module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Function.AEEqFun | {
"line": 620,
"column": 6
} | {
"line": 620,
"column": 12
} | {
"line": 620,
"column": 12
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nf' f g : α →ₘ[μ] β\nhf : ↑f' ≤ᵐ[μ] ↑f\nhg : ↑f' ≤ᵐ[μ] ↑g\na✝ : α\nhaf : ↑f' a✝ ≤ ↑f a✝\nhag : ↑f' a✝ ≤ ↑g a✝\nha_inf : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝\n⊢ ↑... | [
"α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nf' f g : α →ₘ[μ] β\nhf : ↑f' ≤ᵐ[μ] ↑f\nhg : ↑f' ≤ᵐ[μ] ↑g\na✝ : α\nhaf : ↑f' a✝ ≤ ↑f a✝\nhag : ↑f' a✝ ≤ ↑g a✝\nha_inf : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝\n⊢ ↑f' a✝ ≤ ↑f a... | ha_inf | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.AEEqFun | {
"line": 942,
"column": 14
} | {
"line": 942,
"column": 21
} | {
"line": 942,
"column": 22
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : MeasurableSpace α\nμ : Measure α\nβ : Type u_5\ninst✝⁴ : TopologicalSpace β\ninst✝³ : Lattice β\ninst✝² : TopologicalLattice β\ninst✝¹ : AddGroup β\ninst✝ : IsTopologicalAddGroup β\nf : α →ₘ[μ] β\nx : α\nhx_sup : ↑(f ⊔ -f) x = ↑f x ⊔ ↑(-f) x\nhx_neg : ↑(-f) x = (-↑f) x\n⊢ ↑f x ⊔ ... | [
"α : Type u_1\ninst✝⁵ : MeasurableSpace α\nμ : Measure α\nβ : Type u_5\ninst✝⁴ : TopologicalSpace β\ninst✝³ : Lattice β\ninst✝² : TopologicalLattice β\ninst✝¹ : AddGroup β\ninst✝ : IsTopologicalAddGroup β\nf : α →ₘ[μ] β\nx : α\nhx_sup : ↑(f ⊔ -f) x = ↑f x ⊔ ↑(-f) x\nhx_neg : ↑(-f) x = (-↑f) x\n⊢ ↑f x ⊔ (-↑f) x = ↑f... | hx_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 56
} | {
"line": 95,
"column": 0
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\n⊢ eLpNormEssSup 0 μ = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"MeasureTheory.Measure",
"... | [] | simp [eLpNormEssSup, ← bot_eq_zero', essSup_const_bot] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 56
} | {
"line": 95,
"column": 0
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\n⊢ eLpNormEssSup 0 μ = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"MeasureTheory.Measure",
"... | [] | simp [eLpNormEssSup, ← bot_eq_zero', essSup_const_bot] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 56
} | {
"line": 95,
"column": 0
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\n⊢ eLpNormEssSup 0 μ = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"MeasureTheory.Measure",
"... | [] | simp [eLpNormEssSup, ← bot_eq_zero', essSup_const_bot] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity | {
"line": 81,
"column": 2
} | {
"line": 82,
"column": 36
} | {
"line": 83,
"column": 2
} | [
{
"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\nhg : AEStronglyMeasurable g μ\nh : ∀ᵐ (x : α) ∂μ, ‖f... | [
"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\nhg : AEStronglyMeasurable g μ\nh : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ c * ‖... | simp_rw [← ENNReal.rpow_mul, one_div, inv_mul_cancel₀ hp.ne', ENNReal.rpow_one,
← lintegral_const_mul' _ _ this] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.Constructions.Pi | {
"line": 814,
"column": 2
} | {
"line": 814,
"column": 52
} | {
"line": 815,
"column": 2
} | [
{
"pp": "n : ℕ\nα : Fin (n + 1) → Type u\nm : (i : Fin (n + 1)) → MeasurableSpace (α i)\nμ : (i : Fin (n + 1)) → Measure (α i)\ninst✝ : ∀ (i : Fin (n + 1)), SigmaFinite (μ i)\ni : Fin (n + 1)\n⊢ MeasurePreserving (⇑(MeasurableEquiv.piFinSuccAbove α i)) (Measure.pi μ)\n ((μ i).prod (Measure.pi fun j ↦ μ (i.su... | [
"n : ℕ\nα : Fin (n + 1) → Type u\nm : (i : Fin (n + 1)) → MeasurableSpace (α i)\nμ : (i : Fin (n + 1)) → Measure (α i)\ninst✝ : ∀ (i : Fin (n + 1)), SigmaFinite (μ i)\ni : Fin (n + 1)\ne : α i × ((j : Fin n) → α (i.succAbove j)) ≃ᵐ ((j : Fin (n + 1)) → α j) := (MeasurableEquiv.piFinSuccAbove α i).symm\n⊢ MeasurePre... | set e := (MeasurableEquiv.piFinSuccAbove α i).symm | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.Analysis.Convex.SpecificFunctions.Basic | {
