module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Analysis.Calculus.Deriv.Basic | {
"line": 221,
"column": 2
} | {
"line": 223,
"column": 6
} | [
{
"pp": "𝕜 : Type u\ninst✝³ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 F\ninst✝ : TopologicalSpace F\nf : 𝕜 → F\nx : 𝕜\ns : Set 𝕜\nh : ¬DifferentiableWithinAt 𝕜 f s x\n⊢ derivWithin f s x = 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.... | unfold derivWithin
rw [fderivWithin_zero_of_not_differentiableWithinAt h]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.Deriv.Basic | {
"line": 221,
"column": 2
} | {
"line": 223,
"column": 6
} | [
{
"pp": "𝕜 : Type u\ninst✝³ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 F\ninst✝ : TopologicalSpace F\nf : 𝕜 → F\nx : 𝕜\ns : Set 𝕜\nh : ¬DifferentiableWithinAt 𝕜 f s x\n⊢ derivWithin f s x = 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.... | unfold derivWithin
rw [fderivWithin_zero_of_not_differentiableWithinAt h]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.FDeriv.Basic | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 31
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : Module 𝕜 E\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : ContinuousAdd E\ninst✝⁵ : ContinuousSMul 𝕜 E\nF : Type u_3\ninst✝⁴ : AddCommGroup F\ninst✝³ : Module 𝕜 F\ninst✝² : TopologicalSpace F\ninst✝¹ : Conti... | rw [tendsto_nhdsWithin_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.FDeriv.Basic | {
"line": 549,
"column": 36
} | {
"line": 553,
"column": 80
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : TopologicalSpace E\nF : Type u_3\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 F\ninst✝ : TopologicalSpace F\nf : E → F\ns t : Set E\nhf : DifferentiableOn 𝕜 f s\nhf' : Differentiab... | by
intro x hx
obtain (hx | hx) := hx
· exact (hf x hx).differentiableAt (hs.mem_nhds hx) |>.differentiableWithinAt
· exact (hf' x hx).differentiableAt (ht.mem_nhds hx) |>.differentiableWithinAt | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Analytic.CPolynomialDef | {
"line": 72,
"column": 69
} | {
"line": 73,
"column": 62
} | [
{
"pp": "𝕜 : 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\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\nx : E\nn : ℕ\nr : ℝ≥0∞\nfinite : ∀ (m : ℕ), n ≤ m → p m = 0\n... | by
rw [Finset.mem_range, not_lt] at hm; rw [finite m hm]; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Module.Multilinear.Curry | {
"line": 130,
"column": 87
} | {
"line": 130,
"column": 97
} | [
{
"pp": "case h.H\n𝕜 : Type u\nn : ℕ\nEi : Fin n.succ → Type wEi\nG : Type wG\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)\ninst✝² : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : Ei 0 →L[𝕜] ContinuousMultil... | tail_cons, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Analytic.OfScalars | {
"line": 210,
"column": 6
} | {
"line": 210,
"column": 14
} | [
{
"pp": "case neg.hs.refine_1.refine_1\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedRing E\ninst✝ : NormedAlgebra 𝕜 E\nc : ℕ → 𝕜\nr : ℝ≥0\nhr : r ≠ 0\nhc : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 ↑r)\nr' : ℝ≥0\nhr' : r' * r < 1\nhrz : ¬r' = 0\n⊢ ∀ (x : ℕ), ‖c x.succ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Analytic.OfScalars | {
"line": 259,
"column": 8
} | {
"line": 259,
"column": 16
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedRing E\ninst✝ : NormedAlgebra 𝕜 E\nc : ℕ → 𝕜\nhc : ∀ᶠ (n : ℕ) in atTop, c n ≠ 0\nhc' : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 0)\nr' : ℝ≥0\nhrz : ¬r' = 0\n⊢ ∀ (x : ℕ), c x ≠ 0 → ‖‖c x‖ * ↑r' ^ x‖ ≠ 0",
"usedC... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Analytic.OfScalars | {
"line": 259,
"column": 8
} | {
"line": 259,
"column": 16
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedRing E\ninst✝ : NormedAlgebra 𝕜 E\nc : ℕ → 𝕜\nhc : ∀ᶠ (n : ℕ) in atTop, c n ≠ 0\nhc' : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 0)\nr' : ℝ≥0\nhrz : ¬r' = 0\n⊢ ∀ (x : ℕ), c x ≠ 0 → ‖‖c x‖ * ↑r' ^ x‖ ≠ 0",
"usedC... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.OfScalars | {
"line": 259,
"column": 8
} | {
"line": 259,
"column": 16
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedRing E\ninst✝ : NormedAlgebra 𝕜 E\nc : ℕ → 𝕜\nhc : ∀ᶠ (n : ℕ) in atTop, c n ≠ 0\nhc' : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 0)\nr' : ℝ≥0\nhrz : ¬r' = 0\n⊢ ∀ (x : ℕ), c x ≠ 0 → ‖‖c x‖ * ↑r' ^ x‖ ≠ 0",
"usedC... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.OfScalars | {
"line": 268,
"column": 41
} | {
"line": 268,
"column": 49
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedRing E\ninst✝¹ : NormedAlgebra 𝕜 E\nc : ℕ → 𝕜\ninst✝ : NormOneClass E\nhc : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop atTop\nthis : (ofScalars E c).radius ≤ 0\n⊢ (ofScalars E c).radius = 0",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Analytic.OfScalars | {
"line": 268,
"column": 41
} | {
"line": 268,
"column": 49
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedRing E\ninst✝¹ : NormedAlgebra 𝕜 E\nc : ℕ → 𝕜\ninst✝ : NormOneClass E\nhc : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop atTop\nthis : (ofScalars E c).radius ≤ 0\n⊢ (ofScalars E c).radius = 0",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.OfScalars | {
"line": 268,
"column": 41
} | {
"line": 268,
"column": 49
