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 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.Analytic.Composition | {
"line": 1133,
"column": 2
} | {
"line": 1166,
"column": 88
} | {
"line": 1168,
"column": 0
} | [
{
"pp": "n : ℕ\na : Composition n\nb : Composition a.length\ni j : ℕ\nhi : i < b.length\nhj : j < b.blocksFun ⟨i, hi⟩\n⊢ a.sizeUpTo (b.sizeUpTo i + j) = (a.gather b).sizeUpTo i + (a.sigmaCompositionAux b ⟨i, ⋯⟩).sizeUpTo j",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"List.getElem... | [] | induction j with
| zero =>
change
sum (take (b.blocks.take i).sum a.blocks) =
sum (take i (map sum (splitWrtComposition a.blocks b)))
induction i with
| zero => rfl
| succ i IH =>
have A : i < b.length := Nat.lt_of_succ_lt hi
have B : i < List.length (map List.sum (splitWrtCo... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Analysis.Analytic.Composition | {
"line": 1133,
"column": 2
} | {
"line": 1166,
"column": 88
} | {
"line": 1168,
"column": 0
} | [
{
"pp": "n : ℕ\na : Composition n\nb : Composition a.length\ni j : ℕ\nhi : i < b.length\nhj : j < b.blocksFun ⟨i, hi⟩\n⊢ a.sizeUpTo (b.sizeUpTo i + j) = (a.gather b).sizeUpTo i + (a.sigmaCompositionAux b ⟨i, ⋯⟩).sizeUpTo j",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"List.getElem... | [] | induction j with
| zero =>
change
sum (take (b.blocks.take i).sum a.blocks) =
sum (take i (map sum (splitWrtComposition a.blocks b)))
induction i with
| zero => rfl
| succ i IH =>
have A : i < b.length := Nat.lt_of_succ_lt hi
have B : i < List.length (map List.sum (splitWrtCo... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.Composition | {
"line": 1133,
"column": 2
} | {
"line": 1166,
"column": 88
} | {
"line": 1168,
"column": 0
} | [
{
"pp": "n : ℕ\na : Composition n\nb : Composition a.length\ni j : ℕ\nhi : i < b.length\nhj : j < b.blocksFun ⟨i, hi⟩\n⊢ a.sizeUpTo (b.sizeUpTo i + j) = (a.gather b).sizeUpTo i + (a.sigmaCompositionAux b ⟨i, ⋯⟩).sizeUpTo j",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"List.getElem... | [] | induction j with
| zero =>
change
sum (take (b.blocks.take i).sum a.blocks) =
sum (take i (map sum (splitWrtComposition a.blocks b)))
induction i with
| zero => rfl
| succ i IH =>
have A : i < b.length := Nat.lt_of_succ_lt hi
have B : i < List.length (map List.sum (splitWrtCo... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.Constructions | {
"line": 1021,
"column": 4
} | {
"line": 1021,
"column": 32
} | {
"line": 1022,
"column": 4
} | [
{
"pp": "case empty\nα : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nc : E\ns : Set E\nh : ∀ n ∈ ∅, AnalyticWithinAt 𝕜 (f n) s c\n⊢ Ana... | [
"case empty\nα : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nc : E\ns : Set E\nh : ∀ n ∈ ∅, AnalyticWithinAt 𝕜 (f n) s c\n⊢ AnalyticWithinA... | simp only [Finset.sum_empty] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Analytic.Constructions | {
"line": 1057,
"column": 4
} | {
"line": 1057,
"column": 33
} | {
"line": 1058,
"column": 4
} | [
{
"pp": "case empty\nα : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nA : Type u_9\ninst✝¹ : NormedCommRing A\ninst✝ : NormedAlgebra 𝕜 A\nf : α → E → A\nc : E\ns : Set E\nh : ∀ n ∈ ∅, AnalyticWithinAt 𝕜 (f n) s c\n⊢ Analy... | [
"case empty\nα : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nA : Type u_9\ninst✝¹ : NormedCommRing A\ninst✝ : NormedAlgebra 𝕜 A\nf : α → E → A\nc : E\ns : Set E\nh : ∀ n ∈ ∅, AnalyticWithinAt 𝕜 (f n) s c\n⊢ AnalyticWithinAt ... | simp only [Finset.prod_empty] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Analytic.Constructions | {
"line": 1118,
"column": 4
} | {
"line": 1118,
"column": 42
} | {
"line": 1119,
"column": 4
} | [
{
"pp": "case neg\n𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nα : Type u_9\nA : Type u_10\ninst✝¹ : NormedCommRing A\ninst✝ : NormedAlgebra 𝕜 A\nf : α → E → A\nc : E\nh : ∀ (a : α), AnalyticAt 𝕜 (f a) c\nhf : ¬(Function.mulSuppor... | [
"case neg\n𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nα : Type u_9\nA : Type u_10\ninst✝¹ : NormedCommRing A\ninst✝ : NormedAlgebra 𝕜 A\nf : α → E → A\nc : E\nh : ∀ (a : α), AnalyticAt 𝕜 (f a) c\nhf : ¬(Function.mulSupport f).Finite\... | rw [finprod_of_infinite_mulSupport hf] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Analytic.Constructions | {
"line": 1177,
"column": 6
} | {
"line": 1180,
"column": 57
} | {
"line": 1181,
"column": 2
} | [] | [] | _ ≤ pf.radius / ‖u‖ₑ := by
gcongr
exact hf.r_le
_ ≤ _ := pf.div_le_radius_compContinuousLinearMap _ | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Analysis.Calculus.FDeriv.Equiv | {
"line": 177,
"column": 6
} | {
"line": 177,
"column": 31
} | {
"line": 177,
"column": 32
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\niso : E ≃L[𝕜] F\nf : F → G\ns : Set... | [
"𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\niso : E ≃L[𝕜] F\nf : F → G\ns : Set F\nH : Diff... | ← iso.apply_symm_apply y, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.FDeriv.Add | {
"line": 860,
"column": 2
} | {
"line": 860,
"column": 89
} | {
"line": 862,
"column": 0
} | [
{
"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_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nf' : E →L[𝕜] F\nx : E\ns : Set E\na : E\nthis : map (fun x ↦ a + x) (𝓝[s] x) = 𝓝[a +ᵥ s] (a + ... | [] | simp [HasFDerivWithinAt, hasFDerivAtFilter_iff_isLittleOTVS, ← this, Function.comp_def] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.FDeriv.Add | {
"line": 894,
"column": 59
} | {
"line": 895,
"column": 57
} | {
"line": 897,
"column": 0
} | [
{
"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_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nx a : E\n⊢ fderiv 𝕜 (fun x ↦ f (x + a)) x = fderiv 𝕜 f (x + a)",
"ppTerm": "?m.55",
"as... | [] | by
simp [← fderivWithin_univ, fderivWithin_comp_add_right] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.Completion | {
"line": 123,
"column": 4
} | {
"line": 123,
"column": 74
} | {
"line": 124,
"column": 4
} | [
{
"pp": "case mpr\nα : Type u\ninst✝ : PseudoMetricSpace α\ns : Set (Completion α × Completion α)\nε : ℝ\nεpos : ε > 0\nhε : ∀ {a b : Completion α}, dist a b < ε → (a, b) ∈ s\nr : Set (ℝ × ℝ) := {p | dist p.1 p.2 < ε}\nthis : r ∈ 𝓤 ℝ\nT : {x | (dist x.1.1 x.1.2, dist x.2.1 x.2.2) ∈ r} ∈ 𝓤 (Completion α × Comp... | [
"case mpr\nα : Type u\ninst✝ : PseudoMetricSpace α\ns : Set (Completion α × Completion α)\nε : ℝ\nεpos : ε > 0\nhε : ∀ {a b : Completion α}, dist a b < ε → (a, b) ∈ s\nr : Set (ℝ × ℝ) := {p | dist p.1 p.2 < ε}\nthis : r ∈ 𝓤 ℝ\nT :\n ∃ t₁ ∈ 𝓤 (Completion α),\n ∃ t₂ ∈ 𝓤 (Completion α),\n t₁ ×ˢ t₂ ⊆ (fun p... | simp only [uniformity_prod_eq_prod, mem_prod_iff, Filter.mem_map] at T | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.MetricSpace.Completion | {
"line": 137,
"column": 2
} | {
"line": 141,
"column": 20
} | {
"line": 143,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\n⊢ 𝓤 (Completion α) = ⨅ ε, 𝓟 {p | dist p.1 p.2 < ↑ε}",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Real.instIsOrderedRing",
"Eq.mpr",
"Real.partialOrder",
"Real",
"Preorder.to... | [] | ext s; rw [mem_iInf_of_directed]
· simp [Completion.mem_uniformity_dist, subset_def]
· rintro ⟨r, hr⟩ ⟨p, hp⟩
use ⟨min r p, lt_min hr hp⟩
simp +contextual | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Completion | {
"line": 137,
"column": 2
} | {
"line": 141,
"column": 20
} | {
"line": 143,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\n⊢ 𝓤 (Completion α) = ⨅ ε, 𝓟 {p | dist p.1 p.2 < ↑ε}",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Real.instIsOrderedRing",
