module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 256,
"column": 34
} | {
"line": 274,
"column": 99
} | {
"line": 276,
"column": 0
} | [
{
"pp": "n : ℕ\nc : OrderedFinpartition (n + 1)\nhc : range (c.emb 0) ≠ {0}\n⊢ 1 < c.partSize (c.index 0)",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Nat.sub_one_lt_of_lt",
"Iff.mpr",
"Eq.mpr",
"Fintype.card_ofFinset",
"instNeZeroNatHAdd_1",
"Set.fi... | [] | by
have : c.partSize (c.index 0) = Nat.card (range (c.emb (c.index 0))) := by
rw [Nat.card_range_of_injective (c.emb_strictMono _).injective]; simp
rw [this]
rcases eq_or_ne (c.index 0) 0 with h | h
· rw [← h] at hc
have : {0} ⊂ range (c.emb (c.index 0)) := by
apply ssubset_of_subset_of_ne ?_ hc.s... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 994,
"column": 58
} | {
"line": 995,
"column": 83
} | {
"line": 997,
"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\nf : E → F\nx : E\nn : ℕ∞ω\n⊢ ContDiffWithinAt 𝕜 n f {x}ᶜ x ↔ ContDiffAt 𝕜 n f x",
"ppTerm": "?m.39",
"... | [] | by
rw [compl_eq_univ_sdiff, contDiffWithinAt_sdiff_singleton, contDiffWithinAt_univ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Algebra.Exponential | {
"line": 325,
"column": 2
} | {
"line": 326,
"column": 70
} | {
"line": 328,
"column": 0
} | [
{
"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\n⊢ ContinuousOn exp (Metric.eball 0 (expSeries 𝕂 𝔸).radius)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
... | [] | have := FormalMultilinearSeries.continuousOn (p := expSeries 𝕂 𝔸)
simpa only [exp_eq_expSeries_sum 𝕂, expSeries_sum_eq_rat] using this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Algebra.Exponential | {
"line": 325,
"column": 2
} | {
"line": 326,
"column": 70
} | {
"line": 328,
"column": 0
} | [
{
"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\n⊢ ContinuousOn exp (Metric.eball 0 (expSeries 𝕂 𝔸).radius)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
... | [] | have := FormalMultilinearSeries.continuousOn (p := expSeries 𝕂 𝔸)
simpa only [exp_eq_expSeries_sum 𝕂, expSeries_sum_eq_rat] using this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Algebra.Exponential | {
"line": 332,
"column": 4
} | {
"line": 333,
"column": 74
} | {
"line": 335,
"column": 0
} | [
{
"pp": "case neg\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",
"pp... | [] | have h := pos_iff_ne_zero.mpr h
exact (hasFPowerSeriesOnBall_exp_of_radius_pos h).analyticAt_of_mem hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Algebra.Exponential | {
"line": 332,
"column": 4
} | {
"line": 333,
"column": 74
} | {
"line": 335,
"column": 0
} | [
{
"pp": "case neg\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",
"pp... | [] | have h := pos_iff_ne_zero.mpr h
exact (hasFPowerSeriesOnBall_exp_of_radius_pos h).analyticAt_of_mem hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 222,
"column": 2
} | {
"line": 223,
"column": 72
} | {
"line": 225,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nn : ℕ\ns : Set 𝕜\nf : 𝕜 → F\nc : 𝕜\n⊢ iteratedDerivWithin n (fun z ↦ f (c + z)) s = fun x ↦ iteratedDerivWithin n f (c +ᵥ s) (c + x)",
"ppTerm": "?m.68",
"assigned": tru... | [] | ext x
simp [iteratedDerivWithin, ← iteratedFDerivWithin_comp_add_left n c x] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 222,
"column": 2
} | {
"line": 223,
"column": 72
} | {
"line": 225,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nn : ℕ\ns : Set 𝕜\nf : 𝕜 → F\nc : 𝕜\n⊢ iteratedDerivWithin n (fun z ↦ f (c + z)) s = fun x ↦ iteratedDerivWithin n f (c +ᵥ s) (c + x)",
"ppTerm": "?m.68",
"assigned": tru... | [] | ext x
simp [iteratedDerivWithin, ← iteratedFDerivWithin_comp_add_left n c x] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 228,
"column": 58
} | {
"line": 230,
"column": 73
} | {
"line": 232,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nn : ℕ\ns : Set 𝕜\nf : 𝕜 → F\nc : 𝕜\n⊢ iteratedDerivWithin n (fun z ↦ f (z + c)) s = fun x ↦ iteratedDerivWithin n f (c +ᵥ s) (x + c)",
"ppTerm": "?m.68",
"assigned": tru... | [] | by
ext x
simp [iteratedDerivWithin, ← iteratedFDerivWithin_comp_add_right n c x] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Exponential | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 39
} | {
"line": 77,
"column": 2
} | [
{
"pp": "case e'_12\n𝕂 : Type u_1\n𝔸 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕂\ninst✝³ : NormedRing 𝔸\ninst✝² : CharZero 𝕂\ninst✝¹ : NormedAlgebra 𝕂 𝔸\ninst✝ : CompleteSpace 𝔸\nh : 0 < (expSeries 𝕂 𝔸).radius\ne_4✝ : inst✝³.toAddCommGroup = NonUnitalNormedRing.toNormedAddCommGroup.toAddCommGroup\n... | [
