module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 255,
"column": 2
} | {
"line": 255,
"column": 64
} | [
{
"pp": "case h\n𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : TopologicalSpace H\ninst✝ : TopologicalSpace M\nf : OpenPartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\nα : Type u_8\nl : Filter α... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.IsManifold.Basic | {
"line": 359,
"column": 4
} | {
"line": 359,
"column": 19
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nh : IsRCLikeNormedField 𝕜\nthis✝ : RCLike 𝕜 := ⋯\nthis : NormedSpace ℝ E := ⋯\n⊢ Convex ℝ (range ↑I)",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.LocalInvariantProperties | {
"line": 586,
"column": 10
} | {
"line": 586,
"column": 80
} | [
{
"pp": "case mpr.refine_2\nH : Type u_1\ninst✝¹ : TopologicalSpace H\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\ns : Set H\nx : H\nu : Set H\nf : H → H\nhu : IsOpen[inst✝¹] u\nhux : x ∈ u\nh : G.IsLocalStructomorphWithinAt f (s ∩ u) x\nhx : x ∈ s\ne : OpenPartialHomeomorph H H\nheG : e ∈ G\nhef... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.LocalInvariantProperties | {
"line": 590,
"column": 29
} | {
"line": 590,
"column": 76
} | [
{
"pp": "H : Type u_1\ninst✝¹ : TopologicalSpace H\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\ns : Set H\nx : H\nf : H → H\ne' : OpenPartialHomeomorph H H\nhe'G : e' ∈ G\nhe'x : x ∈ e'.source\nh : G.IsLocalStructomorphWithinAt f s x\nhx : ↑e' x ∈ ↑e'.symm ⁻¹' s\n⊢ x ∈ s",
"usedConstants": []... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.LocalInvariantProperties | {
"line": 610,
"column": 8
} | {
"line": 610,
"column": 57
} | [
{
"pp": "case refine_2\nH : Type u_1\ninst✝¹ : TopologicalSpace H\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\ns : Set H\nx : H\nf : H → H\ne' : OpenPartialHomeomorph H H\nhe'G : e' ∈ G\na✝ : s ⊆ f ⁻¹' e'.source\nhfx : f x ∈ e'.source\nh : G.IsLocalStructomorphWithinAt f s x\nhx : x ∈ s\ne : Open... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.IsManifold.Basic | {
"line": 581,
"column": 74
} | {
"line": 581,
"column": 90
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_4\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\n⊢ 𝓘(𝕜, E × F) = 𝓘(𝕜, E).prod 𝓘(𝕜, F)",
"usedConstants": [
"Prod.normedSpace",
"Prod.no... | by ext1 <;> simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 537,
"column": 63
} | {
"line": 537,
"column": 74
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : TopologicalSpace H\ninst✝¹ : TopologicalSpace M\nI : ModelWithCorners 𝕜 E H\ninst✝ : ChartedSpace H M\nx y : M\nhx : y ∈ (extChartAt I x).sou... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.LocalInvariantProperties | {
"line": 682,
"column": 6
} | {
"line": 682,
"column": 48
} | [
{
"pp": "H₁ : Type u_6\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_7\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_8\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroup... | refine ⟨_, hs.inter φ.open_source, ?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 139,
"column": 48
} | {
"line": 139,
"column": 81
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nE' : Type u_5\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝ : Topologic... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 647,
"column": 48
} | {
"line": 647,
"column": 80
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : TopologicalSpace H\ninst✝¹ : TopologicalSpace M\nI : ModelWithCorners 𝕜 E H\ninst✝ : ChartedSpace H M\nx : M\ny : E\nhy : y ∈ (extChartAt I x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 728,
"column": 51
} | {
"line": 728,
"column": 62
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : TopologicalSpace H\ninst✝¹ : TopologicalSpace M\nI : ModelWithCorners 𝕜 E H\ninst✝ : ChartedSpace H M\ns : Set E\nx x' : M\nhx : x' ∈ (extCha... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 304,
"column": 47
} | {
"line": 304,
"column": 58
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝⁴ : NormedAddCom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 304,
"column": 47
} | {
"line": 304,
"column": 63
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝⁴ : NormedAddCom... | simpa using hy.2 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 304,
"column": 47
} | {
"line": 304,
"column": 63
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝⁴ : NormedAddCom... | simpa using hy.2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 304,
"column": 47
} | {
"line": 304,
"column": 63
