toLp p (0, y)‖₊ = ‖y‖₊",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"ENNReal.coe_ne_top._simp_1",
"WithLp",
"GroupWithZero.t... | [] | induction p generalizing hp with
| top =>
simp [prod_nnnorm_eq_sup]
| coe p =>
have hp0 : (p : ℝ) ≠ 0 := mod_cast (zero_lt_one.trans_le <| Fact.out (p := 1 ≤ (p : ℝ≥0∞))).ne'
simp [prod_nnnorm_eq_add, NNReal.zero_rpow hp0, ← NNReal.rpow_mul, mul_inv_cancel₀ hp0] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 494,
"column": 4
} | {
"line": 494,
"column": 83
} | {
"line": 495,
"column": 4
} | [
{
"pp": "case inr\np : ℝ≥0∞\nι : Type u_2\nβ : ι → Type u_4\ninst✝² : Fact (1 ≤ p)\ninst✝¹ : (i : ι) → PseudoEMetricSpace (β i)\ninst✝ : Fintype ι\nx y : WithLp p ((i : ι) → β i)\nh : 1 ≤ p.toReal\npos : 0 < p.toReal\nnonneg : 0 ≤ 1 / p.toReal\n⊢ edist x y ≤ ↑(↑(Fintype.card ι) ^ (1 / p).toReal) * edist x.ofLp ... | [
"case inr\np : ℝ≥0∞\nι : Type u_2\nβ : ι → Type u_4\ninst✝² : Fact (1 ≤ p)\ninst✝¹ : (i : ι) → PseudoEMetricSpace (β i)\ninst✝ : Fintype ι\nx y : WithLp p ((i : ι) → β i)\nh : 1 ≤ p.toReal\npos : 0 < p.toReal\nnonneg : 0 ≤ 1 / p.toReal\ncancel : p.toReal * (1 / p.toReal) = 1\n⊢ edist x y ≤ ↑(↑(Fintype.card ι) ^ (1 ... | have cancel : p.toReal * (1 / p.toReal) = 1 := mul_div_cancel₀ 1 (ne_of_gt pos) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 648,
"column": 2
} | {
"line": 648,
"column": 43
} | {
"line": 648,
"column": 44
} | [
{
"pp": "p : ℝ≥0∞\nι : Type u_2\nβ : ι → Type u_4\nhp : Fact (1 ≤ p)\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → PseudoMetricSpace (β i)\nx y : PiLp p β\ni : ι\n⊢ nndist (x.ofLp i) (y.ofLp i) ≤ nndist x y",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": [... | [
"p : ℝ≥0∞\nι : Type u_2\nβ : ι → Type u_4\nhp : Fact (1 ≤ p)\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → PseudoMetricSpace (β i)\nx y : PiLp p β\ni : ι\n⊢ nndist (x.ofLp i) (y.ofLp i) ≤ nndist x y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 675,
"column": 18
} | {
"line": 675,
"column": 61
} | {
"line": 675,
"column": 62
} | [
{
"pp": "ι : Type u_2\nβ : ι → Type u_4\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → PseudoEMetricSpace (β i)\nx y : WithLp ∞ ((i : ι) → β i)\n⊢ edist x.ofLp y.ofLp ≤ edist x y",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_2\nβ : ι → Type u_4\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → PseudoEMetricSpace (β i)\nx y : WithLp ∞ ((i : ι) → β i)\n⊢ edist x.ofLp y.ofLp ≤ edist x y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 676,
"column": 8
} | {
"line": 677,
"column": 20
} | {
"line": 677,
"column": 21
} | [
{
"pp": "ι : Type u_2\nβ : ι → Type u_4\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → PseudoEMetricSpace (β i)\nx y : WithLp ∞ ((i : ι) → β i)\n⊢ edist x y ≤ edist x.ofLp y.ofLp",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_2\nβ : ι → Type u_4\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → PseudoEMetricSpace (β i)\nx y : WithLp ∞ ((i : ι) → β i)\n⊢ edist x y ≤ edist x.ofLp y.ofLp"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 1052,
"column": 4
} | {
"line": 1052,
"column": 46
} | {
"line": 1052,
"column": 47
} | [
{
"pp": "case refine_1\np : ℝ≥0∞\nhp : Fact (1 ≤ p)\nα : Type u_4\nβ : Type u_5\ninst✝⁶ : SeminormedAddCommGroup α\ninst✝⁵ : SeminormedAddCommGroup β\nR : Type u_6\ninst✝⁴ : SeminormedRing R\ninst✝³ : Module R α\ninst✝² : Module R β\ninst✝¹ : IsBoundedSMul R α\ninst✝ : IsBoundedSMul R β\nthis : PseudoMetricSpac... | [