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedRing E\ninst✝¹ : NormedAlgebra 𝕜 E\nc : ℕ → 𝕜\ninst✝ : NormOneClass E\nhc : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop atTop\nthis : (ofScalars E c).radius ≤ 0\n⊢ (ofScalars E c).radius = 0",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.OfScalars | {
"line": 286,
"column": 6
} | {
"line": 286,
"column": 14
} | [
{
"pp": "case h.hb\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedRing E\ninst✝¹ : NormedAlgebra 𝕜 E\nc : ℕ → 𝕜\ninst✝ : NormOneClass E\nhc✝ : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop atTop\nr : ℝ≥0\nhr : ↑r < (ofScalars E c).radius\nthis : 0 < r\nn : ℕ\nhc : 2 * ↑r⁻¹ ≤ ‖c... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Analytic.OfScalars | {
"line": 286,
"column": 6
} | {
"line": 286,
"column": 14
} | [
{
"pp": "case h.hb\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedRing E\ninst✝¹ : NormedAlgebra 𝕜 E\nc : ℕ → 𝕜\ninst✝ : NormOneClass E\nhc✝ : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop atTop\nr : ℝ≥0\nhr : ↑r < (ofScalars E c).radius\nthis : 0 < r\nn : ℕ\nhc : 2 * ↑r⁻¹ ≤ ‖c... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.OfScalars | {
"line": 286,
"column": 6
} | {
"line": 286,
"column": 14
} | [
{
"pp": "case h.hb\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedRing E\ninst✝¹ : NormedAlgebra 𝕜 E\nc : ℕ → 𝕜\ninst✝ : NormOneClass E\nhc✝ : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop atTop\nr : ℝ≥0\nhr : ↑r < (ofScalars E c).radius\nthis : 0 < r\nn : ℕ\nhc : 2 * ↑r⁻¹ ≤ ‖c... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.OfScalars | {
"line": 317,
"column": 4
} | {
"line": 317,
"column": 12
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedRing E\ninst✝¹ : NormedAlgebra 𝕜 E\nc : ℕ → 𝕜\ninst✝ : NormOneClass E\nr : ℝ≥0∞\nhr : r = ∞\nhc' : Tendsto (fun n ↦ ENNReal.ofReal ‖c n.succ‖ / ENNReal.ofReal ‖c n‖) atTop (𝓝 ∞)\nn : ℕ\nhn : ENNReal.ofReal ‖c n.succ‖ / ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Analytic.Composition | {
"line": 608,
"column": 85
} | {
"line": 637,
"column": 14
} | [
{
"pp": "α : Type u_6\ninst✝ : AddCommMonoid α\nm M N : ℕ\nf : (n : ℕ) × (Fin n → ℕ) → α\ng : (n : ℕ) × Composition n → α\nh : ∀ (e : (n : ℕ) × (Fin n → ℕ)) (he : e ∈ compPartialSumSource m M N), f e = g (compChangeOfVariables m M N e he)\n⊢ ∑ e ∈ compPartialSumSource m M N, f e = ∑ e ∈ compPartialSumTarget m M... | by
apply Finset.sum_bij (compChangeOfVariables m M N)
-- We should show that the correspondence we have set up is indeed a bijection
-- between the index sets of the two sums.
-- 1 - show that the image belongs to `compPartialSumTarget m N N`
· rintro ⟨k, blocks_fun⟩ H
rw [mem_compPartialSumSource_iff] at... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Analytic.Composition | {
"line": 647,
"column": 4
} | {
"line": 647,
"column": 12
} | [
{
"pp": "case h\nm n : ℕ × ℕ\nhmn : m ≤ n\na : (n : ℕ) × Composition n\nha : a ∈ (fun p ↦ compPartialSumTarget 0 p.1 p.2) m\nthis✝ : ∀ i < m.1, i < n.1\nthis : ∀ i < m.2, i < n.2\n⊢ a ∈ (fun p ↦ compPartialSumTarget 0 p.1 p.2) n",
"usedConstants": [
"Nat.instMulZeroClass",
"LinearOrderedCommMono... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Analytic.Inverse | {
"line": 77,
"column": 68
} | {
"line": 77,
"column": 80
} | [
{
"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\np : FormalMultilinearSeries 𝕜 E F\ni : E ≃L[𝕜] F\nx : E\n⊢ p.leftInv i x 0 = ContinuousMultilinearMap.uncu... | rw [leftInv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Analytic.Inverse | {
"line": 77,
"column": 68
} | {
"line": 77,
"column": 80
} | [
{
"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\np : FormalMultilinearSeries 𝕜 E F\ni : E ≃L[𝕜] F\nx : E\n⊢ p.leftInv i x 0 = ContinuousMultilinearMap.uncu... | rw [leftInv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.Inverse | {
"line": 77,
"column": 68
} | {
"line": 77,
"column": 80
} | [
{
"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\np : FormalMultilinearSeries 𝕜 E F\ni : E ≃L[𝕜] F\nx : E\n⊢ p.leftInv i x 0 = ContinuousMultilinearMap.uncu... | rw [leftInv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.Inverse | {
"line": 81,
"column": 79
} | {
"line": 81,
"column": 91
} | [
{
"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\np : FormalMultilinearSeries 𝕜 E F\ni : E ≃L[𝕜] F\nx : E\n⊢ p.leftInv i x 1 = (continuousMultilinearCurryFi... | rw [leftInv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Analytic.Inverse | {
"line": 81,
"column": 79
} | {
"line": 81,
"column": 91
} | [
{
"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\np : FormalMultilinearSeries 𝕜 E F\ni : E ≃L[𝕜] F\nx : E\n⊢ p.leftInv i x 1 = (continuousMultilinearCurryFi... | rw [leftInv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.Inverse | {
"line": 81,
"column": 79
} | {
"line": 81,
"column": 91
} | [
{
"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\np : FormalMultilinearSeries 𝕜 E F\ni : E ≃L[𝕜] F\nx : E\n⊢ p.leftInv i x 1 = (continuousMultilinearCurryFi... | rw [leftInv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.Within | {
"line": 178,
"column": 67
} | {
"line": 196,
"column": 67
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nf : E → F\ns : Set E\nx : E\n⊢ AnalyticWithinAt 𝕜 f s x ↔ ∃ g, f x = g x ∧ EqOn f... | by
classical
simp only [analyticWithinAt_iff_exists_analyticAt]
refine ⟨?_, ?_⟩
· rintro ⟨g, hf, hg⟩
rcases mem_nhdsWithin.1 hf with ⟨u, u_open, xu, hu⟩
let g' := Set.piecewise u g f
refine ⟨g', ?_, ?_, ?_⟩
· have : x ∈ u ∩ insert x s := ⟨xu, by simp⟩
simpa [g', xu, this] using hu this