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝⁴ : NormedAddCom... | simpa using hy.2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.ContMDiff.Basic | {
"line": 176,
"column": 12
} | {
"line": 176,
"column": 23
} | [
{
"pp": "case zero\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\ns : Set M\nn : ℕ∞ω\nf : M →... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 305,
"column": 8
} | {
"line": 305,
"column": 26
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝⁴ : NormedAddCom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.ContMDiff.Basic | {
"line": 177,
"column": 16
} | {
"line": 177,
"column": 27
} | [
{
"pp": "case succ\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\ns : Set M\nn : ℕ∞ω\nf : M →... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 309,
"column": 47
} | {
"line": 309,
"column": 58
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝⁴ : NormedAddCom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 309,
"column": 47
} | {
"line": 309,
"column": 63
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝⁴ : NormedAddCom... | simpa using hy.2 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 309,
"column": 47
} | {
"line": 309,
"column": 63
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝⁴ : NormedAddCom... | simpa using hy.2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 309,
"column": 47
} | {
"line": 309,
"column": 63
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝⁴ : NormedAddCom... | simpa using hy.2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 824,
"column": 2
} | {
"line": 824,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\nI : ModelWithCorners 𝕜 E H\ninst✝⁵ : NormedAdd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 824,
"column": 44
} | {
"line": 824,
"column": 55
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\nI : ModelWithCorners 𝕜 E H\ninst✝⁵ : NormedAdd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.ContMDiff.Basic | {
"line": 297,
"column": 4
} | {
"line": 297,
"column": 15
} | [
{
"pp": "case h.hi\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁸ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁷ : TopologicalSpace M\nE' : Type u_5\ninst✝⁶ : NormedAddCommGroup E'\ninst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.ContMDiff.Basic | {
"line": 307,
"column": 22
} | {
"line": 307,
"column": 33
} | [
{
"pp": "n : ℕ∞ω\nE : Type u_11\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\nH : Type u_12\ninst✝² : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_13\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\nf g : ℝ → M\ns : ℝ\nhf : ContMDiff 𝓘(ℝ, ℝ) I n f\nhg : ContMDiff 𝓘(ℝ, ℝ) I n g... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.ContMDiff.Constructions | {
"line": 110,
"column": 4
} | {
"line": 110,
"column": 97
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nF : Type u_8\ninst✝⁴ ... | exact (extChartAt I p.1).right_inv <| (extChartAt I p.1).map_source (mem_extChartAt_source _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Manifold.ContMDiff.Constructions | {
"line": 110,
"column": 4
} | {
"line": 110,
"column": 97
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nF : Type u_8\ninst✝⁴ ... | exact (extChartAt I p.1).right_inv <| (extChartAt I p.1).map_source (mem_extChartAt_source _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.ContMDiff.Constructions | {
"line": 110,
"column": 4
} | {
"line": 110,
"column": 97
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nF : Type u_8\ninst✝⁴ ... | exact (extChartAt I p.1).right_inv <| (extChartAt I p.1).map_source (mem_extChartAt_source _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Operator.Prod | {
"line": 36,
"column": 49
} | {
"line": 36,
"column": 75
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace 𝕜 F\nx✝ : E × F\ne : E\nf : F\n⊢ ‖(fst 𝕜 E F) (e, f)‖ ≤ 1 * ‖(e, f)‖",
"usedConstants": [
"Nor... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Prod | {
"line": 41,
"column": 49
} | {
"line": 41,
"column": 75
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace 𝕜 F\nx✝ : E × F\ne : E\nf : F\n⊢ ‖(snd 𝕜 E F) (e, f)‖ ≤ 1 * ‖(e, f)‖",
"usedConstants": [
"Nor... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Prod | {
"line": 51,
"column": 8
} | {
"line": 51,
"column": 84
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedAddCommGroup E\ninst✝⁴ : SeminormedAddCommGroup F\ninst✝³ : SeminormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : NormedSpace 𝕜 G\nf : E →L[𝕜] F\ng : E →L... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.ContMDiff.Constructions | {
