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.Normed.Operator.NNNorm | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 38
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_4\nF : Type u_5\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : SeminormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.NNNorm | {
"line": 83,
"column": 2
} | {
"line": 83,
"column": 40
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : NontriviallyNormedField 𝕜₂\ninst✝¹⁰ : NontriviallyNormedField 𝕜₃\ninst✝⁹ : SeminormedAddCommGroup E\ninst✝⁸ : SeminormedAddCommGroup F\ninst✝⁷ : SeminormedAddCommGr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.NNNorm | {
"line": 104,
"column": 2
} | {
"line": 104,
"column": 54
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_4\nF : Type u_5\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : SeminormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 866,
"column": 2
} | {
"line": 877,
"column": 42
} | [
{
"pp": "𝕜 : Type u_2\nF : Type u_7\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nq : Seminorm 𝕜 F\nhq : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ⇑q\n⊢ ∃ C, 0 < C ∧ ∀ (x : F), q x ≤ C * ‖x‖",
"usedConstants": [
"Filter.in... | have hq' : Tendsto q (𝓝 0) (𝓝 0) := map_zero q ▸ hq.tendsto 0
rcases NormedAddGroup.nhds_zero_basis_norm_lt.mem_iff.mp (hq' <| Iio_mem_nhds one_pos)
with ⟨ε, ε_pos, hε⟩
rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
have : 0 < ‖c‖ / ε := by positivity
refine ⟨‖c‖ / ε, this, fun x ↦ ?_⟩
by_cases h... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 866,
"column": 2
} | {
"line": 877,
"column": 42
} | [
{
"pp": "𝕜 : Type u_2\nF : Type u_7\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nq : Seminorm 𝕜 F\nhq : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ⇑q\n⊢ ∃ C, 0 < C ∧ ∀ (x : F), q x ≤ C * ‖x‖",
"usedConstants": [
"Filter.in... | have hq' : Tendsto q (𝓝 0) (𝓝 0) := map_zero q ▸ hq.tendsto 0
rcases NormedAddGroup.nhds_zero_basis_norm_lt.mem_iff.mp (hq' <| Iio_mem_nhds one_pos)
with ⟨ε, ε_pos, hε⟩
rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
have : 0 < ‖c‖ / ε := by positivity
refine ⟨‖c‖ / ε, this, fun x ↦ ?_⟩
by_cases h... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Operator.NNNorm | {
"line": 149,
"column": 17
} | {
"line": 149,
"column": 28
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_4\nF : Type u_5\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : SeminormedAddCommGroup F\ninst✝⁴ : DenselyNormedField 𝕜\ninst✝³ : NontriviallyNormedField 𝕜₂\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\nf : E... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.NNNorm | {
"line": 158,
"column": 4
} | {
"line": 158,
"column": 30
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_4\nF : Type u_5\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : SeminormedAddCommGroup F\ninst✝⁴ : DenselyNormedField 𝕜\ninst✝³ : NontriviallyNormedField 𝕜₂\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsome... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.NNNorm | {
"line": 158,
"column": 4
} | {
"line": 158,
"column": 76
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_4\nF : Type u_5\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : SeminormedAddCommGroup F\ninst✝⁴ : DenselyNormedField 𝕜\ninst✝³ : NontriviallyNormedField 𝕜₂\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsome... | simpa only [mul_one] using f.le_opNorm_of_le (mem_ball_zero_iff.1 hx).le | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Normed.Operator.NNNorm | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 53
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_4\nF : Type u_5\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : SeminormedAddCommGroup F\ninst✝⁴ : DenselyNormedField 𝕜\ninst✝³ : NontriviallyNormedField 𝕜₂\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\nf : E... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.NNNorm | {
"line": 178,
"column": 2
} | {
"line": 178,
"column": 53
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_4\nF : Type u_5\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : SeminormedAddCommGroup F\ninst✝⁴ : DenselyNormedField 𝕜\ninst✝³ : NontriviallyNormedField 𝕜₂\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\nf : E... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.NNNorm | {
"line": 188,
"column": 34
} | {
"line": 188,
"column": 84
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_4\nF : Type u_5\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SeminormedAddCommGroup F\ninst✝⁵ : DenselyNormedField 𝕜\ninst✝⁴ : NontriviallyNormedField 𝕜₂\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝¹ : RingHomIsometric σ₁₂\ninst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.NNNorm | {
"line": 203,
"column": 4
} | {