"line": 49,
"column": 72
} | {
"line": 49,
"column": 79
} | {
"line": 50,
"column": 6
} | [
{
"pp": "x y z : ℝ\nhxy : x < y\nhyz : y < z\nh1 : 0 < y - x\nh2 : x - y < 0\n⊢ rexp y - rexp x = rexp y - rexp (y + (x - y))",
"ppTerm": "?m.157",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.to... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.Convex.SpecificFunctions.Basic | {
"line": 60,
"column": 46
} | {
"line": 60,
"column": 53
} | {
"line": 60,
"column": 53
} | [
{
"pp": "x y z : ℝ\nhxy : x < y\nhyz : y < z\nh1 : 0 < z - y\n⊢ rexp (z - y + y) - rexp y ≤ rexp z - rexp y",
"ppTerm": "?m.295",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithO... | [
"x y z : ℝ\nhxy : x < y\nhyz : y < z\nh1 : 0 < z - y\n⊢ rexp z - rexp y ≤ rexp z - rexp y"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.Convex.SpecificFunctions.Basic | {
"line": 118,
"column": 4
} | {
"line": 123,
"column": 40
} | {
"line": 125,
"column": 0
} | [
{
"pp": "case inr.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\nhs' : 0 < s\n⊢ log (1 + p * s) < log (1 + s) * p",
"ppTerm": "?inr.inr.inr",
"assigned": true,
"usedConstants": [
... | [] | rw [← div_lt_iff₀ hp', ← div_lt_div_iff_of_pos_right hs']
convert! strictConcaveOn_log_Ioi.secant_strict_mono (zero_lt_one' ℝ) hs1 hs2 hs3 hs4 _ using 1
· rw [add_sub_cancel_left, div_div, log_one, sub_zero]
· rw [add_sub_cancel_left, log_one, sub_zero]
· gcongr
exact lt_mul_of_one_lt_left hs' hp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.SpecificFunctions.Basic | {
"line": 118,
"column": 4
} | {
"line": 123,
"column": 40
} | {
"line": 125,
"column": 0
} | [
{
"pp": "case inr.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\nhs' : 0 < s\n⊢ log (1 + p * s) < log (1 + s) * p",
"ppTerm": "?inr.inr.inr",
"assigned": true,
"usedConstants": [
... | [] | rw [← div_lt_iff₀ hp', ← div_lt_div_iff_of_pos_right hs']
convert! strictConcaveOn_log_Ioi.secant_strict_mono (zero_lt_one' ℝ) hs1 hs2 hs3 hs4 _ using 1
· rw [add_sub_cancel_left, div_div, log_one, sub_zero]
· rw [add_sub_cancel_left, log_one, sub_zero]
· gcongr
exact lt_mul_of_one_lt_left hs' hp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Real.ConjExponents | {
"line": 464,
"column": 2
} | {
"line": 465,
"column": 55
} | {
"line": 467,
"column": 0
} | [
{
"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",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"ENNReal.instAdd",
"False",
"Real",
"ENNReal.toReal_add",
"Preorder.toLT",
"eq_f... | [] | simpa [toReal_add, Finiteness.inv_ne_top, hp.1.ne', hq.1.ne']
using congr(ENNReal.toReal $(h.inv_add_inv_eq_inv)) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Data.Real.ConjExponents | {
"line": 556,
"column": 2
} | {
"line": 556,
"column": 22
} | {
"line": 557,
"column": 2
} | [
{
"pp": "case inr.inl\np : ℝ≥0∞\nhp : p ≠ ∞\nh : p.HolderConjugate ∞\n⊢ p * ∞ = p + ∞",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
"ENNReal.instAdd",
"False",
"HMul.hMul",
"eq_false",
"add_top",
"congrArg",
"CommSemiring.toSemiring",
"P... | [
"case inr.inr\np q : ℝ≥0∞\nh : p.HolderConjugate q\nhp : p ≠ ∞\nhq : q ≠ ∞\n⊢ p * q = p + q"
] | · simp [ne_zero p ∞] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.SpecificFunctions.Basic | {
"line": 235,
"column": 57
} | {
"line": 235,
"column": 64
} | {
"line": 236,
"column": 4
} | [
{
"pp": "t : ℝ\nht : -1 ≤ t ∧ t ≤ 1\nx : ℝ\n⊢ rexp (t * x) = rexp ((1 + t) / 2 * x + (1 - t) / 2 * -x)",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.Convex.SpecificFunctions.Basic | {