"line": 441,
"column": 6
} | {
"line": 441,
"column": 17
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁸ : TopologicalSpace M\ninst✝⁷ : ChartedSpace H M\nM' : Type u_16\ninst✝⁶ : Topologic... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.ContMDiff.Constructions | {
"line": 456,
"column": 6
} | {
"line": 456,
"column": 17
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁸ : TopologicalSpace M\ninst✝⁷ : ChartedSpace H M\nM' : Type u_16\ninst✝⁶ : Topologic... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.ContMDiff.Constructions | {
"line": 452,
"column": 4
} | {
"line": 456,
"column": 23
} | [
{
"pp": "case inr\n𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁸ : TopologicalSpace M\ninst✝⁷ : ChartedSpace H M\nM' : Type u_16\ninst✝⁶ :... | have : ContDiffWithinAt 𝕜 n ((extChartAt J (g x)) ∘ g ∘ (extChartAt I x).symm)
(range I) ((extChartAt I (.inr x : M ⊕ M')) (Sum.inr x)) := by
let hg' := hg x
rw [contMDiffAt_iff] at hg'
simpa using hg'.2 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Manifold.ContMDiff.Constructions | {
"line": 478,
"column": 10
} | {
"line": 478,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁸ : TopologicalSpace M\ninst✝⁷ : ChartedSpace H M\nn : WithTop ℕ∞\nE' : Type u_17\nin... | _hx | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Geometry.Manifold.ContMDiff.Constructions | {
"line": 491,
"column": 10
} | {
"line": 491,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM' : Type u_16\ninst✝⁸ : TopologicalSpace M'\ninst✝⁷ : ChartedSpace H M'\nn : WithTop ℕ∞\nE' : Type u_17... | _hx | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Geometry.Manifold.MFDeriv.Defs | {
"line": 157,
"column": 6
} | {
"line": 166,
"column": 47
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nE' : Type u_5\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝ : Topologic... | intro s x u f u_open xu
have : I.symm ⁻¹' (s ∩ u) ∩ Set.range I = I.symm ⁻¹' s ∩ Set.range I ∩ I.symm ⁻¹' u := by
simp only [Set.inter_right_comm, Set.preimage_inter]
rw [DifferentiableWithinAtProp, DifferentiableWithinAtProp, this]
symm
apply differentiableWithinAt_inter
have : u ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.MFDeriv.Defs | {
"line": 157,
"column": 6
} | {
"line": 166,
"column": 47
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nE' : Type u_5\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝ : Topologic... | intro s x u f u_open xu
have : I.symm ⁻¹' (s ∩ u) ∩ Set.range I = I.symm ⁻¹' s ∩ Set.range I ∩ I.symm ⁻¹' u := by
simp only [Set.inter_right_comm, Set.preimage_inter]
rw [DifferentiableWithinAtProp, DifferentiableWithinAtProp, this]
symm
apply differentiableWithinAt_inter
have : u ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 632,
"column": 2
} | {
"line": 632,
"column": 12
} | [
{
"pp": "case coe\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝⁴ : No... | | coe n => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.Geometry.Manifold.MFDeriv.Defs | {
"line": 197,
"column": 50
} | {
"line": 197,
"column": 83
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nE' : Type u_5\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝ : Topologic... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.Algebra.Monoid | {
"line": 268,
"column": 16
} | {
"line": 268,
"column": 38
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nn : ℕ∞ω\nH : Type u_2\ninst✝⁶ : TopologicalSpace H\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nG : Type u_4\ninst✝³ : Monoid G\ninst✝² : TopologicalSpace G\ninst✝¹ : ChartedSpace H G\ninst✝ : C... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.Algebra.Monoid | {
"line": 513,
"column": 2
} | {
"line": 513,
"column": 35
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nn : ℕ∞ω\nH : Type u_2\ninst✝¹¹ : TopologicalSpace H\nE : Type u_3\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nG : Type u_4\ninst✝⁸ : DivInvMonoid G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : ChartedSpace H G\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.Algebra.LieGroup | {
"line": 309,
"column": 2
} | {
"line": 309,
"column": 30
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹³ : NontriviallyNormedField 𝕜\nn : ℕ∞ω\nH : Type u_2\ninst✝¹² : TopologicalSpace H\nE : Type u_3\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nG : Type u_4\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : ChartedSpace H G\ninst✝⁷ : GroupWithZero G... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.Algebra.LieGroup | {
"line": 313,
"column": 2
