module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.Normed.Module.MultipliableUniformlyOn | {
"line": 101,
"column": 74
} | {
"line": 101,
"column": 85
} | {
"line": 101,
"column": 86
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nK : Set α\nu : ι → ℝ\nR : Type u_3\ninst✝³ : NormedCommRing R\ninst✝² : NormOneClass R\ninst✝¹ : CompleteSpace R\ninst✝ : TopologicalSpace α\nf : ι → α → R\nhK : IsCompact K\nhu : Summable u\nh : ∀ᶠ (i : ι) in cofinite, ∀ x ∈ K, ‖f i x‖ ≤ u i\nhcts : ∀ (i : ι), ContinuousOn ... | [
"α : Type u_1\nι : Type u_2\nK : Set α\nu : ι → ℝ\nR : Type u_3\ninst✝³ : NormedCommRing R\ninst✝² : NormOneClass R\ninst✝¹ : CompleteSpace R\ninst✝ : TopologicalSpace α\nf : ι → α → R\nhK : IsCompact K\nhu : Summable u\nh : ∀ᶠ (i : ι) in cofinite, ∀ x ∈ K, ‖f i x‖ ≤ u i\nhcts : ∀ (i : ι), ContinuousOn (f i) K\nhKe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.lpHolder | {
"line": 51,
"column": 20
} | {
"line": 51,
"column": 50
} | {
"line": 51,
"column": 51
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_2\nE : α → Type u_3\nF : α → Type u_4\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\ninst✝³ : (i : α) → NormedSpace 𝕜 (E i)\ninst✝² : (i : α) → NormedAddCommGroup (F i)\ninst✝¹ : (i : α) → NormedSpace 𝕜 (F i)\np✝ q r p : ℝ≥0∞\ninst✝ : Fact... | [
"α : Type u_1\n𝕜 : Type u_2\nE : α → Type u_3\nF : α → Type u_4\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\ninst✝³ : (i : α) → NormedSpace 𝕜 (E i)\ninst✝² : (i : α) → NormedAddCommGroup (F i)\ninst✝¹ : (i : α) → NormedSpace 𝕜 (F i)\np✝ q r p : ℝ≥0∞\ninst✝ : Fact (1 ≤ p)\nT ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.lpHolder | {
"line": 59,
"column": 6
} | {
"line": 59,
"column": 36
} | {
"line": 59,
"column": 37
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_2\nE : α → Type u_3\nF : α → Type u_4\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\ninst✝³ : (i : α) → NormedSpace 𝕜 (E i)\ninst✝² : (i : α) → NormedAddCommGroup (F i)\ninst✝¹ : (i : α) → NormedSpace 𝕜 (F i)\np✝ q r p : ℝ≥0∞\ninst✝ : Fact... | [
"α : Type u_1\n𝕜 : Type u_2\nE : α → Type u_3\nF : α → Type u_4\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\ninst✝³ : (i : α) → NormedSpace 𝕜 (E i)\ninst✝² : (i : α) → NormedAddCommGroup (F i)\ninst✝¹ : (i : α) → NormedSpace 𝕜 (F i)\np✝ q r p : ℝ≥0∞\ninst✝ : Fact (1 ≤ p)\nT ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.lpHolder | {
"line": 93,
"column": 4
} | {
"line": 93,
"column": 81
} | {
"line": 93,
"column": 82
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\nE : ι → Type u_3\nF : ι → Type u_4\nG : ι → Type u_5\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝³ : (i : ι) → NormedAddCommGroup (F i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (F i)\ninst✝¹ : (i : ι) → NormedAdd... | [
"ι : Type u_1\n𝕜 : Type u_2\nE : ι → Type u_3\nF : ι → Type u_4\nG : ι → Type u_5\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝³ : (i : ι) → NormedAddCommGroup (F i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (F i)\ninst✝¹ : (i : ι) → NormedAddCommGroup (G... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.lpHolder | {
"line": 104,
"column": 48
} | {
"line": 104,
"column": 59
} | {
"line": 104,
"column": 60
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\nE : ι → Type u_3\nF : ι → Type u_4\nG : ι → Type u_5\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝³ : (i : ι) → NormedAddCommGroup (F i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (F i)\ninst✝¹ : (i : ι) → NormedAdd... | [
"ι : Type u_1\n𝕜 : Type u_2\nE : ι → Type u_3\nF : ι → Type u_4\nG : ι → Type u_5\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝³ : (i : ι) → NormedAddCommGroup (F i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (F i)\ninst✝¹ : (i : ι) → NormedAddCommGroup (G... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.PiTensorProduct.InjectiveSeminorm | {
"line": 59,
"column": 16
} | {
"line": 59,
"column": 38
} | {
"line": 59,
"column": 39
} | [
{
"pp": "ι : Type uι\ninst✝⁵ : Fintype ι\n𝕜 : Type u𝕜\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : ι → Type uE\ninst✝³ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (E i)\nF : Type u_1\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nx : ⨂[𝕜] (i : ι), E i\nf : Continuo... | [
"ι : Type uι\ninst✝⁵ : Fintype ι\n𝕜 : Type u𝕜\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : ι → Type uE\ninst✝³ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (E i)\nF : Type u_1\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nx : ⨂[𝕜] (i : ι), E i\nf : ContinuousMultilinea... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.PiTensorProduct.InjectiveSeminorm | {
"line": 88,
"column": 2
} | {
"line": 91,
"column": 74
} | {
"line": 93,
"column": 0
} | [
{
"pp": "ι : Type uι\ninst✝³ : Fintype ι\n𝕜 : Type u𝕜\ninst✝² : NontriviallyNormedField 𝕜\nE : ι → Type uE\ninst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\n⊢ BddAbove\n {p |\n ∃ G x x_1, p = (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (to... | [] | use projectiveSeminorm
simp only [mem_upperBounds, Set.mem_ofPred_eq, forall_exists_index]
intro p G _ _ hp x
simpa [hp] using! toDualContinuousMultilinearMap_le_projectiveSeminorm _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.PiTensorProduct.InjectiveSeminorm | {
"line": 88,
"column": 2
} | {
"line": 91,
"column": 74
} | {
"line": 93,
"column": 0
} | [
{
"pp": "ι : Type uι\ninst✝³ : Fintype ι\n𝕜 : Type u𝕜\ninst✝² : NontriviallyNormedField 𝕜\nE : ι → Type uE\ninst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\n⊢ BddAbove\n {p |\n ∃ G x x_1, p = (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (to... | [] | use projectiveSeminorm
simp only [mem_upperBounds, Set.mem_ofPred_eq, forall_exists_index]
intro p G _ _ hp x
simpa [hp] using! toDualContinuousMultilinearMap_le_projectiveSeminorm _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.PiTensorProduct.InjectiveSeminorm | {
"line": 101,
"column": 2
} | {
"line": 102,
"column": 9
} | {
"line": 102,
"column": 10
} | [
{
"pp": "ι : Type uι\ninst✝³ : Fintype ι\n𝕜 : Type u𝕜\ninst✝² : NontriviallyNormedField 𝕜\nE : ι → Type uE\ninst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\nx : ⨂[𝕜] (i : ι), E i\n⊢ injectiveSeminorm x = ⨆ p, ↑p x",
"ppTerm": "?m.77",
"assigned": true,
"us... | [
"ι : Type uι\ninst✝³ : Fintype ι\n𝕜 : Type u𝕜\ninst✝² : NontriviallyNormedField 𝕜\nE : ι → Type uE\ninst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\nx : ⨂[𝕜] (i : ι), E i\n⊢ (sSup\n {p |\n ∃ G x x_1,\n p = (normSeminorm 𝕜 (ContinuousMultilinear... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.PiTensorProduct.ProjectiveSeminorm | {
"line": 80,
"column": 2
} | {
"line": 80,
"column": 19
} | {
"line": 80,
"column": 20
} | [
{
"pp": "ι : Type u_1\ninst✝² : Fintype ι\n𝕜 : Type u_2\nE : ι → Type u_3\ninst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝ : NormedField 𝕜\np : FreeAddMonoid (𝕜 × ((i : ι) → E i))\na : ℝ\nx : 𝕜\nm : (i : ι) → E i\na✝ : (x, m) ∈ FreeAddMonoid.toList p\nh : ‖x‖ * ∏ x, ‖m x‖ = a\n⊢ 0 ≤ a",
"ppTerm":... | [
"ι : Type u_1\ninst✝² : Fintype ι\n𝕜 : Type u_2\nE : ι → Type u_3\ninst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝ : NormedField 𝕜\np : FreeAddMonoid (𝕜 × ((i : ι) → E i))\na : ℝ\nx : 𝕜\nm : (i : ι) → E i\na✝ : (x, m) ∈ FreeAddMonoid.toList p\nh : ‖x‖ * ∏ x, ‖m x‖ = a\n⊢ 0 ≤ ‖x‖ * ∏ x, ‖m x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.PiTensorProduct.ProjectiveSeminorm | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 42
} | {