"line": 203,
"column": 30
} | [
{
"pp": "case inr.refine_1\n𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_4\nF : Type u_5\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SeminormedAddCommGroup F\ninst✝⁵ : DenselyNormedField 𝕜\ninst✝⁴ : NontriviallyNormedField 𝕜₂\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝¹ : RingHom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.NNNorm | {
"line": 205,
"column": 21
} | {
"line": 205,
"column": 32
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_4\nF : Type u_5\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SeminormedAddCommGroup F\ninst✝⁵ : DenselyNormedField 𝕜\ninst✝⁴ : NontriviallyNormedField 𝕜₂\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝¹ : RingHomIsometric σ₁₂\ninst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.NNNorm | {
"line": 211,
"column": 2
} | {
"line": 211,
"column": 53
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_4\nF : Type u_5\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SeminormedAddCommGroup F\ninst✝⁵ : DenselyNormedField 𝕜\ninst✝⁴ : NontriviallyNormedField 𝕜₂\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝¹ : RingHomIsometric σ₁₂\ninst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Germ.Basic | {
"line": 758,
"column": 4
} | {
"line": 758,
"column": 78
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nl : Filter α\nf✝ g✝ h✝ : α → β\ninst✝² : Mul β\ninst✝¹ : LE β\ninst✝ : ExistsMulOfLE β\nx y : l.Germ β\nf g : α → β\nh : f ≤ᶠ[l] g\nc : (x : α) → f x ≤ g x → β\nhc : ∀ (x : α) (hx : f x ≤ g x), g x = f x * c x hx\n⊢ ∃ c, ↑g = ↑f * c",
"usedCon... | refine ⟨ofFun fun x ↦ if hx : f x ≤ g x then c x hx else f x, coe_eq.2 ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Order.Filter.ENNReal | {
"line": 34,
"column": 2
} | {
"line": 34,
"column": 57
} | [
{
"pp": "case h.e'_2.h.e'_3\nf : Filter ℝ\nhf : ¬IsBounded (fun x1 x2 ↦ x1 ≤ x2) f\n⊢ {a | ∀ᶠ (n : ℝ) in f, n ≤ a} = ∅",
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"congrArg",
"Filter.Eventually",
"PartialOrder.toPreorder",
"setOf",
"Preorder.toLE",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.ENNReal | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 57
} | [
{
"pp": "case h.e'_2.h.e'_3\nf : Filter ℝ\nhf : ¬IsBounded (fun x1 x2 ↦ x1 ≥ x2) f\n⊢ {a | ∀ᶠ (n : ℝ) in f, a ≤ n} = ∅",
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"congrArg",
"Filter.Eventually",
"PartialOrder.toPreorder",
"setOf",
"_private.Mathli... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.ENNReal | {
"line": 91,
"column": 45
} | {
"line": 91,
"column": 60
} | [
{
"pp": "ι : Type u_1\nf : Filter ι\nu : ι → ℝ≥0\ninst✝ : f.NeBot\nb : ℝ\nhb : ∀ (a : ℝ), (∀ᶠ (a_1 : ι) in f, ↑(u a_1) ≤ a) → b ≤ a\nx : ℝ\nx✝ : 0 ≤ x\n⊢ (∀ᶠ (a : ι) in f, ↑(u a) ≤ ↑(NNReal.mk x x✝)) → ↑(NNReal.mk ↑b.toNNReal ⋯) ≤ ↑(NNReal.mk x x✝)",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.Pi | {
"line": 94,
"column": 4
} | {
"line": 94,
"column": 64
} | [
{
"pp": "case intro.a.h.h\nι : Type u_1\nα : ι → Type u_2\ninst✝ : Finite ι\nC : (i : ι) → Set (Set (α i))\nhC : ∀ (i : ι), IsCountablySpanning (C i)\nval✝ : Encodable ι\ns : (i : ι) → Set (α i)\nhs : s ∈ univ.pi C\ni : ι\n⊢ MeasurableSet (eval i ⁻¹' s i)",
"usedConstants": [
"measurable_pi_apply",
... | apply @measurable_pi_apply _ _ (fun i => generateFrom (C i)) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Order.Filter.ENNReal | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 57
} | [
{
"pp": "case h.e'_2.h.e'_3\nf : Filter ℝ≥0\nhf : ¬IsBounded (fun x1 x2 ↦ x1 ≤ x2) f\n⊢ {a | ∀ᶠ (n : ℝ≥0) in f, n ≤ a} = ∅",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"_private.Mathlib.Order.Filter.ENNReal.0.NNReal.limsSup_of_not_isBounded._simp_1_1",
"Filter.Ev... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.ENNReal | {
"line": 135,
"column": 2
} | {
"line": 135,
"column": 13
} | [
{
"pp": "case pos\nι : Type u_1\nf : Filter ι\nu : ι → ℝ≥0\nhf : IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) f u\nc r : ℝ\nhr : r < c\n⊢ (∀ᶠ (a : ι) in f, r < ↑(u a)) ↔ 0 ≤ r → ∀ᶠ (a : ι) in f, r < ↑(u a)",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Preorder.toLT",
"Real.i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.ENNReal | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 13
} | [
{
"pp": "case pos\nι : Type u_1\nf : Filter ι\nu : ι → ℝ≥0\nhf✝ : f.NeBot\nhf : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u\nthis : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u\nc r : ℝ\nhr : r < c\n⊢ (∃ᶠ (a : ι) in f, r < ↑(u a)) ↔ 0 ≤ r → ∃ᶠ (a : ι) in f, r < ↑(u a)",
"usedConstants": [
"Eq.mpr",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.ENNReal | {
"line": 176,
"column": 2
} | {
"line": 176,
"column": 24
} | [
{