"case e'_12\n𝕂 : Type u_1\n𝔸 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕂\ninst✝³ : NormedRing 𝔸\ninst✝² : CharZero 𝕂\ninst✝¹ : NormedAlgebra 𝕂 𝔸\ninst✝ : CompleteSpace 𝔸\nh : 0 < (expSeries 𝕂 𝔸).radius\ne_4✝ : inst✝³.toAddCommGroup = NonUnitalNormedRing.toNormedAddCommGroup.toAddCommGroup\ne_5✝ : (Norm... | change x = expSeries 𝕂 𝔸 1 fun _ => x | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 1044,
"column": 12
} | {
"line": 1044,
"column": 14
} | {
"line": 1044,
"column": 15
} | [
{
"pp": "case e_6.inl\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\nm : ℕ\nq : FormalMulti... | [
"case e_6.inl\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\nm : ℕ\nq : FormalMultilinearSeries... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Calculus.LocalExtr.Basic | {
"line": 82,
"column": 2
} | {
"line": 85,
"column": 52
} | {
"line": 87,
"column": 0
} | [
{
"pp": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ns : Set E\nx y : E\nh : ∃ᶠ (t : ℝ) in 𝓝[>] 0, x + t • y ∈ s\n⊢ y ∈ posTangentConeAt s x",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace",
"Iff.mpr",
"Real",
"S... | [] | rw [← NNReal.coe_zero, ← NNReal.map_coe_nhdsGT, frequently_map, frequently_iff_neBot] at h
apply mem_tangentConeAt_of_add_smul_mem (l := 𝓝[>] (0 : ℝ≥0) ⊓ 𝓟 {t | x + (t : ℝ) • y ∈ s})
· exact tendsto_id'.mpr <| inf_le_left.trans <| nhdsGT_le_nhdsNE _
· simp [eventually_inf_principal, NNReal.smul_def] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.LocalExtr.Basic | {
"line": 82,
"column": 2
} | {
"line": 85,
"column": 52
} | {
"line": 87,
"column": 0
} | [
{
"pp": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ns : Set E\nx y : E\nh : ∃ᶠ (t : ℝ) in 𝓝[>] 0, x + t • y ∈ s\n⊢ y ∈ posTangentConeAt s x",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace",
"Iff.mpr",
"Real",
"S... | [] | rw [← NNReal.coe_zero, ← NNReal.map_coe_nhdsGT, frequently_map, frequently_iff_neBot] at h
apply mem_tangentConeAt_of_add_smul_mem (l := 𝓝[>] (0 : ℝ≥0) ⊓ 𝓟 {t | x + (t : ℝ) • y ∈ s})
· exact tendsto_id'.mpr <| inf_le_left.trans <| nhdsGT_le_nhdsNE _
· simp [eventually_inf_principal, NNReal.smul_def] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 309,
"column": 2
} | {
"line": 309,
"column": 49
} | {
"line": 311,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\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\nx : E\nn : ℕ∞ω\ns : Set E\nf g : E → F\nhf : ContDiffWithinAt 𝕜 n f s x\nhg : ContDiffWithinAt 𝕜 n g s x\n⊢ ... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 309,
"column": 2
} | {
"line": 309,
"column": 49
} | {
"line": 311,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\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\nx : E\nn : ℕ∞ω\ns : Set E\nf g : E → F\nhf : ContDiffWithinAt 𝕜 n f s x\nhg : ContDiffWithinAt 𝕜 n g s x\n⊢ ... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 309,
"column": 2
} | {
"line": 309,
"column": 49
} | {
"line": 311,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\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\nx : E\nn : ℕ∞ω\ns : Set E\nf g : E → F\nhf : ContDiffWithinAt 𝕜 n f s x\nhg : ContDiffWithinAt 𝕜 n g s x\n⊢ ... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 314,
"column": 48
} | {
"line": 314,
"column": 95
} | {
"line": 316,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\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\nx : E\nn : ℕ∞ω\nf g : E → F\nhf : ContDiffAt 𝕜 n f x\nhg : ContDiffAt 𝕜 n g x\n⊢ ContDiffAt 𝕜 n (fun x ↦ f ... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 314,
"column": 48
} | {
"line": 314,
"column": 95
} | {
"line": 316,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\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\nx : E\nn : ℕ∞ω\nf g : E → F\nhf : ContDiffAt 𝕜 n f x\nhg : ContDiffAt 𝕜 n g x\n⊢ ContDiffAt 𝕜 n (fun x ↦ f ... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 314,
"column": 48
} | {
"line": 314,
"column": 95
} | {
"line": 316,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\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\nx : E\nn : ℕ∞ω\nf g : E → F\nhf : ContDiffAt 𝕜 n f x\nhg : ContDiffAt 𝕜 n g x\n⊢ ContDiffAt 𝕜 n (fun x ↦ f ... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 320,
"column": 2
} | {
"line": 320,
"column": 49
} | {
"line": 322,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\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\nn : ℕ∞ω\ns : Set E\nf g : E → F\nhf : ContDiffOn 𝕜 n f s\nhg : ContDiffOn 𝕜 n g s\n⊢ ContDiffOn 𝕜 n (fun x ... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 320,
"column": 2
} | {
"line": 320,
"column": 49