"line": 117,
"column": 4
} | [
{
"pp": "ι : Type u_1\ninst✝³ : Fintype ι\n𝕜 : Type u_2\nE : ι → Type u_3\ninst✝² : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝¹ : NormedField 𝕜\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\na : 𝕜\nx : ⨂[𝕜] (i : ι), E i\np : ↑x.lifts\n⊢ ⨅ p, projectiveSeminormAux ↑p ≤ ‖a‖ * projectiveSeminormAux ↑p",
"ppTe... | [
"ι : Type u_1\ninst✝³ : Fintype ι\n𝕜 : Type u_2\nE : ι → Type u_3\ninst✝² : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝¹ : NormedField 𝕜\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\na : 𝕜\nx : ⨂[𝕜] (i : ι), E i\np : ↑x.lifts\n⊢ ⨅ p, projectiveSeminormAux ↑p ≤ ‖a‖ * projectiveSeminormAux ↑p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.PiTensorProduct.ProjectiveSeminorm | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 54
} | {
"line": 157,
"column": 4
} | [
{
"pp": "ι : Type u_1\ninst✝⁵ : Fintype ι\n𝕜 : Type u_2\nE : ι → Type u_3\ninst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : (i : ι) → NormedSpace 𝕜 (E i)\nG : Type u_4\ninst✝¹ : SeminormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ContinuousMultilinearMap 𝕜 E ... | [
"ι : Type u_1\ninst✝⁵ : Fintype ι\n𝕜 : Type u_2\nE : ι → Type u_3\ninst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : (i : ι) → NormedSpace 𝕜 (E i)\nG : Type u_4\ninst✝¹ : SeminormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ContinuousMultilinearMap 𝕜 E G\nx : ⨂[𝕜]... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.lpHolder | {
"line": 157,
"column": 4
} | {
"line": 163,
"column": 19
} | {
"line": 165,
"column": 0
} | [] | [] | _ ≤ (∑ i ∈ s, ‖e i‖ ^ p.toReal) ^ (r.toReal / p.toReal) *
(∑ i ∈ s, ‖f i‖ ^ q.toReal) ^ (r.toReal / q.toReal) := by
apply Real.Lr_rpow_le_Lp_mul_Lq_of_nonneg s hpqr <;> (intros; positivity)
_ ≤ _ := by
gcongr
· exact hCe s
· exact hDf s | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Analysis.Normed.Operator.Perturbation.StrictByFinite | {
"line": 118,
"column": 10
} | {
"line": 118,
"column": 21
} | {
"line": 118,
"column": 22
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : CompleteSpace 𝕜\nE : Type u_2\nF : Type u_3\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : Module 𝕜 E\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜 F\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : IsTopologicalAddGroup E\ninst✝³ : ContinuousSMul 𝕜 E\ninst✝² ... | [
"𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : CompleteSpace 𝕜\nE : Type u_2\nF : Type u_3\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : Module 𝕜 E\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜 F\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : IsTopologicalAddGroup E\ninst✝³ : ContinuousSMul 𝕜 E\ninst✝² : Topologica... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Perturbation.StrictByFinite | {
"line": 120,
"column": 66
} | {
"line": 120,
"column": 77
} | {
"line": 120,
"column": 78
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : CompleteSpace 𝕜\nE : Type u_2\nF : Type u_3\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : Module 𝕜 E\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜 F\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : IsTopologicalAddGroup E\ninst✝³ : ContinuousSMul 𝕜 E\ninst✝² ... | [
"𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : CompleteSpace 𝕜\nE : Type u_2\nF : Type u_3\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : Module 𝕜 E\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜 F\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : IsTopologicalAddGroup E\ninst✝³ : ContinuousSMul 𝕜 E\ninst✝² : Topologica... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Fredholm.Basic | {
"line": 187,
"column": 2
} | {
"line": 187,
"column": 63
} | {
"line": 187,
"column": 64
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : AddCommGroup F\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕜 F\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalSpace F\nu : E →L[𝕜] F\npkg : u.FredholmPackage\n⊢ MapsTo ⇑u ↑pkg.decDom.X₁ ↑pkg... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : AddCommGroup F\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕜 F\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalSpace F\nu : E →L[𝕜] F\npkg : u.FredholmPackage\n⊢ MapsTo (⇑u) (↑pkg.decDom.X₁) (Set.range ⇑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Perturbation.StrictByFinite | {
"line": 214,
"column": 2
} | {
"line": 214,
"column": 13
} | {
"line": 214,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : CompleteSpace 𝕜\nE : Type u_2\nF : Type u_3\ninst✝¹⁰ : AddCommGroup E\ninst✝⁹ : Module 𝕜 E\ninst✝⁸ : AddCommGroup F\ninst✝⁷ : Module 𝕜 F\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : IsTopologicalAddGroup E\ninst✝⁴ : ContinuousSMul 𝕜 E\ninst✝³... | [
"𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : CompleteSpace 𝕜\nE : Type u_2\nF : Type u_3\ninst✝¹⁰ : AddCommGroup E\ninst✝⁹ : Module 𝕜 E\ninst✝⁸ : AddCommGroup F\ninst✝⁷ : Module 𝕜 F\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : IsTopologicalAddGroup E\ninst✝⁴ : ContinuousSMul 𝕜 E\ninst✝³ : Topologic... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Order.UpperLower | {
"line": 136,
"column": 2
} | {
"line": 136,
"column": 30
} | {
"line": 136,
"column": 31
} | [
{
"pp": "ι : Type u_2\ninst✝ : Fintype ι\nx : ι → ℝ\n⊢ MonotoneOn (dist x) (Ici x)",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_2\ninst✝ : Fintype ι\nx : ι → ℝ\n⊢ MonotoneOn (dist x) (Ici x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Order.UpperLower | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 30
} | {
"line": 145,
"column": 31
} | [
{
"pp": "ι : Type u_2\ninst✝ : Fintype ι\nx : ι → ℝ\n⊢ AntitoneOn (dist x) (Iic x)",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_2\ninst✝ : Fintype ι\nx : ι → ℝ\n⊢ AntitoneOn (dist x) (Iic x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 59,
"column": 4
} | {
"line": 59,
"column": 15
} | {
"line": 59,
"column": 16
} | [
{
"pp": "case neg\n𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedAddCommGroup V\ninst✝⁴ : SeminormedAddCommGroup W\ninst✝³ : NormedSpace 𝕜 V\ninst✝² : NormedSpace 𝕜 W\ninst✝¹ : SeparatingDual 𝕜 V\ninst✝ : SeparatingDual 𝕜 W\nf : (V →L[𝕜] V) ≃A[𝕜] W →L[�... | [
"case neg\n𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedAddCommGroup V\ninst✝⁴ : SeminormedAddCommGroup W\ninst✝³ : NormedSpace 𝕜 V\ninst✝² : NormedSpace 𝕜 W\ninst✝¹ : SeparatingDual 𝕜 V\ninst✝ : SeparatingDual 𝕜 W\nf : (V →L[𝕜] V) ≃A[𝕜] W →L[𝕜] W\nhV✝ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 81,
"column": 4
} | {
"line": 81,
"column": 45
} | {
"line": 81,
"column": 46
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedAddCommGroup V\ninst✝⁴ : SeminormedAddCommGroup W\ninst✝³ : NormedSpace 𝕜 V\ninst✝² : NormedSpace 𝕜 W\ninst✝¹ : SeparatingDual 𝕜 V\ninst✝ : SeparatingDual 𝕜 W\nf : (V →L[𝕜] V) ≃A[𝕜] W →L[𝕜] W\nhV :... | [
"𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedAddCommGroup V\ninst✝⁴ : SeminormedAddCommGroup W\ninst✝³ : NormedSpace 𝕜 V\ninst✝² : NormedSpace 𝕜 W\ninst✝¹ : SeparatingDual 𝕜 V\ninst✝ : SeparatingDual 𝕜 W\nf : (V →L[𝕜] V) ≃A[𝕜] W →L[𝕜] W\nhV : Nontrivial ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 116,
"column": 15
} | {
"line": 116,
"column": 26
} | {
"line": 116,
"column": 27
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : CompleteSpace V\ninst✝² : NormedAddCommGroup W\ninst✝¹ : InnerProductSpace 𝕜 W\ninst✝ : CompleteSpace W\ne : V ≃L[𝕜] W\nα α' : 𝕜\nhα : α ≠ 0\nhα2 : α' * α' = α⁻¹\nh... | [
"𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : CompleteSpace V\ninst✝² : NormedAddCommGroup W\ninst✝¹ : InnerProductSpace 𝕜 W\ninst✝ : CompleteSpace W\ne : V ≃L[𝕜] W\nα α' : 𝕜\nhα : α ≠ 0\nhα2 : α' * α' = α⁻¹\nhe : adjoint ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 119,
"column": 15
} | {
"line": 119,
"column": 26
} | {
"line": 119,
"column": 27
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : CompleteSpace V\ninst✝² : NormedAddCommGroup W\ninst✝¹ : InnerProductSpace 𝕜 W\ninst✝ : CompleteSpace W\ne : V ≃L[𝕜] W\nα α' : 𝕜\nhα : α ≠ 0\nhα2 : α' * α' = α⁻¹\nh... | [
"𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : CompleteSpace V\ninst✝² : NormedAddCommGroup W\ninst✝¹ : InnerProductSpace 𝕜 W\ninst✝ : CompleteSpace W\ne : V ≃L[𝕜] W\nα α' : 𝕜\nhα : α ≠ 0\nhα2 : α' * α' = α⁻¹\nhe : adjoint ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ODE.Basic | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 92
} | {
"line": 87,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nγ : ℝ → E\nv : ℝ → E → E\nt₀ ε y : ℝ\nhy : y ∈ Metric.ball t₀ ε\n⊢ HasDerivAt γ (v y (γ y)) y ↔ HasDerivWithinAt γ (v y (γ y)) (Metric.ball t₀ ε) y",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"IsIntegr... | [] | exact ⟨HasDerivAt.hasDerivWithinAt, fun h ↦ h.hasDerivAt (Metric.isOpen_ball.mem_nhds hy)⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 170,
"column": 4
} | {
"line": 170,
"column": 15
} | {
"line": 170,
"column": 16
} | [
{
"pp": "case neg\n𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : CompleteSpace V\ninst✝² : NormedAddCommGroup W\ninst✝¹ : InnerProductSpace 𝕜 W\ninst✝ : CompleteSpace W\nf : (V →L[𝕜] V) ≃⋆ₐ[𝕜] W →L[𝕜] W\nhf : Continuou... | [
"case neg\n𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : CompleteSpace V\ninst✝² : NormedAddCommGroup W\ninst✝¹ : InnerProductSpace 𝕜 W\ninst✝ : CompleteSpace W\nf : (V →L[𝕜] V) ≃⋆ₐ[𝕜] W →L[𝕜] W\nhf : Continuous ⇑f\nh✝¹ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 191,
"column": 23
} | {
"line": 191,
"column": 34
} | {
"line": 191,
"column": 35
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : CompleteSpace V\ninst✝² : NormedAddCommGroup W\ninst✝¹ : InnerProductSpace 𝕜 W\ninst✝ : CompleteSpace W\nf : (V →L[𝕜] V) ≃⋆ₐ[𝕜] W →L[𝕜] W\nhf : Continuous ⇑f\nh✝ :... | [
"𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : CompleteSpace V\ninst✝² : NormedAddCommGroup W\ninst✝¹ : InnerProductSpace 𝕜 W\ninst✝ : CompleteSpace W\nf : (V →L[𝕜] V) ≃⋆ₐ[𝕜] W →L[𝕜] W\nhf : Continuous ⇑f\nh✝ : Nontrivial ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 198,
"column": 15
} | {
"line": 198,
"column": 74
} | {
"line": 199,
"column": 6
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : CompleteSpace V\ninst✝² : NormedAddCommGroup W\ninst✝¹ : InnerProductSpace 𝕜 W\ninst✝ : CompleteSpace W\nf : (V →L[𝕜] V) ≃⋆ₐ[𝕜] W →L[𝕜] W\nhf : Continuous ⇑f\nh✝ :... | [
"𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : CompleteSpace V\ninst✝² : NormedAddCommGroup W\ninst✝¹ : InnerProductSpace 𝕜 W\ninst✝ : CompleteSpace W\nf : (V →L[𝕜] V) ≃⋆ₐ[𝕜] W →L[𝕜] W\nhf : Continuous ⇑f\nh✝ : Nontrivial ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 215,
"column": 4
} | {
"line": 215,
"column": 15
} | {
"line": 215,
"column": 16
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : CompleteSpace V\ninst✝² : NormedAddCommGroup W\ninst✝¹ : InnerProductSpace 𝕜 W\ninst✝ : CompleteSpace W\nf : (V →L[𝕜] V) ≃⋆ₐ[𝕜] W →L[𝕜] W\nhf : Continuous ⇑f\nh✝ :... | [
"𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : CompleteSpace V\ninst✝² : NormedAddCommGroup W\ninst✝¹ : InnerProductSpace 𝕜 W\ninst✝ : CompleteSpace W\nf : (V →L[𝕜] V) ≃⋆ₐ[𝕜] W →L[𝕜] W\nhf : Continuous ⇑f\nh✝ : Nontrivial ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Contracting | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 48
} | {
"line": 72,
"column": 49
} | [
{
"pp": "α : Type u_1\ninst✝ : EMetricSpace α\nK : ℝ≥0\nf : α → α\nhf : ContractingWith K f\nx y : α\nh : edist x y ≠ ∞\nhy : IsFixedPt f y\n⊢ edist x y ≤ edist x (f x) / (1 - ↑K)",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝ : EMetricSpace α\nK : ℝ≥0\nf : α → α\nhf : ContractingWith K f\nx y : α\nh : edist x y ≠ ∞\nhy : IsFixedPt f y\n⊢ edist x y ≤ edist x (f x) / (1 - ↑K)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Contracting | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 66
} | {
"line": 77,
"column": 67
} | [
{
"pp": "α : Type u_1\ninst✝ : EMetricSpace α\nK : ℝ≥0\nf : α → α\nhf : ContractingWith K f\nx y : α\nhx : IsFixedPt f x\nhy : IsFixedPt f y\nh : ¬edist x y = ∞\n⊢ edist x y ≤ 0",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝ : EMetricSpace α\nK : ℝ≥0\nf : α → α\nhf : ContractingWith K f\nx y : α\nhx : IsFixedPt f x\nhy : IsFixedPt f y\nh : ¬edist x y = ∞\n⊢ edist x y ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ODE.DiscreteGronwall | {
"line": 56,
"column": 10
} | {
"line": 56,
"column": 61
} | {
"line": 56,
"column": 61
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\nu b c : ℕ → R\nn₀ : ℕ\nhu : ∀ n ≥ n₀, u (n + 1) ≤ c n * u n + b n\nhc : ∀ n ≥ n₀, 0 ≤ c n\nn k : ℕ\nhk : n₀ ≤ k\nih : u k ≤ u n₀ * ∏ i ∈ Ico n₀ k, c i + ∑ k_1 ∈ Ico n₀ k, b k_1 * ∏ i ∈ Ico (k_1 + 1) k, c i\nhck : 0... | [
"R : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\nu b c : ℕ → R\nn₀ : ℕ\nhu : ∀ n ≥ n₀, u (n + 1) ≤ c n * u n + b n\nhc : ∀ n ≥ n₀, 0 ≤ c n\nn k : ℕ\nhk : n₀ ≤ k\nih : u k ≤ u n₀ * ∏ i ∈ Ico n₀ k, c i + ∑ k_1 ∈ Ico n₀ k, b k_1 * ∏ i ∈ Ico (k_1 + 1) k, c i\nhck : 0 ≤ c k\nj : ... | prod_Ico_succ_top (by have := mem_Ico.mp hj; omega) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.Contracting | {
"line": 255,
"column": 2
} | {
"line": 255,
"column": 47
} | {
"line": 255,
"column": 48
} | [
{
"pp": "α : Type u_1\ninst✝ : MetricSpace α\nK : ℝ≥0\nf : α → α\nhf : ContractingWith K f\nx y : α\nhy : IsFixedPt f y\n⊢ dist x y ≤ dist x (f x) / (1 - ↑K)",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝ : MetricSpace α\nK : ℝ≥0\nf : α → α\nhf : ContractingWith K f\nx y : α\nhy : IsFixedPt f y\n⊢ dist x y ≤ dist x (f x) / (1 - ↑K)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ODE.Gronwall | {
"line": 151,
"column": 77
} | {
"line": 151,
"column": 88
} | {
"line": 151,
"column": 89
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf f' : ℝ → E\nK a b : ℝ\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x\nha : f a = 0\nbound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ K * ‖f x‖\nx : ℝ\nhx : x ∈ Icc a b\n⊢ ∀ x ∈ Ico a b, ‖f' x‖ ≤ K * ‖f x‖ +... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf f' : ℝ → E\nK a b : ℝ\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x\nha : f a = 0\nbound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ K * ‖f x‖\nx : ℝ\nhx : x ∈ Icc a b\n⊢ ∀ (x : ℝ), a ≤ x → x < b → ‖f' x‖ ≤ K * ‖f x‖"
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Contracting | {
"line": 323,
"column": 2
} | {
"line": 323,
"column": 13
} | {
"line": 323,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝² : MetricSpace α\nK : ℝ≥0\nf : α → α\ninst✝¹ : Nonempty α\ninst✝ : CompleteSpace α\nn : ℕ\nhf : ContractingWith K f^[n]\nx : α := fixedPoint f^[n] hf\nhx : f^[n] x = x\nthis : ¬IsFixedPt f x\n⊢ ↑K * dist x (f x) < dist x (f x)",
"ppTerm": "?m.74",
"assigned": false,
"use... | [
"α : Type u_1\ninst✝² : MetricSpace α\nK : ℝ≥0\nf : α → α\ninst✝¹ : Nonempty α\ninst✝ : CompleteSpace α\nn : ℕ\nhf : ContractingWith K f^[n]\nx : α := fixedPoint f^[n] hf\nhx : f^[n] x = x\nthis : ¬IsFixedPt f x\n⊢ ↑K * dist x (f x) < dist x (f x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ODE.DiscreteGronwall | {