"case refine_1\np : ℝ≥0∞\nhp : Fact (1 ≤ p)\nα : Type u_4\nβ : Type u_5\ninst✝⁶ : SeminormedAddCommGroup α\ninst✝⁵ : SeminormedAddCommGroup β\nR : Type u_6\ninst✝⁴ : SeminormedRing R\ninst✝³ : Module R α\ninst✝² : Module R β\ninst✝¹ : IsBoundedSMul R α\ninst✝ : IsBoundedSMul R β\nthis : PseudoMetricSpace (α × β) :=... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 1053,
"column": 4
} | {
"line": 1053,
"column": 46
} | {
"line": 1053,
"column": 47
} | [
{
"pp": "case refine_2\np : ℝ≥0∞\nhp : Fact (1 ≤ p)\nα : Type u_4\nβ : Type u_5\ninst✝⁶ : SeminormedAddCommGroup α\ninst✝⁵ : SeminormedAddCommGroup β\nR : Type u_6\ninst✝⁴ : SeminormedRing R\ninst✝³ : Module R α\ninst✝² : Module R β\ninst✝¹ : IsBoundedSMul R α\ninst✝ : IsBoundedSMul R β\nthis : PseudoMetricSpac... | [
"case refine_2\np : ℝ≥0∞\nhp : Fact (1 ≤ p)\nα : Type u_4\nβ : Type u_5\ninst✝⁶ : SeminormedAddCommGroup α\ninst✝⁵ : SeminormedAddCommGroup β\nR : Type u_6\ninst✝⁴ : SeminormedRing R\ninst✝³ : Module R α\ninst✝² : Module R β\ninst✝¹ : IsBoundedSMul R α\ninst✝ : IsBoundedSMul R β\nthis : PseudoMetricSpace (α × β) :=... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 686,
"column": 4
} | {
"line": 691,
"column": 57
} | {
"line": 693,
"column": 0
} | [
{
"pp": "case inr\np : ℝ≥0∞\n𝕜 : Type u_1\nι : Type u_2\nα : ι → Type u_3\nβ : ι → Type u_4\nhp : Fact (1 ≤ p)\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → SeminormedAddCommGroup (β i)\nx y : PiLp p β\nh : 1 ≤ p.toReal\n⊢ dist x y = ‖-x + y‖",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
... | [] | · have : p ≠ ∞ := by
intro hp
rw [hp, ENNReal.toReal_top] at h
linarith
simp only [dist_eq_sum (zero_lt_one.trans_le h), norm_eq_sum (zero_lt_one.trans_le h),
dist_eq_norm, add_apply, neg_apply, norm_neg_add] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 705,
"column": 2
} | {
"line": 705,
"column": 13
} | {
"line": 705,
"column": 14
} | [
{
"pp": "p : ℝ≥0∞\nι : Type u_2\nβ : ι → Type u_4\nhp : Fact (1 ≤ p)\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → SeminormedAddCommGroup (β i)\nx : PiLp p β\ni : ι\n⊢ ‖x.ofLp i‖ₑ ≤ ‖x‖ₑ",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℝ≥0∞\nι : Type u_2\nβ : ι → Type u_4\nhp : Fact (1 ≤ p)\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → SeminormedAddCommGroup (β i)\nx : PiLp p β\ni : ι\n⊢ ‖x.ofLp i‖ₑ ≤ ‖x‖ₑ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 709,
"column": 2
} | {
"line": 709,
"column": 13
} | {
"line": 709,
"column": 14
} | [
{
"pp": "p : ℝ≥0∞\nι : Type u_2\nβ : ι → Type u_4\nhp : Fact (1 ≤ p)\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → SeminormedAddCommGroup (β i)\nx : PiLp p β\ni : ι\n⊢ ‖x.ofLp i‖₊ ≤ ‖x‖₊",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℝ≥0∞\nι : Type u_2\nβ : ι → Type u_4\nhp : Fact (1 ≤ p)\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → SeminormedAddCommGroup (β i)\nx : PiLp p β\ni : ι\n⊢ ‖x.ofLp i‖₊ ≤ ‖x‖₊"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 713,
"column": 2
} | {
"line": 713,
"column": 13
} | {
"line": 713,
"column": 14
} | [
{