} | {
"line": 313,
"column": 30
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹³ : NontriviallyNormedField 𝕜\nn : ℕ∞ω\nH : Type u_2\ninst✝¹² : TopologicalSpace H\nE : Type u_3\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nG : Type u_4\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : ChartedSpace H G\ninst✝⁷ : GroupWithZero G... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.Algebra.LieGroup | {
"line": 317,
"column": 2
} | {
"line": 317,
"column": 30
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹³ : NontriviallyNormedField 𝕜\nn : ℕ∞ω\nH : Type u_2\ninst✝¹² : TopologicalSpace H\nE : Type u_3\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nG : Type u_4\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : ChartedSpace H G\ninst✝⁷ : GroupWithZero G... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.Algebra.LieGroup | {
"line": 320,
"column": 27
} | {
"line": 320,
"column": 60
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹³ : NontriviallyNormedField 𝕜\nn : ℕ∞ω\nH : Type u_2\ninst✝¹² : TopologicalSpace H\nE : Type u_3\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nG : Type u_4\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : ChartedSpace H G\ninst✝⁷ : GroupWithZero G... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.Algebra.Structures | {
"line": 47,
"column": 22
} | {
"line": 47,
"column": 52
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝⁶ : TopologicalSpace H\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nn : ℕ∞ω\nI : ModelWithCorners 𝕜 E H\nR : Type u_4\ninst✝³ : Ring R\ninst✝² : TopologicalSpace R\ninst✝¹ : ChartedSpace H R\ninst✝ : Con... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.BumpFunction | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 23
} | [
{
"pp": "E : Type uE\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nH : Type uH\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\nc : M\nf : SmoothBumpFunction I c\ninst✝ : FiniteDimensional ℝ E\ns : Set M\nhse : s ⊆ (extCha... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Geometry.Manifold.BumpFunction | {
"line": 212,
"column": 38
} | {
"line": 216,
"column": 59
} | [
{
"pp": "E : Type uE\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nH : Type uH\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\nc : M\nf : SmoothBumpFunction I c\ninst✝ : FiniteDimensional ℝ E\ns : Set M\nhsc : IsClosed[in... | by
rw [f.image_eq_inter_preimage_of_subset_support hs]
refine ContinuousOn.preimage_isClosed_of_isClosed
((continuousOn_extChartAt_symm _).mono f.closedBall_subset) ?_ hsc
exact IsClosed.inter isClosed_closedBall I.isClosed_range | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.BumpFunction | {
"line": 262,
"column": 2
} | {
"line": 262,
"column": 38
} | [
{
"pp": "E : Type uE\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\nH : Type uH\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nc : M\nf : SmoothBumpFunction I c\ninst✝¹ : FiniteDimensional ℝ E\ninst✝ : T2Space M\n⊢ tsuppo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear | {
"line": 130,
"column": 4
} | {
"line": 130,
"column": 57
} | [
{
"pp": "𝕜 : Type u_1\nB : Type u_2\nF : Type u_3\ninst✝⁷ : TopologicalSpace B\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nEB : Type u_4\ninst✝³ : NormedAddCommGroup EB\ninst✝² : NormedSpace 𝕜 EB\nHB : Type u_5\ninst✝¹ : TopologicalSpace HB\ninst✝ : ChartedS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.MFDeriv.Basic | {
"line": 73,
"column": 21
} | {
"line": 73,
"column": 67
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\ns t : Set M\nx : M\nhs : UniqueMDiffWi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear | {
"line": 255,
"column": 4
} | {
"line": 255,
"column": 71
} | [
{
"pp": "𝕜 : Type u_1\nB : Type u_2\nF : Type u_3\ninst✝⁷ : TopologicalSpace B\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nEB : Type u_4\ninst✝³ : NormedAddCommGroup EB\ninst✝² : NormedSpace 𝕜 EB\nHB : Type u_5\ninst✝¹ : TopologicalSpace HB\ninst✝ : ChartedS... | rintro e e' ⟨φ, U, hU, hφ, h2φ, heφ⟩ ⟨φ', U', hU', hφ', h2φ', heφ'⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Topology.VectorBundle.Basic | {
"line": 495,
"column": 2
} | {
"line": 495,
"column": 13
} | [
{
"pp": "R : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁹ : NontriviallyNormedField R\ninst✝⁸ : (x : B) → AddCommMonoid (E x)\ninst✝⁷ : (x : B) → Module R (E x)\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace R F\ninst✝⁴ : TopologicalSpace B\ninst✝³ : TopologicalSpace (TotalSpace F E)\nin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.VectorBundle.Basic | {