} | {
"line": 322,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\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\nn : ℕ∞ω\ns : Set E\nf g : E → F\nhf : ContDiffOn 𝕜 n f s\nhg : ContDiffOn 𝕜 n g s\n⊢ ContDiffOn 𝕜 n (fun x ... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 320,
"column": 2
} | {
"line": 320,
"column": 49
} | {
"line": 322,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\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\nn : ℕ∞ω\ns : Set E\nf g : E → F\nhf : ContDiffOn 𝕜 n f s\nhg : ContDiffOn 𝕜 n g s\n⊢ ContDiffOn 𝕜 n (fun x ... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 325,
"column": 42
} | {
"line": 325,
"column": 89
} | {
"line": 327,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\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\nn : ℕ∞ω\nf g : E → F\nhf : ContDiff 𝕜 n f\nhg : ContDiff 𝕜 n g\n⊢ ContDiff 𝕜 n fun x ↦ f x - g x",
"ppT... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 325,
"column": 42
} | {
"line": 325,
"column": 89
} | {
"line": 327,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\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\nn : ℕ∞ω\nf g : E → F\nhf : ContDiff 𝕜 n f\nhg : ContDiff 𝕜 n g\n⊢ ContDiff 𝕜 n fun x ↦ f x - g x",
"ppT... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 325,
"column": 42
} | {
"line": 325,
"column": 89
} | {
"line": 327,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\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\nn : ℕ∞ω\nf g : E → F\nhf : ContDiff 𝕜 n f\nhg : ContDiff 𝕜 n g\n⊢ ContDiff 𝕜 n fun x ↦ f x - g x",
"ppT... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 444,
"column": 48
} | {
"line": 444,
"column": 50
} | {
"line": 445,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set 𝕜\nf g : 𝕜 → F\nhfg : EqOn f g s\nhs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s\nn : ℕ\nx : 𝕜\nhx : x ∈ s\na : 𝕜\n⊢ a ∈ s → f a = g a",
"ppTerm... | [
"𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set 𝕜\nf g : 𝕜 → F\nhfg : EqOn f g s\nhs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s\nn : ℕ\nx : 𝕜\nhx : x ∈ s\na : 𝕜\nha : a ∈ s\n⊢ f a = g a"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 515,
"column": 2
} | {
"line": 515,
"column": 43
} | {
"line": 517,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nι : Type u_7\nn : ℕ\nx : 𝕜\nf : ι → 𝕜 → F\nI : Finset ι\nhf : ∀ i ∈ I, ContDiffAt 𝕜 (↑n) (f i) x\n⊢ iteratedDeriv n (fun z ↦ ∑ i ∈ I, f i z) x = ∑ i ∈ I, iteratedDeriv n (f i) x... | [] | simpa [sum_fn] using iteratedDeriv_sum hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 515,
"column": 2
} | {
"line": 515,
"column": 43
} | {
"line": 517,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nι : Type u_7\nn : ℕ\nx : 𝕜\nf : ι → 𝕜 → F\nI : Finset ι\nhf : ∀ i ∈ I, ContDiffAt 𝕜 (↑n) (f i) x\n⊢ iteratedDeriv n (fun z ↦ ∑ i ∈ I, f i z) x = ∑ i ∈ I, iteratedDeriv n (f i) x... | [] | simpa [sum_fn] using iteratedDeriv_sum hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 515,
"column": 2
} | {
"line": 515,
"column": 43
} | {
"line": 517,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nι : Type u_7\nn : ℕ\nx : 𝕜\nf : ι → 𝕜 → F\nI : Finset ι\nhf : ∀ i ∈ I, ContDiffAt 𝕜 (↑n) (f i) x\n⊢ iteratedDeriv n (fun z ↦ ∑ i ∈ I, f i z) x = ∑ i ∈ I, iteratedDeriv n (f i) x... | [] | simpa [sum_fn] using iteratedDeriv_sum hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.FundThmCalculus | {
"line": 51,
"column": 77
} | {
"line": 51,
"column": 79
} | {
"line": 51,
"column": 79
} | [
{
"pp": "X : Type u_1\nE : Type u_2\nι : Type u_3\ninst✝⁴ : MeasurableSpace X\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure X\nl : Filter X\ninst✝ : l.IsMeasurablyGenerated\nf : X → E\nb : E\nh : Tendsto f (l ⊓ ae μ) (𝓝 b)\nhfm : StronglyMeasurableAtFilter f l ... | [
"X : Type u_1\nE : Type u_2\nι : Type u_3\ninst✝⁴ : MeasurableSpace X\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure X\nl : Filter X\ninst✝ : l.IsMeasurablyGenerated\nf : X → E\nb : E\nh : Tendsto f (l ⊓ ae μ) (𝓝 b)\nhfm : StronglyMeasurableAtFilter f l μ\nhμ : μ.Fi... | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.Deriv.MeanValue | {
"line": 243,
"column": 4
} | {
"line": 243,
"column": 30
} | {
"line": 244,
"column": 4
} | [
{
"pp": "f : ℝ → ℝ\na : ℝ\nhf : ∀ (i : ℝ), True → ∀ᶠ (x : ℝ) in 𝓝[<] a, deriv f x ∈ Iic i\nf' : ℝ → ℝ := f ∘ Neg.neg\n⊢ deriv f' =ᶠ[𝓝[>] (-a)] -deriv f ∘ Neg.neg",
"ppTerm": "?m.147",
"assigned": true,
"usedConstants": [
"Real",
"trivial",
"Function.comp",
"Real.instOne",
... | [