"pp": "p : ℝ≥0∞\nι : Type u_2\nβ : ι → Type u_4\nhp : Fact (1 ≤ p)\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → SeminormedAddCommGroup (β i)\nx : PiLp p β\ni : ι\n⊢ ‖x.ofLp i‖ ≤ ‖x‖",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℝ≥0∞\nι : Type u_2\nβ : ι → Type u_4\nhp : Fact (1 ≤ p)\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → SeminormedAddCommGroup (β i)\nx : PiLp p β\ni : ι\n⊢ ‖x.ofLp i‖ ≤ ‖x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 137,
"column": 2
} | {
"line": 137,
"column": 28
} | {
"line": 139,
"column": 0
} | [
{
"pp": "a r : ℝ\nhr : 0 ≤ r\n⊢ volume.real (ball a r) = 2 * r",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"MeasureTheory.Measure",
"HMul.hMul",
"... | [] | simp [measureReal_def, hr] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 137,
"column": 2
} | {
"line": 137,
"column": 28
} | {
"line": 139,
"column": 0
} | [
{
"pp": "a r : ℝ\nhr : 0 ≤ r\n⊢ volume.real (ball a r) = 2 * r",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"MeasureTheory.Measure",
"HMul.hMul",
"... | [] | simp [measureReal_def, hr] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 137,
"column": 2
} | {
"line": 137,
"column": 28
} | {
"line": 139,
"column": 0
} | [
{
"pp": "a r : ℝ\nhr : 0 ≤ r\n⊢ volume.real (ball a r) = 2 * r",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"MeasureTheory.Measure",
"HMul.hMul",
"... | [] | simp [measureReal_def, hr] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 28
} | {
"line": 148,
"column": 0
} | [
{
"pp": "a r : ℝ\nhr : 0 ≤ r\n⊢ volume.real (closedBall a r) = 2 * r",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"MeasureTheory.Measure",
"HMul.hMul",
... | [] | simp [measureReal_def, hr] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 28
} | {
"line": 148,
"column": 0
} | [
{
"pp": "a r : ℝ\nhr : 0 ≤ r\n⊢ volume.real (closedBall a r) = 2 * r",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"MeasureTheory.Measure",
"HMul.hMul",
... | [] | simp [measureReal_def, hr] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 28
} | {
"line": 148,
"column": 0
} | [
{
"pp": "a r : ℝ\nhr : 0 ≤ r\n⊢ volume.real (closedBall a r) = 2 * r",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"MeasureTheory.Measure",
"HMul.hMul",
... | [] | simp [measureReal_def, hr] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 234,
"column": 2
} | {
"line": 234,
"column": 24
} | {
"line": 234,
"column": 25
} | [
{
"pp": "p : ℝ → Prop\na : ℝ\nh : ∀ᶠ (x : ℝ) in 𝓝 a, p x\nl u : ℝ\nhx : a ∈ Ioo l u\nhs : Ioo l u ⊆ {x | p x}\n⊢ 0 < volume (Ioo l u)",
"ppTerm": "?m.74",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"Eq.mpr",
"sub_pos._simp_1",
... | [
"p : ℝ → Prop\na : ℝ\nh : ∀ᶠ (x : ℝ) in 𝓝 a, p x\nl u : ℝ\nhx : a ∈ Ioo l u\nhs : Ioo l u ⊆ {x | p x}\n⊢ l < u"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 340,
"column": 2
} | {
"line": 340,
"column": 29
} | {
"line": 340,
"column": 30
} | [
{
"pp": "a : ℝ\nh : a ≠ 0\n⊢ ofReal |a| • Measure.map (fun x ↦ x * a) volume = volume",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Real",
"instHSMul",
"MeasureTheory.Measure",
"No... | [
"a : ℝ\nh : a ≠ 0\n⊢ ofReal |a| • Measure.map (fun x ↦ a * x) volume = volume"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 344,
"column": 2
} | {
"line": 344,
"column": 29
} | {
"line": 344,
"column": 30
} | [
{
"pp": "a : ℝ\nh : a ≠ 0\n⊢ Measure.map (fun x ↦ x * a) volume = ofReal |a⁻¹| • volume",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Real",
"instHSMul",
"MeasureTheory.Measure",
"... | [