"line": 87,
"column": 8
} | {
"line": 87,
"column": 29
} | {
"line": 87,
"column": 30
} | [
{
"pp": "case hbc\nu b c : ℕ → ℝ\nn₀ : ℕ\nhun₀ : 0 ≤ u n₀\nhu : ∀ n ≥ n₀, u (n + 1) ≤ (1 + c n) * u n + b n\nhc : ∀ n ≥ n₀, 0 ≤ c n\nhb : ∀ n ≥ n₀, 0 ≤ b n\nn : ℕ\nhn : n₀ ≤ n\n⊢ ∏ i ∈ Ico n₀ n, (1 + c i) ≤ rexp (∑ i ∈ Ico n₀ n, c i)",
"ppTerm": "?hbc",
"assigned": true,
"usedConstants": [
"Eq... | [
"case hbc\nu b c : ℕ → ℝ\nn₀ : ℕ\nhun₀ : 0 ≤ u n₀\nhu : ∀ n ≥ n₀, u (n + 1) ≤ (1 + c n) * u n + b n\nhc : ∀ n ≥ n₀, 0 ≤ c n\nhb : ∀ n ≥ n₀, 0 ≤ b n\nn : ℕ\nhn : n₀ ≤ n\n⊢ ∏ i ∈ Ico n₀ n, (1 + c i) ≤ ∏ x ∈ Ico n₀ n, rexp (c x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 172,
"column": 63
} | {
"line": 172,
"column": 74
} | {
"line": 172,
"column": 75
} | [
{
"pp": "case mk.mk\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ntmin tmax : ℝ\nt₀ : ↑(Icc tmin tmax)\nx₀ : E\na r L : ℝ≥0\ntoFun✝¹ : ↑(Icc tmin tmax) → E\nlipschitzWith✝¹ : LipschitzWith L toFun✝¹\nmem_closedBall₀✝¹ : toFun✝¹ t₀ ∈ closedBall x₀ ↑r\ntoFun✝ : ↑(Icc tmin tmax) → E\nlipschitzWith✝ : LipschitzWith ... | [
"case mk.mk\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ntmin tmax : ℝ\nt₀ : ↑(Icc tmin tmax)\nx₀ : E\na r L : ℝ≥0\ntoFun✝¹ : ↑(Icc tmin tmax) → E\nlipschitzWith✝¹ : LipschitzWith L toFun✝¹\nmem_closedBall₀✝¹ : toFun✝¹ t₀ ∈ closedBall x₀ ↑r\ntoFun✝ : ↑(Icc tmin tmax) → E\nlipschitzWith✝ : LipschitzWith L toFun✝\nme... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 180,
"column": 2
} | {
"line": 180,
"column": 13
} | {
"line": 180,
"column": 14
} | [
{
"pp": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ntmin tmax : ℝ\nt₀ : ↑(Icc tmin tmax)\nx₀ : E\nL : ℝ≥0\nα : FunSpace t₀ x₀ 0 L\n⊢ α.toFun t₀ = x₀",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_1\ninst✝ : NormedAddCommGroup E\ntmin tmax : ℝ\nt₀ : ↑(Icc tmin tmax)\nx₀ : E\nL : ℝ≥0\nα : FunSpace t₀ x₀ 0 L\n⊢ α.toFun t₀ = x₀"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ODE.Transform | {
"line": 40,
"column": 2
} | {
"line": 41,
"column": 94
} | {
"line": 42,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nγ : ℝ → E\nv : ℝ → E → E\ns : Set ℝ\nhγ : IsIntegralCurveOn γ v s\ndt t : ℝ\nht : t ∈ -dt +ᵥ s\n⊢ HasDerivWithinAt (γ ∘ fun x ↦ x + dt) ((v ∘ fun x ↦ x + dt) t ((γ ∘ fun x ↦ x + dt) t)) (-dt +ᵥ s) t",
"ppTerm": "?m.32",
"assi... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nγ : ℝ → E\nv : ℝ → E → E\ns : Set ℝ\nhγ : IsIntegralCurveOn γ v s\ndt t : ℝ\nht : t ∈ -dt +ᵥ s\n⊢ HasDerivWithinAt γ (v (t + dt) ((fun x ↦ γ (x + dt)) t)) s (t + dt)"
] | rw [comp_apply, hasDerivWithinAt_iff_hasFDerivWithinAt, Function.comp_def,
hasFDerivWithinAt_comp_add_right, ← hasDerivWithinAt_iff_hasFDerivWithinAt, vadd_neg_vadd] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.ODE.Transform | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 13
} | {
"line": 85,
"column": 14
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nγ : ℝ → E\nv : ℝ → E → E\nhγ : IsIntegralCurveOn γ v univ\ndt : ℝ\n⊢ IsIntegralCurveOn (γ ∘ fun x ↦ x + dt) (v ∘ fun x ↦ x + dt) univ",
"ppTerm": "?m.43",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nγ : ℝ → E\nv : ℝ → E → E\nhγ : IsIntegralCurveOn γ v univ\ndt : ℝ\n⊢ IsIntegralCurveOn (γ ∘ fun x ↦ x + dt) (v ∘ fun x ↦ x + dt) univ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ODE.ExistUnique | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 29
} | {
"line": 121,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E → E\ntmin tmax : ℝ\nt₀ : ↑(Icc tmin tmax)\nx₀ : E\na r L K : ℝ≥0\nhf : IsPicardLindelof f t₀ x₀ a r L K\nα : E → ℝ → E\nhα1 : ∀ x ∈ closedBall x₀ ↑r, α x ↑t₀ = x ∧ ∀ t ∈ Icc tmin tmax, HasDerivWith... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E → E\ntmin tmax : ℝ\nt₀ : ↑(Icc tmin tmax)\nx₀ : E\na r L K : ℝ≥0\nhf : IsPicardLindelof f t₀ x₀ a r L K\nα : E → ℝ → E\nhα1 : ∀ x ∈ closedBall x₀ ↑r, α x ↑t₀ = x ∧ ∀ t ∈ Icc tmin tmax, HasDerivWithinAt (α x) (... | refine ⟨uncurry α, hα1, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Polynomial.Fourier | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 13
} | {
"line": 52,
"column": 14
} | [
{
"pp": "p : ℂ[X]\n⊢ Integrable (⇑(toAddCircle p)) haarAddCircle",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℂ[X]\n⊢ Integrable (⇑(toAddCircle p)) haarAddCircle"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Polynomial.Basic | {
"line": 180,
"column": 10
} | {
"line": 180,
"column": 37
} | {
"line": 180,
"column": 38
} | [
{
"pp": "case pos.inl\n𝕜 : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhQ : Q ≠ 0\nh : Tendsto (fun x ↦ eval x P / eval x Q) atTop (𝓝 0)\nhP0 : P.leadingCoeff = 0\n⊢ P.degree < Q.degree",
"ppTerm": "?pos.inl✝",
"as... | [
"case pos.inl\n𝕜 : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhQ : Q ≠ 0\nh : Tendsto (fun x ↦ eval x P / eval x Q) atTop (𝓝 0)\nhP0 : P.leadingCoeff = 0\n⊢ degree 0 < Q.degree"
] | leadingCoeff_eq_zero.1 hP0, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Polynomial.Fourier | {
"line": 99,
"column": 6
} | {
"line": 99,
"column": 17
} | {
"line": 99,
"column": 18
} | [
{
"pp": "case inl\np : ℂ[X]\nthis :\n ∑' (i : ℤ), ‖if 0 ≤ i then p.coeff i.natAbs else 0‖ ^ 2 =\n ∫ (t : AddCircle (2 * π)), ‖↑↑((ContinuousMap.toLp 2 haarAddCircle ℂ) (toAddCircle p)) t‖ ^ 2 ∂haarAddCircle\nb : ℕ\nhb : ↑b ∉ Finset.map { toFun := Nat.cast, inj' := ⋯ } p.support\n⊢ ‖if 0 ≤ ↑b then p.coeff (↑... | [
"case inl\np : ℂ[X]\nthis :\n ∑' (i : ℤ), ‖if 0 ≤ i then p.coeff i.natAbs else 0‖ ^ 2 =\n ∫ (t : AddCircle (2 * π)), ‖↑↑((ContinuousMap.toLp 2 haarAddCircle ℂ) (toAddCircle p)) t‖ ^ 2 ∂haarAddCircle\nb : ℕ\nhb : ↑b ∉ Finset.map { toFun := Nat.cast, inj' := ⋯ } p.support\n⊢ p.coeff b = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Polynomial.Basic | {
"line": 280,
"column": 4
} | {
"line": 280,
"column": 29
} | {
"line": 281,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhQ : Q ≠ 0\nh : Tendsto (fun x ↦ eval x P / eval x Q) atBot (𝓝 0)\n⊢ Q.comp (-X) ≠ 0",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Nor... | [
"𝕜 : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhQ : Q ≠ 0\nh : Tendsto (fun x ↦ eval x P / eval x Q) atBot (𝓝 0)\n⊢ ¬(Q = 0 ∨ eval ((-X).coeff 0) Q = 0 ∧ -X = C ((-X).coeff 0))"
] | rw [Ne, comp_eq_zero_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Polynomial.Basic | {
"line": 295,
"column": 4
} | {
"line": 295,
"column": 29
} | {
"line": 296,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhdeg : (Q.comp (-X)).degree < (P.comp (-X)).degree\nhQ : Q ≠ 0\n⊢ Q.comp (-X) ≠ 0",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"NormedC... | [
"𝕜 : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhdeg : (Q.comp (-X)).degree < (P.comp (-X)).degree\nhQ : Q ≠ 0\n⊢ ¬(Q = 0 ∨ eval ((-X).coeff 0) Q = 0 ∧ -X = C ((-X).coeff 0))"
] | rw [Ne, comp_eq_zero_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Polynomial.Basic | {
"line": 318,
"column": 4
} | {
"line": 318,
"column": 20
} | {
"line": 318,
"column": 21
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh : P.degree ≤ Q.degree\nhp : P = 0\n⊢ (fun x ↦ eval x P) =O[atTop] fun x ↦ eval x Q",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
... | [
"case pos\n𝕜 : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh : P.degree ≤ Q.degree\nhp : P = 0\n⊢ (fun x ↦ 0) =O[atTop] fun x ↦ eval x Q"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.GaussNorm | {