"line": 235,
"column": 57
} | {
"line": 235,
"column": 64
} | {
"line": 236,
"column": 4
} | [
{
"pp": "t : ℝ\nht : -1 ≤ t ∧ t ≤ 1\nx : ℝ\n⊢ rexp (t * x) = rexp ((1 + t) / 2 * x + (1 - t) / 2 * -x)",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
... | [] | ring_nf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.SpecificFunctions.Basic | {
"line": 235,
"column": 57
} | {
"line": 235,
"column": 64
} | {
"line": 236,
"column": 4
} | [
{
"pp": "t : ℝ\nht : -1 ≤ t ∧ t ≤ 1\nx : ℝ\n⊢ rexp (t * x) = rexp ((1 + t) / 2 * x + (1 - t) / 2 * -x)",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
... | [] | ring_nf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Monovary | {
"line": 376,
"column": 47
} | {
"line": 376,
"column": 80
} | {
"line": 376,
"column": 80
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁷ : Ring α\ninst✝⁶ : LinearOrder α\ninst✝⁵ : IsStrictOrderedRing α\ninst✝⁴ : AddCommGroup β\ninst✝³ : LinearOrder β\ninst✝² : IsOrderedAddMonoid β\ninst✝¹ : Module α β\ninst✝ : IsStrictOrderedModule α β\nf : ι → α\ng : ι → β\ns : Set ι\n⊢ MonovaryOn f (⇑Or... | [
"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁷ : Ring α\ninst✝⁶ : LinearOrder α\ninst✝⁵ : IsStrictOrderedRing α\ninst✝⁴ : AddCommGroup β\ninst✝³ : LinearOrder β\ninst✝² : IsOrderedAddMonoid β\ninst✝¹ : Module α β\ninst✝ : IsStrictOrderedModule α β\nf : ι → α\ng : ι → β\ns : Set ι\n⊢ (∀ ⦃i : ι⦄, i ∈ s → ∀ ⦃j : ι⦄... | monovaryOn_iff_forall_smul_nonneg | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 138,
"column": 42
} | {
"line": 138,
"column": 73
} | {
"line": 139,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np : ℝ\nhp0 : 0 ≤ p\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhf_zero : ∫⁻ (a : α), f a ^ p ∂μ = 0\nh_mul_zero : f * g =ᵐ[μ] 0 * g\n⊢ f * g =ᵐ[μ] 0",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
... | [] | by rwa [zero_mul] at h_mul_zero | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.MeanInequalities | {
"line": 176,
"column": 6
} | {
"line": 178,
"column": 25
} | {
"line": 179,
"column": 6
} | [
{
"pp": "ι : Type u\ns : Finset ι\nw z : ι → ℝ\nx : ℝ\nhw : ∀ i ∈ s, 0 ≤ w i\nhw' : ∑ i ∈ s, w i = 1\nhz : ∀ i ∈ s, 0 ≤ z i\nhx : ∀ i ∈ s, w i ≠ 0 → z i = x\n⊢ 0 ≤ x",
"ppTerm": "?m.106",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"FloorRing.t... | [
"ι : Type u\ns : Finset ι\nw z : ι → ℝ\nx : ℝ\nhw : ∀ i ∈ s, 0 ≤ w i\nhw' : ∑ i ∈ s, w i = 1\nhz : ∀ i ∈ s, 0 ≤ z i\nhx : ∀ i ∈ s, w i ≠ 0 → z i = x\nthis : ∑ i ∈ s, w i ≠ 0\n⊢ 0 ≤ x"
] | have : (∑ i ∈ s, w i) ≠ 0 := by
rw [hw']
exact one_ne_zero | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.MeanInequalities | {
"line": 210,
"column": 4
} | {
"line": 219,
"column": 22
} | {
"line": 220,
"column": 4
} | [
{
"pp": "case pos\nι : Type u\ns : Finset ι\nw z : ι → ℝ\nhw : ∀ i ∈ s, 0 < w i\nhw' : ∑ i ∈ s, w i = 1\nhz : ∀ i ∈ s, 0 ≤ z i\ni : ι\nhis : i ∈ s\nhzi : z i = 0\nhwi : w i ≠ 0\n⊢ 0 = ∑ i ∈ s, w i * z i ↔ ∀ j ∈ s, z j = ∑ i ∈ s, w i * z i",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
... | [
"case pos\nι : Type u\ns : Finset ι\nw z : ι → ℝ\nhw : ∀ i ∈ s, 0 < w i\nhw' : ∑ i ∈ s, w i = 1\nhz : ∀ i ∈ s, 0 ≤ z i\ni : ι\nhis : i ∈ s\nhzi : z i = 0\nhwi : w i ≠ 0\n⊢ z i ^ w i = 0"
] | · constructor
· intro h
rw [← h]
intro j hj
apply eq_zero_of_ne_zero_of_mul_left_eq_zero (ne_of_lt (hw j hj)).symm
apply (sum_eq_zero_iff_of_nonneg ?_).mp h.symm j hj
exact fun i hi => (mul_nonneg_iff_of_pos_left (hw i hi)).mpr (hz i hi)
· intro h
convert! h i... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.Mul | {
"line": 179,
"column": 13
} | {
"line": 179,
"column": 25
} | {
"line": 179,
"column": 25