"f : ℝ → ℝ\na : ℝ\nhf : ∀ᶠ (x : ℝ) in 𝓝[<] a, deriv f x ∈ Iic (-1)\nf' : ℝ → ℝ := f ∘ Neg.neg\n⊢ deriv f' =ᶠ[𝓝[>] (-a)] -deriv f ∘ Neg.neg"
] | specialize hf (-1) trivial | Lean.Elab.Tactic.evalSpecialize | Lean.Parser.Tactic.specialize |
Mathlib.Analysis.SpecialFunctions.Log.Deriv | {
"line": 353,
"column": 2
} | {
"line": 361,
"column": 64
} | {
"line": 362,
"column": 2
} | [
{
"pp": "x : ℝ\nh : |x| < 1\n⊢ Tendsto (fun n ↦ ∑ i ∈ Finset.range n, x ^ (i + 1) / (↑i + 1)) atTop (𝓝 (-log (1 - x)))",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"squeeze_zero",
"GroupWithZero.toMonoidWithZero",
"NormedCommRing.toS... | [
"x : ℝ\nh : |x| < 1\n⊢ Summable fun n ↦ x ^ (n + 1) / (↑n + 1)"
] | · show Tendsto (fun n : ℕ => ∑ i ∈ range n, x ^ (i + 1) / (i + 1)) atTop (𝓝 (-log (1 - x)))
rw [tendsto_iff_norm_sub_tendsto_zero]
simp only [norm_eq_abs, sub_neg_eq_add]
refine squeeze_zero (fun n => abs_nonneg _) (abs_log_sub_add_sum_range_le h) ?_
suffices Tendsto (fun t : ℕ => |x| ^ (t + 1) / (1 - ... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.Gauge | {
"line": 145,
"column": 8
} | {
"line": 145,
"column": 48
} | {
"line": 145,
"column": 48
} | [
{
"pp": "case refine_1\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\na : ℝ\nhs₁ : Convex ℝ s\nhs₀ : 0 ∈ s\nhs₂ : Absorbent ℝ s\nha : 0 ≤ a\nx : E\nh : gauge s x ≤ a\nr : ℝ\nhr : a < r\nhr' : 0 < r\nδ : ℝ\nδ_pos : 0 < δ\nhδr : δ < r\nhδ : x ∈ δ • s\n⊢ (r⁻¹ * δ) • δ⁻¹ • x ∈ s",
"ppTer... | [
"case refine_1\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\na : ℝ\nhs₁ : Convex ℝ s\nhs₀ : 0 ∈ s\nhs₂ : Absorbent ℝ s\nha : 0 ≤ a\nx : E\nh : gauge s x ≤ a\nr : ℝ\nhr : a < r\nhr' : 0 < r\nδ : ℝ\nδ_pos : 0 < δ\nhδr : δ < r\nhδ : δ⁻¹ • x ∈ s\n⊢ (r⁻¹ * δ) • δ⁻¹ • x ∈ s"
] | mem_smul_set_iff_inv_smul_mem₀ δ_pos.ne' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Cone.Extension | {
"line": 118,
"column": 16
} | {
"line": 118,
"column": 23
} | {
"line": 118,
"column": 24
} | [
{
"pp": "E : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : PointedCone ℝ E\np : E →ₗ.[ℝ] ℝ\nhp_nonneg : ∀ (x : ↥p.domain), ↑x ∈ s → 0 ≤ ↑p x\nhp_dense : ∀ (y : E), ∃ x, ↑x + y ∈ s\nS : Set (E →ₗ.[ℝ] ℝ) := {p | ∀ (x : ↥p.domain), ↑x ∈ s → 0 ≤ ↑p x}\nc : Set (E →ₗ.[ℝ] ℝ)\nhcs : c ⊆ S\n⊢ IsChain (fun ... | [
"E : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : PointedCone ℝ E\np : E →ₗ.[ℝ] ℝ\nhp_nonneg : ∀ (x : ↥p.domain), ↑x ∈ s → 0 ≤ ↑p x\nhp_dense : ∀ (y : E), ∃ x, ↑x + y ∈ s\nS : Set (E →ₗ.[ℝ] ℝ) := {p | ∀ (x : ↥p.domain), ↑x ∈ s → 0 ≤ ↑p x}\nc : Set (E →ₗ.[ℝ] ℝ)\nhcs : c ⊆ S\nc_chain : IsChain (fun x1 x... | c_chain | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Convex.Gauge | {
"line": 231,
"column": 2
} | {
"line": 231,
"column": 21
} | {
"line": 232,
"column": 2
} | [
{
"pp": "E : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\nhs : Convex ℝ s\nh₀ : 0 ∈ s\nabsorbs : Absorbent ℝ s\na : ℝ\n⊢ Convex ℝ {x | gauge s x ≤ a}",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Real.partialOrder",
"Real.instLE",
"Real",
"ga... | [
"case pos\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\nhs : Convex ℝ s\nh₀ : 0 ∈ s\nabsorbs : Absorbent ℝ s\na : ℝ\nha : 0 ≤ a\n⊢ Convex ℝ {x | gauge s x ≤ a}",
"case neg\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\nhs : Convex ℝ s\nh₀ : 0 ∈ s\nabsorbs : Absorbe... | by_cases ha : 0 ≤ a | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 307,
"column": 4
} | {
"line": 307,
"column": 18
} | {
"line": 308,
"column": 4
} | [
{
"pp": "case pos\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\nf : E → F\nK : Set (E →L[𝕜] F)\nhK : IsComplete K\nP : ∀ {n : ℕ}, 0 < (1 / 2) ^ n\nc : 𝕜\nhc : 1... | [
"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\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 : ... | · simp [y_pos] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 360,
"column": 2
} | {
"line": 382,
"column": 67
} | {
"line": 384,
"column": 0
} | [
{
"pp": "α : 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\nint_f : ∫⁻ (x : α), ↑(f x) ∂μ ≠ ∞\nε : ℝ≥0∞\nε0 : ε ≠ 0\n⊢ ∃ g, (∀ (x : α), g x ≤ f x) ∧ UpperSemicontinuous g ∧ ∫⁻ (x : α), ↑(f x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ... | [] | obtain ⟨fs, fs_le_f, int_fs⟩ :
∃ fs : α →ₛ ℝ≥0, (∀ x, fs x ≤ f x) ∧ (∫⁻ x, f x ∂μ) ≤ (∫⁻ x, fs x ∂μ) + ε / 2 := by
have := ENNReal.lt_add_right int_f (ENNReal.half_pos ε0).ne'
conv_rhs at this => rw [lintegral_eq_nnreal (fun x => (f x : ℝ≥0∞)) μ]
rw [ENNReal.biSup_add'] at this <;> [skip; exact ⟨0, fun ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 360,
"column": 2
} | {
"line": 382,
"column": 67
} | {
"line": 384,
"column": 0
} | [
{