"line": 77,
"column": 4
} | {
"line": 77,
"column": 23
} | {
"line": 77,
"column": 24
} | [
{
"pp": "case h\nR : Type u_1\ninst✝ : Semiring R\nv : R → ℝ\nc : ℝ\nf : R⟦X⟧\nh : HasGaussNorm v c f\nx✝ : ℝ\ny : ℕ\nhy : v ((coeff y) f) * c ^ y = x✝\n⊢ v ((MvPowerSeries.coeff ((Finsupp.uniqueEquiv ()).symm y)) f) * c ^ ((Finsupp.uniqueEquiv ()).symm y) PUnit.unit = x✝",
"ppTerm": "?h",
"assigned": t... | [
"case h\nR : Type u_1\ninst✝ : Semiring R\nv : R → ℝ\nc : ℝ\nf : R⟦X⟧\nh : HasGaussNorm v c f\nx✝ : ℝ\ny : ℕ\nhy : v ((coeff y) f) * c ^ y = x✝\n⊢ v ((coeff y) f) * c ^ y = x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 452,
"column": 2
} | {
"line": 468,
"column": 64
} | {
"line": 470,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nf : ℝ → E → E\ntmin tmax : ℝ\nt₀ : ↑(Icc tmin tmax)\nx₀ : E\na r L K : ℝ≥0\ninst✝ : CompleteSpace E\nhf : IsPicardLindelof f t₀ x₀ a r L K\n⊢ ∃ L',\n ∀ (x y : E) (hx : x ∈ closedBall x₀ ↑r) (hy : y ∈ closedBall x₀ ↑r) (α β : FunS... | [] | obtain ⟨m, C, h⟩ := exists_contractingWith_iterate_next hf
let L' := (∑ i ∈ Finset.range m, (K * max (tmax - t₀) (t₀ - tmin)) ^ i / i !) * (1 - C)⁻¹
have hL' : 0 ≤ L' := by
have : 0 ≤ max (tmax - t₀) (t₀ - tmin) := le_max_of_le_left <| sub_nonneg_of_le t₀.2.2
positivity
refine ⟨.mk L' hL', fun x y hx hy α... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 452,
"column": 2
} | {
"line": 468,
"column": 64
} | {
"line": 470,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nf : ℝ → E → E\ntmin tmax : ℝ\nt₀ : ↑(Icc tmin tmax)\nx₀ : E\na r L K : ℝ≥0\ninst✝ : CompleteSpace E\nhf : IsPicardLindelof f t₀ x₀ a r L K\n⊢ ∃ L',\n ∀ (x y : E) (hx : x ∈ closedBall x₀ ↑r) (hy : y ∈ closedBall x₀ ↑r) (α β : FunS... | [] | obtain ⟨m, C, h⟩ := exists_contractingWith_iterate_next hf
let L' := (∑ i ∈ Finset.range m, (K * max (tmax - t₀) (t₀ - tmin)) ^ i / i !) * (1 - C)⁻¹
have hL' : 0 ≤ L' := by
have : 0 ≤ max (tmax - t₀) (t₀ - tmin) := le_max_of_le_left <| sub_nonneg_of_le t₀.2.2
positivity
refine ⟨.mk L' hL', fun x y hx hy α... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 557,
"column": 2
} | {
"line": 557,
"column": 12
} | {
"line": 557,
"column": 13
} | [
{
"pp": "case coe\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E → E\nα : ℝ → E\nu : Set E\nt₀ tmin tmax : ℝ\nht₀ : t₀ ∈ Icc tmin tmax\nhα : ContinuousOn α (Icc tmin tmax)\nhmem : ∀ t ∈ Icc tmin tmax, α t ∈ u\nx₀ : E\nheqon : ∀ t ∈ Icc tmin tmax, α t =... | [] | | coe n => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Polynomial.GaussNorm | {
"line": 184,
"column": 53
} | {
"line": 187,
"column": 37
} | {
"line": 188,
"column": 2
} | [
{
"pp": "R : Type u_1\nF : Type u_2\ninst✝³ : Semiring R\ninst✝² : FunLike F R ℝ\nv : F\ninst✝¹ : ZeroHomClass F R ℝ\ninst✝ : NonnegHomClass F R ℝ\nhna : IsNonarchimedean ⇑v\nc : ℝ\nhc : 0 ≤ c\np q : R[X]\nh✝¹ : p ≠ 0\nh✝ : q ≠ 0\nhpq : p + q ≠ 0\ni : ℕ\na✝ : i ∈ (p + q).support\n⊢ v ((p + q).coeff i) * c ^ i ≤... | [] | by
rw [coeff_add]
gcongr
exact hna (p.coeff i) (q.coeff i) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Polynomial.Norm | {
"line": 90,
"column": 6
} | {
"line": 90,
"column": 27
} | {
"line": 90,
"column": 28
} | [
{
"pp": "A : Type u_1\ninst✝ : SeminormedRing A\np : A[X]\n⊢ p.supNorm ∈ Set.range fun x ↦ ‖p.coeff x‖",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"SeminormedRing.toNorm",
"Real",
"SeminormedRing.toRing",
"Membership.mem",
... | [
"A : Type u_1\ninst✝ : SeminormedRing A\np : A[X]\n⊢ ∃ y, ‖p.coeff y‖ = p.supNorm"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Polynomial.Norm | {
"line": 90,
"column": 52
} | {
"line": 90,
"column": 81
} | {
"line": 90,
"column": 82
} | [
{
"pp": "A : Type u_1\ninst✝ : SeminormedRing A\np : A[X]\n⊢ p.supNorm ∈ upperBounds (Set.range fun x ↦ ‖p.coeff x‖)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"SeminormedRing.toNorm",
"Real.instLE",
"Real",
"Preorder.toLE",
... | [
"A : Type u_1\ninst✝ : SeminormedRing A\np : A[X]\n⊢ ∀ (a : ℕ), ‖p.coeff a‖ ≤ p.supNorm"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.GaussNorm | {
"line": 148,
"column": 6
} | {
"line": 148,
"column": 43
} | {
"line": 148,
"column": 44
} | [
{
"pp": "R : Type u_1\nσ : Type u_2\nv : R → ℝ\nc : σ → ℝ\ninst✝ : Semiring R\nf g : MvPowerSeries σ R\nhc : 0 ≤ c\nvNonneg : ∀ (a : R), v a ≥ 0\nvMul : ∀ (a b : R), v (a * b) ≤ v a * v b\nvna : IsNonarchimedean v\nvZero : v 0 = 0\nhbfd : HasGaussNorm v c f\nhbgd : HasGaussNorm v c g\nt : σ →₀ ℕ\nk : (σ →₀ ℕ) ×... | [
"R : Type u_1\nσ : Type u_2\nv : R → ℝ\nc : σ → ℝ\ninst✝ : Semiring R\nf g : MvPowerSeries σ R\nhc : 0 ≤ c\nvNonneg : ∀ (a : R), v a ≥ 0\nvMul : ∀ (a b : R), v (a * b) ≤ v a * v b\nvna : IsNonarchimedean v\nvZero : v 0 = 0\nhbfd : HasGaussNorm v c f\nhbgd : HasGaussNorm v c g\nt : σ →₀ ℕ\nk : (σ →₀ ℕ) × (σ →₀ ℕ)\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Polynomial.Order | {
"line": 98,
"column": 6
} | {
"line": 98,
"column": 17
} | {
"line": 98,
"column": 18
} | [
{
"pp": "P : ℝ[X]\nx : ℝ\nhroots : ∀ (y : ℝ), P.IsRoot y → x < y\nhlc : 0 ≤ P.leadingCoeff\nhroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x\nh : Even P.natDegree\nhlc' : 0 ≤ (P.comp (-X)).leadingCoeff\nthis : 0 < eval (-x) (P.comp (-X))\n⊢ 0 < eval x P",
"ppTerm": "?m.187",
"assigned": true,
"u... | [
"P : ℝ[X]\nx : ℝ\nhroots : ∀ (y : ℝ), P.IsRoot y → x < y\nhlc : 0 ≤ P.leadingCoeff\nhroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x\nh : Even P.natDegree\nhlc' : 0 ≤ (P.comp (-X)).leadingCoeff\nthis : 0 < eval (-x) (P.comp (-X))\n⊢ 0 < eval x P"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Polynomial.Order | {
"line": 105,
"column": 28
} | {
"line": 105,
"column": 39
} | {
"line": 105,
"column": 40
} | [
{
"pp": "P : ℝ[X]\nx : ℝ\nhroots : ∀ (y : ℝ), P.IsRoot y → x < y\nhlc : 0 ≤ P.leadingCoeff\nhroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x\nh : Odd P.natDegree\nhlc' : 0 ≤ -(P.comp (-X)).leadingCoeff\nthis : eval (-x) (P.comp (-X)) < 0\n⊢ eval x P < 0",
"ppTerm": "?m.249",
"assigned": false,
"... | [
"P : ℝ[X]\nx : ℝ\nhroots : ∀ (y : ℝ), P.IsRoot y → x < y\nhlc : 0 ≤ P.leadingCoeff\nhroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x\nh : Odd P.natDegree\nhlc' : 0 ≤ -(P.comp (-X)).leadingCoeff\nthis : eval (-x) (P.comp (-X)) < 0\n⊢ eval x P < 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Polynomial.MahlerMeasure | {
"line": 129,
"column": 4
} | {
"line": 129,
"column": 41
} | {
"line": 129,
"column": 42
} | [
{
"pp": "case pos\np q : ℂ[X]\nhpq : p * q = 0\n⊢ (p * q).mahlerMeasure = p.mahlerMeasure * q.mahlerMeasure",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"Real",
"HMul.hMul",
"Polynomial.mahlerMeasure_eq_zero_iff._sim... | [
"case pos\np q : ℂ[X]\nhpq : p * q = 0\n⊢ p = 0 ∨ q = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.GaussNorm | {
"line": 186,
"column": 4
} | {
"line": 186,
"column": 36
} | {
"line": 186,
"column": 37
} | [
{
"pp": "case inr\nα : Type u_3\nS : Type u_4\ninst✝¹ : LinearOrder S\ninst✝ : AddCommGroup α\nf : α → S\nna : IsNonarchimedean f\nNeg : ∀ (a : α), f a = f (-a)\na b : α\nhne : f a ≠ f b\nH : ∀ {a b : α}, f a ≠ f b → f a > f b → f (a + b) = max (f a) (f b)\nhab : ¬f a > f b\n⊢ f (a + b) = max (f a) (f b)",
... | [
"case inr\nα : Type u_3\nS : Type u_4\ninst✝¹ : LinearOrder S\ninst✝ : AddCommGroup α\nf : α → S\nna : IsNonarchimedean f\nNeg : ∀ (a : α), f a = f (-a)\na b : α\nhne : f a ≠ f b\nH : ∀ {a b : α}, f a ≠ f b → f a > f b → f (a + b) = max (f a) (f b)\nhab : ¬f a > f b\n⊢ f (a + b) = max (f a) (f b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.GaussNorm | {