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Analytic.Constructions | {
"line": 276,
"column": 40
} | {
"line": 276,
"column": 76
} | [
{
"pp": "case a.left.h\n𝕜 : Type u_2\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\nG : Type u_5\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\np : FormalMultilinear... | ContinuousMultilinearMap.opNorm_prod | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Analytic.Constructions | {
"line": 276,
"column": 40
} | {
"line": 276,
"column": 76
} | [
{
"pp": "case a.right.h\n𝕜 : Type u_2\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\nG : Type u_5\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\np : FormalMultilinea... | ContinuousMultilinearMap.opNorm_prod | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Analytic.Constructions | {
"line": 286,
"column": 38
} | {
"line": 286,
"column": 74
} | [
{
"pp": "case a\n𝕜 : Type u_2\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\nG : Type u_5\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\np : FormalMultilinearSeries ... | ContinuousMultilinearMap.opNorm_prod | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.FDeriv.Equiv | {
"line": 326,
"column": 4
} | {
"line": 326,
"column": 25
} | [
{
"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\nh : HasFDerivWithinAt f f' s x\nhf' : ∃ C, AntilipschitzWith C... | obtain ⟨C, hC⟩ := hf' | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Calculus.FDeriv.Equiv | {
"line": 334,
"column": 7
} | {
"line": 334,
"column": 24
} | [
{
"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\nh : HasFDerivWithinAt f f' s x\nhf' : ∃ C, ∀ (z : E), ‖z‖ ≤ C ... | le_principal_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 607,
"column": 50
} | {
"line": 607,
"column": 58
} | [
{
"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\ny x✝ : E\na✝ : x✝ ∈ {x}ᶜ\n⊢ x✝ ∈ insert x s ↔ x✝ ∈ s",
"usedConstants": ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 790,
"column": 37
} | {
"line": 794,
"column": 71
} | [
{
"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\nf : E → F\nn : ℕ∞ω\np : E → FormalMultilinearSeries 𝕜 E F\nm : ℕ\nh : ↑m ≤ n\nhf : HasFTaylorSeriesUpTo n f p\n... | by
refine (hf.tsupport_mono zero_le h).trans_eq ?_
rw [← funext hf.zero_eq]
refine tsupport_comp_eq (g := ContinuousMultilinearMap.curry0) (fun {x} ↦ ?_) _ |>.symm
exact (continuousMultilinearCurryFin0 _ _ _).map_eq_zero_iff (x := x) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 988,
"column": 25
} | {
"line": 988,
"column": 43
} | [
{
"pp": "case succ\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\ns : Set 𝕜\nn : ℕ\nih : ∀ (a : 𝕜), iteratedFDerivWithin 𝕜 n (fun x ↦ f (-x)) s a = (-1) ^ n • iteratedFDerivWithin 𝕜 n f (-s) (-a)\na : 𝕜\nih' : iteratedFDe... | ← pow_succ (-1) n, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 238,
"column": 4
} | {
"line": 238,
"column": 22
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type v\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0∞\nf : E → F\nx : E\ns : Set E\ninst✝ : CompleteSpace F\nh : HasFPow... | · simpa using hz.1 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 289,
"column": 4
} | {
"line": 289,
"column": 39
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type v\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E → F\ns : Set E\ninst✝ : CompleteSpace F\nn : WithTop ℕ∞\nh : AnalyticOnNhd 𝕜 f s\nm : ℕ\n... | apply HasFDerivAt.hasFDerivWithinAt | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 320,
"column": 2
} | {
"line": 320,
"column": 37
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0∞\nf : E → F\nx : E\ns : Set E\nh : HasFPowerSeriesWithinOnBall f p s... | let F' := UniformSpace.Completion F | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.Calculus.Deriv.Pow | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 57
} | [
{
"pp": "case neg\n𝕜 : Type u_1\n𝔸 : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedRing 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸\nf : 𝕜 → 𝔸\nx : 𝕜\ns : Set 𝕜\nh : DifferentiableWithinAt 𝕜 f s x\nn : ℕ\nhsx : ¬UniqueDiffWithinAt 𝕜 s x\n⊢ derivWithin (fun x ↦ f x ^ n) s x = ∑ i ∈ Finset.range n, f ... | · simp [derivWithin_zero_of_not_uniqueDiffWithinAt hsx] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.Deriv.Pow | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 57
} | [
{
"pp": "case neg\n𝕜 : Type u_1\n𝔸 : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedCommRing 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸\nf : 𝕜 → 𝔸\nx : 𝕜\ns : Set 𝕜\nh : DifferentiableWithinAt 𝕜 f s x\nn : ℕ\nhsx : ¬UniqueDiffWithinAt 𝕜 s x\n⊢ derivWithin (f ^ n) s x = ↑n * f x ^ (n - 1) * derivWith... | · simp [derivWithin_zero_of_not_uniqueDiffWithinAt hsx] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 57
} | [
{
"pp": "case neg\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf g : 𝕜 → F\nx : 𝕜\ns : Set 𝕜\nhf : DifferentiableWithinAt 𝕜 f s x\nhg : DifferentiableWithinAt 𝕜 g s x\nhsx : ¬UniqueDiffWithinAt 𝕜 s x\n⊢ derivWithin (fun y ↦ f y + g... | · simp [derivWithin_zero_of_not_uniqueDiffWithinAt hsx] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 230,
"column": 2
} | {
"line": 230,
"column": 57
} | [
{
"pp": "case neg\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nx : 𝕜\ns : Set 𝕜\nι : Type u_1\nu : Finset ι\nA : ι → 𝕜 → F\nh : ∀ i ∈ u, DifferentiableWithinAt 𝕜 (A i) s x\nhsx : ¬UniqueDiffWithinAt 𝕜 s x\n⊢ derivWithin (fun y ↦ ∑ i... | · simp [derivWithin_zero_of_not_uniqueDiffWithinAt hsx] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 236,