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nn : ℕ\n⊢ ConvexOn 𝕜 (Ioi 0) fun x ↦ x ^ ↑n",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"Set.Ioi",
"instSMulOfMul",
"congrArg",
... | [
"𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nn : ℕ\n⊢ ConvexOn 𝕜 (Ioi 0) fun x ↦ x ^ n"
] | zpow_natCast | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 206,
"column": 44
} | {
"line": 206,
"column": 79
} | {
"line": 207,
"column": 8
} | [
{
"pp": "α : Type u_2\nι : Type u_3\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : ι → α → ℝ≥0∞\ni₀ : ι\ns : Finset ι\nhi₀ : i₀ ∉ s\nih :\n (∀ i ∈ s, AEMeasurable (f i) μ) →\n ∀ {p : ι → ℝ},\n ∑ i ∈ s, p i = 1 →\n (∀ i ∈ s, 0 ≤ p i) → ∫⁻ (a : α), ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ (a : α), f... | [] | by rw [h2p i hi, ENNReal.rpow_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 210,
"column": 44
} | {
"line": 210,
"column": 79
} | {
"line": 211,
"column": 4
} | [
{
"pp": "α : Type u_2\nι : Type u_3\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : ι → α → ℝ≥0∞\ni₀ : ι\ns : Finset ι\nhi₀ : i₀ ∉ s\nih :\n (∀ i ∈ s, AEMeasurable (f i) μ) →\n ∀ {p : ι → ℝ},\n ∑ i ∈ s, p i = 1 →\n (∀ i ∈ s, 0 ≤ p i) → ∫⁻ (a : α), ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ (a : α), f... | [] | by rw [h2p i hi, ENNReal.rpow_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.MeanInequalities | {
"line": 690,
"column": 4
} | {
"line": 691,
"column": 49
} | {
"line": 692,
"column": 4
} | [
{
"pp": "case refine_1\nι : Type u\nf g : ι → ℝ≥0\np q r : ℝ\nhpqr : p.HolderTriple q r\nhf : Summable fun i ↦ f i ^ p\nhg : Summable fun i ↦ g i ^ q\ns : Finset ι\nhp : 0 < p\nhq : 0 < q\nhr : 0 < r\n⊢ (∑ i ∈ s, f i ^ p) ^ (r / p) ≤ (∑' (i : ι), f i ^ p) ^ (r / p)",
"ppTerm": "?refine_1",
"assigned": t... | [
"case refine_2\nι : Type u\nf g : ι → ℝ≥0\np q r : ℝ\nhpqr : p.HolderTriple q r\nhf : Summable fun i ↦ f i ^ p\nhg : Summable fun i ↦ g i ^ q\ns : Finset ι\nhp : 0 < p\nhq : 0 < q\nhr : 0 < r\n⊢ (∑ i ∈ s, g i ^ q) ^ (r / q) ≤ (∑' (i : ι), g i ^ q) ^ (r / q)"
] | · gcongr
exact hf.sum_le_tsum _ (fun _ _ => zero_le) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.MeanInequalities | {
"line": 698,
"column": 2
} | {
"line": 698,
"column": 72
} | {
"line": 699,
"column": 2
} | [
{
"pp": "ι : Type u\nf g : ι → ℝ≥0\np q r : ℝ\nhpqr : p.HolderTriple q r\nhf : Summable fun i ↦ f i ^ p\nhg : Summable fun i ↦ g i ^ q\nH₁ : ∀ (s : Finset ι), ∑ i ∈ s, (f i * g i) ^ r ≤ (∑' (i : ι), f i ^ p) ^ (r / p) * (∑' (i : ι), g i ^ q) ^ (r / q)\nbdd : BddAbove (Set.range fun s ↦ ∑ i ∈ s, (f i * g i) ^ r)... | [
"ι : Type u\nf g : ι → ℝ≥0\np q r : ℝ\nhpqr : p.HolderTriple q r\nhf : Summable fun i ↦ f i ^ p\nhg : Summable fun i ↦ g i ^ q\nH₁ : ∀ (s : Finset ι), ∑ i ∈ s, (f i * g i) ^ r ≤ (∑' (i : ι), f i ^ p) ^ (r / p) * (∑' (i : ι), g i ^ q) ^ (r / q)\nbdd : BddAbove (Set.range fun s ↦ ∑ i ∈ s, (f i * g i) ^ r)\nH₂ : Summa... | have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.MeanInequalities | {
"line": 836,
"column": 2
} | {
"line": 836,
"column": 72
} | {
"line": 837,
"column": 2
} | [
{
"pp": "ι : Type u\nf g : ι → ℝ≥0\np : ℝ\nhp : 1 ≤ p\nhf : Summable fun i ↦ f i ^ p\nhg : Summable fun i ↦ g i ^ p\npos : 0 < p\nH₁ :\n ∀ (s : Finset ι), ∑ i ∈ s, (f i + g i) ^ p ≤ ((∑' (i : ι), f i ^ p) ^ (1 / p) + (∑' (i : ι), g i ^ p) ^ (1 / p)) ^ p\nbdd : BddAbove (Set.range fun s ↦ ∑ i ∈ s, (f i + g i) ^... | [
"ι : Type u\nf g : ι → ℝ≥0\np : ℝ\nhp : 1 ≤ p\nhf : Summable fun i ↦ f i ^ p\nhg : Summable fun i ↦ g i ^ p\npos : 0 < p\nH₁ :\n ∀ (s : Finset ι), ∑ i ∈ s, (f i + g i) ^ p ≤ ((∑' (i : ι), f i ^ p) ^ (1 / p) + (∑' (i : ι), g i ^ p) ^ (1 / p)) ^ p\nbdd : BddAbove (Set.range fun s ↦ ∑ i ∈ s, (f i + g i) ^ p)\nH₂ : Su... | have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 416,