"pp": "α : 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\nint_f : ∫⁻ (x : α), ↑(f x) ∂μ ≠ ∞\nε : ℝ≥0∞\nε0 : ε ≠ 0\n⊢ ∃ g, (∀ (x : α), g x ≤ f x) ∧ UpperSemicontinuous g ∧ ∫⁻ (x : α), ↑(f x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ... | [] | obtain ⟨fs, fs_le_f, int_fs⟩ :
∃ fs : α →ₛ ℝ≥0, (∀ x, fs x ≤ f x) ∧ (∫⁻ x, f x ∂μ) ≤ (∫⁻ x, fs x ∂μ) + ε / 2 := by
have := ENNReal.lt_add_right int_f (ENNReal.half_pos ε0).ne'
conv_rhs at this => rw [lintegral_eq_nnreal (fun x => (f x : ℝ≥0∞)) μ]
rw [ENNReal.biSup_add'] at this <;> [skip; exact ⟨0, fun ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 394,
"column": 2
} | {
"line": 414,
"column": 37
} | {
"line": 416,
"column": 0
} | [
{
"pp": "α : 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\nεpos : 0 < ε\n⊢ ∃ g,\n (∀ (x : α), g x ≤ f x) ∧\n UpperSemicontinuous g ∧ Integrable (fun x ↦ ↑(g x)) μ... | [] | lift ε to ℝ≥0 using εpos.le
rw [NNReal.coe_pos, ← ENNReal.coe_pos] at εpos
have If : (∫⁻ x, f x ∂μ) < ∞ := hasFiniteIntegral_iff_ofNNReal.1 fint.hasFiniteIntegral
rcases exists_upperSemicontinuous_le_lintegral_le f If.ne εpos.ne' with ⟨g, gf, gcont, gint⟩
have Ig : (∫⁻ x, g x ∂μ) < ∞ := by
refine lt_of_le_o... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 394,
"column": 2
} | {
"line": 414,
"column": 37
} | {
"line": 416,
"column": 0
} | [
{
"pp": "α : 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\nεpos : 0 < ε\n⊢ ∃ g,\n (∀ (x : α), g x ≤ f x) ∧\n UpperSemicontinuous g ∧ Integrable (fun x ↦ ↑(g x)) μ... | [] | lift ε to ℝ≥0 using εpos.le
rw [NNReal.coe_pos, ← ENNReal.coe_pos] at εpos
have If : (∫⁻ x, f x ∂μ) < ∞ := hasFiniteIntegral_iff_ofNNReal.1 fint.hasFiniteIntegral
rcases exists_upperSemicontinuous_le_lintegral_le f If.ne εpos.ne' with ⟨g, gf, gcont, gint⟩
have Ig : (∫⁻ x, g x ∂μ) < ∞ := by
refine lt_of_le_o... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Gauge | {
"line": 387,
"column": 2
} | {
"line": 387,
"column": 49
} | {
"line": 389,
"column": 0
} | [
{
"pp": "E : Type u_2\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx : E\nhx : x ∈ interior s\nH₁ : Tendsto (fun r ↦ r⁻¹ • x) (𝓝[<] 1) (𝓝 x)\nH₂ : ∀ᶠ (r : ℝ) in 𝓝[<] 1, x ∈ r • s ∧ 0 < r ∧ r < 1\nr : ℝ\nhxr : x ∈ r • s\nhr₀ : 0 < r\nhr₁ : ... | [] | exact (gauge_le_of_mem hr₀.le hxr).trans_lt hr₁ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 113,
"column": 12
} | {
"line": 113,
"column": 14
} | {
"line": 113,
"column": 15
} | [
{
"pp": "E : 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 : ... | [
"E : 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 : Set E := x₀ ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 519,
"column": 6
} | {
"line": 519,
"column": 21
} | {
"line": 520,
"column": 4
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nr x : ℝ\nhr : 0 < r\nε : ℝ\nL₁ L₂ : F\nh₁ : x ∈ A f L₁ r ε\nh₂ : x ∈ A f L₂ r ε\n⊢ ‖(r / 2) • (L₁ - L₂)‖ = ‖f (x + r / 2) - f x - (x + r / 2 - x) • L₂ - (f (x + r / 2) - f x - (x + r / 2 - x) • L₁)‖",
"ppTerm": "?m.212... | [] | simp [smul_sub] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 519,
"column": 6
} | {
"line": 519,
"column": 21
} | {
"line": 520,
"column": 4
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nr x : ℝ\nhr : 0 < r\nε : ℝ\nL₁ L₂ : F\nh₁ : x ∈ A f L₁ r ε\nh₂ : x ∈ A f L₂ r ε\n⊢ ‖(r / 2) • (L₁ - L₂)‖ = ‖f (x + r / 2) - f x - (x + r / 2 - x) • L₂ - (f (x + r / 2) - f x - (x + r / 2 - x) • L₁)‖",
"ppTerm": "?m.212... | [] | simp [smul_sub] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 519,
"column": 6
} | {
"line": 519,
"column": 21
} | {
"line": 520,
"column": 4
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nr x : ℝ\nhr : 0 < r\nε : ℝ\nL₁ L₂ : F\nh₁ : x ∈ A f L₁ r ε\nh₂ : x ∈ A f L₂ r ε\n⊢ ‖(r / 2) • (L₁ - L₂)‖ = ‖f (x + r / 2) - f x - (x + r / 2 - x) • L₂ - (f (x + r / 2) - f x - (x + r / 2 - x) • L₁)‖",
"ppTerm": "?m.212... | [] | simp [smul_sub] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 382,
"column": 86
} | {
"line": 384,
"column": 94
} | {
"line": 386,
"column": 0
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_5\ninst✝³ : NormedAddCommGroup E\na b : ℝ\nμ : Measure ℝ\nf : ℝ → 𝕜\ng : ℝ → E\ninst✝² : NormedRing 𝕜\ninst✝¹ : Module 𝕜 E\ninst✝ : IsBoundedSMul 𝕜 E\nhg : IntervalIntegrable g μ a b\nhf : ContinuousOn f [[a, b]]\n⊢ IntervalIntegrable (fun x ↦ f x • g x) μ a b",
"ppTer... | [] | by
rw [intervalIntegrable_iff] at hg ⊢
exact hg.continuousOn_smul_of_subset hf isCompact_uIcc measurableSet_Ioc Ioc_subset_Icc_self | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 400,
"column": 2
} | {
"line": 400,
"column": 38
} | {
"line": 401,
"column": 2
} | [
{