"Eq.mpr",
"Real.partialOrder",
"Real",
"Preorder.to... | [] | ext s; rw [mem_iInf_of_directed]
· simp [Completion.mem_uniformity_dist, subset_def]
· rintro ⟨r, hr⟩ ⟨p, hp⟩
use ⟨min r p, lt_min hr hp⟩
simp +contextual | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.FDeriv.Prod | {
"line": 514,
"column": 2
} | {
"line": 514,
"column": 69
} | {
"line": 515,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nn : ℕ\nF' : Fin n.succ → Type u_6\ninst✝¹ : (i : Fin n.succ) → NormedAddCommGroup (F' i)\ninst✝ : (i : Fin n.succ) → NormedSpace 𝕜 (F' i)\nφ : E → F' 0\nφs : E → (i : Fin n) → F'... | [
"𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nn : ℕ\nF' : Fin n.succ → Type u_6\ninst✝¹ : (i : Fin n.succ) → NormedAddCommGroup (F' i)\ninst✝ : (i : Fin n.succ) → NormedSpace 𝕜 (F' i)\nφ : E → F' 0\nφs : E → (i : Fin n) → F' i.succ\nφ' ... | dsimp [ContinuousLinearMap.comp, LinearMap.comp, Function.comp_def] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 237,
"column": 2
} | {
"line": 237,
"column": 48
} | {
"line": 238,
"column": 2
} | [
{
"pp": "case e'_12\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nn : ℕ∞ω\np : E → FormalMultilinearSeries 𝕜 E F\nh : HasFTaylorSeriesUp... | [
"case e'_12\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nn : ℕ∞ω\np : E → FormalMultilinearSeries 𝕜 E F\nh : HasFTaylorSeriesUpToOn n f p s... | change (p x 1) (snoc 0 y) = (p x 1) (cons y v) | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 373,
"column": 4
} | {
"line": 373,
"column": 86
} | {
"line": 375,
"column": 0
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nN : ℕ∞ω\np : E → FormalMultilinearSeries 𝕜 E F\nhN : ∞ ≤ N\nh :\n (∀ x ∈ s, (p x 0).curr... | [] | exact ⟨h.1, h.2.1, (h.2.2).of_le (m := n) (natCast_le_of_coe_top_le_withTop hN n)⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 479,
"column": 4
} | {
"line": 482,
"column": 23
} | {
"line": 483,
"column": 4
} | [
{
"pp": "case succ\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nhs : UniqueDiffOn 𝕜 s\nn : ℕ\nIH :\n ∀ {x : E},\n x ∈ s →\n ∀ (m : Fi... | [
"case succ\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nhs : UniqueDiffOn 𝕜 s\nn : ℕ\nIH :\n ∀ {x : E},\n x ∈ s →\n ∀ (m : Fin (n + 1) → ... | have A : ∀ y ∈ s, iteratedFDerivWithin 𝕜 n.succ f s y =
(I ∘ iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) y := fun y hy ↦ by
ext m
simp [IH hy m, I] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 642,
"column": 2
} | {
"line": 642,
"column": 33
} | {
"line": 643,
"column": 2
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nn : ℕ∞ω\np : E → FormalMultilinearSeries 𝕜 E F\nh : HasFTaylorSeriesUpToOn n f p s... | [] | induction m generalizing x with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 593,
"column": 20
} | {
"line": 593,
"column": 40
} | {
"line": 593,
"column": 40
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nι : Type u_2\nE : ι → Type u_3\ninst✝³ : (i : ι) → NormedAddCommGroup (E i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹ : Fintype ι\nf : ContinuousMultilinearMap 𝕜 E F\nx : (i... | [
"𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nι : Type u_2\nE : ι → Type u_3\ninst✝³ : (i : ι) → NormedAddCommGroup (E i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹ : Fintype ι\nf : ContinuousMultilinearMap 𝕜 E F\nx : (i : ι) → E i\... | Fintype.card_eq_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 656,
"column": 2
} | {
"line": 658,
"column": 45
} | {
"line": 659,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nι : Type u_2\nE : ι → Type u_3\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝³ : Fintype ι\nG : Type u_4\ninst✝² : NormedAddCommGroup G... | [
"𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nι : Type u_2\nE : ι → Type u_3\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝³ : Fintype ι\nG : Type u_4\ninst✝² : NormedAddCommGroup G\ninst✝¹ : N... | convert!
ContinuousMultilinearMap.hasStrictFDerivAt_uncurry (f x, (g · x)) |>.comp x
(hf.prodMk (hasStrictFDerivAt_pi.2 hg)) | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 64,
"column": 2
} | {
"line": 69,
"column": 93
} | {
"line": 70,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nE : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nG : Type u_1\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nx : 𝕜\nB : E →L[𝕜] F →L[𝕜] G\... | [
"case neg\n𝕜 : Type u\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nE : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nG : Type u_1\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nx : 𝕜\nB : E →L[𝕜] F →L[𝕜] G\nu : 𝕜 → E\... | · by_cases hxv : x ∈ tsupport v
· simpa using (B.hasFDerivAt_of_bilinear (hu hxv).hasFDerivAt (hv hxu).hasFDerivAt).hasDerivAt
· have hx : x ∉ tsupport fun x ↦ B (u x) (v x) :=
mt (closure_mono (fun x ↦ mt fun h ↦ by simp [h]) ·) hxv
convert! HasDerivAt.of_notMem_tsupport hx
simp [(hv hxu).u... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.Deriv.Slope | {
"line": 166,
"column": 2
} | {
"line": 173,
"column": 49
} | {
"line": 175,
"column": 0
} | [
{
"pp": "𝕜 : Type u\ninst✝³ : NontriviallyNormedField 𝕜\nx : 𝕜\ns : Set 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\ninst✝ : OrderTopology 𝕜\ng : 𝕜 → 𝕜\ng' : 𝕜\nhx : AccPt x (𝓟 s)\nhd : HasDerivWithinAt g g' s x\nhg : MonotoneOn g s\n⊢ 0 ≤ g'",
"ppTerm": "?m.25",
"assigned": tru... | [] | have : (𝓝[s \ {x}] x).NeBot := accPt_principal_iff_nhdsWithin.mp hx
have h'g : MonotoneOn g (insert x s) :=
hg.insert_of_continuousWithinAt hx.clusterPt hd.continuousWithinAt
have : Tendsto (slope g x) (𝓝[s \ {x}] x) (𝓝 g') := hasDerivWithinAt_iff_tendsto_slope.mp hd
apply ge_of_tendsto this
filter_upwar... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.Deriv.Slope | {
"line": 166,
"column": 2
} | {
"line": 173,
"column": 49
} | {
"line": 175,
"column": 0
} | [
{
"pp": "𝕜 : Type u\ninst✝³ : NontriviallyNormedField 𝕜\nx : 𝕜\ns : Set 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\ninst✝ : OrderTopology 𝕜\ng : 𝕜 → 𝕜\ng' : 𝕜\nhx : AccPt x (𝓟 s)\nhd : HasDerivWithinAt g g' s x\nhg : MonotoneOn g s\n⊢ 0 ≤ g'",
"ppTerm": "?m.25",
"assigned": tru... | [] | have : (𝓝[s \ {x}] x).NeBot := accPt_principal_iff_nhdsWithin.mp hx
have h'g : MonotoneOn g (insert x s) :=
hg.insert_of_continuousWithinAt hx.clusterPt hd.continuousWithinAt
have : Tendsto (slope g x) (𝓝[s \ {x}] x) (𝓝 g') := hasDerivWithinAt_iff_tendsto_slope.mp hd
apply ge_of_tendsto this
filter_upwar... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 93,
"column": 2
} | {
"line": 95,
"column": 68
} | {
"line": 97,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\np : FormalMultilinearSeries 𝕜 𝕜 E\nf : 𝕜 → E\nz₀ : 𝕜\nn : ℕ\nhp : HasFPowerSeriesAt f p z₀\n⊢ HasFPowerSeriesAt ((swap dslope z₀)^[n] f) (fslope^[n] p) z₀",
"ppTerm": "?m.5... | [] | induction n generalizing f p with
| zero => exact hp
| succ n ih => simpa using ih (has_fpower_series_dslope_fslope hp) | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 93,
"column": 2
} | {
"line": 95,
"column": 68
} | {
"line": 97,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\np : FormalMultilinearSeries 𝕜 𝕜 E\nf : 𝕜 → E\nz₀ : 𝕜\nn : ℕ\nhp : HasFPowerSeriesAt f p z₀\n⊢ HasFPowerSeriesAt ((swap dslope z₀)^[n] f) (fslope^[n] p) z₀",
"ppTerm": "?m.5... | [] | induction n generalizing f p with
| zero => exact hp
| succ n ih => simpa using ih (has_fpower_series_dslope_fslope hp) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 93,