"line": 189,
"column": 4
} | {
"line": 189,
"column": 44
} | {
"line": 189,
"column": 45
} | [
{
"pp": "case inl\nα : Type u_3\nS : Type u_4\ninst✝¹ : LinearOrder S\ninst✝ : AddCommGroup α\nf : α → S\nna : IsNonarchimedean f\nNeg : ∀ (a : α), f a = f (-a)\na b : α\nhne : f a ≠ f b\nhab : f a > f b\nh : f (a + b + -b) ≤ f (a + b)\n⊢ max (f a) (f b) ≤ f (a + b)",
"ppTerm": "?inl",
"assigned": true,... | [
"case inl\nα : Type u_3\nS : Type u_4\ninst✝¹ : LinearOrder S\ninst✝ : AddCommGroup α\nf : α → S\nna : IsNonarchimedean f\nNeg : ∀ (a : α), f a = f (-a)\na b : α\nhne : f a ≠ f b\nhab : f a > f b\nh : f (a + b + -b) ≤ f (a + b)\n⊢ f a ≤ f (a + b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.GaussNorm | {
"line": 190,
"column": 35
} | {
"line": 190,
"column": 54
} | {
"line": 190,
"column": 55
} | [
{
"pp": "α : Type u_3\nS : Type u_4\ninst✝¹ : LinearOrder S\ninst✝ : AddCommGroup α\nf : α → S\nna : IsNonarchimedean f\nNeg : ∀ (a : α), f a = f (-a)\na b : α\nhne : f a ≠ f b\nhab : f a > f b\nh : f (a + b + -b) ≤ f (-b)\n⊢ f (-b) < f (a + b + -b)",
"ppTerm": "?m.119",
"assigned": true,
"usedConst... | [
"α : Type u_3\nS : Type u_4\ninst✝¹ : LinearOrder S\ninst✝ : AddCommGroup α\nf : α → S\nna : IsNonarchimedean f\nNeg : ∀ (a : α), f a = f (-a)\na b : α\nhne : f a ≠ f b\nhab : f a > f b\nh : f (a + b + -b) ≤ f (-b)\n⊢ f (-b) < f a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.ContinuousMap | {
"line": 40,
"column": 28
} | {
"line": 40,
"column": 39
} | {
"line": 40,
"column": 40
} | [
{
"pp": "X : Type u_1\n𝕜 : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : RCLike 𝕜\nf g : C(X, ℝ)\nhfg : realToRCLike 𝕜 f = realToRCLike 𝕜 g\nx : X\n⊢ f x = g x",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_1\n𝕜 : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : RCLike 𝕜\nf g : C(X, ℝ)\nhfg : realToRCLike 𝕜 f = realToRCLike 𝕜 g\nx : X\n⊢ f x = g x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Rat.NatSqrt.Real | {
"line": 25,
"column": 2
} | {
"line": 25,
"column": 36
} | {
"line": 25,
"column": 37
} | [
{
"pp": "x prec : ℕ\nh : 0 < prec\nthis✝¹ : x.ratSqrt prec ^ 2 ≤ ↑x\nthis✝ : ↑(x.ratSqrt prec) ^ 2 ≤ ↑x\nthis : √(↑(x.ratSqrt prec) ^ 2) ≤ √↑x\n⊢ 0 ≤ ↑(x.ratSqrt prec)",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"Eq.mpr",
"Real.instLE",
"Real",
... | [
"x prec : ℕ\nh : 0 < prec\nthis✝¹ : x.ratSqrt prec ^ 2 ≤ ↑x\nthis✝ : ↑(x.ratSqrt prec) ^ 2 ≤ ↑x\nthis : √(↑(x.ratSqrt prec) ^ 2) ≤ √↑x\n⊢ 0 ≤ x.ratSqrt prec"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Rat.NatSqrt.Real | {
"line": 36,
"column": 25
} | {
"line": 36,
"column": 36
} | {
"line": 36,
"column": 37
} | [
{
"pp": "x prec : ℕ\nh : 0 < prec\nthis✝¹ : ↑x < (x.ratSqrt prec + 1 / ↑prec) ^ 2\nthis✝ : ↑x < ↑((x.ratSqrt prec + 1 / ↑prec) ^ 2)\nthis : √↑x < √(↑(x.ratSqrt prec + 1 / ↑prec) ^ 2)\n⊢ 0 ≤ ↑(x.ratSqrt prec)",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"Eq.m... | [
"x prec : ℕ\nh : 0 < prec\nthis✝¹ : ↑x < (x.ratSqrt prec + 1 / ↑prec) ^ 2\nthis✝ : ↑x < ↑((x.ratSqrt prec + 1 / ↑prec) ^ 2)\nthis : √↑x < √(↑(x.ratSqrt prec + 1 / ↑prec) ^ 2)\n⊢ 0 ≤ x.ratSqrt prec"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Polynomial.MahlerMeasure | {
"line": 336,
"column": 8
} | {
"line": 336,
"column": 20
} | {
"line": 336,
"column": 20
} | [
{
"pp": "p : ℂ[X]\nthis✝¹ : IsFiniteMeasure (volume.restrict (uIoc 0 (2 * π)))\nthis✝ : NeZero (volume (uIoc 0 (2 * π)))\nhp : p ≠ 0\nthis : ∀ᵐ (θ : ℝ) ∂volume.restrict (uIoc 0 (2 * π)), 0 < ‖eval (circleMap 0 1 θ) p‖\nhlogAe :\n ∀ᵐ (θ : ℝ) ∂volume.restrict (uIoc 0 (2 * π)), rexp (log ‖eval (circleMap 0 1 θ) p... | [
"p : ℂ[X]\nthis✝¹ : IsFiniteMeasure (volume.restrict (uIoc 0 (2 * π)))\nthis✝ : NeZero (volume (uIoc 0 (2 * π)))\nhp : p ≠ 0\nthis : ∀ᵐ (θ : ℝ) ∂volume.restrict (uIoc 0 (2 * π)), 0 < ‖eval (circleMap 0 1 θ) p‖\nhlogAe :\n ∀ᵐ (θ : ℝ) ∂volume.restrict (uIoc 0 (2 * π)), rexp (log ‖eval (circleMap 0 1 θ) p‖) = ‖eval (... | rw [sqrt_sq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Real.Hyperreal | {
"line": 380,
"column": 6
} | {
"line": 380,
"column": 17
} | {
"line": 380,
"column": 18
} | [
{
"pp": "case refine_2\nx : ℝ*\nr : ℝ\nh : x.IsSt r\ns : ℝ\nhs : s < r\n⊢ coeRingHom s ≤ x",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Hyperreal.instField",
"Eq.mpr",
"Real",
"congrArg",
"PartialOrder.toPreorder",
"OrderRingHom.instFunLike",
... | [
"case refine_2\nx : ℝ*\nr : ℝ\nh : x.IsSt r\ns : ℝ\nhs : s < r\n⊢ ↑s ≤ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Real.Hyperreal | {
"line": 381,
"column": 6
} | {
"line": 381,
"column": 17
} | {
"line": 381,
"column": 18
} | [
{
"pp": "case refine_3\nx : ℝ*\nr : ℝ\nh : x.IsSt r\ns : ℝ\nhs : s > r\n⊢ x ≤ coeRingHom s",
"ppTerm": "?refine_3",
"assigned": true,
"usedConstants": [
"Hyperreal.instField",
"Eq.mpr",
"Real",
"congrArg",
"PartialOrder.toPreorder",
"OrderRingHom.instFunLike",
... | [
"case refine_3\nx : ℝ*\nr : ℝ\nh : x.IsSt r\ns : ℝ\nhs : s > r\n⊢ x ≤ ↑s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Real.Irrational | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 13
} | {
"line": 146,
"column": 14
} | [
{
"pp": "⊢ Irrational √2",
"ppTerm": "?m.3",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ Irrational √2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Real.Irrational | {
"line": 194,
"column": 48
} | {
"line": 194,
"column": 79
} | {
"line": 194,
"column": 80
} | [
{
"pp": "x : ℝ\nh : Irrational x\n⊢ x ≠ 1",
"ppTerm": "?m.4",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℝ\nh : Irrational x\n⊢ x ≠ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Real.Hyperreal | {
"line": 424,
"column": 2
} | {
"line": 426,
"column": 36
} | {
"line": 427,
"column": 2
} | [
{
"pp": "case refine_1\nx : ℝ*\nh : x.InfinitePos\n⊢ 0 < x ∧ ArchimedeanClass.mk x < 0",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"ArchimedeanOrder.of",
"Hyperreal.instField",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"instHSM... | [
"case refine_2\nx : ℝ*\nx✝ : 0 < x ∧ ArchimedeanClass.mk x < 0\nr : ℝ\nhx : 0 < x\nhx' : ArchimedeanClass.mk x < 0\n⊢ ↑r < x"
] | · have hx : 0 < x := h 0
refine ⟨h 0, fun n ↦ ?_⟩
simpa [abs_of_pos hx] using! h n | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.Real.Irrational | {
"line": 270,
"column": 2
} | {
"line": 270,
"column": 45
} | {
"line": 270,
"column": 46
} | [
{
"pp": "q : ℚ\nx : ℝ\nh : Irrational x\n⊢ Irrational (x - ↑q)",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Real.instSub",
"AddMonoid.toAddZeroClass",
"Real.instRatCast",
"sub_eq_add_neg",
"HSub.hSub",
"... | [
"q : ℚ\nx : ℝ\nh : Irrational x\n⊢ Irrational (x + -↑q)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Real.Irrational | {
"line": 272,
"column": 2
} | {
"line": 272,
"column": 35
} | {
"line": 272,
"column": 36
} | [
{
"pp": "q : ℚ\nx : ℝ\nh : Irrational x\n⊢ Irrational (↑q - x)",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Real.instSub",
"AddMonoid.toAddZeroClass",
"Real.instRatCast",
"sub_eq_add_neg",
"HSub.hSub",
"... | [
"q : ℚ\nx : ℝ\nh : Irrational x\n⊢ Irrational (↑q + -x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Real.Irrational | {
"line": 274,
"column": 28
} | {
"line": 274,
"column": 71
} | {
"line": 274,
"column": 72
} | [
{
"pp": "q : ℚ\nx : ℝ\nh : Irrational (x - ↑q)\n⊢ Irrational (x + ↑(-q))",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"DivisionRing.toRatCast",
"congrArg",
"Real.instRatCast",
"Rat",
"id",
"Rat.... | [