"pp": "case inr\nε : Type u_3\ninst✝² : TopologicalSpace ε\ninst✝¹ : ENormedAddMonoid ε\nf : ℝ → ε\na b : ℝ\ninst✝ : PseudoMetrizableSpace ε\nhf : IntervalIntegrable f volume a b\nc : ℝ\nh : ‖f (min a b)‖ₑ ≠ ∞\nh' : ‖f (c * min (a / c) (b / c))‖ₑ ≠ ∞\nhc : c ≠ 0\n⊢ IntervalIntegrable (fun x ↦ f (c * x)) volum... | [
"case inr\nε : Type u_3\ninst✝² : TopologicalSpace ε\ninst✝¹ : ENormedAddMonoid ε\nf : ℝ → ε\na b : ℝ\ninst✝ : PseudoMetrizableSpace ε\nhf : IntegrableOn f [[a, b]] volume\nc : ℝ\nh : ‖f (min a b)‖ₑ ≠ ∞\nh' : ‖f (c * min (a / c) (b / c))‖ₑ ≠ ∞\nhc : c ≠ 0\n⊢ IntervalIntegrable (fun x ↦ f (c * x)) volume (a / c) (b ... | rw [intervalIntegrable_iff' h] at hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 436,
"column": 2
} | {
"line": 436,
"column": 38
} | {
"line": 437,
"column": 2
} | [
{
"pp": "ε : Type u_3\ninst✝² : TopologicalSpace ε\ninst✝¹ : ENormedAddMonoid ε\nf : ℝ → ε\na✝ b✝ : ℝ\ninst✝ : PseudoMetrizableSpace ε\nc a b : ℝ\nhf : IntervalIntegrable f volume a b\nh : ‖f (min a b)‖ₑ ≠ ∞\nh' : ‖f (min (a - c) (b - c) + c)‖ₑ ≠ ∞\nhab : a ≤ b\n⊢ IntervalIntegrable (fun x ↦ f (x + c)) volume (... | [
"ε : Type u_3\ninst✝² : TopologicalSpace ε\ninst✝¹ : ENormedAddMonoid ε\nf : ℝ → ε\na✝ b✝ : ℝ\ninst✝ : PseudoMetrizableSpace ε\nc a b : ℝ\nhf : IntegrableOn f [[a, b]] volume\nh : ‖f (min a b)‖ₑ ≠ ∞\nh' : ‖f (min (a - c) (b - c) + c)‖ₑ ≠ ∞\nhab : a ≤ b\n⊢ IntervalIntegrable (fun x ↦ f (x + c)) volume (a - c) (b - c... | rw [intervalIntegrable_iff' h] at hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 449,
"column": 4
} | {
"line": 449,
"column": 71
} | {
"line": 450,
"column": 4
} | [
{
"pp": "ε : Type u_3\ninst✝² : TopologicalSpace ε\ninst✝¹ : ENormedAddMonoid ε\nf : ℝ → ε\na b : ℝ\ninst✝ : PseudoMetrizableSpace ε\nc : ℝ\nh : ‖f (min a b + c)‖ₑ ≠ ∞\nhf : IntervalIntegrable f volume (a + c) (b + c)\n⊢ IntervalIntegrable (fun x ↦ f (x + c)) volume a b",
"ppTerm": "?m.45",
"assigned": ... | [
"ε : Type u_3\ninst✝² : TopologicalSpace ε\ninst✝¹ : ENormedAddMonoid ε\nf : ℝ → ε\na b : ℝ\ninst✝ : PseudoMetrizableSpace ε\nc : ℝ\nh : ‖f (min a b + c)‖ₑ ≠ ∞\nhf : IntervalIntegrable f volume (a + c) (b + c)\nthis : ‖f (min (a + c) (b + c))‖ₑ ≠ ∞\n⊢ IntervalIntegrable (fun x ↦ f (x + c)) volume a b"
] | have : ‖f (min (a + c) (b + c))‖ₑ ≠ ⊤ := by rwa [min_add_add_right] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 400,
"column": 84
} | {
"line": 419,
"column": 63
} | {
"line": 421,
"column": 0
} | [
{
"pp": "ι : Type u_1\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nca cb : E\nla la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nua va ub vb : ι → ℝ\ninst✝² : IsLocallyFiniteMeasure μ\ninst✝¹ : FTCFilter a la la'\ninst✝ : FTCFilter b lb lb'\nhab : IntervalInte... | [] | by
have := FTCFilter.meas_gen la; have := FTCFilter.meas_gen lb
refine
((measure_integral_sub_linear_isLittleO_of_tendsto_ae hmeas_a ha_lim hua hva).neg_left.add_add
(measure_integral_sub_linear_isLittleO_of_tendsto_ae hmeas_b hb_lim hub hvb)).congr'
?_ EventuallyEq.rfl
have A : ∀ᶠ t in lt, In... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hst\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ Disjoint (Ioc b a ∪ Ioc c b) (Ioc a c)",
"ppTerm": "?hst",
"assigned": tru... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hst\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ Disjoint (Ioc b a ∪ Ioc c b) (Ioc a c)",
"ppTerm": "?hst",
"assigned": tru... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hst\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ Disjoint (Ioc b a ∪ Ioc c b) (Ioc a c)",
"ppTerm": "?hst",
"assigned": tru... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case ht\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ MeasurableSet (Ioc a c)",
"ppTerm": "?ht",
"assigned": true,
"usedConst... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case ht\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ MeasurableSet (Ioc a c)",
"ppTerm": "?ht",
"assigned": true,
"usedConst... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case ht\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ MeasurableSet (Ioc a c)",
"ppTerm": "?ht",
"assigned": true,
"usedConst... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hfs\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ IntegrableOn f (Ioc b a ∪ Ioc c b) μ",
"ppTerm": "?hfs",
"assigned": true,... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hfs\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ IntegrableOn f (Ioc b a ∪ Ioc c b) μ",
"ppTerm": "?hfs",