"column": 94
} | {
"line": 419,
"column": 44
} | {
"line": 420,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhp0 : 0 ≤ p\nhp1 : p ≤ 1\nhp : 0 < p\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ + ∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)",
"ppTerm": "?m.193",
"assigned": true,
"usedC... | [] | by
rw [← lintegral_add_left' (hf.pow_const p)]
gcongr with a
exact rpow_add_le_add_rpow _ _ hp0 hp1 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.ConvergenceInMeasure | {
"line": 209,
"column": 34
} | {
"line": 209,
"column": 50
} | {
"line": 209,
"column": 50
} | [
{
"pp": "case neg\nα : Type u_1\nE : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : PseudoEMetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), Measurable fun a ↦ edist (f n a) (g a)\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))\nε : ℝ≥0∞\nhε : 0 < ε\nδ ... | [
"case neg\nα : Type u_1\nE : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : PseudoEMetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), Measurable fun a ↦ edist (f n a) (g a)\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))\nε : ℝ≥0∞\nhε : 0 < ε\nδ : ℝ≥0\nhδ : ... | ← NNReal.coe_pos | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Operator.NormedSpace | {
"line": 305,
"column": 2
} | {
"line": 305,
"column": 41
} | {
"line": 306,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_5\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : NormedSpace 𝕜 E\nx : E\nh : x ≠ 0\n⊢ ‖coord 𝕜 x h‖ = ‖x‖⁻¹",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Norm.... | [
"𝕜 : Type u_1\nE : Type u_5\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : NormedSpace 𝕜 E\nx : E\nh : x ≠ 0\nhx : 0 < ‖x‖\n⊢ ‖coord 𝕜 x h‖ = ‖x‖⁻¹"
] | have hx : 0 < ‖x‖ := norm_pos_iff.mpr h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Operator.NormedSpace | {
"line": 305,
"column": 2
} | {
"line": 308,
"column": 83
} | {
"line": 310,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_5\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : NormedSpace 𝕜 E\nx : E\nh : x ≠ 0\n⊢ ‖coord 𝕜 x h‖ = ‖x‖⁻¹",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Norm.... | [] | have hx : 0 < ‖x‖ := norm_pos_iff.mpr h
haveI : Nontrivial (𝕜 ∙ x) := Submodule.nontrivial_span_singleton h
exact ContinuousLinearMap.homothety_norm _ fun y =>
homothety_inverse _ hx _ (LinearEquiv.toSpanNonzeroSingleton_homothety 𝕜 x h) _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Operator.NormedSpace | {
"line": 305,
"column": 2
} | {
"line": 308,
"column": 83
} | {
"line": 310,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_5\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : NormedSpace 𝕜 E\nx : E\nh : x ≠ 0\n⊢ ‖coord 𝕜 x h‖ = ‖x‖⁻¹",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Norm.... | [] | have hx : 0 < ‖x‖ := norm_pos_iff.mpr h
haveI : Nontrivial (𝕜 ∙ x) := Submodule.nontrivial_span_singleton h
exact ContinuousLinearMap.homothety_norm _ fun y =>
homothety_inverse _ hx _ (LinearEquiv.toSpanNonzeroSingleton_homothety 𝕜 x h) _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Idempotent | {
"line": 94,
"column": 95
} | {
"line": 98,
"column": 72
} | {
"line": 100,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : TopologicalSpace M\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsTopologicalAddGroup M\nf T : M →L[R] M\nhT : IsUnit T\nhf : IsIdempotentElem f\n⊢ Commute f T ↔ Submodule.map (↑T) (↑f).range = (↑f).range ∧ Submodule.map (↑T) (↑f).ker = (↑... | [] | by
have := hT.map ContinuousLinearMap.toLinearMapRingHom
lift T to (M →L[R] M)ˣ using hT
simpa [Commute, SemiconjBy, Module.End.mul_eq_comp, ← toLinearMap_comp] using!