"column": 2
} | {
"line": 95,
"column": 68
} | {
"line": 97,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\np : FormalMultilinearSeries 𝕜 𝕜 E\nf : 𝕜 → E\nz₀ : 𝕜\nn : ℕ\nhp : HasFPowerSeriesAt f p z₀\n⊢ HasFPowerSeriesAt ((swap dslope z₀)^[n] f) (fslope^[n] p) z₀",
"ppTerm": "?m.5... | [] | induction n generalizing f p with
| zero => exact hp
| succ n ih => simpa using ih (has_fpower_series_dslope_fslope hp) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.Deriv.ZPow | {
"line": 166,
"column": 73
} | {
"line": 167,
"column": 61
} | {
"line": 169,
"column": 0
} | [
{
"pp": "𝕜 : Type u\ninst✝ : NontriviallyNormedField 𝕜\nk : ℕ\nc d : 𝕜\n⊢ (deriv^[k] fun x ↦ (c * x - d)⁻¹) = fun x ↦ (-1) ^ k * ↑k ! * c ^ k * (c * x - d) ^ (-1 - ↑k)",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"NegZeroClas... | [] | by
simpa [sub_eq_add_neg] using iter_deriv_inv_linear k c (-d) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 112,
"column": 29
} | {
"line": 112,
"column": 31
} | {
"line": 112,
"column": 32
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\np : FormalMultilinearSeries 𝕜 𝕜 E\nf : 𝕜 → E\nz₀ : 𝕜\nhp : HasFPowerSeriesAt f p z₀\nh : p ≠ 0\nh2 : ContinuousAt ((swap dslope z₀)^[p.order] f) z₀\nh3 : ∀ᶠ (z : 𝕜) in 𝓝 z₀, ... | [
"𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\np : FormalMultilinearSeries 𝕜 𝕜 E\nf : 𝕜 → E\nz₀ : 𝕜\nhp : HasFPowerSeriesAt f p z₀\nh : p ≠ 0\nh2 : ContinuousAt ((swap dslope z₀)^[p.order] f) z₀\nh3 : ∀ᶠ (z : 𝕜) in 𝓝 z₀, (swap dslope... | e1 | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 290,
"column": 2
} | {
"line": 291,
"column": 73
} | {
"line": 292,
"column": 2
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nU : Set 𝕜\nA : Type u_3\ninst✝⁶ : NormedRing A\ninst✝⁵ : IsDomain A\ninst✝⁴ : NormedAlgebra 𝕜 A\nB : Type u_4\ninst✝³ : NormedAddCommGroup B\ninst✝² : NormedSpace 𝕜 B\ninst✝¹ : Module A B\ninst✝ : IsTorsionFree A B\nf : 𝕜 → A\ng : 𝕜 → B... | [
"case neg\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nU : Set 𝕜\nA : Type u_3\ninst✝⁶ : NormedRing A\ninst✝⁵ : IsDomain A\ninst✝⁴ : NormedAlgebra 𝕜 A\nB : Type u_4\ninst✝³ : NormedAddCommGroup B\ninst✝² : NormedSpace 𝕜 B\ninst✝¹ : Module A B\ninst✝ : IsTorsionFree A B\nf : 𝕜 → A\ng : 𝕜 → B\nhf : Analy... | have : ∃ᶠ w in 𝓝[≠] z, w ∈ U :=
frequently_mem_iff_neBot.mpr <| hU.preperfect_of_nontrivial hU'' z hz | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.ContDiff.Basic | {
"line": 397,
"column": 4
} | {
"line": 406,
"column": 65
} | {
"line": 407,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nf : E → F\nn : ℕ∞ω\nx : G... | [] | obtain ⟨u, hu, p, hp, h'p⟩ := hf
refine ⟨g ⁻¹' u, ?_, _, hp.compContinuousLinearMap g, ?_⟩
· refine g.continuous.continuousWithinAt.tendsto_nhdsWithin ?_ hu
exact (mapsTo_singleton.2 <| mem_singleton _).union_union (mapsTo_preimage _ _)
· intro i
change AnalyticOn 𝕜 (fun x ↦
ContinuousM... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Basic | {
"line": 397,
"column": 4
} | {
"line": 406,
"column": 65
} | {
"line": 407,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nf : E → F\nn : ℕ∞ω\nx : G... | [] | obtain ⟨u, hu, p, hp, h'p⟩ := hf
refine ⟨g ⁻¹' u, ?_, _, hp.compContinuousLinearMap g, ?_⟩
· refine g.continuous.continuousWithinAt.tendsto_nhdsWithin ?_ hu
exact (mapsTo_singleton.2 <| mem_singleton _).union_union (mapsTo_preimage _ _)
· intro i
change AnalyticOn 𝕜 (fun x ↦
ContinuousM... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Basic | {
"line": 476,
"column": 12
} | {
"line": 476,
"column": 76
} | {
"line": 476,
"column": 76
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\ng : G ≃ₗᵢ[𝕜] E\nf : E → ... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\ng : G ≃ₗᵢ[𝕜] E\nf : E → F\nhs : Uniq... | ContinuousMultilinearMap.norm_compContinuous_linearIsometryEquiv | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.ContDiff.Comp | {
"line": 643,
"column": 8
} | {
"line": 643,
"column": 30
} | {
"line": 643,
"column": 30
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nx₀ : E\nm n : ℕ∞ω\nf : E ... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nx₀ : E\nm n : ℕ∞ω\nf : E → F → G\ng :... | contDiffWithinAt_infty | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.ContDiff.Comp | {
"line": 703,
"column": 4
} | {
"line": 706,
"column": 36
} | {
"line": 708,
"column": 0
} | [
{
"pp": "case succ\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx₀ : E\nn : ℕ∞ω\nhf : ContDiffWithinAt 𝕜 n f s x₀\nhs : UniqueDiffOn 𝕜 s\... | [] | rw [Nat.cast_succ, add_comm _ 1, ← add_assoc] at hmn
exact ((hi hmn).fderivWithin_right hs le_rfl hx₀s).continuousLinearMap_comp
((continuousMultilinearCurryLeftEquiv 𝕜 (fun _ : Fin (i + 1) ↦ E) F).symm :
_ →L[𝕜] E [×(i + 1)]→L[𝕜] F) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Comp | {
"line": 703,
"column": 4
} | {
"line": 706,
"column": 36
} | {
"line": 708,
"column": 0
} | [
{
"pp": "case succ\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx₀ : E\nn : ℕ∞ω\nhf : ContDiffWithinAt 𝕜 n f s x₀\nhs : UniqueDiffOn 𝕜 s\... | [] | rw [Nat.cast_succ, add_comm _ 1, ← add_assoc] at hmn
exact ((hi hmn).fderivWithin_right hs le_rfl hx₀s).continuousLinearMap_comp
((continuousMultilinearCurryLeftEquiv 𝕜 (fun _ : Fin (i + 1) ↦ E) F).symm :
_ →L[𝕜] E [×(i + 1)]→L[𝕜] F) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.Deriv.Inverse | {
"line": 103,
"column": 14
} | {
"line": 103,
"column": 80
} | {
"line": 103,
"column": 80
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\nf' : F\ns : Set 𝕜\nx : 𝕜\nh : HasDerivWithinAt f f' s x\nhf' : f' ≠ 0\nx✝¹ x✝ : 𝕜\n⊢ dist x✝¹ x✝ ≤\n ↑‖f'‖₊⁻¹ * dist ((ContinuousLinearMap.toSpanSingleton 𝕜 f') x✝¹)... | [] | by simp [dist_eq_norm_sub, ← sub_smul, norm_smul]; field_simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.Deriv.Inverse | {
"line": 108,
"column": 14
} | {
"line": 108,
"column": 80
} | {
"line": 108,
"column": 80
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\nf' : F\ns : Set 𝕜\nx : 𝕜\nc : F\nh : HasDerivWithinAt f f' s x\nhf' : f' ≠ 0\nx✝¹ x✝ : 𝕜\n⊢ dist x✝¹ x✝ ≤\n ↑‖f'‖₊⁻¹ * dist ((ContinuousLinearMap.toSpanSingleton 𝕜 f... | [] | by simp [dist_eq_norm_sub, ← sub_smul, norm_smul]; field_simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.Deriv.Inverse | {
"line": 113,
"column": 14
} | {
"line": 113,
"column": 80
} | {
"line": 113,
"column": 80
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\nf' : F\ns : Set 𝕜\nx : 𝕜\nh : HasDerivWithinAt f f' s x\nhf' : f' ≠ 0\nt : Set F\nht : ¬AccPt (f x) (𝓟 t)\nx✝¹ x✝ : 𝕜\n⊢ dist x✝¹ x✝ ≤\n ↑‖f'‖₊⁻¹ * dist ((Continuous... | [] | by simp [dist_eq_norm_sub, ← sub_smul, norm_smul]; field_simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 177,
"column": 9
} | {
"line": 177,
"column": 19
} | {
"line": 178,
"column": 2
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nn : ℕ∞ω\nhn : ∞ ≠ ∞\n⊢ ContDiffWithinAt 𝕜 ∞ f s x ↔\n ∃ u ∈ 𝓝[insert x s] x, ∃... | [] | simp at hn | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 177,
"column": 9
} | {
"line": 177,
"column": 19
} | {
"line": 178,
"column": 2
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nn : ℕ∞ω\nhn : ∞ ≠ ∞\n⊢ ContDiffWithinAt 𝕜 ∞ f s x ↔\n ∃ u ∈ 𝓝[insert x s] x, ∃... | [] | simp at hn | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 177,
"column": 9
} | {
"line": 177,
"column": 19
} | {
"line": 178,
"column": 2