"a : ℝ\nh : a ≠ 0\n⊢ Measure.map (fun x ↦ a * x) volume = ofReal |a⁻¹| • volume"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 382,
"column": 8
} | {
"line": 382,
"column": 35
} | {
"line": 382,
"column": 36
} | [
{
"pp": "ι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nD : ι → ℝ\nh : (Matrix.diagonal D).det ≠ 0\ns : ι → Set ℝ\nhs : ∀ (i : ι), MeasurableSet (s i)\nthis : (⇑(toLin' (Matrix.diagonal D)) ⁻¹' univ.pi fun i ↦ s i) = univ.pi fun i ↦ (fun x ↦ D i * x) ⁻¹' s i\ni : ι\nA : D i ≠ 0\n⊢ ofReal |D i| * volum... | [
"ι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nD : ι → ℝ\nh : (Matrix.diagonal D).det ≠ 0\ns : ι → Set ℝ\nhs : ∀ (i : ι), MeasurableSet (s i)\nthis : (⇑(toLin' (Matrix.diagonal D)) ⁻¹' univ.pi fun i ↦ s i) = univ.pi fun i ↦ (fun x ↦ D i * x) ⁻¹' s i\ni : ι\nA : D i ≠ 0\n⊢ ofReal |D i| * (ofReal |(D i)⁻¹|... | volume_preimage_mul_left A, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 1267,
"column": 4
} | {
"line": 1267,
"column": 44
} | {
"line": 1267,
"column": 45
} | [
{
"pp": "case refine_1\np : ℝ≥0∞\nι : Type u_2\nα : ι → Type u_3\ninst✝⁵ : Fact (1 ≤ p)\ninst✝⁴ : Fintype ι\ninst✝³ : (i : ι) → SeminormedAddCommGroup (α i)\nR : Type u_5\ninst✝² : SeminormedRing R\ninst✝¹ : (i : ι) → Module R (α i)\ninst✝ : ∀ (i : ι), IsBoundedSMul R (α i)\nthis : PseudoMetricSpace ((i : ι) → ... | [
"case refine_1\np : ℝ≥0∞\nι : Type u_2\nα : ι → Type u_3\ninst✝⁵ : Fact (1 ≤ p)\ninst✝⁴ : Fintype ι\ninst✝³ : (i : ι) → SeminormedAddCommGroup (α i)\nR : Type u_5\ninst✝² : SeminormedRing R\ninst✝¹ : (i : ι) → Module R (α i)\ninst✝ : ∀ (i : ι), IsBoundedSMul R (α i)\nthis : PseudoMetricSpace ((i : ι) → α i) := pseu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 1268,
"column": 4
} | {
"line": 1268,
"column": 44
} | {
"line": 1268,
"column": 45
} | [
{
"pp": "case refine_2\np : ℝ≥0∞\nι : Type u_2\nα : ι → Type u_3\ninst✝⁵ : Fact (1 ≤ p)\ninst✝⁴ : Fintype ι\ninst✝³ : (i : ι) → SeminormedAddCommGroup (α i)\nR : Type u_5\ninst✝² : SeminormedRing R\ninst✝¹ : (i : ι) → Module R (α i)\ninst✝ : ∀ (i : ι), IsBoundedSMul R (α i)\nthis : PseudoMetricSpace ((i : ι) → ... | [
"case refine_2\np : ℝ≥0∞\nι : Type u_2\nα : ι → Type u_3\ninst✝⁵ : Fact (1 ≤ p)\ninst✝⁴ : Fintype ι\ninst✝³ : (i : ι) → SeminormedAddCommGroup (α i)\nR : Type u_5\ninst✝² : SeminormedRing R\ninst✝¹ : (i : ι) → Module R (α i)\ninst✝ : ∀ (i : ι), IsBoundedSMul R (α i)\nthis : PseudoMetricSpace ((i : ι) → α i) := pseu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Haar.OfBasis | {
"line": 87,
"column": 6
} | {
"line": 87,
"column": 47
} | {
"line": 87,
"column": 48
} | [
{
"pp": "case refine_1\nι : Type u_1\nι' : Type u_2\nE : Type u_3\ninst✝³ : Fintype ι\ninst✝² : Fintype ι'\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\nv : ι → E\ne : ι' ≃ ι\nK : (ι' → ℝ) ≃ (ι → ℝ) := Equiv.piCongrLeft' (fun _a ↦ ℝ) e\nx : ι' → ℝ\nh : (∀ (i : ι), 0 ≤ x (e.symm i)) ∧ ∀ (i : ι), x (e.symm i) ≤ 1... | [
"case refine_1\nι : Type u_1\nι' : Type u_2\nE : Type u_3\ninst✝³ : Fintype ι\ninst✝² : Fintype ι'\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\nv : ι → E\ne : ι' ≃ ι\nK : (ι' → ℝ) ≃ (ι → ℝ) := Equiv.piCongrLeft' (fun _a ↦ ℝ) e\nx : ι' → ℝ\nh : (∀ (i : ι), 0 ≤ x (e.symm i)) ∧ ∀ (i : ι), x (e.symm i) ≤ 1\ni : ι'\n⊢ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Haar.OfBasis | {
"line": 88,
"column": 6
} | {
"line": 88,
"column": 47
} | {
"line": 88,
"column": 48
} | [
{