LinearMap.IsIdempotentElem.commute_iff_of_isUnit this hf.toLinearMap | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Module.Multilinear.Topology | {
"line": 212,
"column": 2
} | {
"line": 212,
"column": 80
} | {
"line": 213,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nι✝ : Type u_2\nE : ι✝ → Type u_3\nF : Type u_4\ninst✝⁷ : NormedField 𝕜\ninst✝⁶ : (i : ι✝) → TopologicalSpace (E i)\ninst✝⁵ : (i : ι✝) → AddCommGroup (E i)\ninst✝⁴ : (i : ι✝) → Module 𝕜 (E i)\ninst✝³ : AddCommGroup F\ninst✝² : Module 𝕜 F\ninst✝¹ : TopologicalSpace F\ninst✝ : IsTopologi... | [
"case refine_1\n𝕜 : Type u_1\nι✝ : Type u_2\nE : ι✝ → Type u_3\nF : Type u_4\ninst✝⁷ : NormedField 𝕜\ninst✝⁶ : (i : ι✝) → TopologicalSpace (E i)\ninst✝⁵ : (i : ι✝) → AddCommGroup (E i)\ninst✝⁴ : (i : ι✝) → Module 𝕜 (E i)\ninst✝³ : AddCommGroup F\ninst✝² : Module 𝕜 F\ninst✝¹ : TopologicalSpace F\ninst✝ : IsTopol... | refine (UniformOnFun.hasBasis_nhds_zero_of_basis _ ?_ ?_ h).comap DFunLike.coe | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 178,
"column": 92
} | {
"line": 180,
"column": 82
} | {
"line": 182,
"column": 0
} | [
{
"pp": "α : Type u_1\nε : Type u_3\nmα : MeasurableSpace α\nf : α → ε\ns t : Set α\nμ : Measure α\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\nh : IntegrableOn f s μ\n⊢ IntegrableOn f s (μ.restrict t)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"MeasureTheory.Integra... | [] | by
dsimp only [IntegrableOn] at h ⊢
exact h.mono_measure <| Measure.restrict_mono_measure Measure.restrict_le_self _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 139,
"column": 59
} | {
"line": 144,
"column": 53
} | {
"line": 146,
"column": 0
} | [
{
"pp": "𝕜 : Type u\nι : Type v\nE : ι → Type wE\nG : Type wG\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹ : SeminormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : MultilinearMap 𝕜 E G\nhf : Continuous[Pi.topologicalSpa... | [] | by
classical
rw [← inseparable_zero_iff_norm] at hi ⊢
have : Inseparable (update m i 0) m := inseparable_pi.2 <|
(forall_update_iff m fun i a ↦ Inseparable a (m i)).2 ⟨hi.symm, fun _ _ ↦ rfl⟩
simpa only [map_update_zero] using this.symm.map hf | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 586,
"column": 45
} | {
"line": 594,
"column": 47
} | {
"line": 596,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NormedAddCommGroup β\nR : Type u_8\ninst✝² : NormedRing R\ninst✝¹ : Module R β\ninst✝ : IsBoundedSMul R β\nf : α → β\nhf : Integrable f μ\ng : α → R\ng_aestronglyMeasurable : AEStronglyMeasurable g μ\ness_sup_g : essSup (fun x ↦... | [] | by
rw [← memLp_one_iff_integrable] at *
refine ⟨g_aestronglyMeasurable.smul hf.1, ?_⟩
have hg' : eLpNorm g ∞ μ ≠ ∞ := by rwa [eLpNorm_exponent_top]
calc
eLpNorm (fun x : α => g x • f x) 1 μ ≤ _ := by
simpa using! MeasureTheory.eLpNorm_smul_le_mul_eLpNorm hf.1 g_aestronglyMeasurable
(p := ∞) (q... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 756,
"column": 2
} | {
"line": 756,
"column": 77
} | {
"line": 758,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetrizableSpace α\ninst✝² : OpensMeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : PseudoMetrizableSpace β\nf : α → β\ns : Set α\nμ : Measure α\nhf : ContinuousOn f s\nhs : MeasurableSet s\nh's : IsSep... | [] | exact mem_of_superset (self_mem_ae_restrict hs) (subset_preimage_image _ _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 830,
"column": 19
} | {
"line": 832,
"column": 19
} | {
"line": 833,
"column": 2
} | [
{
"pp": "𝕜 : Type u\nι : Type v\nι' : Type v'\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nG : Type wG\nG' : Type wG'\ninst✝¹⁰ : Fintype ι'\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝⁷ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁶ : (i : ι) → SeminormedAd... | [] | by
ext
simp [smul_add] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.LocallyIntegrable | {
"line": 288,
"column": 2
} | {
"line": 288,
"column": 36
} | {
"line": 289,
"column": 2
} | [
{