"assigned": true,... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hfs\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ IntegrableOn f (Ioc b a ∪ Ioc c b) μ",
"ppTerm": "?hfs",
"assigned": true,... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hft\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ IntegrableOn f (Ioc a c) μ",
"ppTerm": "?hft",
"assigned": true,
"used... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hft\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ IntegrableOn f (Ioc a c) μ",
"ppTerm": "?hft",
"assigned": true,
"used... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hft\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ IntegrableOn f (Ioc a c) μ",
"ppTerm": "?hft",
"assigned": true,
"used... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hst\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ Disjoint (Ioc b a) (Ioc c b)",
"ppTerm": "?hst✝",
"assigned": true,
"u... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hst\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ Disjoint (Ioc b a) (Ioc c b)",
"ppTerm": "?hst✝",
"assigned": true,
"u... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hst\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ Disjoint (Ioc b a) (Ioc c b)",
"ppTerm": "?hst✝",
"assigned": true,
"u... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case ht\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ MeasurableSet (Ioc c b)",
"ppTerm": "?ht✝",
"assigned": true,
"usedCons... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case ht\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ MeasurableSet (Ioc c b)",
"ppTerm": "?ht✝",
"assigned": true,
"usedCons... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case ht\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ MeasurableSet (Ioc c b)",
"ppTerm": "?ht✝",
"assigned": true,
"usedCons... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hfs\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ IntegrableOn f (Ioc b a) μ",
"ppTerm": "?hfs✝",
"assigned": true,
"use... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hfs\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ IntegrableOn f (Ioc b a) μ",
"ppTerm": "?hfs✝",
"assigned": true,
"use... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hfs\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ IntegrableOn f (Ioc b a) μ",
"ppTerm": "?hfs✝",
"assigned": true,
"use... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hft\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ IntegrableOn f (Ioc c b) μ",
"ppTerm": "?hft✝",
"assigned": true,
"use... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hft\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ IntegrableOn f (Ioc c b) μ",
"ppTerm": "?hft✝",
"assigned": true,
"use... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hft\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ IntegrableOn f (Ioc c b) μ",
"ppTerm": "?hft✝",
"assigned": true,
"use... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hst\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ Disjoint (Ioc a b ∪ Ioc b c) (Ioc c a)",
"ppTerm": "?hst✝",
"assigned": tr... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hst\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ Disjoint (Ioc a b ∪ Ioc b c) (Ioc c a)",
"ppTerm": "?hst✝",
"assigned": tr... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hst\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ Disjoint (Ioc a b ∪ Ioc b c) (Ioc c a)",
"ppTerm": "?hst✝",
"assigned": tr... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case ht\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ MeasurableSet (Ioc c a)",
"ppTerm": "?ht✝",
"assigned": true,
"usedCons... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case ht\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ MeasurableSet (Ioc c a)",
"ppTerm": "?ht✝",
"assigned": true,
"usedCons... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case ht\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ MeasurableSet (Ioc c a)",
"ppTerm": "?ht✝",
"assigned": true,
"usedCons... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hfs\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ IntegrableOn f (Ioc a b ∪ Ioc b c) μ",
"ppTerm": "?hfs✝",
"assigned": true... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hfs\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ IntegrableOn f (Ioc a b ∪ Ioc b c) μ",
"ppTerm": "?hfs✝",
"assigned": true... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hfs\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ IntegrableOn f (Ioc a b ∪ Ioc b c) μ",
"ppTerm": "?hfs✝",