"line": 301,
"column": 45
} | {
"line": 305,
"column": 51
} | [
{
"pp": "n : ℕ∞ω\n𝕜 : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B → Type u_6\ninst✝¹⁸ : NontriviallyNormedField 𝕜\nEB : Type u_7\ninst✝¹⁷ : NormedAddCommGroup EB\ninst✝¹⁶ : NormedSpace 𝕜 EB\nHB : Type u_8\ninst✝¹⁵ : TopologicalSpace HB\nIB : ModelWithCorners 𝕜 EB HB\ninst✝¹⁴ : T... | by
constructor
intro e e' he he'
rw [contMDiffOn_zero_iff]
exact VectorBundle.continuousOn_coordChange' e e' | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Compactness.Paracompact | {
"line": 134,
"column": 6
} | {
"line": 134,
"column": 73
} | [
{
"pp": "X : Type v\nY : Type w\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : ParacompactSpace Y\ne : X → Y\nhe : IsClosedEmbedding e\nα : Type v\ns : α → Set X\nho : ∀ (a : α), IsOpen[inst✝²] (s a)\nU : α → Set Y\nhUo : ∀ (a : α), IsOpen[inst✝¹] (U a)\nhU : ∀ (a : α), e ⁻¹' U a = s a\nhu :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Paracompact | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 71
} | [
{
"pp": "X : Type v\nY : Type w\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : ParacompactSpace Y\ne : X → Y\nhe : IsClosedEmbedding e\nα : Type v\ns : α → Set X\nho : ∀ (a : α), IsOpen[inst✝²] (s a)\nU : α → Set Y\nhUo : ∀ (a : α), IsOpen[inst✝¹] (U a)\nhU : ∀ (a : α), e ⁻¹' U a = s a\nhu :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Paracompact | {
"line": 221,
"column": 6
} | {
"line": 221,
"column": 39
} | [
{
"pp": "X : Type v\ninst✝³ : TopologicalSpace X\ninst✝² : WeaklyLocallyCompactSpace X\ninst✝¹ : SigmaCompactSpace X\ninst✝ : T2Space X\nι : X → Type u\np : (x : X) → ι x → Prop\nB : (x : X) → ι x → Set X\ns : Set X\nhs : IsClosed[inst✝³] s\nhB : ∀ x ∈ s, (𝓝 x).HasBasis (p x) (B x)\nK' : CompactExhaustion X :=... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection | {
"line": 163,
"column": 4
} | {
"line": 163,
"column": 39
} | [
{
"pp": "case empty\n𝕜 : Type u_1\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁰ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : ChartedSpace H M\nF : Type u_5\ninst✝⁷ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.Paracompact | {
"line": 92,
"column": 6
} | {
"line": 92,
"column": 49
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\npow_pos : ∀ (k : ℕ), 0 < 2⁻¹ ^ k\nhpow_le : ∀ {m n : ℕ}, m ≤ n → 2⁻¹ ^ n ≤ 2⁻¹ ^ m\nh2pow : ∀ (n : ℕ), 2 * 2⁻¹ ^ (n + 1) = 2⁻¹ ^ n\nι : Type u_1\ns : ι → Set α\nho : ∀ (a : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (s a)\nhcov : ∀ (x : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection | {
"line": 220,
"column": 6
} | {
"line": 220,
"column": 17
} | [
{
"pp": "case h\n𝕜 : Type u_1\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁰ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : ChartedSpace H M\nF : Type u_5\ninst✝⁷ : No... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.VectorBundle.Basic | {
"line": 616,
"column": 4
} | {
"line": 616,
"column": 40
} | [
{
"pp": "n : ℕ∞ω\n𝕜 : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B → Type u_6\ninst✝³⁸ : NontriviallyNormedField 𝕜\nEB : Type u_7\ninst✝³⁷ : NormedAddCommGroup EB\ninst✝³⁶ : NormedSpace 𝕜 EB\nHB : Type u_8\ninst✝³⁵ : TopologicalSpace HB\nIB : ModelWithCorners 𝕜 EB HB\ninst✝³⁴ : T... | refine ContMDiffOn.clm_prodMap ?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection | {
"line": 229,
"column": 4
} | {
"line": 229,
"column": 15
} | [
{
"pp": "case h\n𝕜 : Type u_1\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁰ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : ChartedSpace H M\nF : Type u_5\ninst✝⁷ : No... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection | {
"line": 265,
"column": 4
} | {
"line": 265,
"column": 15
} | [
{
"pp": "case h\n𝕜 : Type u_1\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁰ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : ChartedSpace H M\nF : Type u_5\ninst✝⁷ : No... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.PartitionOfUnity | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 72
} | [
{
"pp": "ι : Type u\nX : Type v\ninst✝ : TopologicalSpace X\ns : Set X\nf : PartitionOfUnity ι X s\nx : X\nhx : x ∈ s\nH : ∀ (i : ι), (f i) x ≤ 0\n⊢ ∑ᶠ (i : ι), (f i) x ≠ 1",
"usedConstants": [
"Eq.mpr",
"finsum_zero",
"Real.partialOrder",
"Real",
"Real.instZero",
"congrA... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.PartitionOfUnity | {
"line": 204,
"column": 29
} | {
"line": 204,
"column": 49
} | [
{