"pp": "X : Type u_1\nε : Type u_3\ninst✝⁴ : MeasurableSpace X\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\nf : X → ε\nμ : Measure X\ns : Set X\ninst✝ : OpensMeasurableSpace X\nhf : LocallyIntegrable f (μ.restrict s)\nx : X\na✝ : x ∈ s\n⊢ IntegrableAtFilter f (𝓝[s] x)... | [
"X : Type u_1\nε : Type u_3\ninst✝⁴ : MeasurableSpace X\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\nf : X → ε\nμ : Measure X\ns : Set X\ninst✝ : OpensMeasurableSpace X\nhf : LocallyIntegrable f (μ.restrict s)\nx : X\na✝ : x ∈ s\nt : Set X\nht_mem : t ∈ 𝓝 x\nht_int : Integ... | obtain ⟨t, ht_mem, ht_int⟩ := hf x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 999,
"column": 2
} | {
"line": 1002,
"column": 14
} | {
"line": 1004,
"column": 0
} | [
{
"pp": "case refine_2\n𝕜 : Type u\nι : Type v\nι' : Type v'\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nG : Type wG\nG' : Type wG'\ninst✝¹⁰ : Fintype ι'\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝⁷ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁶ : (i : ι)... | [] | · intro f
suffices ‖f.flipLinear‖ ≤ ‖f‖ by simpa
apply MultilinearMap.mkContinuousLinear_norm_le
positivity | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Algebra.Module.FiniteDimension | {
"line": 754,
"column": 2
} | {
"line": 754,
"column": 22
} | {
"line": 755,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : CompleteSpace 𝕜\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : IsTopologicalAddGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : ContinuousSMul 𝕜 E\nA B : Submodule 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 ↥A\nhA : A.ClosedC... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : CompleteSpace 𝕜\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : IsTopologicalAddGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : ContinuousSMul 𝕜 E\nA B : Submodule 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 ↥A\ninst✝ : T2Space ↥A\nhB : B... | obtain ⟨p, hp⟩ := hA | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 429,
"column": 23
} | {
"line": 429,
"column": 49
} | {
"line": 430,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nE : Type u_4\nF : Type u_5\n𝕜 : Type u_6\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\np : ℝ≥0∞\nμ : Measure α\ninst✝² : NormedRing 𝕜\ninst✝¹ : Module 𝕜 E\ninst✝ : IsBoundedSMul 𝕜 E\nx y : 𝕜\nf : ↥(simpleFunc E ... | [] | ext1; exact mul_smul _ _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 429,
"column": 23
} | {
"line": 429,
"column": 49
} | {
"line": 430,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nE : Type u_4\nF : Type u_5\n𝕜 : Type u_6\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\np : ℝ≥0∞\nμ : Measure α\ninst✝² : NormedRing 𝕜\ninst✝¹ : Module 𝕜 E\ninst✝ : IsBoundedSMul 𝕜 E\nx y : 𝕜\nf : ↥(simpleFunc E ... | [] | ext1; exact mul_smul _ _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LocallyIntegrable | {
"line": 538,
"column": 2
} | {
"line": 542,
"column": 59
} | {
"line": 544,
"column": 0
} | [
{
"pp": "case mpr\nX : Type u_1\nε : Type u_3\ninst✝⁸ : MeasurableSpace X\ninst✝⁷ : TopologicalSpace X\ninst✝⁶ : TopologicalSpace ε\ninst✝⁵ : ContinuousENorm ε\nf : X → ε\nμ : Measure X\ninst✝⁴ : PseudoMetrizableSpace ε\na : X\ninst✝³ : LinearOrder X\ninst✝² : CompactIccSpace X\ninst✝¹ : NoMinOrder X\ninst✝ : O... | [] | · intro ⟨hbot, ⟨s, hsl, hs⟩, hlocal⟩
obtain ⟨s', ⟨hs'_mono, hs'⟩⟩ := mem_nhdsLT_iff_exists_Ioo_subset.mp hsl
refine (integrableOn_union.mpr ⟨?_, hs.mono hs' le_rfl⟩).mono Iio_subset_Iic_union_Ioo le_rfl
exact integrableOn_Iic_iff_integrableAtFilter_atBot.mpr
⟨hbot, hlocal.mono_set (Iic_subset_Iio.mpr ... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 141,
"column": 65
} | {
"line": 143,
"column": 36
} | {
"line": 145,
"column": 0
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : PartialOrder F\ninst✝ : IsOrderedModule ℝ F\ns : Set α\nx : F\nhx : 0 ≤ x\n⊢ 0 ≤ (weightedSMul μ s) x",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [... | [] | by
simp only [weightedSMul, _root_.id, coe_id', smul_apply]