"assigned": true... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hft\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ IntegrableOn f (Ioc c a) μ",
"ppTerm": "?hft✝",
"assigned": true,
"use... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hft\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ IntegrableOn f (Ioc c a) μ",
"ppTerm": "?hft✝",
"assigned": true,
"use... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hft\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ IntegrableOn f (Ioc c a) μ",
"ppTerm": "?hft✝",
"assigned": true,
"use... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hst\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ Disjoint (Ioc a b) (Ioc b c)",
"ppTerm": "?hst✝",
"assigned": true,
"u... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hst\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ Disjoint (Ioc a b) (Ioc b c)",
"ppTerm": "?hst✝",
"assigned": true,
"u... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hst\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ Disjoint (Ioc a b) (Ioc b c)",
"ppTerm": "?hst✝",
"assigned": true,
"u... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case ht\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ MeasurableSet (Ioc b c)",
"ppTerm": "?ht✝",
"assigned": true,
"usedCons... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case ht\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ MeasurableSet (Ioc b c)",
"ppTerm": "?ht✝",
"assigned": true,
"usedCons... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case ht\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ MeasurableSet (Ioc b c)",
"ppTerm": "?ht✝",
"assigned": true,
"usedCons... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hfs\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ IntegrableOn f (Ioc a b) μ",
"ppTerm": "?hfs✝",
"assigned": true,
"use... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hfs\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ IntegrableOn f (Ioc a b) μ",
"ppTerm": "?hfs✝",
"assigned": true,
"use... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hfs\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ IntegrableOn f (Ioc a b) μ",
"ppTerm": "?hfs✝",
"assigned": true,
"use... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hft\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ IntegrableOn f (Ioc b c) μ",
"ppTerm": "?hft✝",
"assigned": true,
"use... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hft\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ IntegrableOn f (Ioc b c) μ",
"ppTerm": "?hft✝",
"assigned": true,
"use... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1093,
"column": 4
} | {
"line": 1093,
"column": 54
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "case hft\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhab : IntervalIntegrable f μ a b\nhbc : IntervalIntegrable f μ b c\nhac : IntervalIntegrable f μ a c\n⊢ IntegrableOn f (Ioc b c) μ",
"ppTerm": "?hft✝",
"assigned": true,
"use... | [] | simp [*, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1201,
"column": 2
} | {
"line": 1201,
"column": 72
} | {
"line": 1202,
"column": 2
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhf : IntegrableOn f (Ici a) μ\nhab : a ≤ b\nha : IntegrableOn f (Ici b) μ\n⊢ ∫ (x : ℝ) in Ici a, f x ∂μ - ∫ (x : ℝ) in Ici b, f x ∂μ = ∫ (x : ℝ) in Ico a b, f x ∂μ",
"ppTerm": "?m.73",
"assi... | [
"E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhf : IntegrableOn f (Ici a) μ\nhab : a ≤ b\nha : IntegrableOn f (Ici b) μ\nh : IntegrableOn f (Ico a b) μ\n⊢ ∫ (x : ℝ) in Ici a, f x ∂μ - ∫ (x : ℝ) in Ici b, f x ∂μ = ∫ (x : ℝ) in Ico a b, f x ∂μ"
] | have h : IntegrableOn f (Ico a b) μ := hf.mono_set Ico_subset_Ici_self | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Algebra.Order.Floor | {
"line": 153,
"column": 4
} | {
"line": 153,
"column": 34
} | {
"line": 153,
"column": 34
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : Ring α\ninst✝⁴ : LinearOrder α\ninst✝³ : FloorRing α\ninst✝² : TopologicalSpace α\ninst✝¹ : IsStrictOrderedRing α\ninst✝ : OrderClosedTopology α\nn : ℤ\n⊢ pure (IntCast.intCast (n - 1)) ≤ 𝓝[≤] (↑n - 1)",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"... | [
"α : Type u_1\ninst✝⁵ : Ring α\ninst✝⁴ : LinearOrder α\ninst✝³ : FloorRing α\ninst✝² : TopologicalSpace α\ninst✝¹ : IsStrictOrderedRing α\ninst✝ : OrderClosedTopology α\nn : ℤ\n⊢ pure (IntCast.intCast (n - 1)) ≤ 𝓝[≤] ↑(n - 1)"
] | rw [← @cast_one α, ← cast_sub] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
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