"pp": "α : Type u_1\nf : Filter α\nu : α → ℝ≥0∞\na : ℝ≥0∞\nha_top : a ≠ ∞\n⊢ limsup (fun x ↦ u x * a) f = a * limsup u f",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"CommSemiring.toNonUnitalCommSemiring",
"congrArg",
"CommSemiring.toSemiring",
"id",
"CommMagma.toM... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.ENNReal | {
"line": 188,
"column": 2
} | {
"line": 188,
"column": 24
} | [
{
"pp": "α : Type u_1\nf : Filter α\nu : α → ℝ≥0∞\na : ℝ≥0∞\nha₀ : a ≠ 0\nha_top : a ≠ ∞\n⊢ liminf (fun x ↦ u x * a) f = a * liminf u f",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Filter.liminf",
"CommSemiring.toNonUnitalCommSemiring",
"congrArg",
"CommSemiring.toSemirin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ConditionalProbability | {
"line": 187,
"column": 2
} | {
"line": 189,
"column": 23
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\n⊢ μ ≪ μ[|univ]",
"usedConstants": [
"Eq.mpr",
"False",
"instHSMul",
"MeasureTheory.Measure",
"instSMulOfMul",
"congrArg",
"CommSemiring.toSemiring",
"Set.univ",
"ENNR... | rw [cond, restrict_univ]
refine absolutelyContinuous_smul ?_
simp [measure_ne_top] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.ConditionalProbability | {
"line": 187,
"column": 2
} | {
"line": 189,
"column": 23
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\n⊢ μ ≪ μ[|univ]",
"usedConstants": [
"Eq.mpr",
"False",
"instHSMul",
"MeasureTheory.Measure",
"instSMulOfMul",
"congrArg",
"CommSemiring.toSemiring",
"Set.univ",
"ENNR... | rw [cond, restrict_univ]
refine absolutelyContinuous_smul ?_
simp [measure_ne_top] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.ENNReal | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 24
} | [
{
"pp": "α : Type u_1\nf : Filter α\ninst✝ : f.NeBot\nu : α → ℝ≥0∞\na : ℝ≥0∞\nha_top : a ≠ ∞\n⊢ liminf (fun x ↦ u x * a) f = a * liminf u f",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Filter.liminf",
"CommSemiring.toNonUnitalCommSemiring",
"congrArg",
"CommSemiring.toSem... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ConditionalProbability | {
"line": 216,
"column": 61
} | {
"line": 216,
"column": 76
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\ns : Set Ω\nhms : MeasurableSet s\nμ : Measure Ω\nt : Set Ω\n⊢ (μ s)⁻¹ • μ (t ∩ s) = (μ s)⁻¹ * μ (s ∩ t)",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"MeasureTheory.Measure",
"instSMulOfMul",
"HMul.hMul",
"congrArg",
"Comm... | Set.inter_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.ConditionalProbability | {
"line": 219,
"column": 59
} | {
"line": 219,
"column": 74
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\ns t : Set Ω\nht : MeasurableSet t\nμ : Measure Ω\n⊢ (μ s)⁻¹ • μ (t ∩ s) = (μ s)⁻¹ * μ (s ∩ t)",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"MeasureTheory.Measure",
"instSMulOfMul",
"HMul.hMul",
"congrArg",
"CommSemiring.t... | Set.inter_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Filter.ENNReal | {
"line": 220,
"column": 2
} | {
"line": 220,
"column": 24
} | [
{
"pp": "α : Type u_1\nf : Filter α\ninst✝ : CountableInterFilter f\nu : α → ℝ≥0∞\na : ℝ≥0∞\n⊢ limsup (fun x ↦ u x * a) f = a * limsup u f",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"CommSemiring.toNonUnitalCommSemiring",
"congrArg",
"CommSemiring.toSemiring",
"id",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ConditionalProbability | {
"line": 222,
"column": 2
} | {
"line": 222,
"column": 20
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nμ : Measure Ω\ns : Set Ω\nhs₀ : μ s ≠ 0\nhs : μ s ≠ ∞\n⊢ μ[s | s] = 1",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"MeasureTheory.Measure",
"instSMulOfMul",
"HMul.hMul",
"MeasureTheory.Measure.restrict_apply_self",
"congr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.ENNReal | {
"line": 227,
"column": 6
} | {
"line": 228,
"column": 90
} | [
{
"pp": "α : Type u_1\nf : Filter α\ninst✝ : CountableInterFilter f\nu v : α → ℝ≥0∞\n⊢ limsup (u * v) f ≤ limsup (fun x ↦ limsup u f * v x) f",
"usedConstants": [
"le_rfl",
"HMul.hMul",
"CommSemiring.toNonUnitalCommSemiring",
"CommSemiring.toSemiring",
"Filter.isBounded_le_of_t... | refine limsup_le_limsup ?_
filter_upwards [@eventually_le_limsup _ f _ u] with x hx using mul_le_mul' hx le_rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.ENNReal | {
"line": 227,
"column": 6
} | {
"line": 228,
"column": 90
} | [
{
"pp": "α : Type u_1\nf : Filter α\ninst✝ : CountableInterFilter f\nu v : α → ℝ≥0∞\n⊢ limsup (u * v) f ≤ limsup (fun x ↦ limsup u f * v x) f",
"usedConstants": [
"le_rfl",
"HMul.hMul",
"CommSemiring.toNonUnitalCommSemiring",
"CommSemiring.toSemiring",
"Filter.isBounded_le_of_t... | refine limsup_le_limsup ?_
filter_upwards [@eventually_le_limsup _ f _ u] with x hx using mul_le_mul' hx le_rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.ENNReal | {
"line": 255,
"column": 28
} | {
"line": 255,
"column": 39
} | [
{