"q : ℚ\nx : ℝ\nh : Irrational (x - ↑q)\n⊢ Irrational (x + -↑q)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Real.Irrational | {
"line": 276,
"column": 31
} | {
"line": 276,
"column": 64
} | {
"line": 276,
"column": 65
} | [
{
"pp": "q : ℚ\nx : ℝ\nh : Irrational (↑q - x)\n⊢ Irrational (↑q + -x)",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"q : ℚ\nx : ℝ\nh : Irrational (↑q - x)\n⊢ Irrational (↑q + -x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Real.Irrational | {
"line": 278,
"column": 2
} | {
"line": 278,
"column": 37
} | {
"line": 278,
"column": 38
} | [
{
"pp": "x : ℝ\nh : Irrational x\nm : ℤ\n⊢ Irrational (x - ↑m)",
"ppTerm": "?m.5",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℝ\nh : Irrational x\nm : ℤ\n⊢ Irrational (x - ↑m)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Real.Irrational | {
"line": 280,
"column": 2
} | {
"line": 280,
"column": 37
} | {
"line": 280,
"column": 38
} | [
{
"pp": "x : ℝ\nh : Irrational x\nm : ℤ\n⊢ Irrational (↑m - x)",
"ppTerm": "?m.5",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℝ\nh : Irrational x\nm : ℤ\n⊢ Irrational (↑m - x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Real.Irrational | {
"line": 416,
"column": 32
} | {
"line": 416,
"column": 68
} | {
"line": 416,
"column": 69
} | [
{
"pp": "x : ℝ\nhx : Irrational x\na b : ℤ\np_nonzero : C a * X + C b ≠ 0\nx_is_root : (aeval x) (C a * X + C b) = 0\n⊢ ↑a * x = -↑b",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Eq.mpr",
"Real",
"HMul.hMul",
... | [
"x : ℝ\nhx : Irrational x\na b : ℤ\np_nonzero : C a * X + C b ≠ 0\nx_is_root : (aeval x) (C a * X + C b) = 0\n⊢ ↑a * x + ↑b = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ArithmeticGeometricMean | {
"line": 86,
"column": 4
} | {
"line": 86,
"column": 47
} | {
"line": 86,
"column": 48
} | [
{
"pp": "case inr\nx y : ℝ≥0\nthis : ∀ {x y : ℝ≥0}, x ≤ y → dist (sqrt (x * y)) ((x + y) / 2) ≤ dist x y / 2\nh : ¬x ≤ y\n⊢ dist (sqrt (x * y)) ((x + y) / 2) ≤ dist x y / 2",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [
"case inr\nx y : ℝ≥0\nthis : ∀ {x y : ℝ≥0}, x ≤ y → dist (sqrt (x * y)) ((x + y) / 2) ≤ dist x y / 2\nh : ¬x ≤ y\n⊢ dist ((x + y) / 2) (sqrt (x * y)) ≤ dist x y / 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ArithmeticGeometricMean | {
"line": 126,
"column": 2
} | {
"line": 129,
"column": 55
} | {
"line": 130,
"column": 2
} | [
{
"pp": "x y : ℝ≥0\nh : x ≠ y\nn m : ℕ\n⊢ (x.agmSequences y n).1 < (x.agmSequences y m).2",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"NNReal.agmSequences",
"Preorder.toLT",
"PartialOrder.toPreorder",
"Preorder.toLE",
"NNReal",
"NNReal.agmSequences_fs... | [
"x y : ℝ≥0\nh : x ≠ y\nn m : ℕ\n⊢ ∀ {k : ℕ}, (x.agmSequences y k).1 < (x.agmSequences y k).2"
] | suffices ∀ {k}, (agmSequences x y k).1 < (agmSequences x y k).2 by
obtain h | h := le_total n m
· exact (agmSequences_fst_monotone h).trans_lt this
· exact this.trans_le (agmSequences_snd_antitone h) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Analysis.SpecialFunctions.ArithmeticGeometricMean | {
"line": 180,
"column": 4
} | {
"line": 180,
"column": 36
} | {
"line": 180,
"column": 37
} | [
{
"pp": "case inr\nx y : ℝ≥0\nthis : ∀ {x y : ℝ≥0}, x ≤ y → x.agm y ≤ max x y\nh : ¬x ≤ y\n⊢ x.agm y ≤ max x y",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PartialOrder.toPreorder",
"Preorder.toLE",
"NNReal.agm",
"SemilatticeInf.toPartialOrder",
... | [
"case inr\nx y : ℝ≥0\nthis : ∀ {x y : ℝ≥0}, x ≤ y → x.agm y ≤ max x y\nh : ¬x ≤ y\n⊢ x.agm y ≤ x ∨ x.agm y ≤ y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ArithmeticGeometricMean | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 36
} | {
"line": 212,
"column": 37
} | [
{
"pp": "case inr\nx y : ℝ≥0\nthis : ∀ {x y : ℝ≥0}, x ≤ y → min x y ≤ x.agm y\nh : ¬x ≤ y\n⊢ min x y ≤ x.agm y",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"PartialOrder.toPreorder",
"Preorder.toLE",
"NNReal.agm",
... | [
"case inr\nx y : ℝ≥0\nthis : ∀ {x y : ℝ≥0}, x ≤ y → min x y ≤ x.agm y\nh : ¬x ≤ y\n⊢ x ≤ x.agm y ∨ y ≤ x.agm y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ArithmeticGeometricMean | {
"line": 250,
"column": 2
} | {
"line": 250,
"column": 13
} | {
"line": 250,
"column": 14
} | [
{
"pp": "x y : ℝ≥0\n⊢ x.agm y = (sqrt (x * y)).agm ((x + y) / 2)",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x y : ℝ≥0\n⊢ x.agm y = (sqrt (x * y)).agm ((x + y) / 2)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ArithmeticGeometricMean | {
"line": 276,
"column": 4
} | {
"line": 276,
"column": 36
} | {
"line": 276,
"column": 37
} | [
{
"pp": "case inr\nx y : ℝ≥0\nhx : 0 < x\nhy : 0 < y\nhn : x ≠ y\nthis : ∀ {x y : ℝ≥0}, 0 < x → 0 < y → x ≠ y → x < y → min x y < x.agm y\nh : ¬x < y\n⊢ min x y < x.agm y",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Lattice.toSemilatticeSup",... | [
"case inr\nx y : ℝ≥0\nhx : 0 < x\nhy : 0 < y\nhn : x ≠ y\nthis : ∀ {x y : ℝ≥0}, 0 < x → 0 < y → x ≠ y → x < y → min x y < x.agm y\nh : ¬x < y\n⊢ x < x.agm y ∨ y < x.agm y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ArithmeticGeometricMean | {
"line": 284,
"column": 4
} | {
"line": 284,
"column": 36
} | {
"line": 284,
"column": 37
} | [
{
"pp": "case inr\nx y : ℝ≥0\nhn : x ≠ y\nthis : ∀ {x y : ℝ≥0}, x ≠ y → x < y → x.agm y < max x y\nh : ¬x < y\n⊢ x.agm y < max x y",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"lt_sup_iff._simp_3",
"PartialOrder.toPreorder",
"NNRea... | [
"case inr\nx y : ℝ≥0\nhn : x ≠ y\nthis : ∀ {x y : ℝ≥0}, x ≠ y → x < y → x.agm y < max x y\nh : ¬x < y\n⊢ x.agm y < x ∨ x.agm y < y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ArithmeticGeometricMean | {
"line": 285,
"column": 6
} | {
"line": 285,
"column": 23
} | {
"line": 285,
"column": 23
} | [
{
"pp": "x✝ y✝ x y : ℝ≥0\nhn : x ≠ y\nh : x < y\n⊢ x.agm y < max x y",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"NNReal.agm",
"SemilatticeSup.toMax",
"NNReal.instSemilatticeSup"... | [
"x✝ y✝ x y : ℝ≥0\nhn : x ≠ y\nh : x < y\n⊢ x.agm y < y"
] | max_eq_right h.le | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arctan | {
"line": 123,
"column": 81
} | {
"line": 139,
"column": 91
} | {
"line": 141,
"column": 0
} | [
{
"pp": "z : ℂ\nhz : ‖z‖ < 1\n⊢ HasSum (fun n ↦ (-1) ^ n * z ^ (2 * n + 1) / ↑(2 * n + 1)) z.arctan",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Iff.mpr",
"Even.add_one._simp_1",
"Norm.norm",
"Mathlib.Tactic.Ring.C... | [] | by
have := ((hasSum_taylorSeries_log (z := z * I) (by simpa)).add
(hasSum_taylorSeries_neg_log (z := z * I) (by simpa))).mul_left (-I / 2)
simp_rw [← add_div, ← add_one_mul, hasSum_arctan_aux hz] at this
replace := (Nat.divModEquiv 2).symm.hasSum_iff.mpr this
dsimp [Function.comp_def] at this
simp_rw [← m... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Choose | {
"line": 34,
"column": 12
} | {
"line": 34,
"column": 23
} | {
"line": 34,
"column": 24
} | [
{
"pp": "case zero\n⊢ (fun n ↦ ↑(n.descFactorial 0)) ~[atTop] fun n ↦ ↑n ^ 0",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"MulOne.toOne",
"Real",
"Monoid.toMulOneClass",
"congrArg",
"AddGroupWithO... | [
"case zero\n⊢ (fun n ↦ 1) ~[atTop] fun n ↦ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Choose | {
"line": 41,
"column": 4
} | {
"line": 41,
"column": 15
} | {
"line": 41,
"column": 16
} | [
{
"pp": "case succ\nk : ℕ\nh : (fun n ↦ ↑(n.descFactorial k)) ~[atTop] fun n ↦ ↑n ^ k\nhz : ∀ᶠ (x : ℕ) in atTop, ↑x ≠ 0\n⊢ Tendsto (fun n ↦ ((fun n ↦ ↑(n - k)) / Nat.cast) (n + k)) atTop (𝓝 1)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminorm... | [