"pp": "ι : Type u\nX : Type v\ninst✝ : TopologicalSpace X\ns : Set X\nρ : PartitionOfUnity ι X s\nx₀ : X\nhx₀ : x₀ ∈ s\nI : Finset ι\nhI : ρ.finsupport x₀ ⊆ I\n⊢ ∑ x ∈ I \\ ρ.finsupport x₀, (ρ x) x₀ + ∑ x ∈ ρ.finsupport x₀, (ρ x) x₀ = 1",
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
... | ρ.sum_finsupport hx₀ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.PartitionOfUnity | {
"line": 471,
"column": 2
} | {
"line": 471,
"column": 13
} | [
{
"pp": "ι : Type u\nX : Type v\ninst✝² : TopologicalSpace X\ns : Set X\ninst✝¹ : LocallyCompactSpace X\ninst✝ : T2Space X\nhs : IsCompact s\nU : ι → Set X\nho : ∀ (i : ι), IsOpen[inst✝²] (U i)\nhf : LocallyFinite U\nhU : s ⊆ ⋃ i, U i\n⊢ ∃ f, f.IsSubordinate U ∧ ∀ (i : ι), HasCompactSupport ⇑(f i)",
"usedCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.PartitionOfUnity | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 90
} | [
{
"pp": "ι : Type u_1\nX : Type u_2\ninst✝ : EMetricSpace X\nK U : ι → Set X\nhK : ∀ (i : ι), IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (K i)\nhU : ∀ (i : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (U i)\nhKU : ∀ (i : ι), K i ⊆ U i\nhfin : LocallyFinite K\n⊢ ∃ δ, (∀ (x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.PartitionOfUnity | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 48
} | [
{
"pp": "ι : Type u_1\nX : Type u_2\ninst✝ : EMetricSpace X\nK U : ι → Set X\nhK : ∀ (i : ι), IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (K i)\nhU : ∀ (i : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (U i)\nhKU : ∀ (i : ι), K i ⊆ U i\nhfin : LocallyFinite K\nδ : C(X, ℝ≥0... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ShrinkingLemma | {
"line": 319,
"column": 2
} | {
"line": 330,
"column": 34
} | [
{
"pp": "case refine_4\nι : Type u_1\nX : Type u_2\ninst✝² : TopologicalSpace X\nu : ι → Set X\ns : Set X\ninst✝¹ : T2Space X\ninst✝ : LocallyCompactSpace X\nv : PartialRefinement u s fun w ↦ IsCompact (closure w)\nhs : IsCompact s\ni : ι\nhi : i ∉ v.carrier\nsi : Set X := s ∩ (⋃ j, ⋃ (_ : j ≠ i), v.toFun j)ᶜ\n... | · intro j hj
rw [mem_insert_iff] at hj
by_cases h : j = i
· rw [← h]
simp only [update_self]
exact hvi.2.2.2
· apply hj.elim
· intro hji
exact False.elim (h hji)
· intro hjmemv
rw [update_of_ne h]
exact v.pred_of_mem hjmemv | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Distribution.AEEqOfIntegralContDiff | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 58
} | [
{
"pp": "E : Type u_1\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace ℝ E\ninst✝¹¹ : FiniteDimensional ℝ E\nF : Type u_2\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\ninst✝⁸ : CompleteSpace F\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_4\ninst✝⁶ : Topol... | have := ChartedSpace.secondCountable_of_sigmaCompact H M | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Distribution.AEEqOfIntegralContDiff | {
"line": 104,
"column": 10
} | {
"line": 104,
"column": 21
} | [
{
"pp": "E : Type u_1\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace ℝ E\ninst✝¹¹ : FiniteDimensional ℝ E\nF : Type u_2\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\ninst✝⁸ : CompleteSpace F\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_4\ninst✝⁶ : Topol... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.AEEqOfIntegral | {
"line": 219,
"column": 4
} | {
"line": 219,
"column": 15
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ\nhf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = 0\nt : Set α\nht : MeasurableSet t\nhμt : μ t ≠ ∞\nh_neg : 0 ≤ᶠ[ae (μ.restrict t)]... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.AEEqOfIntegral | {
"line": 252,
"column": 4
} | {
"line": 252,
"column": 45
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : α → E\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ\nhg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → Int... | exact sub_eq_zero.mpr (hfg_zero s hs hμs) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Manifold.PartitionOfUnity | {
"line": 384,
"column": 4
} | {
"line": 384,
"column": 69
} | [
{
"pp": "case refine_3\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nH : Type uH\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : FiniteDimensional ℝ E\ns : Set M\nU : M → Set M\ninst✝¹ : T2Space M\ni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.PartitionOfUnity | {
"line": 463,
"column": 2
} | {
"line": 463,
"column": 13
} | [
{