"pp": "α : Type u_1\nf : Filter α\nu : α → ℝ\nh₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u\nh₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u\nr : ℝ≥0\nh : ∀ y > ↑r, ∀ᶠ (a : α) in f, u a < y\nx : ℝ≥0\nhx : some x > ↑r\n⊢ ↑x > ↑r",
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.ENNReal | {
"line": 257,
"column": 6
} | {
"line": 257,
"column": 39
} | [
{
"pp": "case h.inl\nα : Type u_1\nf : Filter α\nu : α → ℝ\nh₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u\nh₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u\nr : ℝ≥0\nh : ∀ y > ↑r, ∀ᶠ (a : α) in f, u a < y\nx : ℝ≥0\nhx : some x > ↑r\na : α\nha : u a < ↑x\nha₀ : u a ≤ 0\n⊢ ENNReal.ofReal (u a) < some x",
"used... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ConditionalProbability | {
"line": 275,
"column": 42
} | {
"line": 275,
"column": 57
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\ns t : Set Ω\nhms : MeasurableSet s\nhmt : MeasurableSet t\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\n⊢ μ[t | s] = (μ s)⁻¹ * μ (t ∩ s)",
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"MeasureTheory.Measure",
"HMul.hMul",
"CommSemir... | Set.inter_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Filter.ENNReal | {
"line": 261,
"column": 38
} | {
"line": 261,
"column": 56
} | [
{
"pp": "α : Type u_1\nf : Filter α\nu : α → ℝ\nh₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u\nh₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u\nr : ℝ≥0\nh : ∀ y > ↑r, ∀ᶠ (a : α) in f, ENNReal.ofReal (u a) < y\nx : ℝ\nhx : x > ↑r\nthis : 0 < x\n⊢ ENNReal.ofReal x > ↑r",
"usedConstants": [
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.ENNReal | {
"line": 290,
"column": 38
} | {
"line": 290,
"column": 56
} | [
{
"pp": "α : Type u_1\nf : Filter α\nu : α → ℝ≥0∞\nh₁ : ∀ᶠ (a : α) in f, u a ≠ ∞\nh₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f fun a ↦ (u a).toReal\nhf : f.NeBot\nthis✝ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f fun a ↦ (u a).toReal\nr : ℝ\nhr : 0 ≤ r\nh : ∀ y > ENNReal.ofReal r, ∀ᶠ (a : α) in f, u a < y\nx : ℝ\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.ENNReal | {
"line": 294,
"column": 32
} | {
"line": 294,
"column": 66
} | [
{
"pp": "α : Type u_1\nf : Filter α\nu : α → ℝ≥0∞\nh₁ : ∀ᶠ (a : α) in f, u a ≠ ∞\nh₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f fun a ↦ (u a).toReal\nhf : f.NeBot\nthis : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f fun a ↦ (u a).toReal\nr : ℝ\nhr : 0 ≤ r\nh : ∀ y > r, ∀ᶠ (a : α) in f, (u a).toReal < y\nx : ℝ≥0\nhx :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Bilinear | {
"line": 166,
"column": 18
} | {
"line": 166,
"column": 46
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nF : Type u_6\nG : Type u_8\ninst✝¹⁰ : SeminormedAddCommGroup E\ninst✝⁹ : SeminormedAddCommGroup F\ninst✝⁸ : SeminormedAddCommGroup G\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NontriviallyNormedField 𝕜₂\ninst✝⁵ : NontriviallyNormedField ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Bilinear | {
"line": 267,
"column": 20
} | {
"line": 267,
"column": 46
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁹ : SeminormedAddCommGroup E\ninst✝¹⁸ : SeminormedAddCommGroup Eₗ\ninst✝¹⁷ : SeminormedAddCommGroup F\ninst✝¹⁶ : SeminormedAddCommGroup Fₗ\ninst✝¹⁵... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.UniformOn | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 90
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t : Set Ω\nhsf : s.Finite\nh : #⋯.toFinset = #hsf.toFinset\n⊢ ⋯.toFinset = hsf.toFinset",
"usedConstants": [
"Eq.ge",
"Set.Finite.inter_of_left",
"Set.inter_subset_left",
"Set.instInter",
"... | exact Finset.eq_of_subset_of_card_le (Set.Finite.toFinset_mono s.inter_subset_left) h.ge | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.EssSup | {
"line": 357,
"column": 6
} | {
"line": 357,
"column": 81
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0\nhf : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (ae μ) f\nr : ℝ≥0∞\n⊢ r ≤ ⨅ a ∈ fun x ↦ (map f (ae μ)).sets {x_1 | (fun x_2 ↦ (fun x1 x2 ↦ x1 ≤ x2) x_2 x) x_1}, ↑a ↔\n r ≤ essSup (fun x ↦ ↑(f x)) μ",
"usedConstants": [
"MeasureTheo... | simp [essSup, limsup, limsSup, eventually_map, ENNReal.forall_ennreal]; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.EssSup | {
"line": 357,
"column": 6
} | {
"line": 357,
"column": 81
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0\nhf : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (ae μ) f\nr : ℝ≥0∞\n⊢ r ≤ ⨅ a ∈ fun x ↦ (map f (ae μ)).sets {x_1 | (fun x_2 ↦ (fun x1 x2 ↦ x1 ≤ x2) x_2 x) x_1}, ↑a ↔\n r ≤ essSup (fun x ↦ ↑(f x)) μ",
"usedConstants": [
"MeasureTheo... | simp [essSup, limsup, limsSup, eventually_map, ENNReal.forall_ennreal]; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.SpecialFunctions.Basic | {