"case succ\nk : ℕ\nh : (fun n ↦ ↑(n.descFactorial k)) ~[atTop] fun n ↦ ↑n ^ k\nhz : ∀ᶠ (x : ℕ) in atTop, ↑x ≠ 0\n⊢ Tendsto (fun n ↦ ↑n / (↑n + ↑k)) atTop (𝓝 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.CompareExp | {
"line": 76,
"column": 4
} | {
"line": 76,
"column": 31
} | {
"line": 76,
"column": 32
} | [
{
"pp": "l : Filter ℂ\nhre : Tendsto re l atTop\nhim : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun z ↦ |z.im|\n⊢ im =O[l] fun z ↦ z.re ^ 0",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Real",
"Monoid.toMulOneClass",
"congrArg",
... | [
"l : Filter ℂ\nhre : Tendsto re l atTop\nhim : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun z ↦ |z.im|\n⊢ im =O[l] fun z ↦ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.BinaryEntropy | {
"line": 145,
"column": 10
} | {
"line": 145,
"column": 21
} | {
"line": 145,
"column": 21
} | [
{
"pp": "p : ℝ\nh : p ≠ 2⁻¹\nthis : ∀ {p : ℝ}, p ≠ 2⁻¹ → p < 2⁻¹ → binEntropy p < log 2\nhp : ¬p < 2⁻¹\n⊢ 1 - p < 2⁻¹",
"ppTerm": "?m.71",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"sub_lt_comm",
"congrArg",
"Real.instInv",
... | [
"p : ℝ\nh : p ≠ 2⁻¹\nthis : ∀ {p : ℝ}, p ≠ 2⁻¹ → p < 2⁻¹ → binEntropy p < log 2\nhp : ¬p < 2⁻¹\n⊢ 1 - 2⁻¹ < p"
] | sub_lt_comm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.CompareExp | {
"line": 188,
"column": 54
} | {
"line": 188,
"column": 65
} | {
"line": 188,
"column": 66
} | [
{
"pp": "l : Filter ℂ\nhl : IsExpCmpFilter l\na : ℂ\nb : ℝ\nhb : b < 0\n⊢ (fun z ↦ cexp (↑b * z)) =o[l] fun z ↦ z ^ a",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"l : Filter ℂ\nhl : IsExpCmpFilter l\na : ℂ\nb : ℝ\nhb : b < 0\n⊢ (fun z ↦ cexp (↑b * z)) =o[l] fun z ↦ z ^ a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.CompareExp | {
"line": 195,
"column": 2
} | {
"line": 195,
"column": 33
} | {
"line": 195,
"column": 34
} | [
{
"pp": "l : Filter ℂ\nb₁ b₂ : ℝ\nhl : IsExpCmpFilter l\nhb : b₁ < b₂\nm n : ℕ\n⊢ (fun z ↦ z ^ m * cexp (↑b₁ * z)) =o[l] fun z ↦ z ^ n * cexp (↑b₂ * z)",
"ppTerm": "?m.34",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"l : Filter ℂ\nb₁ b₂ : ℝ\nhl : IsExpCmpFilter l\nhb : b₁ < b₂\nm n : ℕ\n⊢ (fun z ↦ z ^ m * cexp (↑b₁ * z)) =o[l] fun z ↦ z ^ n * cexp (↑b₂ * z)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.CompareExp | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 33
} | {
"line": 202,
"column": 34
} | [
{
"pp": "l : Filter ℂ\nb₁ b₂ : ℝ\nhl : IsExpCmpFilter l\nhb : b₁ < b₂\nm n : ℤ\n⊢ (fun z ↦ z ^ m * cexp (↑b₁ * z)) =o[l] fun z ↦ z ^ n * cexp (↑b₂ * z)",
"ppTerm": "?m.34",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"l : Filter ℂ\nb₁ b₂ : ℝ\nhl : IsExpCmpFilter l\nhb : b₁ < b₂\nm n : ℤ\n⊢ (fun z ↦ z ^ m * cexp (↑b₁ * z)) =o[l] fun z ↦ z ^ n * cexp (↑b₂ * z)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.BinaryEntropy | {
"line": 227,
"column": 4
} | {
"line": 227,
"column": 44
} | {
"line": 227,
"column": 45
} | [
{
"pp": "case inr.inl\nq : ℕ\nhp₀✝ : 0 ≤ 1\nhp₁ : 1 ≤ 1\nhp₀ : 0 < 1\n⊢ 0 ≤ qaryEntropy q 1",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Real.qaryEntropy",
"Real.instLE",
"Real",
"HMul.hMul",
"Real.instZero",
"Rea... | [
"case inr.inl\nq : ℕ\nhp₀✝ : 0 ≤ 1\nhp₁ : 1 ≤ 1\nhp₀ : 0 < 1\n⊢ 0 ≤ log ↑(↑q - 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.BinaryEntropy | {
"line": 279,
"column": 4
} | {
"line": 279,
"column": 69
} | {
"line": 280,
"column": 4
} | [
{
"pp": "case hf\n⊢ Tendsto (fun x ↦ log (1 - x)) (𝓝[<] 1) atBot",
"ppTerm": "?hf",
"assigned": true,
"usedConstants": [
"Real",
"Set.Ioi",
"Real.instZero",
"nhdsWithin",
"PseudoMetricSpace.toUniformSpace",
"Real.tendsto_log_nhdsGT_zero",
"Real.log",
... | [
"case hf\nthis : Tendsto log (𝓝[>] 0) atBot\n⊢ Tendsto (fun x ↦ log (1 - x)) (𝓝[<] 1) atBot"
] | have : Tendsto log (𝓝[>] 0) atBot := Real.tendsto_log_nhdsGT_zero | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SpecialFunctions.BinaryEntropy | {
"line": 358,
"column": 8
} | {
"line": 358,
"column": 98
} | {
"line": 360,
"column": 0
} | [
{
"pp": "case neg.inl\nq : ℕ\np : ℝ\nis_x_where_nondiff : ¬(p ≠ 0 ∧ p ≠ 1)\nh : DifferentiableAt ℝ (deriv (qaryEntropy q)) p\ncontAt : ContinuousAt (deriv (qaryEntropy q)) p\nh✝ : p = 0\n⊢ False",
"ppTerm": "?neg.inl✝",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing"... | [] | simp_all [not_continuousAt_deriv_qaryEntropy_zero, not_continuousAt_deriv_qaryEntropy_one] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.SpecialFunctions.BinaryEntropy | {
"line": 358,
"column": 8
} | {
"line": 358,
"column": 98
} | {
"line": 360,
"column": 0
} | [
{
"pp": "case neg.inr\nq : ℕ\np : ℝ\nis_x_where_nondiff : ¬(p ≠ 0 ∧ p ≠ 1)\nh : DifferentiableAt ℝ (deriv (qaryEntropy q)) p\ncontAt : ContinuousAt (deriv (qaryEntropy q)) p\nh✝ : p = 1\n⊢ False",
"ppTerm": "?neg.inr✝",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing"... | [] | simp_all [not_continuousAt_deriv_qaryEntropy_zero, not_continuousAt_deriv_qaryEntropy_one] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Real.Pi.Irrational | {
"line": 176,
"column": 4
} | {
"line": 176,
"column": 28
} | {
"line": 176,
"column": 29
} | [
{
"pp": "case refine_2\nn : ℕ\n⊢ ((monomial 2) (-4)).natDegree + (sinPoly n).natDegree ≤ n + 2",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Nat.instIsOrderedAddMonoid",
"Semiring.toModule",
"Polynomial.instNeg",
"AddLef... | [
"case refine_2\nn : ℕ\n⊢ (sinPoly n).natDegree ≤ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.ExpLog.Order | {
"line": 60,
"column": 4
} | {
"line": 60,
"column": 20
} | {
"line": 60,
"column": 21
} | [
{
"pp": "case pos\nA : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\nha : IsStrictlyPositive a\n⊢ Tendsto (fun i ↦ if a ∈ {b | IsStrictlyPositive b} then cfc (fun x ↦ i⁻¹ * (x ^ i - 1)) a else 0) (𝓝[>] 0)\n (𝓝 (if a ∈ {b | IsStrictlyPositive b} then log a els... | [
"case pos\nA : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\nha : IsStrictlyPositive a\n⊢ Tendsto (fun i ↦ cfc (fun x ↦ i⁻¹ * (x ^ i - 1)) a) (𝓝[>] 0) (𝓝 (log a))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.FrullaniIntegral | {
"line": 127,
"column": 25
} | {
"line": 127,
"column": 43
} | {
"line": 127,
"column": 43
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nL : E\ninst✝ : CompleteSpace E\nhf : LocallyIntegrableOn f (Ioi 0) volume\nha : 0 < a\nhb : 0 < b\nhL : Tendsto f (𝓝[>] 0) (𝓝 L)\nδ : ℝ\nhδ : δ > 0\nhev : {x | dist (f x) L < δ} ∈ 𝓝[>] 0\n⊢ ∀ᶠ (t : ℝ) in 𝓝[{x... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nL : E\ninst✝ : CompleteSpace E\nhf : LocallyIntegrableOn f (Ioi 0) volume\nha : 0 < a\nhb : 0 < b\nhL : Tendsto f (𝓝[>] 0) (𝓝 L)\nδ : ℝ\nhδ : δ > 0\nhev : ∃ ε > 0, ball 0 ε ∩ Ioi 0 ⊆ {x | dist (f x) L < δ}\n⊢ ∀ᶠ (t : ℝ) in... | mem_nhdsWithin_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.RingInverseOrder | {
"line": 93,
"column": 12
} | {
"line": 93,
"column": 23
} | {
"line": 93,
"column": 24
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nx : A\nxpos : IsStrictlyPositive x\ny : A\nypos : IsStrictlyPositive y\na b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nz : A := (conjSqrt x⁻¹ʳ) y\nzpos : IsStrictlyPositive z\nxinvpos : IsStrictlyPositive x⁻¹ʳ... | [
"A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nx : A\nxpos : IsStrictlyPositive x\ny : A\nypos : IsStrictlyPositive y\na b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nz : A := (conjSqrt x⁻¹ʳ) y\nzpos : IsStrictlyPositive z\nxinvpos : IsStrictlyPositive x⁻¹ʳ\nhsp : IsSt... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.