"pp": "ι : Type uι\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nH : Type uH\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : FiniteDimensional ℝ E\ns : Set M\nfs : SmoothBumpCovering ι I M s\ninst✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.AEEqOfIntegralContDiff | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 24
} | [
{
"pp": "E : Type u_1\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace ℝ E\ninst✝¹¹ : FiniteDimensional ℝ E\nF : Type u_2\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\ninst✝⁸ : CompleteSpace F\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_4\ninst✝⁶ : Topol... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.AEEqOfIntegralContDiff | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 20
} | [
{
"pp": "E : Type u_1\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace ℝ E\ninst✝¹¹ : FiniteDimensional ℝ E\nF : Type u_2\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\ninst✝⁸ : CompleteSpace F\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_4\ninst✝⁶ : Topol... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.AEEqOfIntegral | {
"line": 352,
"column": 2
} | {
"line": 357,
"column": 42
} | [
{
"pp": "case refine_2\nα : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhm : m ≤ m0\nf : α → E\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ\nhf_zero : ∀ (s : Set α), Measurabl... | · intro s hs hμs
rw [restrict_trim hm (μ.restrict t) hs, Measure.restrict_restrict (hm s hs)]
rw [← restrict_trim hm μ ht_meas, Measure.restrict_apply hs,
trim_measurableSet_eq hm (hs.inter ht_meas)] at hμs
rw [← integral_trim hm hf_meas_m]
exact hf_zero _ (hs.inter ht_meas) hμs | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.AEEqOfIntegral | {
"line": 384,
"column": 4
} | {
"line": 384,
"column": 23
} | [
{
"pp": "E : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\nβ : Type u_3\ninst✝² : TopologicalSpace β\ninst✝¹ : MeasurableSpace β\ninst✝ : BorelSpace β\nμ : Measure β\nf : β → E\nhf : Integrable f μ\nh'f : ∀ (s : Set β), IsClosed[inst✝²] s → ∫ (x : β) in s, f x ∂μ =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.PartitionOfUnity | {
"line": 699,
"column": 2
} | {
"line": 700,
"column": 9
} | [
{
"pp": "ι : Type uι\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nH : Type uH\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\ninst✝⁴ : FiniteDimensional ℝ E\nn : ℕ∞\nM : Type u_1\ninst✝³ : EMetricSpace M\ninst✝² : ChartedSpace H M\ninst✝¹ : IsManifold I ∞ M\ninst✝ : SigmaComp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.AEEqOfIntegralContDiff | {
"line": 187,
"column": 2
} | {
"line": 187,
"column": 27
} | [
{
"pp": "case h\nE : Type u_1\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace ℝ E\ninst✝¹¹ : FiniteDimensional ℝ E\nF : Type u_2\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\ninst✝⁸ : CompleteSpace F\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_4\ninst✝⁶... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Rademacher | {
"line": 86,
"column": 70
} | {
"line": 86,
"column": 81
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nC : ℝ≥0\nf : E → ℝ\nμ : Measure E\ninst✝¹ : FiniteDimensional ℝ E\ninst✝ : μ.IsAddHaarMeasure\nhf : LipschitzWith C f\nv : E\nL : ℝ →L[ℝ] E := ContinuousLinearMap.smulRight 1 v\np :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.TangentCone.ProperSpace | {
"line": 44,
"column": 61
} | {
"line": 44,
"column": 72
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : ProperSpace E\ns : Set E\nx : E\nhx : AccPt x (𝓟 s)\nu : ℕ → ℝ\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nv : ℕ → E\nhvx : ∀ (n : ℕ), v n ≠ x\nhvu : ∀ (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.PartitionOfUnity | {
"line": 774,
"column": 6
} | {
"line": 774,
"column": 36
} | [
{
"pp": "case refine_1.refine_1.h\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\nH : Type uH\ninst✝⁶ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : IsManifold I ∞ M\ninst✝¹ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.TangentCone.ProperSpace | {
"line": 62,
"column": 32
} | {
"line": 62,
"column": 58
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : ProperSpace E\ns : Set E\nx : E\nhx : AccPt x (𝓟 s)\nu : ℕ → ℝ\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nv : ℕ → E\nhvx : ∀ (n : ℕ), v n ≠ x\nhvu : ∀ (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Haar.Disintegration | {
"line": 93,
"column": 4
} | {
"line": 93,
"column": 49
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : CompleteSpace 𝕜\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : MeasurableSpace E\ninst✝⁸ : BorelSpace E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : MeasurableSpace F\ninst✝⁴ : BorelSpace F\nin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Haar.Disintegration | {