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nn : ℕ∞ω\nhn : ∞ ≠ ∞\n⊢ ContDiffWithinAt 𝕜 ∞ f s x ↔\n ∃ u ∈ 𝓝[insert x s] x, ∃... | [] | simp at hn | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 397,
"column": 12
} | {
"line": 397,
"column": 32
} | {
"line": 397,
"column": 32
} | [
{
"pp": "case e'_12\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nn : ℕ∞ω\nhn : n ≠ ∞\nh'n : n + 1 ≠ ∞\nu : Set E\nhu : u ∈ 𝓝[insert x s... | [
"case e'_12\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nn : ℕ∞ω\nhn : n ≠ ∞\nh'n : n + 1 ≠ ∞\nu : Set E\nhu : u ∈ 𝓝[insert x s] x\nhf : n ... | ← Hp'.zero_eq y hy.1 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 779,
"column": 2
} | {
"line": 779,
"column": 38
} | {
"line": 781,
"column": 0
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nn : ℕ\nc : OrderedFinpartition n\np : (i : Fin c.length) → E [×c.partSize i]→L[𝕜] F\nm : Fin c.le... | [] | · simp [h, applyOrderedFinpartition] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 1208,
"column": 62
} | {
"line": 1208,
"column": 88
} | {
"line": 1208,
"column": 88
} | [
{
"pp": "case mp\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nn : ℕ∞\n⊢ ContDiffOn 𝕜 (↑n) f univ → HasFTaylorSeriesUpToOn (↑n) f (ftaylorSeries 𝕜 f) uni... | [
"case mp\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nn : ℕ∞\n⊢ ContDiffOn 𝕜 (↑n) f univ → HasFTaylorSeriesUpToOn (↑n) f (ftaylorSeriesWithin 𝕜 f univ) univ... | ← ftaylorSeriesWithin_univ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Algebra.Exponential | {
"line": 330,
"column": 2
} | {
"line": 330,
"column": 41
} | {
"line": 331,
"column": 2
} | [
{
"pp": "𝕂 : Type u_1\n𝔸 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕂\ninst✝³ : NormedRing 𝔸\ninst✝² : NormedAlgebra 𝕂 𝔸\ninst✝¹ : CompleteSpace 𝔸\ninst✝ : CharZero 𝕂\nx : 𝔸\nhx : x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius\n⊢ AnalyticAt 𝕂 exp x",
"ppTerm": "?m.36",
"assigned": true,
"us... | [
"case pos\n𝕂 : Type u_1\n𝔸 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕂\ninst✝³ : NormedRing 𝔸\ninst✝² : NormedAlgebra 𝕂 𝔸\ninst✝¹ : CompleteSpace 𝔸\ninst✝ : CharZero 𝕂\nx : 𝔸\nhx : x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius\nh : (expSeries 𝕂 𝔸).radius = 0\n⊢ AnalyticAt 𝕂 exp x",
"case neg\n𝕂 : Ty... | by_cases h : (expSeries 𝕂 𝔸).radius = 0 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Analysis.Normed.Algebra.Exponential | {
"line": 341,
"column": 2
} | {
"line": 343,
"column": 94
} | {
"line": 344,
"column": 2
} | [
{
"pp": "𝕂 : Type u_1\n𝔸 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕂\ninst✝³ : NormedRing 𝔸\ninst✝² : NormedAlgebra 𝕂 𝔸\ninst✝¹ : CompleteSpace 𝔸\ninst✝ : CharZero 𝕂\nx y : 𝔸\nhxy : Commute x y\nhx : x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius\nhy : y ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius\n⊢ exp... | [
"𝕂 : Type u_1\n𝔸 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕂\ninst✝³ : NormedRing 𝔸\ninst✝² : NormedAlgebra 𝕂 𝔸\ninst✝¹ : CompleteSpace 𝔸\ninst✝ : CharZero 𝕂\nx y : 𝔸\nhxy : Commute x y\nhx : x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius\nhy : y ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius\n⊢ (fun x ↦ ∑' (n ... | rw [exp_eq_tsum 𝕂,
tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_norm
(norm_expSeries_summable_of_mem_ball' x hx) (norm_expSeries_summable_of_mem_ball' y hy)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 287,
"column": 12
} | {
"line": 287,
"column": 19
} | {
"line": 287,
"column": 20
} | [
{
"pp": "case zero\n𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : UniqueDiffOn 𝕜 s\nk : ℕ\n⊢ iteratedDerivWithin k (fun x ↦ x ^ 0) s x = ↑(Nat.descFactorial 0 k) * x ^ (0 - k)",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"NormedCommRing.to... | [
"case zero.zero\n𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : UniqueDiffOn 𝕜 s\n⊢ iteratedDerivWithin 0 (fun x ↦ x ^ 0) s x = ↑(Nat.descFactorial 0 0) * x ^ (0 - 0)",
"case zero.succ\n𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : ... | cases k | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 858,
"column": 2
} | {
"line": 858,
"column": 17
} | {
"line": 861,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type uF\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nn : ℕ∞ω\ninst✝ : CompleteSpace E\ne : E ≃L[𝕜] F\n⊢ ContDiffAt 𝕜 n inverse ↑e",
"ppTerm": "?m.63",
"... | [
"𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type uF\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nn : ℕ∞ω\ninst✝ : CompleteSpace E\ne : E ≃L[𝕜] F\na✝ : Nontrivial E\n⊢ ContDiffAt 𝕜 n inverse ↑e"
] | nontriviality E | Mathlib.Tactic.Nontriviality.elabNontriviality | Mathlib.Tactic.Nontriviality.nontriviality |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 857,
"column": 46
} | {
"line": 871,
"column": 20
} | {
"line": 873,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type uF\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nn : ℕ∞ω\ninst✝ : CompleteSpace E\ne : E ≃L[𝕜] F\n⊢ ContDiffAt 𝕜 n inverse ↑e",
"ppTerm": "?m.63",
"... | [] | by
nontriviality E
-- first, we use the lemma `inverse_eq_ringInverse` to rewrite in terms of `Ring.inverse` in the
-- ring `E →L[𝕜] E`
let O₁ : (E →L[𝕜] E) → F →L[𝕜] E := fun f => f.comp (e.symm : F →L[𝕜] E)
let O₂ : (E →L[𝕜] F) → E →L[𝕜] E := fun f => (e.symm : F →L[𝕜] E).comp f
have : ContinuousLi... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Convex.Cone.Basic | {
"line": 315,
"column": 28
} | {
"line": 315,
"column": 49
} | {
"line": 315,
"column": 49
} | [
{
"pp": "R : Type u_2\nG : Type u_3\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : AddCommGroup G\ninst✝ : SMul R G\nC : ConvexCone R G\n⊢ C.Blunt → ¬C.Flat",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"AddCommGroup.toAddCommMonoid",
... | [
"R : Type u_2\nG : Type u_3\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : AddCommGroup G\ninst✝ : SMul R G\nC : ConvexCone R G\n⊢ ¬C.Pointed → ¬C.Flat"
] | blunt_iff_not_pointed | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.LocallyConvex.AbsConvex | {
"line": 139,
"column": 78
} | {
"line": 141,
"column": 40
} | {
"line": 143,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : SeminormedRing 𝕜\ninst✝² : SMul 𝕜 E\ninst✝¹ : AddCommMonoid E\ninst✝ : PartialOrder 𝕜\ns : Set E\n⊢ ((absConvexHull 𝕜) s).Nonempty ↔ s.Nonempty",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ChainCompletePartialOrder... | [] | by
rw [nonempty_iff_ne_empty, nonempty_iff_ne_empty, Ne, Ne]
exact not_congr absConvexHull_eq_empty | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 211,
"column": 62
} | {
"line": 211,
"column": 75
} | {
"line": 211,
"column": 75
} | [
{
"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_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nx : E\nhx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}\ne : ℕ\nt... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 211,
"column": 62
} | {
"line": 211,
"column": 75
} | {
"line": 211,
"column": 75
} | [
{
"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_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nx : E\nhx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}\ne : ℕ\nt... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.LocallyConvex.AbsConvex | {
"line": 353,
"column": 2
} | {
"line": 353,
"column": 33
} | {
"line": 354,
"column": 2
} | [
{
"pp": "E : Type u_2\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ns : Set E\ninst✝² : UniformSpace E\ninst✝¹ : IsUniformAddGroup E\nlcs : LocallyConvexSpace ℝ E\ninst✝ : ContinuousSMul ℝ E\nhs : TotallyBounded s\n⊢ TotallyBounded ((convexHull ℝ) (s ∪ -s))",