"line": 58,
"column": 2
} | {
"line": 59,
"column": 9
} | [
{
"pp": "α : Type u_1\nx✝ : MeasurableSpace α\nf : α → ℝ\nμ : Measure α\nt : ℝ\nht : t ≠ 0\nhf : AEMeasurable (fun x ↦ rexp (t * f x)) μ\n⊢ AEMeasurable f μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Holder | {
"line": 78,
"column": 56
} | {
"line": 78,
"column": 67
} | [
{
"pp": "p q r : ℝ≥0∞\ninst✝ : p.HolderTriple q r\n⊢ 1 / p + 1 / q = 1 / r",
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"DivInvMonoid.toInv",
"instHDiv",
"congrArg",
"id",
"HDiv.hDiv",
"instHAdd",
"Inv.inv",
"HAdd.hAdd",
"congr",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Holder | {
"line": 99,
"column": 16
} | {
"line": 99,
"column": 27
} | [
{
"pp": "p q r : ℝ≥0∞\ninst✝ : p.HolderTriple q r\nhr : r ≠ 0\n⊢ r⁻¹ < ∞",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"PartialOrder.toPreorder",
"ENNReal.inv_lt_top._simp_1",
"id",
"Inv.inv",
"LT.lt",
"ENNReal",
"ENNReal.instPartialOrder",
"Zero... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.Pi | {
"line": 303,
"column": 2
} | {
"line": 303,
"column": 45
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_3\ninst✝² : Fintype ι\ninst✝¹ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ninst✝ : ∀ (i : ι), SigmaFinite (μ i)\nf : (i : ι) → α i\n⊢ (Measure.pi μ) {f} = ∏ i, (μ i) {f i}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.Pi | {
"line": 555,
"column": 2
} | {
"line": 556,
"column": 9
} | [
{
"pp": "case h\nι✝ : Type u_1\nι' : Type u_2\nα : ι✝ → Type u_3\ninst✝⁷ : Fintype ι✝\nm : (i : ι✝) → OuterMeasure (α i)\ninst✝⁶ : (i : ι✝) → MeasurableSpace (α i)\nμ✝ : (i : ι✝) → Measure (α i)\ninst✝⁵ : ∀ (i : ι✝), SigmaFinite (μ✝ i)\nι : Type u_4\ninst✝⁴ : Fintype ι\nX : ι → Type u_5\ninst✝³ : (i : ι) → Pseu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.Pi | {
"line": 867,
"column": 2
} | {
"line": 867,
"column": 69
} | [
{
"pp": "α : Type u\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\n⊢ MeasurePreserving (⇑MeasurableEquiv.finTwoArrow) (Measure.pi fun x ↦ μ) (μ.prod μ)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.SMul | {
"line": 102,
"column": 2
} | {
"line": 103,
"column": 9
} | [
{
"pp": "case inr\nα : Type u_1\nF : Type u_2\nm : MeasurableSpace α\nq : ℝ\nμ : Measure α\ninst✝³ : NormedAddCommGroup F\n𝕜 : Type u_3\ninst✝² : NormedDivisionRing 𝕜\ninst✝¹ : Module 𝕜 F\ninst✝ : NormSMulClass 𝕜 F\nf : α → F\nc : 𝕜\nhq_pos : 0 < q\nhc : c ≠ 0\n⊢ eLpNorm' f q μ ≤ eLpNorm' (c • f) q μ / ‖c‖... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.SMul | {
"line": 115,
"column": 2
} | {
"line": 116,
"column": 9
} | [
{
"pp": "case inr\nα : Type u_1\nF : Type u_2\nm : MeasurableSpace α\ninst✝³ : NormedAddCommGroup F\n𝕜 : Type u_3\ninst✝² : NormedDivisionRing 𝕜\ninst✝¹ : Module 𝕜 F\ninst✝ : NormSMulClass 𝕜 F\nc : 𝕜\nf : α → F\np : ℝ≥0∞\nμ : Measure α\nhc : c ≠ 0\n⊢ eLpNorm f p μ ≤ eLpNorm (c • f) p μ / ‖c‖ₑ",
"usedCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.SMul | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 38
} | [
{
"pp": "α : Type u_1\nF : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nn : ℕ\nf : α → F\n⊢ eLpNorm (n • f) p μ = ↑n * eLpNorm f p μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity | {
"line": 74,
"column": 37
} | {
"line": 74,
"column": 58
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nε' : Type u_6\ninst✝³ : TopologicalSpace ε'\ninst✝² : ContinuousENorm ε'\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\nf : α → ε\nc : ℝ≥0∞\ng : α → ε'\np : ℝ\nhg : AEStronglyMeasurable g μ\nh : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ c *... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity | {
"line": 76,
"column": 6
} | {
"line": 76,
"column": 58
} | [
{
"pp": "case pos\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nε' : Type u_6\ninst✝³ : TopologicalSpace ε'\ninst✝² : ContinuousENorm ε'\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\nf : α → ε\nc : ℝ≥0∞\ng : α → ε'\np : ℝ\nhg : AEStronglyMeasurable g μ\nh : ∀ᵐ (x : α) ∂μ, ‖f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 52,
"column": 4
} | {
"line": 52,
"column": 58
} | [
{
"pp": "case hfq\nα : Type u_1\nε : Type u_2\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : ENorm ε\nf : α → ε\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ∞\nhfp : eLpNorm f p μ < ∞\n⊢ eLpNorm' f p.toReal μ < ∞",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 60,
"column": 4
} | {
"line": 60,
"column": 77
} | [
{
"pp": "α : Type u_1\nε : Type u_2\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : ENorm ε\nf : α → ε\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ∞\nh : ∫⁻ (a : α), ‖f a‖ₑ ^ p.toReal ∂μ < ∞\nhp' : 0 < p.toReal\nthis : 0 < 1 / p.toReal\n⊢ eLpNorm f p μ < ∞",
"usedConstants": [
"Eq.mpr",
"Real... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity | {
"line": 174,
"column": 4
} | {
"line": 174,
"column": 21
} | [
{
"pp": "case pos\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nε' : Type u_6\ninst✝³ : TopologicalSpace ε'\ninst✝² : ContinuousENorm ε'\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\nf : α → ε\nc : ℝ≥0∞\ng : α → ε'\np : ℝ≥0∞\nhg : AEStronglyMeasurable g μ\nh : ∀ᵐ (x : α) ∂μ,... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity | {
"line": 175,
"column": 4
} | {
"line": 175,
"column": 21
} | [
{
"pp": "case neg\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nε' : Type u_6\ninst✝³ : TopologicalSpace ε'\ninst✝² : ContinuousENorm ε'\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\nf : α → ε\nc : ℝ≥0∞\ng : α → ε'\np : ℝ≥0∞\nhg : AEStronglyMeasurable g μ\nh : ∀ᵐ (x : α) ∂μ,... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.SpecificFunctions.Basic | {
"line": 109,
"column": 27
} | {
"line": 109,
"column": 40
} | [
{
"pp": "case inr.inr\ns : ℝ\nhs✝ : -1 ≤ s\nhs' : s ≠ 0\np : ℝ\nhp : 1 < p\nhp' : 0 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhs3 : 1 + s ≠ 1\nhs4 : 1 + p * s ≠ 1\n⊢ 1 + p * s < rexp (log (1 + s) * p)",
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"congrArg",
... | ← exp_log hs2 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 233,
"column": 2
} | {
"line": 233,
"column": 63
} | [
{
"pp": "case neg.inr\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nε'' : Type u_8\ninst✝¹ : TopologicalSpace ε''\ninst✝ : ESeminormedAddMonoid ε''\nc : ε''\nhc' : ‖c‖ₑ ≠ ∞\np : ℝ≥0∞\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ∞\nhp : 0 < p.toReal\nhμ : ¬μ = 0\nhc : ¬‖c‖ₑ = 0\nhμ_ne_top : μ Set.univ ≠ ∞\n⊢ ‖c‖... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 286,
"column": 32
} | {
"line": 286,
"column": 71
} | [
{
"pp": "α : Type u_1\nF : Type u_5\nG : Type u_6\nm0 : MeasurableSpace α\nq : ℝ\nμ : Measure α\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → F\ng : α → G\nhq : 0 ≤ q\nh : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ ‖g x‖\n⊢ ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ ‖g x‖ₑ",
"usedConstants": [
"MeasureTheory.ae",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity | {
"line": 231,
"column": 17
} | {
"line": 231,
"column": 28
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nR : Type u_5\ninst✝² : NormedAddCommGroup R\ninst✝¹ : StarAddMonoid R\ninst✝ : NormedStarGroup R\np : ℝ≥0∞\nf : α → R\nhf : MemLp f p μ\n⊢ eLpNorm (star f) p μ < ∞",
"usedConstants": [
"Eq.mpr",
"Pi.instStarForall",
"Preorder.toL... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.SpecificFunctions.Basic | {
"line": 150,
"column": 27
} | {
"line": 150,
"column": 40
} | [
{
"pp": "case inr\ns : ℝ\nhs✝ : -1 ≤ s\nhs' : s ≠ 0\np : ℝ\nhp1 : 0 < p\nhp2 : p < 1\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhs3 : 1 + s ≠ 1\nhs4 : 1 + p * s ≠ 1\n⊢ rexp (log (1 + s) * p) < 1 + p * s",
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"congrArg",
"R... | ← exp_log hs2 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 331,
"column": 28
} | {
"line": 331,
"column": 67
} | [
{
"pp": "α : Type u_1\nF : Type u_5\nG : Type u_6\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → F\ng : α → G\nh : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ ‖g x‖\n⊢ ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ ‖g x‖ₑ",
"usedConstants": [
"MeasureTheory.ae",
"Norm... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 116,
"column": 56
} | {
"line": 116,
"column": 67
} | [
{
"pp": "p q r : ℝ\nh : p.HolderTriple q r\n⊢ 1 / p + 1 / q = 1 / r",
"usedConstants": [
"Eq.mpr",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"congrArg",
"Real.instDivInvMonoid",
"id",
"HDiv.hDiv",
"Real.instAdd",
"Real.instOne",
"instHAdd",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 124,
"column": 23
} | {
"line": 124,
"column": 34
} | [
{
"pp": "p q r : ℝ\nh : p.HolderTriple q r\n⊢ r < p",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 161,
"column": 41
} | {
"line": 161,
"column": 52
} | [
{
"pp": "p q : ℝ\nh : p.HolderConjugate q\n⊢ p⁻¹ - 1 = -q⁻¹",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 56
} | [
{
"pp": "p q : ℝ\nh : p.HolderConjugate q\n⊢ p * q = p + q",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 336,
"column": 28
} | {
"line": 336,
"column": 67
} | [
{
"pp": "α : Type u_1\nε : Type u_2\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝¹ : ENorm ε\nε' : Type u_7\ninst✝ : ENorm ε'\nf : α → ε\ng : α → ε'\nh : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ ‖g x‖ₑ\n⊢ ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ ‖g x‖ₑ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 189,
"column": 27
} | {
"line": 189,
"column": 38
} | [
{
"pp": "a b : ℝ\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ a⁻¹⁻¹ + b⁻¹⁻¹ = 1⁻¹",