"line": 101,
"column": 4
} | {
"line": 101,
"column": 49
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : CompleteSpace 𝕜\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : MeasurableSpace E\ninst✝⁸ : BorelSpace E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : MeasurableSpace F\ninst✝⁴ : BorelSpace F\nin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.PartitionOfUnity | {
"line": 817,
"column": 45
} | {
"line": 817,
"column": 56
} | [
{
"pp": "E : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\nH : Type uH\ninst✝⁶ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : IsManifold I ∞ M\ninst✝¹ : SigmaCompactSpace M\ninst✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.PartitionOfUnity | {
"line": 812,
"column": 4
} | {
"line": 818,
"column": 24
} | [
{
"pp": "case neg\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\nH : Type uH\ninst✝⁶ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : IsManifold I ∞ M\ninst✝¹ : SigmaCompactSpac... | · have : 0 < g x := by
classical
apply lt_of_le_of_ne (g_pos x) (Ne.symm ?_)
rw [← mem_support, g_supp]
contrapose xs
exact h.trans f_supp.symm.subset (by simpa using xs)
linarith [f_pos x] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Manifold.PartitionOfUnity | {
"line": 825,
"column": 4
} | {
"line": 825,
"column": 46
} | [
{
"pp": "E : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\nH : Type uH\ninst✝⁶ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : IsManifold I ∞ M\ninst✝¹ : SigmaCompactSpace M\ninst✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.TangentCone.Seq | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : NormedDivisionRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕜 E\nα : Type u_3\nl : Filter α\nc : α → 𝕜\nd : α → E\ny : E\nhc : Tendsto c l (Bornology.cobounded 𝕜)\nhd : Tendsto (fun n ↦ c n • d n) l (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SuccPred.IntervalSucc | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 13
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : OrderBot α\ninst✝² : SuccOrder α\ninst✝¹ : IsSuccArchimedean α\ninst✝ : LinearOrder β\nf : α → β\nhf : ∀ (a : α), f ⊥ ≤ f a\nh2f : ¬BddAbove (range f)\n⊢ ⋃ a, Ico (f a) (f (succ a)) = Ici (f ⊥)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SuccPred.IntervalSucc | {
"line": 70,
"column": 2
} | {
"line": 70,
"column": 13
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : OrderBot α\ninst✝² : SuccOrder α\ninst✝¹ : IsSuccArchimedean α\ninst✝ : LinearOrder β\nf : α → β\nhf : ∀ (a : α), f ⊥ ≤ f a\nh2f : ¬BddAbove (range f)\n⊢ ⋃ a, Ioc (f a) (f (succ a)) = Ioi (f ⊥)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SuccPred.IntervalSucc | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 13
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : LinearOrder α\ninst✝¹ : PredOrder α\ninst✝ : Preorder β\nf : α → β\nhf : Monotone f\n⊢ Pairwise (Disjoint on fun n ↦ Ioc (f (pred n)) (f n))",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SuccPred.IntervalSucc | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 13
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : LinearOrder α\ninst✝¹ : PredOrder α\ninst✝ : Preorder β\nf : α → β\nhf : Monotone f\n⊢ Pairwise (Disjoint on fun n ↦ Ico (f (pred n)) (f n))",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SuccPred.IntervalSucc | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 13
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : LinearOrder α\ninst✝¹ : PredOrder α\ninst✝ : Preorder β\nf : α → β\nhf : Monotone f\n⊢ Pairwise (Disjoint on fun n ↦ Ioo (f (pred n)) (f n))",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.TangentCone.Seq | {
"line": 130,
"column": 6
} | {
"line": 130,
"column": 66
} | [
{
"pp": "case mp.inr\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : Set E\nx y : E\nhy₀ : y ≠ 0\nc : ℕ → 𝕜\nd : ℕ → E\nhds : ∀ᶠ (n : ℕ) in atTop, x + d n ∈ s\nhd₀' : ∀ᶠ (n : ℕ) in atTop, d n ≠ 0\nhd₀ : Tendsto (fun x ↦ ‖d x‖) atTo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Rademacher | {
"line": 261,
"column": 2
} | {
"line": 261,
"column": 17
} | [
{
"pp": "case h\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nC : ℝ≥0\nf : E → ℝ\nμ : Measure E\ninst✝¹ : FiniteDimensional ℝ E\ninst✝ : μ.IsAddHaarMeasure\nhf : LipschitzWith C f\ns : Set E\nhs : s.Countable\nB : Basis ↑(Basis.ofVecto... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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