"ppTerm": "?m.32",
"assigned": true,
"u... | [
"E : Type u_2\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ns : Set E\ninst✝² : UniformSpace E\ninst✝¹ : IsUniformAddGroup E\nlcs : LocallyConvexSpace ℝ E\ninst✝ : ContinuousSMul ℝ E\nhs : TotallyBounded s\n⊢ TotallyBounded (s ∪ -s)"
] | apply totallyBounded_convexHull | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 243,
"column": 4
} | {
"line": 243,
"column": 42
} | {
"line": 244,
"column": 4
} | [
{
"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_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nhK : IsComplete K\nP : ∀ {n : ℕ}, 0 < (1 / 2) ^ n\nc : 𝕜\nhc : 1 < ‖c‖\nx ... | [
"𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nhK : IsComplete K\nP : ∀ {n : ℕ}, 0 < (1 / 2) ^ n\nc : 𝕜\nhc : 1 < ‖c‖\nx : E\nhx : x ... | intro e p q e' p' q' hp hq hp' hq' he' | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.LocallyConvex.AbsConvexOpen | {
"line": 120,
"column": 2
} | {
"line": 125,
"column": 87
} | {
"line": 127,
"column": 2
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : Module 𝕜 E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : IsScalarTower ℝ 𝕜 E\ninst✝³ : ContinuousSMul ℝ E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyCon... | [
"case refine_2\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : Module 𝕜 E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : IsScalarTower ℝ 𝕜 E\ninst✝³ : ContinuousSMul ℝ E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace 𝕜 ... | refine
⟨mem_iInter₂.mpr fun _ _ => by simp [hr],
isOpen_biInter_finset fun S _ => ?_,
balanced_iInter₂ fun _ _ => Seminorm.balanced_ball_zero _ _,
convex_iInter₂ fun _ _ => (convex_of_nonneg_surjective_algebraMap _
(fun _ => RCLike.nonneg_iff_exists_ofReal.mp) (Seminorm.convex_ball _ _ _) ... | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 263,
"column": 84
} | {
"line": 264,
"column": 21
} | {
"line": 265,
"column": 6
} | [
{
"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_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nhK : IsComplete K\nP : ∀ {n : ℕ}, 0 < (1 / 2) ^ n\nc : 𝕜\nhc : 1 < ‖c‖\nx ... | [] | by
congr 1; abel | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Gauge | {
"line": 191,
"column": 6
} | {
"line": 191,
"column": 34
} | {
"line": 192,
"column": 6
} | [
{
"pp": "E : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\nhs : Convex ℝ s\nabsorbs : Absorbent ℝ s\nε : ℝ\nhε : 0 < ε\na : ℝ\nha : 0 < a\nx : E\nhx : x ∈ s\nha' : a < gauge s ((fun x ↦ a • x) x) + ε / 2\nb : ℝ\nhb : 0 < b\ny : E\nhy : y ∈ s\nhb' : b < gauge s ((fun x ↦ b • x) y) + ε / 2\n⊢ ... | [
"E : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\nhs : Convex ℝ s\nabsorbs : Absorbent ℝ s\nε : ℝ\nhε : 0 < ε\na : ℝ\nha : 0 < a\nx : E\nhx : x ∈ s\nha' : a < gauge s ((fun x ↦ a • x) x) + ε / 2\nb : ℝ\nhb : 0 < b\ny : E\nhy : y ∈ s\nhb' : b < gauge s ((fun x ↦ b • x) y) + ε / 2\n⊢ a • x + b • ... | rw [hs.add_smul ha.le hb.le] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Module.HahnBanach | {
"line": 51,
"column": 4
} | {
"line": 51,
"column": 41
} | {
"line": 52,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : IsRCLikeNormedField 𝕜\nE : Type u_2\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\np : Subspace 𝕜 E\nf : StrongDual 𝕜 ↥p\n⊢ Continuous ⇑(‖f‖₊ • normSeminorm 𝕜 E)",
"ppTerm": "?m.58",
"assigned": true,
"usedConst... | [] | exact continuous_norm.const_smul ‖f‖₊ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Module.HahnBanach | {
"line": 53,
"column": 51
} | {
"line": 53,
"column": 86
} | {
"line": 55,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : IsRCLikeNormedField 𝕜\nE : Type u_2\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\np : Subspace 𝕜 E\nf : StrongDual 𝕜 ↥p\ng : StrongDual 𝕜 E\nhg : ∀ (x : ↥p), g ↑x = ↑f x\nhl : ∀ (x : E), ‖g x‖ ≤ (‖f‖₊ • normSeminorm 𝕜 E) ... | [] | by simpa [hg x] using g.le_opNorm x | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 42
} | {
"line": 106,
"column": 2
} | [
{
"pp": "case inr.inr\nE : Type u_2\ninst✝⁴ : TopologicalSpace E\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns t : Set E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousSMul ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsOpen s\nht : Convex ℝ t\ndisj : Disjoint s t\na₀ : E\nha₀ : a₀ ∈ s\nb₀ : E\nhb₀ : b₀ ∈ t\nx₀ : E :=... | [
"case inr.inr\nE : Type u_2\ninst✝⁴ : TopologicalSpace E\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns t : Set E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousSMul ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsOpen s\nht : Convex ℝ t\ndisj : Disjoint s t\na₀ : E\nha₀ : a₀ ∈ s\nb₀ : E\nhb₀ : b₀ ∈ t\nx₀ : E := b₀ - a₀\nC ... | have : Convex ℝ C := (hs₁.sub ht).vadd _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 542,
"column": 62
} | {
"line": 542,
"column": 75
} | {
"line": 542,
"column": 75
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nx : ℝ\nhx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}\ne : ℕ\nthis : 0 < (1 / 2) ^ e\nR : ℝ\nR_pos : R > 0\nhR : ∀ r ∈ Ioo 0 R, x ∈ A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 542,
"column": 62
} | {
"line": 542,
"column": 75
} | {
"line": 542,
"column": 75
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nx : ℝ\nhx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}\ne : ℕ\nthis : 0 < (1 / 2) ^ e\nR : ℝ\nR_pos : R > 0\nhR : ∀ r ∈ Ioo 0 R, x ∈ A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.LocallyConvex.SeparatingDual | {
"line": 138,
"column": 68
} | {
"line": 142,
"column": 64
} | {
"line": 144,
"column": 0
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝¹² : Field R\ninst✝¹¹ : AddCommGroup V\ninst✝¹⁰ : TopologicalSpace R\ninst✝⁹ : TopologicalSpace V\ninst✝⁸ : IsTopologicalRing R\ninst✝⁷ : Module R V\ninst✝⁶ : SeparatingDual R V\nW : Type u_3\ninst✝⁵ : AddCommGroup W\ninst✝⁴ : TopologicalSpace W\ninst✝³ : Module R W\nin... | [] | by
obtain ⟨v, hv⟩ := exists_ne (0 : V)
obtain ⟨w, hw⟩ := exists_ne (0 : W)
obtain ⟨ψ, hψ⟩ := exists_ne_zero (R := R) hv
exact ⟨ψ.smulRight w, 0, DFunLike.ne_iff.mpr ⟨v, by simp_all⟩⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 572,
"column": 4
} | {
"line": 572,
"column": 42
} | {
"line": 573,
"column": 4
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nhK : IsComplete K\nP : ∀ {n : ℕ}, 0 < (1 / 2) ^ n\nx : ℝ\nhx : x ∈ D f K\nn : ℕ → ℕ\nL : ℕ → ℕ → ℕ → F\nhn :\n ∀ (e p q : ℕ),\n n e ≤ p →\n n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / ... | [
"F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nhK : IsComplete K\nP : ∀ {n : ℕ}, 0 < (1 / 2) ^ n\nx : ℝ\nhx : x ∈ D f K\nn : ℕ → ℕ\nL : ℕ → ℕ → ℕ → F\nhn :\n ∀ (e p q : ℕ),\n n e ≤ p →\n n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A ... | intro e p q e' p' q' hp hq hp' hq' he' | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 592,
"column": 84
} | {
"line": 593,
"column": 21
} | {
"line": 594,
"column": 6
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nhK : IsComplete K\nP : ∀ {n : ℕ}, 0 < (1 / 2) ^ n\nx : ℝ\nhx : x ∈ D f K\nn : ℕ → ℕ\nL : ℕ → ℕ → ℕ → F\nhn :\n ∀ (e p q : ℕ),\n n e ≤ p →\n n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / ... | [] | by
congr 1; abel | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 323,
"column": 4
} | {
"line": 324,
"column": 53
} | {
"line": 325,
"column": 4