"column": 2
} | {
"line": 236,
"column": 57
} | [
{
"pp": "case neg\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nx : 𝕜\ns : Set 𝕜\nι : Type u_1\nu : Finset ι\nA : ι → 𝕜 → F\nh : ∀ i ∈ u, DifferentiableWithinAt 𝕜 (A i) s x\nhsx : ¬UniqueDiffWithinAt 𝕜 s x\n⊢ derivWithin (∑ i ∈ u, A ... | · simp [derivWithin_zero_of_not_uniqueDiffWithinAt hsx] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 275,
"column": 2
} | {
"line": 275,
"column": 57
} | [
{
"pp": "case neg\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\nx : 𝕜\ns : Set 𝕜\nhsx : ¬UniqueDiffWithinAt 𝕜 s x\n⊢ derivWithin (-f) s x = -derivWithin f s x",
"usedConstants": [
"NegZeroClass.toNeg",
"Pi.i... | · simp [derivWithin_zero_of_not_uniqueDiffWithinAt hsx] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Analytic.Uniqueness | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 48
} | [
{
"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\np : FormalMultilinearSeries 𝕜 E F\nx : E\nn k : ℕ\nhk : ∀ m < k, ∀ (y : E), ((p m) fun x ↦ y) = 0\npsum_eq ... | exact h.continuousMultilinearMap_apply_eq_zero | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Analytic.Uniqueness | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 37
} | [
{
"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\nU : Set E\nhf : AnalyticOnNhd 𝕜 f U\nhU : IsPreconnected U\nz₀ : E\nh₀ : z₀ ∈ U\nhfz₀ : f =ᶠ[𝓝 ... | let F' := UniformSpace.Completion F | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 57
} | [
{
"pp": "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\ns : Set 𝕜\nB : E →L[𝕜]... | · simp [derivWithin_zero_of_not_uniqueDiffWithinAt hsx] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 57
} | [
{
"pp": "case neg\n𝕜 : Type u\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nf : 𝕜 → F\nx : 𝕜\ns : Set 𝕜\n𝕜' : Type u_2\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : NormedAlgebra 𝕜 𝕜'\ninst✝² : Module 𝕜' F\ninst✝¹ : IsBoundedSMul 𝕜' F\ninst✝ : IsScalar... | · simp [derivWithin_zero_of_not_uniqueDiffWithinAt hsx] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 162,
"column": 2
} | {
"line": 162,
"column": 57
} | [
{
"pp": "case neg\n𝕜 : Type u\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nx : 𝕜\ns : Set 𝕜\n𝕜' : Type u_2\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : NormedAlgebra 𝕜 𝕜'\ninst✝² : Module 𝕜' F\ninst✝¹ : IsBoundedSMul 𝕜' F\ninst✝ : IsScalarTower 𝕜 𝕜'... | · simp [derivWithin_zero_of_not_uniqueDiffWithinAt hsx] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 57
} | [
{
"pp": "case neg\n𝕜 : Type u\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nf : 𝕜 → F\nx : 𝕜\ns : Set 𝕜\nR : Type u_2\ninst✝³ : Monoid R\ninst✝² : DistribMulAction R F\ninst✝¹ : SMulCommClass 𝕜 R F\ninst✝ : ContinuousConstSMul R F\nc : R\nhf : D... | · simp [derivWithin_zero_of_not_uniqueDiffWithinAt hsx] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.AffineSpace.Slope | {
"line": 171,
"column": 2
} | {
"line": 171,
"column": 47
} | [
{
"pp": "k : Type u_1\nE : Type u_2\ninst✝⁷ : Field k\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module k E\ninst✝⁴ : LinearOrder k\ninst✝³ : IsStrictOrderedRing k\ninst✝² : PartialOrder E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : PosSMulMono k E\nf : k → E\nx y : k\nhxy : x ≤ y\n⊢ slope f x y < 0 ↔ f y < f x",
"used... | simpa using slope_pos_iff_of_le (f := -f) hxy | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.AffineSpace.Slope | {
"line": 171,
"column": 2
} | {
"line": 171,
"column": 47
} | [
{
"pp": "k : Type u_1\nE : Type u_2\ninst✝⁷ : Field k\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module k E\ninst✝⁴ : LinearOrder k\ninst✝³ : IsStrictOrderedRing k\ninst✝² : PartialOrder E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : PosSMulMono k E\nf : k → E\nx y : k\nhxy : x ≤ y\n⊢ slope f x y < 0 ↔ f y < f x",
"used... | simpa using slope_pos_iff_of_le (f := -f) hxy | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.Slope | {
"line": 171,
"column": 2
} | {
"line": 171,
"column": 47
} | [
{
"pp": "k : Type u_1\nE : Type u_2\ninst✝⁷ : Field k\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module k E\ninst✝⁴ : LinearOrder k\ninst✝³ : IsStrictOrderedRing k\ninst✝² : PartialOrder E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : PosSMulMono k E\nf : k → E\nx y : k\nhxy : x ≤ y\n⊢ slope f x y < 0 ↔ f y < f x",
"used... | simpa using slope_pos_iff_of_le (f := -f) hxy | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 214,
"column": 2
} | {
"line": 214,
"column": 57
} | [
{
"pp": "case neg\n𝕜 : Type u\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nx : 𝕜\ns : Set 𝕜\n𝕝 : Type u_3\ninst✝³ : DivisionSemiring 𝕝\ninst✝² : Module 𝕝 F\ninst✝¹ : SMulCommClass 𝕜 𝕝 F\ninst✝ : ContinuousConstSMul 𝕝 F\nc : 𝕝\nf : 𝕜 → F\n... | · simp [derivWithin_zero_of_not_uniqueDiffWithinAt hsx] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 283,
"column": 2
} | {
"line": 283,
"column": 57
} | [
{
"pp": "case neg\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\ns : Set 𝕜\n𝔸 : Type u_3\ninst✝¹ : NormedRing 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸\nc d : 𝕜 → 𝔸\nhc : DifferentiableWithinAt 𝕜 c s x\nhd : DifferentiableWithinAt 𝕜 d s x\nhsx : ¬UniqueDiffWithinAt 𝕜 s x\n⊢ derivWithin (fun y ↦ c y * d... | · simp [derivWithin_zero_of_not_uniqueDiffWithinAt hsx] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 322,