"usedConstants": [
"Eq.mpr",
"Real",
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"inv_one",
"congrArg",
"Real.instInv",
"D... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 210,
"column": 42
} | {
"line": 210,
"column": 53
} | [
{
"pp": "a b : ℝ\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ (1 / a).HolderConjugate (1 / b)",
"usedConstants": [
"Eq.mpr",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"congrArg",
"Real.instDivInvMonoid",
"id",
"HDiv.hDiv",
"Real.instOne",
"Inv.inv",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 321,
"column": 27
} | {
"line": 321,
"column": 38
} | [
{
"pp": "⊢ 2⁻¹ + 2⁻¹ = 1⁻¹",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"inv_one",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"NNRea... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 349,
"column": 2
} | {
"line": 349,
"column": 71
} | [
{
"pp": "p q : ℝ≥0\nh : p.HolderConjugate q\n⊢ p * q = p + q",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 396,
"column": 2
} | {
"line": 396,
"column": 57
} | [
{
"pp": "α : Type u_1\nF : Type u_5\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup F\nf : α → F\nC : ℝ≥0\nhfC : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ C\n⊢ eLpNorm f p μ ≤ C • μ Set.univ ^ p.toReal⁻¹",
"usedConstants": [
"Real",
"instHSMul",
"MeasureTheory.Measure",
"C... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 398,
"column": 31
} | {
"line": 398,
"column": 42
} | [
{
"pp": "α : Type u_1\nF : Type u_5\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup F\nf : α → F\nC : ℝ≥0\nhfC : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ C\nx✝ : α\nhx : ‖f x✝‖₊ ≤ C\n⊢ ‖f x✝‖ₑ ≤ ↑C",
"usedConstants": [
"Eq.mpr",
"ENNReal.ofNNReal",
"SeminormedAddGroup.toNNNorm"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 364,
"column": 27
} | {
"line": 364,
"column": 38
} | [
{
"pp": "a b : ℝ≥0\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ a⁻¹⁻¹ + b⁻¹⁻¹ = 1⁻¹",
"usedConstants": [
"Eq.mpr",
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"inv_one",
"congrArg",
"NNReal.instInv",
"DivisionMon... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 369,
"column": 58
} | {
"line": 369,
"column": 69
} | [
{
"pp": "a : ℝ≥0\nha₀ : 0 < a\nha₁ : a < 1\n⊢ a⁻¹⁻¹ + (1 - a)⁻¹⁻¹ = 1",
"usedConstants": [
"Eq.mpr",
"DivisionCommMonoid.toDivisionMonoid",
"congrArg",
"HSub.hSub",
"NNReal.instInv",
"id",
"Distrib.toAdd",
"NNReal",
"DivisionMonoid.toInvolutiveInv",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 387,
"column": 42
} | {
"line": 387,
"column": 53
} | [
{
"pp": "a b : ℝ≥0\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ (1 / a).HolderConjugate (1 / b)",
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"GroupWithZero.toDivInvMonoid",
"congrArg",
"DivisionSemiring.toGroupWithZero",
"id",
"HDiv.hDiv",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 397,
"column": 2
} | {
"line": 397,
"column": 13
} | [
{
"pp": "p q : ℝ\nh : p.HolderConjugate q\n⊢ p.toNNReal.HolderConjugate q.toNNReal",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 445,
"column": 2
} | {
"line": 446,
"column": 24
} | [
{
"pp": "p q r : ℝ\nh : p.HolderTriple q r\n⊢ (ENNReal.ofReal p).HolderTriple (ENNReal.ofReal q) (ENNReal.ofReal r)",
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"_private.Mathlib.Data.Real.ConjExponents.0.Real.HolderTriple.ennrealOfReal._simp_1_1",
"ENNReal.ofReal",
"id",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 450,
"column": 2
} | {
"line": 450,
"column": 13
} | [
{
"pp": "p q : ℝ\nh : p.HolderConjugate q\n⊢ (ENNReal.ofReal p).HolderConjugate (ENNReal.ofReal q)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 457,
"column": 2
} | {
"line": 457,
"column": 41
} | [
{
"pp": "p q r : ℝ≥0∞\nh : p.toReal.HolderTriple q.toReal r.toReal\nhp : 0 < p ∧ p < ∞\nhq : 0 < q ∧ q < ∞\nhr : 0 < r ∧ r < ∞\n⊢ p.HolderTriple q r",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 464,
"column": 2
} | {
"line": 465,
"column": 9
} | [
{
"pp": "p q r : ℝ≥0∞\nhp : 0 < p ∧ p < ∞\nhq : 0 < q ∧ q < ∞\nh : p.HolderTriple q r\n⊢ p.toReal⁻¹ + q.toReal⁻¹ = r.toReal⁻¹",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 474,
"column": 19
} | {
"line": 474,
"column": 60
} | [
{
"pp": "p q r : ℝ≥0∞\nh : p.toNNReal.HolderTriple q.toNNReal r.toNNReal\n⊢ p.toReal.HolderTriple q.toReal r.toReal",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.SpecificFunctions.Basic | {
"line": 209,
"column": 4
} | {
"line": 209,
"column": 15
} | [
{
"pp": "case inl\nhp : 1 ≤ 1\n⊢ ConvexOn ℝ (Ici 0) fun x ↦ x ^ 1",
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real.partialOrder",
"Real",
"instSMulOfMul",
"Set.Ici",
"Real.instZero",
"congrArg",
"Real.semiring",
"id",
"Real.rpow_one",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 504,
"column": 2
} | {
"line": 504,