"pp": "case refine_2\nι : Type u_1\nι' : Type u_2\nE : Type u_3\ninst✝³ : Fintype ι\ninst✝² : Fintype ι'\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\nv : ι → E\ne : ι' ≃ ι\nK : (ι' → ℝ) ≃ (ι → ℝ) := Equiv.piCongrLeft' (fun _a ↦ ℝ) e\nx : ι' → ℝ\nh : (∀ (i : ι), 0 ≤ x (e.symm i)) ∧ ∀ (i : ι), x (e.symm i) ≤ 1... | [
"case refine_2\nι : Type u_1\nι' : Type u_2\nE : Type u_3\ninst✝³ : Fintype ι\ninst✝² : Fintype ι'\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\nv : ι → E\ne : ι' ≃ ι\nK : (ι' → ℝ) ≃ (ι → ℝ) := Equiv.piCongrLeft' (fun _a ↦ ℝ) e\nx : ι' → ℝ\nh : (∀ (i : ι), 0 ≤ x (e.symm i)) ∧ ∀ (i : ι), x (e.symm i) ≤ 1\ni : ι'\n⊢ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Haar.OfBasis | {
"line": 108,
"column": 4
} | {
"line": 108,
"column": 25
} | {
"line": 109,
"column": 4
} | [
{
"pp": "ι : Type u_1\ninst✝ : Fintype ι\nb : OrthonormalBasis ι ℝ ℝ\ne : ι ≃ Fin 1\nB : parallelepiped ⇑(b.reindex e) = parallelepiped ⇑b\nF : ℝ → Fin 1 → ℝ := fun t _i ↦ t\n⊢ Icc 0 1 = F '' Icc 0 1",
"ppTerm": "?m.224",
"assigned": true,
"usedConstants": [
"Set.Subset.antisymm",
"Real"... | [
"case h₁\nι : Type u_1\ninst✝ : Fintype ι\nb : OrthonormalBasis ι ℝ ℝ\ne : ι ≃ Fin 1\nB : parallelepiped ⇑(b.reindex e) = parallelepiped ⇑b\nF : ℝ → Fin 1 → ℝ := ⋯\n⊢ Icc 0 1 ⊆ F '' Icc 0 1",
"case h₂\nι : Type u_1\ninst✝ : Fintype ι\nb : OrthonormalBasis ι ℝ ℝ\ne : ι ≃ Fin 1\nB : parallelepiped ⇑(b.reindex e) = ... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 438,
"column": 4
} | {
"line": 438,
"column": 34
} | {
"line": 439,
"column": 4
} | [
{
"pp": "ι : Type u_1\ninst✝ : Fintype ι\nf : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : LinearMap.det f ≠ 0\n⊢ Measure.map (⇑f) volume = ofReal |(LinearMap.det f)⁻¹| • volume",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Pi.Function.module",
"Real",
"Algebra.to_smulCommClass",
"... | [
"ι : Type u_1\ninst✝ : Fintype ι\nf : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : LinearMap.det f ≠ 0\nM : Matrix ι ι ℝ := LinearMap.toMatrix' f\n⊢ Measure.map (⇑f) volume = ofReal |(LinearMap.det f)⁻¹| • volume"
] | let M := LinearMap.toMatrix' f | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 443,
"column": 4
} | {
"line": 443,
"column": 17
} | {
"line": 445,
"column": 0
} | [
{
"pp": "ι : Type u_1\ninst✝ : Fintype ι\nf : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : LinearMap.det f ≠ 0\nM : Matrix ι ι ℝ := ⋯\nA : LinearMap.det f = M.det\nB : f = toLin' M\n⊢ M.det ≠ 0",
"ppTerm": "?m.88",
"assigned": true,
"usedConstants": [
"Pi.Function.module",
"Real",
"MonoidHom.instFunL... | [] | rwa [A] at hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 456,
"column": 2
} | {
"line": 456,
"column": 79
} | {
"line": 456,
"column": 80
} | [
{
"pp": "α : Type u_1\nf g : α → ℝ\ns : Set α\n⊢ regionBetween f g s ⊆ s ×ˢ univ",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Eq.mpr",
"Real",
"SProd.sprod",
"congrArg",
"regionBetween",
"Set.univ",
"setOf",
"Set.pro... | [
"α : Type u_1\nf g : α → ℝ\ns : Set α\n⊢ ∀ (a : α × ℝ), a.1 ∈ s ∧ a.2 ∈ Ioo (f a.1) (g a.1) → a.1 ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 462,
"column": 43
} | {
"line": 468,
"column": 25
} | {
"line": 470,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nf g : α → ℝ\ns : Set α\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\n⊢ MeasurableSet (regionBetween f g s)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLT",