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : ∫⁻ (x : α), ↑((piecewise s hs (const α c) (const α 0)) x) ∂μ ≠ ∞\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : ¬c = 0\nμs_lt_to... | [
"case neg\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : ∫⁻ (x : α), ↑((piecewise s hs (const α c) (const α 0)) x) ∂μ ≠ ∞\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : ¬c = 0\nμs_lt_top : μ s < ∞\... | obtain ⟨F, Fs, F_closed, μF⟩ : ∃ (F : _), F ⊆ s ∧ IsClosed F ∧ μ s < μ F + ε / c :=
hs.exists_isClosed_lt_add μs_lt_top.ne this.ne' | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap | {
"line": 299,
"column": 86
} | {
"line": 303,
"column": 13
} | {
"line": 304,
"column": 4
} | [
{
"pp": "X : Type u_1\nE : Type u_3\ninst✝² : MeasurableSpace X\nμ : Measure X\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : X → ℝ≥0\nhf : AEMeasurable f μ\ng : X → E\nf' : X → ℝ≥0 := AEMeasurable.mk f hf\n⊢ (∫ (x : X), g x ∂μ.withDensity fun x ↦ ↑(f x)) = ∫ (x : X), g x ∂μ.withDensity fun x ↦ ↑(... | [] | by
congr 1
apply withDensity_congr_ae
filter_upwards [hf.ae_eq_mk] with x hx
rw [hx] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 412,
"column": 6
} | {
"line": 416,
"column": 25
} | {
"line": 417,
"column": 4
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nf : α → ℝ≥0\nfint : Integrable (fun x ↦ ↑(f x)) μ\nε : ℝ≥0\nεpos : 0 < ↑ε\nIf : ∫⁻ (x : α), ↑(f x) ∂μ < ∞\ng : α → ℝ≥0\ngf : ∀ (x : α), g x ≤ f x\ngcont :... | [] | rw [sub_le_iff_le_add]
convert! ENNReal.toReal_mono _ gint
· simp
· rw [ENNReal.toReal_add Ig.ne ENNReal.coe_ne_top]; simp
· simpa using Ig.ne | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 412,
"column": 6
} | {
"line": 416,
"column": 25
} | {
"line": 417,
"column": 4
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nf : α → ℝ≥0\nfint : Integrable (fun x ↦ ↑(f x)) μ\nε : ℝ≥0\nεpos : 0 < ↑ε\nIf : ∫⁻ (x : α), ↑(f x) ∂μ < ∞\ng : α → ℝ≥0\ngf : ∀ (x : α), g x ≤ f x\ngcont :... | [] | rw [sub_le_iff_le_add]
convert! ENNReal.toReal_mono _ gint
· simp
· rw [ENNReal.toReal_add Ig.ne ENNReal.coe_ne_top]; simp
· simpa using Ig.ne | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 515,
"column": 40
} | {
"line": 515,
"column": 84
} | {
"line": 515,
"column": 84
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : μ.WeaklyRegular\ninst✝ : SigmaFinite μ\nf : α → ℝ\nhf : Integrable f μ\nε : ℝ\nεpos : 0 < ε\ng : α → EReal\ng_lt_f : ∀ (x : α), ↑(-f x) < g x\ngcont : LowerSemicontinuous g\ng_integrabl... | [] | by simpa only [EReal.coe_neg] using g_lt_f x | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 348,
"column": 8
} | {
"line": 354,
"column": 21
} | {
"line": 355,
"column": 6
} | [
{
"pp": "case refine_3.inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b₀ b₁ b₂ : ℝ\nμ : Measure ℝ\nf : ℝ → E\nhb₀ : μ {b₀} = 0\nh_int : IntervalIntegrable f μ (min a b₁) (max a b₂)\nh₀ : b₀ ∈ Icc b₁ b₂\nh₁₂ : b₁ ≤ b₂\nmin₁₂ : min b₁ b₂ = b₁\nh_int' : ∀ {x : ℝ}, x ∈ Icc b₁ b₂ → Inte... | [] | have : ∀ᶠ x in 𝓝[Icc b₁ b₂] b₀, {t : ℝ | t ≤ x}.indicator f x₀ = f x₀ := by
apply mem_nhdsWithin_of_mem_nhds
apply Eventually.mono (Ioi_mem_nhds hx₀)
intro x hx
simp [hx.le]
apply continuousWithinAt_const.congr_of_eventuallyEq this
simp [hx₀.le] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 348,
"column": 8
} | {
"line": 354,
"column": 21
} | {
"line": 355,
"column": 6
} | [
{
"pp": "case refine_3.inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b₀ b₁ b₂ : ℝ\nμ : Measure ℝ\nf : ℝ → E\nhb₀ : μ {b₀} = 0\nh_int : IntervalIntegrable f μ (min a b₁) (max a b₂)\nh₀ : b₀ ∈ Icc b₁ b₂\nh₁₂ : b₁ ≤ b₂\nmin₁₂ : min b₁ b₂ = b₁\nh_int' : ∀ {x : ℝ}, x ∈ Icc b₁ b₂ → Inte... | [] | have : ∀ᶠ x in 𝓝[Icc b₁ b₂] b₀, {t : ℝ | t ≤ x}.indicator f x₀ = f x₀ := by
apply mem_nhdsWithin_of_mem_nhds
apply Eventually.mono (Ioi_mem_nhds hx₀)
intro x hx
simp [hx.le]
apply continuousWithinAt_const.congr_of_eventuallyEq this
simp [hx₀.le] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 866,
"column": 46
} | {
"line": 866,
"column": 73
} | {
"line": 866,
"column": 74
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_5\nF : Type u_6\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nμ : Measure ℝ\ninst✝³ : RCLike 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\na b : ℝ\nφ : ℝ → F →L[𝕜] E\nhφ : IntervalIntegrable φ μ a b\nv : F\n⊢ ((if a ≤ b th... | [
"𝕜 : Type u_2\nE : Type u_5\nF : Type u_6\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nμ : Measure ℝ\ninst✝³ : RCLike 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\na b : ℝ\nφ : ℝ → F →L[𝕜] E\nhφ : IntervalIntegrable φ μ a b\nv : F\n⊢ ((if a ≤ b then 1 else -1... | ← integral_apply hφ.def' v, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 960,
"column": 39
} | {
"line": 960,
"column": 65
} | {
"line": 962,
"column": 0
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b : ℝ\nf : ℝ → E\nd : ℝ\nA : MeasurableEmbedding fun x ↦ x + d\n⊢ ∫ (x : ℝ) in a + d..b + d, f x ∂Measure.map (fun x ↦ x + d) volume = ∫ (x : ℝ) in a + d..b + d, f x",
"ppTerm": "?m.109",
"assigned": true,
"usedConstant... | [] | rw [map_add_right_eq_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 960,
"column": 39
} | {
"line": 960,
"column": 65
} | {
"line": 962,
"column": 0
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b : ℝ\nf : ℝ → E\nd : ℝ\nA : MeasurableEmbedding fun x ↦ x + d\n⊢ ∫ (x : ℝ) in a + d..b + d, f x ∂Measure.map (fun x ↦ x + d) volume = ∫ (x : ℝ) in a + d..b + d, f x",
"ppTerm": "?m.109",
"assigned": true,
"usedConstant... | [] | rw [map_add_right_eq_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 960,
"column": 39
} | {
"line": 960,
"column": 65
} | {
"line": 962,
"column": 0
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b : ℝ\nf : ℝ → E\nd : ℝ\nA : MeasurableEmbedding fun x ↦ x + d\n⊢ ∫ (x : ℝ) in a + d..b + d, f x ∂Measure.map (fun x ↦ x + d) volume = ∫ (x : ℝ) in a + d..b + d, f x",
"ppTerm": "?m.109",
"assigned": true,
"usedConstant... | [] | rw [map_add_right_eq_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 641,
"column": 59
} | {
"line": 643,
"column": 76
} | {
"line": 645,
"column": 0
} | [
{
"pp": "E : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E\nc : E\na b : ℝ\nhf : IntervalIntegrable f volume a b\nhmeas : StronglyMeasurableAtFilter f (𝓝 a) volume\nha : Tendsto f (𝓝 a ⊓ ae volume) (𝓝 c)\n⊢ HasStrictDerivAt (fun u ↦ ∫ (x : ℝ) in u..b, f... | [] | by
simpa only [← integral_symm] using
(integral_hasStrictDerivAt_of_tendsto_ae_right hf.symm hmeas ha).fun_neg | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 999,
"column": 4
} | {
"line": 1000,
"column": 98
} | {
"line": 1003,
"column": 4
} | [
{
"pp": "g' g φ : ℝ → ℝ\na b : ℝ\nhab : a ≤ b\nhcont : ContinuousOn g (Icc a b)\nhderiv : ∀ x ∈ Ico a b, HasDerivWithinAt g (g' x) (Ioi x) x\nφint : IntegrableOn φ (Icc a b) volume\nhφg : ∀ x ∈ Ico a b, g' x ≤ φ x\nε : ℝ\nεpos : 0 < ε\nG' : ℝ → EReal\nf_lt_G' : ∀ (x : ℝ), ↑(φ x) < G' x\nG'cont : LowerSemicontin... | [
"g' g φ : ℝ → ℝ\na b : ℝ\nhab : a ≤ b\nhcont : ContinuousOn g (Icc a b)\nhderiv : ∀ x ∈ Ico a b, HasDerivWithinAt g (g' x) (Ioi x) x\nφint : IntegrableOn φ (Icc a b) volume\nhφg : ∀ x ∈ Ico a b, g' x ≤ φ x\nε : ℝ\nεpos : 0 < ε\nG' : ℝ → EReal\nf_lt_G' : ∀ (x : ℝ), ↑(φ x) < G' x\nG'cont : LowerSemicontinuous G'\nG'i... | obtain ⟨y, g'_lt_y', y_lt_G'⟩ : ∃ y : ℝ, (g' t : EReal) < y ∧ (y : EReal) < G' t :=
EReal.lt_iff_exists_real_btwn.1 ((EReal.coe_le_coe_iff.2 (hφg t ht.2)).trans_lt (f_lt_G' t)) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 1197,
"column": 4
} | {
"line": 1197,
"column": 59
} | {
"line": 1198,