"column": 2
} | {
"line": 322,
"column": 57
} | [
{
"pp": "case neg\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\ns : Set 𝕜\n𝔸 : Type u_3\ninst✝¹ : NormedRing 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸\nc : 𝕜 → 𝔸\nhc : DifferentiableWithinAt 𝕜 c s x\nd : 𝔸\nhsx : ¬UniqueDiffWithinAt 𝕜 s x\n⊢ derivWithin (fun y ↦ c y * d) s x = derivWithin c s x * d",
... | · simp [derivWithin_zero_of_not_uniqueDiffWithinAt hsx] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.Deriv.Slope | {
"line": 106,
"column": 4
} | {
"line": 106,
"column": 58
} | [
{
"pp": "case neg\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\ns t : Set 𝕜\nh : s ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (s ∩ t)\nx : 𝕜\nH : UniqueDiffWithinAt 𝕜 s x\nH' : ¬DifferentiableWithinAt 𝕜... | rw [derivWithin_zero_of_not_differentiableWithinAt H'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 374,
"column": 2
} | {
"line": 374,
"column": 57
} | [
{
"pp": "case neg\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\ns : Set 𝕜\n𝔸 : Type u_3\ninst✝¹ : NormedRing 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸\nd : 𝕜 → 𝔸\nc : 𝔸\nhd : DifferentiableWithinAt 𝕜 d s x\nhsx : ¬UniqueDiffWithinAt 𝕜 s x\n⊢ derivWithin (fun y ↦ c * d y) s x = c * derivWithin d s x",
... | · simp [derivWithin_zero_of_not_uniqueDiffWithinAt hsx] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 471,
"column": 2
} | {
"line": 471,
"column": 57
} | [
{
"pp": "case neg\n𝕜 : Type u\ninst✝³ : NontriviallyNormedField 𝕜\nx : 𝕜\ns : Set 𝕜\nι : Type u_2\ninst✝² : DecidableEq ι\n𝔸' : Type u_3\ninst✝¹ : NormedCommRing 𝔸'\ninst✝ : NormedAlgebra 𝕜 𝔸'\nu : Finset ι\nf : ι → 𝕜 → 𝔸'\nhf : ∀ i ∈ u, DifferentiableWithinAt 𝕜 (f i) s x\nhsx : ¬UniqueDiffWithinAt �... | · simp [derivWithin_zero_of_not_uniqueDiffWithinAt hsx] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 631,
"column": 2
} | {
"line": 631,
"column": 57
} | [
{
"pp": "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\nx : 𝕜\ns : Set 𝕜\nG : Type u_2\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nc : 𝕜 → F →... | · simp [derivWithin_zero_of_not_uniqueDiffWithinAt hsx] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 655,
"column": 2
} | {
"line": 655,
"column": 57
} | [
{
"pp": "case neg\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nx : 𝕜\ns : Set 𝕜\nG : Type u_2\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nc : 𝕜 → F →L[𝕜] G\nu : 𝕜 → F\nhc : DifferentiableWithinAt 𝕜 c s x\nhu : Differ... | · simp [derivWithin_zero_of_not_uniqueDiffWithinAt hsx] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.Deriv.Inv | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 57
} | [
{
"pp": "case neg\n𝕜 : Type u\ninst✝ : NontriviallyNormedField 𝕜\nx : 𝕜\ns : Set 𝕜\nc : 𝕜 → 𝕜\nhc : DifferentiableWithinAt 𝕜 c s x\nhx : c x ≠ 0\nhsx : ¬UniqueDiffWithinAt 𝕜 s x\n⊢ derivWithin (fun x ↦ (c x)⁻¹) s x = -derivWithin c s x / c x ^ 2",
"usedConstants": [
"NormedCommRing.toNormedRin... | · simp [derivWithin_zero_of_not_uniqueDiffWithinAt hsx] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.Deriv.Inv | {
"line": 221,
"column": 2
} | {
"line": 221,
"column": 57
} | [
{
"pp": "case neg\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\ns : Set 𝕜\n𝕜' : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝕜'\nc d : 𝕜 → 𝕜'\nhc : DifferentiableWithinAt 𝕜 c s x\nhd : DifferentiableWithinAt 𝕜 d s x\nhx : d x ≠ 0\nhsx : ¬UniqueDiffWithinAt 𝕜 s x\... | · simp [derivWithin_zero_of_not_uniqueDiffWithinAt hsx] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.Deriv.Comp | {
"line": 141,
"column": 2
} | {
"line": 141,
"column": 57
} | [
{
"pp": "case neg\n𝕜 : Type u\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nx : 𝕜\ns : Set 𝕜\n𝕜' : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜'\ninst✝² : NormedAlgebra 𝕜 𝕜'\ninst✝¹ : NormedSpace 𝕜' F\ninst✝ : IsScalarTower 𝕜 𝕜' F\nt' : Set... | · simp [derivWithin_zero_of_not_uniqueDiffWithinAt hsx] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 109,
"column": 6
} | {
"line": 109,
"column": 31
} | [
{
"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\n⊢ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z ≠ 0",
"usedConstants": [
"Eq.mpr",
... | eventually_nhdsWithin_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 160,
"column": 8
} | {
"line": 160,
"column": 33
} | [
{
"pp": "case h\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nz₀ : 𝕜\nm n : ℤ\nhm : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = (z - z₀) ^ m • g z\nhn : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z :... | eventually_nhdsWithin_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.Deriv.Comp | {
"line": 305,
"column": 2
} | {
"line": 305,
"column": 57
} | [
{
"pp": "case neg\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\ns : Set 𝕜\n𝕜' : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝕜'\ns' : Set 𝕜'\nh : 𝕜 → 𝕜'\nh₂ : 𝕜' → 𝕜'\nhh₂ : DifferentiableWithinAt 𝕜' h₂ s' (h x)\nhh : DifferentiableWithinAt 𝕜 h s x\nhs : MapsTo... | · simp [derivWithin_zero_of_not_uniqueDiffWithinAt hsx] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.Deriv.Comp | {
"line": 320,
"column": 12
} | {
"line": 320,
"column": 37