"Real.lattice",
"Meas... | [] | by
dsimp only [regionBetween, Ioo, mem_setOf_eq, setOf_and]
refine
MeasurableSet.inter ?_
((measurableSet_lt (hf.comp measurable_fst) measurable_snd).inter
(measurableSet_lt measurable_snd (hg.comp measurable_fst)))
exact measurable_fst hs | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 509,
"column": 2
} | {
"line": 509,
"column": 13
} | {
"line": 509,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ\nhf : Measurable f\n⊢ MeasurableSet {p | p.2 = f p.1}",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"MeasurableSet",
"measurableSet_setOf._simp_1",
"Measurable",
"setOf",
... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ\nhf : Measurable f\n⊢ Measurable fun p ↦ p.2 = f p.1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 176,
"column": 4
} | {
"line": 176,
"column": 72
} | {
"line": 176,
"column": 73
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : Submodule ℝ E\nhs : s ≠ ⊤\n⊢ ∃ x, x ∉ s",
"ppTerm": "?m.30",
"assigned": false,
"usedConst... | [
"E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : Submodule ℝ E\nhs : s ≠ ⊤\n⊢ ∃ x, x ∉ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 192,
"column": 4
} | {
"line": 192,
"column": 38
} | {
"line": 192,
"column": 39
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : Submodule ℝ E\nhs : s ≠ ⊤\nx : E\nhx : x ∉ s\nc : ℝ\ncpos : 0 < c\ncone : c < 1\nA✝ : Bornology.IsBoun... | [
"E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : Submodule ℝ E\nhs : s ≠ ⊤\nx : E\nhx : x ∉ s\nc : ℝ\ncpos : 0 < c\ncone : c < 1\nA✝ : Bornology.IsBounded (range f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 206,
"column": 2
} | {
"line": 207,
"column": 63
} | {
"line": 207,
"column": 64
} | [
{
"pp": "case inr\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : AffineSubspace ℝ E\nhs : ¬s.direction = ⊤\nx : E\nhx : x ∈ s\n⊢ μ ↑s = 0",
"ppTerm": "?i... | [
"case inr\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : AffineSubspace ℝ E\nhs : ¬s.direction = ⊤\nx : E\nhx : x ∈ s\n⊢ μ ↑s = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 566,
"column": 4
} | {
"line": 566,
"column": 72
} | {
"line": 568,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ\nf_mble : AEMeasurable f μ\ng_mble : AEMeasurable g μ\ns : Set α\ns_mble : NullMeasurableSet s μ\n⊢ NullMeasurableSet (fun p ↦ Real.lt✝ p.2 (g p.1)) (μ.prod volume)",
"ppTerm": "?refine_2",
"assigned": true,
... | [] | exact nullMeasurableSet_lt measurable_snd.aemeasurable (by fun_prop) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 566,
"column": 4
} | {
"line": 566,
"column": 72
} | {
"line": 568,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ\nf_mble : AEMeasurable f μ\ng_mble : AEMeasurable g μ\ns : Set α\ns_mble : NullMeasurableSet s μ\n⊢ NullMeasurableSet (fun p ↦ Real.lt✝ p.2 (g p.1)) (μ.prod volume)",
"ppTerm": "?refine_2",
"assigned": true,
... | [] | exact nullMeasurableSet_lt measurable_snd.aemeasurable (by fun_prop) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 566,
"column": 4
} | {
"line": 566,
"column": 72
} | {
"line": 568,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ\nf_mble : AEMeasurable f μ\ng_mble : AEMeasurable g μ\ns : Set α\ns_mble : NullMeasurableSet s μ\n⊢ NullMeasurableSet (fun p ↦ Real.lt✝ p.2 (g p.1)) (μ.prod volume)",
"ppTerm": "?refine_2",
"assigned": true,