"column": 4
} | [
{
"pp": "case inr\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\na b : ℝ\ninst✝ : CompleteSpace E\nf : ℝ → E\nhcont : ContinuousOn f [[a, b]]\nhderiv : ∀ x ∈ uIoo a b, DifferentiableAt ℝ f x\nhint : IntervalIntegrable (deriv f) volume a b\nhab : b ≤ a\n⊢ ∫ (y : ℝ) in a..b, deriv f y = f... | [
"case inr\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\na b : ℝ\ninst✝ : CompleteSpace E\nf : ℝ → E\nhint : IntervalIntegrable (deriv f) volume a b\nhab : b ≤ a\nhcont : ContinuousOn f (Icc b a)\nhderiv : ∀ x ∈ Ioo b a, DifferentiableAt ℝ f x\n⊢ ∫ (y : ℝ) in a..b, deriv f y = f b - f a"
] | simp only [uIcc_of_ge, hab, uIoo_of_ge] at hcont hderiv | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 95,
"column": 95
} | {
"line": 96,
"column": 90
} | {
"line": 98,
"column": 0
} | [
{
"pp": "c : ℂ\nR : ℝ\n⊢ circleMap c R '' Ioc 0 (2 * π) = sphere c |R|",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"NormedCommRing.toSeminormedCommRing",
"range_circleMap",
"Real",
"Real.instArchimedean",
"Real.pi",
"... | [] | by
rw [← range_circleMap, ← (periodic_circleMap c R).image_Ioc Real.two_pi_pos 0, zero_add] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 209,
"column": 2
} | {
"line": 210,
"column": 51
} | {
"line": 211,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nc : ℂ\nR : ℝ\nι : Type u_3\ns : Finset ι\nf : ι → ℂ → E\nh : ∀ i ∈ s, CircleIntegrable (f i) c R\n⊢ CircleIntegrable (∑ i ∈ s, f i) c R",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Pi.addCommMonoid... | [
"E : Type u_1\ninst✝ : NormedAddCommGroup E\nc : ℂ\nR : ℝ\nι : Type u_3\ns : Finset ι\nf : ι → ℂ → E\nh : ∀ i ∈ s, CircleIntegrable (f i) c R\n⊢ IntervalIntegrable (∑ i ∈ s, fun θ ↦ f i (circleMap c R θ)) volume 0 (2 * π)"
] | rw [CircleIntegrable, (by aesop : (fun θ ↦ (∑ i ∈ s, f i) (circleMap c R θ))
= ∑ i ∈ s, fun θ ↦ f i (circleMap c R θ))] at * | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 295,
"column": 8
} | {
"line": 295,
"column": 35
} | {
"line": 295,
"column": 35
} | [
{
"pp": "E✝ : Type u_1\ninst✝¹ : NormedAddCommGroup E✝\nf✝ : ℝ → E✝\nT✝ : ℝ\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\nT t : ℝ\nh₁f : Periodic f T\nhT✝ : T ≠ 0\nh₂f : IntervalIntegrable f volume t (t + T)\na₁ a₂ : ℝ\nhT : 0 < T\nn₁ : ℕ\nhn₁ : (t - min a₁ a₂) / T ≤ ↑n₁\nn₂ : ℕ\nhn₂ : (max a₁ a₂ - t)... | [
"E✝ : Type u_1\ninst✝¹ : NormedAddCommGroup E✝\nf✝ : ℝ → E✝\nT✝ : ℝ\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\nT t : ℝ\nh₁f : Periodic f T\nhT✝ : T ≠ 0\nh₂f : IntervalIntegrable f volume t (t + T)\na₁ a₂ : ℝ\nhT : 0 < T\nn₁ : ℕ\nhn₁ : (t - min a₁ a₂) / T ≤ ↑n₁\nn₂ : ℕ\nhn₂ : (max a₁ a₂ - t) / T ≤ ↑n₂\n... | Set.uIcc_subset_uIcc_iff_le | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.BoxIntegral.DivergenceTheorem | {
"line": 101,
"column": 4
} | {
"line": 101,
"column": 17
} | {
"line": 101,
"column": 17
} | [
{
"pp": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nn : ℕ\ninst✝ : CompleteSpace E\nI : Box (Fin (n + 1))\ni : Fin (n + 1)\nf : (Fin (n + 1) → ℝ) → E\nf' : (Fin (n + 1) → ℝ) →L[ℝ] E\nhfc : ContinuousOn f (Box.Icc I)\nx : Fin (n + 1) → ℝ\nhxI : x ∈ Box.Icc I\na : E\nε : ℝ\nh0 : 0 < ε\nc... | [
"E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nn : ℕ\ninst✝ : CompleteSpace E\nI : Box (Fin (n + 1))\ni : Fin (n + 1)\nf : (Fin (n + 1) → ℝ) → E\nf' : (Fin (n + 1) → ℝ) →L[ℝ] E\nhfc : ContinuousOn f (Box.Icc I)\nx : Fin (n + 1) → ℝ\nhxI : x ∈ Box.Icc I\na : E\nε : ℝ\nh0 : 0 < ε\nc : ℝ≥0\nhc :... | clear_value g | Lean.Elab.Tactic.evalClearValue | Lean.Parser.Tactic.clearValue |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 412,
"column": 4
} | {
"line": 412,
"column": 35
} | {
"line": 412,
"column": 36
} | [
{
"pp": "T : ℝ\ng : ℝ → ℝ\nhg : Periodic g T\nh_int : IntervalIntegrable g volume 0 T\nhT : 0 < T\nt : ℝ\nh'_int : ∀ (a₁ a₂ : ℝ), IntervalIntegrable g volume a₁ a₂ := intervalIntegrable₀ hg (LT.lt.ne' hT) h_int\nε : ℝ := Int.fract (t / T) * T\n⊢ sInf ((fun t ↦ ∫ (x : ℝ) in 0..t, g x) '' Icc 0 T) + ⌊t / T⌋ • ∫ (... | [
"T : ℝ\ng : ℝ → ℝ\nhg : Periodic g T\nh_int : IntervalIntegrable g volume 0 T\nhT : 0 < T\nt : ℝ\nh'_int : ∀ (a₁ a₂ : ℝ), IntervalIntegrable g volume a₁ a₂ := intervalIntegrable₀ hg (LT.lt.ne' hT) h_int\nε : ℝ := Int.fract (t / T) * T\n⊢ sInf ((fun t ↦ ∫ (x : ℝ) in 0..t, g x) '' Icc 0 T) + ⌊t / T⌋ • ∫ (x : ℝ) in 0.... | hg.intervalIntegral_add_eq ε 0, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 428,
"column": 57
} | {
"line": 428,
"column": 88
} | {
"line": 428,
"column": 89
} | [
{
"pp": "T : ℝ\ng : ℝ → ℝ\nhg : Periodic g T\nh_int : IntervalIntegrable g volume 0 T\nhT : 0 < T\nt : ℝ\nh'_int : ∀ (a₁ a₂ : ℝ), IntervalIntegrable g volume a₁ a₂ := intervalIntegrable₀ hg (LT.lt.ne' hT) h_int\nε : ℝ := Int.fract (t / T) * T\n⊢ (∫ (x : ℝ) in 0..ε, g x) + ⌊t / T⌋ • ∫ (x : ℝ) in ε..ε + T, g x ≤\... | [
"T : ℝ\ng : ℝ → ℝ\nhg : Periodic g T\nh_int : IntervalIntegrable g volume 0 T\nhT : 0 < T\nt : ℝ\nh'_int : ∀ (a₁ a₂ : ℝ), IntervalIntegrable g volume a₁ a₂ := intervalIntegrable₀ hg (LT.lt.ne' hT) h_int\nε : ℝ := Int.fract (t / T) * T\n⊢ (∫ (x : ℝ) in 0..ε, g x) + ⌊t / T⌋ • ∫ (x : ℝ) in 0..0 + T, g x ≤\n sSup ((... | hg.intervalIntegral_add_eq ε 0, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.BoxIntegral.DivergenceTheorem | {
"line": 112,
"column": 8
} | {
"line": 112,
"column": 66
} | {
"line": 113,
"column": 6
} | [
{
"pp": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nn : ℕ\ninst✝ : CompleteSpace E\nI : Box (Fin (n + 1))\ni : Fin (n + 1)\nf' : (Fin (n + 1) → ℝ) →L[ℝ] E\nx : Fin (n + 1) → ℝ\nhxI : x ∈ Box.Icc I\na : E\nε : ℝ\nh0 : 0 < ε\nc : ℝ≥0\nhc : I.distortion ≤ c\ne : ℝ → (Fin n → ℝ) → Fin (n +... | [] | exact dist_le_diam_of_mem I.isCompact_Icc.isBounded hy hxI | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.BoxIntegral.DivergenceTheorem | {
"line": 188,
"column": 61
} | {
"line": 188,
"column": 63
} | {
"line": 188,
"column": 64
} | [
{
"pp": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nn : ℕ\ninst✝ : CompleteSpace E\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) → ℝ) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E\ns : Set (Fin (n + 1) → ℝ)\nhs : s.Countable\nHs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x\nHd : ... | [
"E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nn : ℕ\ninst✝ : CompleteSpace E\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) → ℝ) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E\ns : Set (Fin (n + 1) → ℝ)\nhs : s.Countable\nHs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x\nHd : ∀ x ∈ Box.Ic... | y₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 680,
"column": 4
} | {
"line": 680,
"column": 22
} | {
"line": 681,
"column": 4
} | [
{
"pp": "case right\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nμ : Measure ℝ\nf : ℝ → E\ninst✝ : NoAtoms μ\na₀ : ℝ\nhf : IntegrableOn f (Iio a₀) μ\na : ℝ\nha : a ≤ a₀\n⊢ ContinuousWithinAt (fun b ↦ ∫ (x : ℝ) in Iio b, f x ∂μ) (Iic a₀ ∩ Ici a) a",
"ppTerm": "?right",
"assigne... | [
"case right\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nμ : Measure ℝ\nf : ℝ → E\ninst✝ : NoAtoms μ\na₀ : ℝ\nhf : IntegrableOn f (Iio a₀) μ\na : ℝ\nha : a ≤ a₀\n⊢ ContinuousWithinAt (fun b ↦ ∫ (x : ℝ) in Iio b, f x ∂μ) (Icc a a₀) a"
] | rw [Iic_inter_Ici] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.BoxIntegral.DivergenceTheorem | {
"line": 229,
"column": 6
} | {
"line": 229,
"column": 82
} | {
"line": 230,
"column": 6
} | [
{