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\n𝕜' : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝕜'\nh : 𝕜 → 𝕜'\nh₂ : 𝕜' → 𝕜'\nhh : DifferentiableAt 𝕜 h x\nhh₂ : DifferentiableAt 𝕜' h₂ (h x)\n⊢ deriv (h₂ ∘ h) x = deriv h₂ (h x) * deriv h x",
"usedConst... | exact deriv_comp x hh₂ hh | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.Deriv.Comp | {
"line": 414,
"column": 2
} | {
"line": 414,
"column": 57
} | [
{
"pp": "case neg\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → F\nx : 𝕜\ns : Set 𝕜\nl : F → E\nt : Set F\nhl : DifferentiableWithinAt 𝕜 l t (f x)\nhf : Dif... | · simp [derivWithin_zero_of_not_uniqueDiffWithinAt hsx] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 226,
"column": 2
} | {
"line": 226,
"column": 10
} | [
{
"pp": "case h\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nU : Set 𝕜\nhf : AnalyticOnNhd 𝕜 f U\nhU : IsPreconnected U\nx : 𝕜\nhx : x ∈ U\nhx2 : (U \\ {x | ¬f x = 0})ᶜ ∉ 𝓝[≠] x\nnh : ∀ᶠ (x : 𝕜) in 𝓝[≠] x, ¬(fun z ↦... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.MeanValue | {
"line": 86,
"column": 4
} | {
"line": 86,
"column": 29
} | [
{
"pp": "f f' : ℝ → ℝ\na b : ℝ\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ x ∈ Ico a b, ∀ (r : ℝ), f' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r\nB B' : ℝ → ℝ\nha : f a ≤ B a\nhB : ContinuousOn B (Icc a b)\nhB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x\nbound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x\... | simp only [s, inter_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.MeanValue | {
"line": 538,
"column": 40
} | {
"line": 538,
"column": 53
} | [
{
"pp": "E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\n𝕜 : Type u_3\nG : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : IsRCLikeNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : E → G\nC : ℝ\ns : Set E\nx y : E\nf' : E → E ... | congr 1; abel | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.MeanValue | {
"line": 538,
"column": 40
} | {
"line": 538,
"column": 53
} | [
{
"pp": "E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\n𝕜 : Type u_3\nG : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : IsRCLikeNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : E → G\nC : ℝ\ns : Set E\nx y : E\nf' : E → E ... | congr 1; abel | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.MeanValue | {
"line": 629,
"column": 22
} | {
"line": 629,
"column": 35
} | [
{
"pp": "E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\n𝕜 : Type u_3\nG : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : IsRCLikeNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → G\ns : Set E\nhs : IsOpen[PseudoMetricS... | Pi.zero_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.ContDiff.Comp | {
"line": 602,
"column": 4
} | {
"line": 602,
"column": 53
} | [
{
"pp": "case refine_1\n𝕜 : 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\nn : ℕ∞ω\nf... | have := mem_of_mem_nhdsWithin (mem_insert _ _) hv | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.ContDiff.Basic | {
"line": 637,
"column": 2
} | {
"line": 637,
"column": 31
} | [
{
"pp": "𝕜 : 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\nf : E → F\nn : ℕ∞ω\nι : Type u_5\ns : ι → Set E\nhf : ∀ (i : ι), ContDiffOn 𝕜 n f (s i)\nhs : ∀ (i : ι), Is... | rw [← contDiffOn_univ, ← hs'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.Deriv.Inverse | {
"line": 158,
"column": 53
} | {
"line": 160,
"column": 28
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\nx : 𝕜\nt : Set F\nht : ¬AccPt (f x) (𝓟 t)\nh : ∃ᶠ (y : 𝕜) in 𝓝[≠] x, f y ∈ t\n⊢ deriv f x = 0",
"usedConstants": [
"derivWithin_zero_of_frequently_mem",
... | by
rw [← derivWithin_univ, derivWithin_zero_of_frequently_mem t ht]
rwa [← compl_eq_univ_diff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 212,
"column": 2
} | {
"line": 213,
"column": 11
} | [
{
"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\nh : ContDiffWithinAt 𝕜 n f s x\nthis : ContDiffWithinAt 𝕜 0 f s x\n⊢ Con... | simp only [ContDiffWithinAt, nonpos_iff_eq_zero, Nat.cast_eq_zero, forall_eq, CharP.cast_eq_zero]
at this | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.IteratedDeriv.Defs | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 68
} | [
{
"pp": "case h\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nn : ℕ\nf : 𝕜 → F\nx : 𝕜\n⊢ iteratedDerivWithin n f univ x = iteratedDeriv n f x",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"Norme... | rw [iteratedDerivWithin, iteratedDeriv, iteratedFDerivWithin_univ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 152,
"column": 61
} | {
"line": 160,
"column": 41
} | [
{
"pp": "n : ℕ\n⊢ Injective (embSigma n)",
"usedConstants": [
"Nat.sub_one_lt_of_lt",
"Eq.mpr",
"OrderedFinpartition.mk.injEq",
"Fin.mk.injEq",
"ChainCompletePartialOrder.instOfCompleteLattice",
"StrictMono",
"and_true",
"CompleteBooleanAlgebra.toCompleteDistr... | by
rintro ⟨plength, psize, -, pemb, -, -, -, -⟩ ⟨qlength, qsize, -, qemb, -, -, -, -⟩
intro hpq
simp_all only [Sigma.mk.inj_iff, true_and, mk.injEq, Fin.mk.injEq, embSigma]
have : plength = qlength := hpq.1
subst this
simp_all only [Sigma.mk.inj_iff, heq_eq_eq, true_and, and_true]
ext i
exact mk.inj_iff... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 553,
"column": 6
} | {
"line": 569,
"column": 13
} | [
{
"pp": "case h.e'_3\n𝕜 : 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\ns : Set E\nt : Set F\nq... | rcases eq_or_ne i (c.index 0) with rfl | hi
-- We do not yet replace `omega` with `lia` here, as it is measurably slower.