"column": 13
} | [
{
"pp": "p q : ℝ≥0∞\nhp : 1 < p.toReal\nh : p.HolderConjugate q\nhq : 0 < q.toReal\n⊢ p.toReal.HolderConjugate q.toReal",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 512,
"column": 19
} | {
"line": 512,
"column": 60
} | [
{
"pp": "p q : ℝ≥0∞\nh : p.toNNReal.HolderConjugate q.toNNReal\n⊢ p.toReal.HolderConjugate q.toReal",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 557,
"column": 2
} | {
"line": 558,
"column": 39
} | [
{
"pp": "case inr.inr\np q : ℝ≥0∞\nh : p.HolderConjugate q\nhp : p ≠ ∞\nhq : q ≠ ∞\n⊢ p * q = p + q",
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"HMul.hMul",
"ENNReal.instAddCommMonoid",
"congrArg",
"CommSemiring.toSemiring",
"id",
"ENNReal.instCommSemiri... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 571,
"column": 27
} | {
"line": 571,
"column": 48
} | [
{
"pp": "a b : ℝ≥0∞\nhab : a + b = 1\n⊢ a⁻¹⁻¹ + b⁻¹⁻¹ = 1⁻¹",
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"inv_one",
"congrArg",
"id",
"semiOutParam",
"instHAdd",
"Inv.inv",
"HAdd.hAdd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 571,
"column": 24
} | {
"line": 571,
"column": 52
} | [
{
"pp": "a b : ℝ≥0∞\nhab : a + b = 1\n⊢ a⁻¹⁻¹ + b⁻¹⁻¹ = 1⁻¹",
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"inv_one",
"congrArg",
"id",
"semiOutParam",
"instHAdd",
"Inv.inv",
"HAdd.hAdd... | by simpa [inv_inv] using hab | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Real.ConjExponents | {
"line": 577,
"column": 2
} | {
"line": 577,
"column": 13
} | [
{
"pp": "a : ℝ≥0∞\nha : 1 ≤ a\n⊢ a.HolderConjugate (1 - a⁻¹)⁻¹",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 607,
"column": 33
} | {
"line": 607,
"column": 58
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ENormedAddMonoid ε\ns : Set α\nf : α → ε\nhsf : Function.support f ⊆ s\nhp0 : ¬p = 0\nhp_top : ¬p = ∞\n⊢ ¬p.toReal ≤ 0",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"R... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 608,
"column": 4
} | {
"line": 608,
"column": 22
} | [
{
"pp": "case neg.e_a.hsf\nα : Type u_1\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ENormedAddMonoid ε\ns : Set α\nf : α → ε\nhsf : Function.support f ⊆ s\nhp0 : ¬p = 0\nhp_top : ¬p = ∞\nthis : ¬p.toReal ≤ 0\n⊢ (Function.support fun x ↦ ‖f x‖ₑ ^ p.toReal)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 834,
"column": 37
} | {
"line": 834,
"column": 62
} | [
{
"pp": "case pos\nα : Type u_1\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ENormedAddMonoid ε\nf : α → ε\nhf : AEStronglyMeasurable f μ\nh0 : p ≠ 0\nh_top : p = ∞\n⊢ eLpNormEssSup f μ = 0 ↔ f =ᶠ[ae μ] 0",
"usedConstants": [
"MeasureTheory.ae",
... | eLpNormEssSup_eq_zero_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Monovary | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 38
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : PartialOrder α\ninst✝² : CommGroup β\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedMonoid β\ns : Set ι\nf : ι → α\ng : ι → β\n⊢ MonovaryOn f g⁻¹ s ↔ AntivaryOn f g s",
"usedConstants": [
"instIsRightCancelMulOfMulRightReflectLE",
"IsLeftCa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Monovary | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 38
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : PartialOrder α\ninst✝² : CommGroup β\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedMonoid β\ns : Set ι\nf : ι → α\ng : ι → β\n⊢ AntivaryOn f g⁻¹ s ↔ MonovaryOn f g s",
"usedConstants": [
"instIsRightCancelMulOfMulRightReflectLE",
"IsLeftCa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Monovary | {
"line": 106,
"column": 2
} | {
"line": 106,
"column": 34
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : PartialOrder α\ninst✝² : CommGroup β\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedMonoid β\nf : ι → α\ng : ι → β\n⊢ Monovary f g⁻¹ ↔ Antivary f g",
"usedConstants": [
"instIsRightCancelMulOfMulRightReflectLE",
"IsLeftCancelMul.mulLeftRefl... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Monovary | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 34
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : PartialOrder α\ninst✝² : CommGroup β\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedMonoid β\nf : ι → α\ng : ι → β\n⊢ Antivary f g⁻¹ ↔ Monovary f g",
"usedConstants": [
"instIsRightCancelMulOfMulRightReflectLE",
"IsLeftCancelMul.mulLeftRefl... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Slope | {
"line": 288,
"column": 2
} | {
"line": 288,
"column": 33
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y : 𝕜\nhy : y ∈ s\nhxy : x < y\nhxy' : f y < f x\nthis : StrictMonoOn (f ∘ ⇑(-AffineMap.id 𝕜 𝕜)) (⇑(-AffineMap.id 𝕜 𝕜) ⁻¹' s ∩ Set.Ici (-x))\n⊢ StrictAntiOn f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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