"pp": "case refine_3.refine_2\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nn : ℕ\ninst✝ : CompleteSpace E\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) → ℝ) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E\ns : Set (Fin (n + 1) → ℝ)\nhs : s.Countable\nHs : ∀ x ∈ s, ContinuousWithinA... | [
"case refine_3.refine_2\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nn : ℕ\ninst✝ : CompleteSpace E\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) → ℝ) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E\ns : Set (Fin (n + 1) → ℝ)\nhs : s.Countable\nHs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc... | refine (mul_le_mul_of_nonneg_right ?_ (half_pos ε0).le).trans_eq (one_mul _) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Integral.Prod | {
"line": 624,
"column": 28
} | {
"line": 624,
"column": 48
} | {
"line": 624,
"column": 49
} | [
{
"pp": "E : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\nmE : MeasurableSpace E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\nmF : MeasurableSpace F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace ℝ G\nμ : Measure E\ninst✝² : IsProbabilityMeasu... | [
"E : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\nmE : MeasurableSpace E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\nmF : MeasurableSpace F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace ℝ G\nμ : Measure E\ninst✝² : IsProbabilityMeasure μ\nν : Me... | integral_prod _ hLμ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.PolynomialGaloisGroup | {
"line": 360,
"column": 2
} | {
"line": 360,
"column": 42
} | {
"line": 361,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\np : F[X]\ninst✝ : CharZero F\np_irr : Irreducible p\np_deg : Nat.Prime p.natDegree\nhp : p.degree ≠ 0\nα : p.SplittingField := rootOfSplits ⋯ ⋯\nhα : IsIntegral F α\n⊢ p.natDegree ∣ finrank F p.SplittingField",
"ppTerm": "?m.71",
"assigned": true,
"usedConsta... | [
"case h\nF : Type u_1\ninst✝¹ : Field F\np : F[X]\ninst✝ : CharZero F\np_irr : Irreducible p\np_deg : Nat.Prime p.natDegree\nhp : p.degree ≠ 0\nα : p.SplittingField := ⋯\nhα : IsIntegral F α\n⊢ finrank F p.SplittingField = p.natDegree * finrank (↥F⟮α⟯) p.SplittingField"
] | use Module.finrank F⟮α⟯ p.SplittingField | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.Polynomial.Vieta | {
"line": 123,
"column": 34
} | {
"line": 123,
"column": 45
} | {
"line": 123,
"column": 45
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : R[X]\nhroots : p.roots.card = p.natDegree\nk : ℕ\nh : k ≤ p.natDegree\n⊢ k ≤ p.roots.card",
"ppTerm": "?m.96",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.roots",
"congrArg",
"CommSemiring.toSe... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : R[X]\nhroots : p.roots.card = p.natDegree\nk : ℕ\nh : k ≤ p.natDegree\n⊢ k ≤ p.natDegree"
] | rw [hroots] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Maps.Proper.CompactlyGenerated | {
"line": 46,
"column": 11
} | {
"line": 46,
"column": 46
} | {
"line": 47,
"column": 4
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\ninst✝ : CompactlyCoherentSpace Y\nf : X → Y\n⊢ IsProperMap f ↔ Continuous f ∧ Tendsto f (cocompact X) (cocompact Y)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.m... | [
"X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\ninst✝ : CompactlyCoherentSpace Y\nf : X → Y\n⊢ (Continuous f ∧ ∀ ⦃K : Set Y⦄, IsCompact K → IsCompact (f ⁻¹' K)) ↔\n Continuous f ∧ Tendsto f (cocompact X) (cocompact Y)"
] | isProperMap_iff_isCompact_preimage, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Polynomial.Vieta | {
"line": 124,
"column": 55
} | {
"line": 124,
"column": 66
} | {
"line": 126,
"column": 0
} | [
{
"pp": "case e'_5\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : R[X]\nhroots : p.roots.card = p.natDegree\nk : ℕ\nh : k ≤ p.natDegree\nthis : k ≤ p.roots.card\n⊢ p.natDegree - k = p.roots.card - k",
"ppTerm": "?e'_5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynom... | [] | rw [hroots] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.Vieta | {
"line": 124,
"column": 55
} | {
"line": 124,
"column": 66
} | {
"line": 126,
"column": 0
} | [
{
"pp": "case e'_6\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : R[X]\nhroots : p.roots.card = p.natDegree\nk : ℕ\nh : k ≤ p.natDegree\nthis : k ≤ p.roots.card\n⊢ p.natDegree - k = p.roots.card - k",
"ppTerm": "?e'_6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynom... | [] | rw [hroots] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 406,
"column": 45
} | {
"line": 406,
"column": 74
} | {
"line": 406,
"column": 74
} | [
{
"pp": "τ : Type u_2\nσ : Type u_5\nR : Type u_6\ninst✝⁴ : CommSemiring R\ninst✝³ : Fintype σ\ninst✝² : Fintype τ\ninst✝¹ : DecidableEq σ\ninst✝ : DecidableEq τ\nn : ℕ\nμ : n.Partition\ne : σ ≃ τ\nx✝ : { x // Nat.Partition.ofSym x = μ }\n⊢ (Multiset.map (⇑(rename ⇑e) ∘ X) ↑↑x✝).prod = (Multiset.map X ↑↑((μ.ofS... | [
"τ : Type u_2\nσ : Type u_5\nR : Type u_6\ninst✝⁴ : CommSemiring R\ninst✝³ : Fintype σ\ninst✝² : Fintype τ\ninst✝¹ : DecidableEq σ\ninst✝ : DecidableEq τ\nn : ℕ\nμ : n.Partition\ne : σ ≃ τ\nx✝ : { x // Nat.Partition.ofSym x = μ }\n⊢ (Multiset.map (⇑(rename ⇑e) ∘ X) ↑↑x✝).prod =\n (Multiset.map X\n ↑↑({ to... | Nat.Partition.ofSymShapeEquiv | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Polynomial | {
"line": 186,
"column": 2
} | {
"line": 187,
"column": 49
} | {
"line": 188,
"column": 2
} | [
{
"pp": "case inr.inr\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\nB : ℝ\ni : ℕ\nh1 : p.Monic\nh2 : (map f p).Splits\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B\nhB : 0 ≤ B\nhi : i ≤ (map f p).natDegree\n⊢ (Multiset.map (fun x ↦ ‖x‖) (Multiset.map prod (powersetCard (... | [
"F : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\nB : ℝ\ni : ℕ\nh1 : p.Monic\nh2 : (map f p).Splits\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B\nhB : 0 ≤ B\nhi : i ≤ (map f p).natDegree\n⊢ ∀ r ∈ Multiset.map (fun x ↦ ‖x‖) (Multiset.map prod (powersetCard ((map f p).natDegree -... | · rw [Multiset.map_map, card_map, card_powersetCard, ← Splits.natDegree_eq_card_roots h2,
Nat.choose_symm hi, mul_comm, nsmul_eq_mul] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.NumberField.InfinitePlace.Embeddings | {
"line": 101,
"column": 2
} | {
"line": 103,
"column": 11
} | {
"line": 105,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝⁴ : Field K\ninst✝³ : NumberField K\nA : Type u_2\ninst✝² : NormedField A\ninst✝¹ : IsAlgClosed A\ninst✝ : NormedAlgebra ℚ A\nB : ℝ\nx : K\nh : ∀ (φ : K →+* A), ‖φ x‖ ≤ B\ni : ℕ\nhx : IsIntegral ℚ x\nz : A\nhz : z ∈ (map (algebraMap ℚ A) (minpoly ℚ x)).roots\n⊢ ‖z‖ ≤ B",
"ppTerm"... | [] | rw [← Multiset.mem_toFinset] at hz
obtain ⟨φ, rfl⟩ := (range_eval_eq_rootSet_minpoly K A x).symm.subset hz
exact h φ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.InfinitePlace.Embeddings | {
"line": 101,
"column": 2
} | {
"line": 103,
"column": 11
} | {
"line": 105,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝⁴ : Field K\ninst✝³ : NumberField K\nA : Type u_2\ninst✝² : NormedField A\ninst✝¹ : IsAlgClosed A\ninst✝ : NormedAlgebra ℚ A\nB : ℝ\nx : K\nh : ∀ (φ : K →+* A), ‖φ x‖ ≤ B\ni : ℕ\nhx : IsIntegral ℚ x\nz : A\nhz : z ∈ (map (algebraMap ℚ A) (minpoly ℚ x)).roots\n⊢ ‖z‖ ≤ B",
"ppTerm"... | [] | rw [← Multiset.mem_toFinset] at hz
obtain ⟨φ, rfl⟩ := (range_eval_eq_rootSet_minpoly K A x).symm.subset hz
exact h φ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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