· simp only [↓reduceDIte, update_self, succ_mk, cast_mk, val_pred]
have A := c.one_lt_partSize_index_zero hc
rw [Nat.sub_add_cancel]
· congr; omega
· rw [... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 553,
"column": 6
} | {
"line": 569,
"column": 13
} | [
{
"pp": "case h.e'_3\n𝕜 : 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\ns : Set E\nt : Set F\nq... | rcases eq_or_ne i (c.index 0) with rfl | hi
-- We do not yet replace `omega` with `lia` here, as it is measurably slower.
· simp only [↓reduceDIte, update_self, succ_mk, cast_mk, val_pred]
have A := c.one_lt_partSize_index_zero hc
rw [Nat.sub_add_cancel]
· congr; omega
· rw [... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Algebra.Exponential | {
"line": 567,
"column": 4
} | {
"line": 570,
"column": 78
} | [
{
"pp": "case insert\n𝔸 : Type u_1\ninst✝² : NormedRing 𝔸\ninst✝¹ : NormedAlgebra ℚ 𝔸\ninst✝ : CompleteSpace 𝔸\nι : Type u_3\nf : ι → 𝔸\na : ι\ns : Finset ι\nha : a ∉ s\nih : ∀ (h : (↑s).Pairwise (Commute on f)), exp (∑ i ∈ s, f i) = s.noncommProd (fun i ↦ exp (f i)) ⋯\nh : (↑(insert a s)).Pairwise (Commut... | rw [Finset.noncommProd_insert_of_notMem _ _ _ _ ha, Finset.sum_insert ha, exp_add_of_commute,
ih (h.mono <| Finset.subset_insert _ _)]
refine Commute.sum_right _ _ _ fun i hi => ?_
exact h.of_refl (Finset.mem_insert_self _ _) (Finset.mem_insert_of_mem hi) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Algebra.Exponential | {
"line": 567,
"column": 4
} | {
"line": 570,
"column": 78
} | [
{
"pp": "case insert\n𝔸 : Type u_1\ninst✝² : NormedRing 𝔸\ninst✝¹ : NormedAlgebra ℚ 𝔸\ninst✝ : CompleteSpace 𝔸\nι : Type u_3\nf : ι → 𝔸\na : ι\ns : Finset ι\nha : a ∉ s\nih : ∀ (h : (↑s).Pairwise (Commute on f)), exp (∑ i ∈ s, f i) = s.noncommProd (fun i ↦ exp (f i)) ⋯\nh : (↑(insert a s)).Pairwise (Commut... | rw [Finset.noncommProd_insert_of_notMem _ _ _ _ ha, Finset.sum_insert ha, exp_add_of_commute,
ih (h.mono <| Finset.subset_insert _ _)]
refine Commute.sum_right _ _ _ fun i hi => ?_
exact h.of_refl (Finset.mem_insert_self _ _) (Finset.mem_insert_of_mem hi) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Algebra.Exponential | {
"line": 564,
"column": 2
} | {
"line": 570,
"column": 78
} | [
{
"pp": "𝔸 : Type u_1\ninst✝² : NormedRing 𝔸\ninst✝¹ : NormedAlgebra ℚ 𝔸\ninst✝ : CompleteSpace 𝔸\nι : Type u_3\ns : Finset ι\nf : ι → 𝔸\nh : (↑s).Pairwise (Commute on f)\n⊢ exp (∑ i ∈ s, f i) = s.noncommProd (fun i ↦ exp (f i)) ⋯",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Normed... | induction s using Finset.induction_on with
| empty => simp
| insert a s ha ih =>
rw [Finset.noncommProd_insert_of_notMem _ _ _ _ ha, Finset.sum_insert ha, exp_add_of_commute,
ih (h.mono <| Finset.subset_insert _ _)]
refine Commute.sum_right _ _ _ fun i hi => ?_
exact h.of_refl (Finset.mem_insert_s... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Analysis.Normed.Algebra.Exponential | {
"line": 564,
"column": 2
} | {
"line": 570,
"column": 78
} | [
{
"pp": "𝔸 : Type u_1\ninst✝² : NormedRing 𝔸\ninst✝¹ : NormedAlgebra ℚ 𝔸\ninst✝ : CompleteSpace 𝔸\nι : Type u_3\ns : Finset ι\nf : ι → 𝔸\nh : (↑s).Pairwise (Commute on f)\n⊢ exp (∑ i ∈ s, f i) = s.noncommProd (fun i ↦ exp (f i)) ⋯",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Normed... | induction s using Finset.induction_on with
| empty => simp
| insert a s ha ih =>
rw [Finset.noncommProd_insert_of_notMem _ _ _ _ ha, Finset.sum_insert ha, exp_add_of_commute,
ih (h.mono <| Finset.subset_insert _ _)]
refine Commute.sum_right _ _ _ fun i hi => ?_
exact h.of_refl (Finset.mem_insert_s... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Algebra.Exponential | {
"line": 564,
"column": 2
} | {
"line": 570,
"column": 78
} | [
{
"pp": "𝔸 : Type u_1\ninst✝² : NormedRing 𝔸\ninst✝¹ : NormedAlgebra ℚ 𝔸\ninst✝ : CompleteSpace 𝔸\nι : Type u_3\ns : Finset ι\nf : ι → 𝔸\nh : (↑s).Pairwise (Commute on f)\n⊢ exp (∑ i ∈ s, f i) = s.noncommProd (fun i ↦ exp (f i)) ⋯",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Normed... | induction s using Finset.induction_on with
| empty => simp
| insert a s ha ih =>
rw [Finset.noncommProd_insert_of_notMem _ _ _ _ ha, Finset.sum_insert ha, exp_add_of_commute,
ih (h.mono <| Finset.subset_insert _ _)]
refine Commute.sum_right _ _ _ fun i hi => ?_
exact h.of_refl (Finset.mem_insert_s... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.Deriv.MeanValue | {
"line": 178,
"column": 4
} | {
"line": 178,
"column": 43
} | [
{
"pp": "f : ℝ → ℝ\na : ℝ\nhf : Tendsto (derivWithin f (Ioi a)) (𝓝[>] a) atTop\nhcont_at_a : ContinuousWithinAt f (Ici a) a\nhdiff : DifferentiableWithinAt ℝ f (Ioi a) a\n⊢ False",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"Set.Ioi",
"NormedSpace.toIsBoundedS... | replace hdiff := hdiff.hasDerivWithinAt | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.Analysis.Calculus.Deriv.MeanValue | {
"line": 291,
"column": 2
} | {
"line": 292,
"column": 63
} | [
{