... | [] | exact nullMeasurableSet_lt measurable_snd.aemeasurable (by fun_prop) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 596,
"column": 4
} | {
"line": 596,
"column": 72
} | {
"line": 598,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ\nf_mble : AEMeasurable f μ\ng_mble : AEMeasurable g μ\ns : Set α\ns_mble : NullMeasurableSet s μ\n⊢ NullMeasurableSet (fun p ↦ Real.lt✝ p.2 (g p.1)) (μ.prod volume)",
"ppTerm": "?refine_2",
"assigned": true,
... | [] | exact nullMeasurableSet_lt measurable_snd.aemeasurable (by fun_prop) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 596,
"column": 4
} | {
"line": 596,
"column": 72
} | {
"line": 598,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ\nf_mble : AEMeasurable f μ\ng_mble : AEMeasurable g μ\ns : Set α\ns_mble : NullMeasurableSet s μ\n⊢ NullMeasurableSet (fun p ↦ Real.lt✝ p.2 (g p.1)) (μ.prod volume)",
"ppTerm": "?refine_2",
"assigned": true,
... | [] | exact nullMeasurableSet_lt measurable_snd.aemeasurable (by fun_prop) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 596,
"column": 4
} | {
"line": 596,
"column": 72
} | {
"line": 598,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ\nf_mble : AEMeasurable f μ\ng_mble : AEMeasurable g μ\ns : Set α\ns_mble : NullMeasurableSet s μ\n⊢ NullMeasurableSet (fun p ↦ Real.lt✝ p.2 (g p.1)) (μ.prod volume)",
"ppTerm": "?refine_2",
"assigned": true,
... | [] | exact nullMeasurableSet_lt measurable_snd.aemeasurable (by fun_prop) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 638,
"column": 8
} | {
"line": 638,
"column": 83
} | {
"line": 638,
"column": 84
} | [
{
"pp": "μ : Measure ℝ\ninst✝ : NullSingletonClass μ\ns : Set ℝ\np : ℝ → Prop\nh : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ.restrict (s ∩ Ioo a b), p x\nT : ↑s × ↑s → Set ℝ := fun p ↦ Ioo ↑p.1 ↑p.2\nu : Set ℝ := ⋃ i, T i\nhfinite : (s \\ u).Finite\nA : Set (↑s × ↑s)\nA_count : A.Countable\nhA : ⋃ i ∈ ... | [
"μ : Measure ℝ\ninst✝ : NullSingletonClass μ\ns : Set ℝ\np : ℝ → Prop\nh : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ.restrict (s ∩ Ioo a b), p x\nT : ↑s × ↑s → Set ℝ := fun p ↦ Ioo ↑p.1 ↑p.2\nu : Set ℝ := ⋃ i, T i\nhfinite : (s \\ u).Finite\nA : Set (↑s × ↑s)\nA_count : A.Countable\nhA : ⋃ i ∈ A, T i = ⋃ i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 304,
"column": 4
} | {
"line": 304,
"column": 20
} | {
"line": 304,
"column": 21
} | [
{
"pp": "case inr\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nf : E →ₗ[ℝ] E\ns : Set E\nhf : LinearMap.det f = 0\n⊢ μ (⇑f '' s) = ENNReal.ofReal |LinearMap.de... | [
"case inr\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nf : E →ₗ[ℝ] E\ns : Set E\nhf : LinearMap.det f = 0\n⊢ μ (⇑f '' s) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 681,
"column": 6
} | {
"line": 681,
"column": 81
} | {
"line": 681,
"column": 82
} | [
{
"pp": "μ : Measure ℝ\ninst✝ : NullSingletonClass μ\ns : Set ℝ\np : ℝ → Prop\nh : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x\nT : ↑s × ↑s → Set ℝ := fun p ↦ Ioo ↑p.1 ↑p.2\nu : Set ℝ := ⋃ i, T i\nhfinite : (s \\ u).Finite\nA : Set (↑s × ↑s)\nA_count : A.Countable\nhA : ⋃ i ∈ A, T ... | [
"μ : Measure ℝ\ninst✝ : NullSingletonClass μ\ns : Set ℝ\np : ℝ → Prop\nh : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x\nT : ↑s × ↑s → Set ℝ := fun p ↦ Ioo ↑p.1 ↑p.2\nu : Set ℝ := ⋃ i, T i\nhfinite : (s \\ u).Finite\nA : Set (↑s × ↑s)\nA_count : A.Countable\nhA : ⋃ i ∈ A, T i = ⋃ i, T i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Box.Basic | {