"pp": "D : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : DifferentiableOn ℝ f (interior D)\nC : ℝ\nhf'_gt : ∀ x ∈ interior D, C < deriv f x\nx : ℝ\nhx : x ∈ D\ny : ℝ\nhy : y ∈ D\nhxy : x < y\nhxyD : Icc x y ⊆ D\nhxyD' : Ioo x y ⊆ interior D\n⊢ C * (y - x) < f y - f x",
"usedConstants": [... | obtain ⟨a, a_mem, ha⟩ : ∃ a ∈ Ioo x y, deriv f a = (f y - f x) / (y - x) :=
exists_deriv_eq_slope f hxy (hf.mono hxyD) (hf'.mono hxyD') | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Calculus.Deriv.MeanValue | {
"line": 338,
"column": 23
} | {
"line": 338,
"column": 37
} | [
{
"pp": "D : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : DifferentiableOn ℝ f (interior D)\nC : ℝ\nlt_hf' : ∀ x ∈ interior D, deriv f x < C\nx✝ : ℝ\nhx✝ : x✝ ∈ D\ny : ℝ\nhy : y ∈ D\nhxy : x✝ < y\nx : ℝ\nhx : x ∈ interior D\n⊢ -C < -deriv f x",
"usedConstants": [
"IsRightCancelAdd.a... | neg_lt_neg_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 273,
"column": 52
} | {
"line": 273,
"column": 60
} | [
{
"pp": "case funProp.discharger\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nn✝ : ℕ\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : UniqueDiffOn 𝕜 s\n𝔸 : Type u_5\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕜 𝔸\ninst✝² : Module 𝔸 F\n... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 273,
"column": 52
} | {
"line": 273,
"column": 60
} | [
{
"pp": "case funProp.discharger\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nn✝ : ℕ\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : UniqueDiffOn 𝕜 s\n𝔸 : Type u_5\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕜 𝔸\ninst✝² : Module 𝔸 F\n... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 273,
"column": 52
} | {
"line": 273,
"column": 60
} | [
{
"pp": "case funProp.discharger\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nn✝ : ℕ\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : UniqueDiffOn 𝕜 s\n𝔸 : Type u_5\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕜 𝔸\ninst✝² : Module 𝔸 F\n... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 273,
"column": 52
} | {
"line": 273,
"column": 60
} | [
{
"pp": "case funProp.discharger\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nn✝ : ℕ\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : UniqueDiffOn 𝕜 s\n𝔸 : Type u_5\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕜 𝔸\ninst✝² : Module 𝔸 F\n... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 273,
"column": 52
} | {
"line": 273,
"column": 60
} | [
{
"pp": "case funProp.discharger\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nn✝ : ℕ\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : UniqueDiffOn 𝕜 s\n𝔸 : Type u_5\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕜 𝔸\ninst✝² : Module 𝔸 F\n... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 273,
"column": 52
} | {
"line": 273,
"column": 60
} | [
{
"pp": "case funProp.discharger\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nn✝ : ℕ\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : UniqueDiffOn 𝕜 s\n𝔸 : Type u_5\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕜 𝔸\ninst✝² : Module 𝔸 F\n... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 273,
"column": 52
} | {
"line": 273,
"column": 60
} | [
{
"pp": "case funProp.discharger\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nn✝ : ℕ\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : UniqueDiffOn 𝕜 s\n𝔸 : Type u_5\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕜 𝔸\ninst✝² : Module 𝔸 F\n... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 273,
"column": 52
} | {
"line": 273,
"column": 60
} | [
{
"pp": "case funProp.discharger\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nn✝ : ℕ\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : UniqueDiffOn 𝕜 s\n𝔸 : Type u_5\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕜 𝔸\ninst✝² : Module 𝔸 F\n... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 273,
"column": 52
} | {
"line": 273,
"column": 60
} | [
{
"pp": "case funProp.discharger\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nn✝ : ℕ\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : UniqueDiffOn 𝕜 s\n𝔸 : Type u_5\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕜 𝔸\ninst✝² : Module 𝔸 F\n... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 273,
"column": 52
} | {
"line": 273,
"column": 60
} | [
{
"pp": "case funProp.discharger\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nn✝ : ℕ\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : UniqueDiffOn 𝕜 s\n𝔸 : Type u_5\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕜 𝔸\ninst✝² : Module 𝔸 F\n... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 273,
"column": 52
} | {
"line": 273,
"column": 60
} | [
{
"pp": "case funProp.discharger\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nn✝ : ℕ\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : UniqueDiffOn 𝕜 s\n𝔸 : Type u_5\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕜 𝔸\ninst✝² : Module 𝔸 F\n... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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