"line": 149,
"column": 12
} | {
"line": 149,
"column": 67
} | {
"line": 149,
"column": 68
} | [
{
"pp": "ι : Type u_1\nI J : Box ι\ntfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J\nx✝ : ↑I ⊆ ↑J\nh : ↑I ⊆ ↑J := x✝\n⊢ Icc I.lower I.upper ⊆ Icc J.lower J.upper",
"ppTerm": "?m.36",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_1\nI J : Box ι\ntfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J\nx✝ : ↑I ⊆ ↑J\nh : ↑I ⊆ ↑J := x✝\n⊢ Icc I.lower I.upper ⊆ Icc J.lower J.upper"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Box.Basic | {
"line": 289,
"column": 23
} | {
"line": 289,
"column": 34
} | {
"line": 289,
"column": 35
} | [
{
"pp": "case pos\nι : Type u_1\nl u : ι → ℝ\nh : ∀ (i : ι), l i < u i\n⊢ ↑{ lower := l, upper := u, lower_lt_upper := h } = ⊥ ↔ ∃ i, u i ≤ l i",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"not_exists._simp_1",
"False",
"Real.instLE",
"Real",
... | [
"case pos\nι : Type u_1\nl u : ι → ℝ\nh : ∀ (i : ι), l i < u i\n⊢ ∀ (x : ι), l x < u x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Box.Basic | {
"line": 289,
"column": 23
} | {
"line": 289,
"column": 34
} | {
"line": 289,
"column": 35
} | [
{
"pp": "case neg\nι : Type u_1\nl u : ι → ℝ\nh : ¬∀ (i : ι), l i < u i\n⊢ ⊥ = ⊥ ↔ ∃ i, u i ≤ l i",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"WithBot",
"congrArg",
"true_iff",
"Exists",
"id",
"Bot.b... | [
"case neg\nι : Type u_1\nl u : ι → ℝ\nh : ¬∀ (i : ι), l i < u i\n⊢ ∃ i, u i ≤ l i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.Box.Basic | {
"line": 471,
"column": 2
} | {
"line": 472,
"column": 18
} | {
"line": 472,
"column": 19
} | [
{
"pp": "ι : Type u_1\ninst✝ : Fintype ι\nI : Box ι\ni : ι\nA : I.lower i - I.upper i < 0\n⊢ dist I.lower I.upper ≤ ↑I.distortion * (I.upper i - I.lower i)",
"ppTerm": "?m.39",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_1\ninst✝ : Fintype ι\nI : Box ι\ni : ι\nA : I.lower i - I.upper i < 0\n⊢ dist I.lower I.upper ≤ ↑I.distortion * (I.upper i - I.lower i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 510,
"column": 41
} | {
"line": 513,
"column": 51
} | {
"line": 515,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : FiniteDimensional ℝ E\nμ : Measure E\ninst✝¹ : μ.IsAddHaarMeasure\ninst✝ : Nontrivial E\nx : E\nr : ℝ\n⊢ μ (closedBall x r) = μ (ball x r)",
"ppTerm": "?m.20",
"ass... | [] | by
by_cases! h : r < 0
· rw [Metric.closedBall_eq_empty.mpr h, Metric.ball_eq_empty.mpr h.le]
rw [addHaar_closedBall μ x h, addHaar_ball μ x h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.BoxIntegral.Box.SubboxInduction | {
"line": 130,
"column": 39
} | {
"line": 130,
"column": 59
} | {
"line": 130,
"column": 60
} | [
{
"pp": "ι : Type u_1\np : Box ι → Prop\nI : Box ι\nH_nhds :\n ∀ z ∈ Box.Icc I,\n ∃ U ∈ 𝓝[Box.Icc I] z,\n ∀ J ≤ I,\n ∀ (m : ℕ),\n z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J\nhpI : ¬p I\ns : Box ι → Set ι\nhs : ∀ J ≤ I, ¬p ... | [
"ι : Type u_1\np : Box ι → Prop\nI : Box ι\nH_nhds :\n ∀ z ∈ Box.Icc I,\n ∃ U ∈ 𝓝[Box.Icc I] z,\n ∀ J ≤ I,\n ∀ (m : ℕ),\n z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J\nhpI : ¬p I\ns : Box ι → Set ι\nhs : ∀ J ≤ I, ¬p J → ¬p (J.sp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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