exact smul_nonneg toReal_nonneg hx | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 338,
"column": 4
} | {
"line": 338,
"column": 17
} | {
"line": 340,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : PartialOrder F\ninst✝¹ : IsOrderedAddMonoid F\ninst✝ : IsOrderedModule ℝ F\nf : α →ₛ F\nhf : 0 ≤ᵐ[μ] ⇑f\ny : α\nhy : ¬0 ≤ f y\nx : α\nhx : f x = f y\n⊢ ¬0 x ≤ f ... | [] | exact hx ▸ hy | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 354,
"column": 4
} | {
"line": 354,
"column": 17
} | {
"line": 355,
"column": 4
} | [
{
"pp": "case pos.inl\nα : Type u_1\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : PartialOrder F\ninst✝¹ : IsOrderedAddMonoid F\ninst✝ : IsOrderedModule ℝ F\nν : Measure α\nf : α →ₛ F\nhf : 0 ≤ᵐ[ν] ⇑f\nhμν : μ ≤ ν\nhfν : Integrable (⇑f) ν\... | [
"case pos.inr\nα : Type u_1\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : PartialOrder F\ninst✝¹ : IsOrderedAddMonoid F\ninst✝ : IsOrderedModule ℝ F\nν : Measure α\nf : α →ₛ F\nhf : 0 ≤ᵐ[ν] ⇑f\nhμν : μ ≤ ν\nhfν : Integrable (⇑f) ν\nx : α\nhx✝ ... | · simp [← hx] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.FinMeasAdditive | {
"line": 496,
"column": 39
} | {
"line": 496,
"column": 75
} | {
"line": 496,
"column": 75
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nc : ℝ\nf : α →ₛ E\nhf : Integrable (⇑f) μ\nb : E\nx✝ : b ∈... | [] | by rw [map_smul (T (f ⁻¹' {b})) c b] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 203,
"column": 2
} | {
"line": 204,
"column": 66
} | {
"line": 206,
"column": 0
} | [
{
"pp": "α : Type u_1\nG : Type u_5\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nm : MeasurableSpace α\nμ : Measure α\nf : α → G\nh : ¬Integrable f μ\n⊢ ∫ (a : α), f a ∂μ = 0",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"MeasureTheory.setT... | [] | simp only [integral_eq_setToFun]
exact setToFun_undef (dominatedFinMeasAdditive_weightedSMul μ) h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 203,
"column": 2
} | {
"line": 204,
"column": 66
} | {
"line": 206,
"column": 0
} | [
{
"pp": "α : Type u_1\nG : Type u_5\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nm : MeasurableSpace α\nμ : Measure α\nf : α → G\nh : ¬Integrable f μ\n⊢ ∫ (a : α), f a ∂μ = 0",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"MeasureTheory.setT... | [] | simp only [integral_eq_setToFun]
exact setToFun_undef (dominatedFinMeasAdditive_weightedSMul μ) h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.FinMeasAdditive | {
"line": 621,
"column": 10
} | {
"line": 621,
"column": 35
} | {
"line": 622,
"column": 2
} | [
{
"pp": "case neg\nα : Type u_1\nF : Type u_3\nF' : Type u_4\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : NormedSpace ℝ F'\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\nhs_empty : s.Nonempty\nhs_un... | [] | exact Set.mem_singleton x | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.FinMeasAdditive | {
"line": 621,
"column": 10
} | {
"line": 621,
"column": 35
} | {
"line": 622,
"column": 2
} | [
{
"pp": "case neg\nα : Type u_1\nF : Type u_3\nF' : Type u_4\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : NormedSpace ℝ F'\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\nhs_empty : s.Nonempty\nhs_un... | [] | exact Set.mem_singleton x | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.FinMeasAdditive | {
"line": 621,
"column": 10
} | {
"line": 621,
"column": 35
} | {
"line": 622,
"column": 2
} | [
{
"pp": "case neg\nα : Type u_1\nF : Type u_3\nF' : Type u_4\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : NormedSpace ℝ F'\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\nhs_empty : s.Nonempty\nhs_un... | [] | exact Set.mem_singleton x | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 488,
"column": 4
} | {
"line": 494,
"column": 70
} | {
"line": 495,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : 0 ≤ᵐ[μ] f\nhfm : AEStronglyMeasurable f μ\nhfi : Integrable f μ\n⊢ (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal - (∫⁻ (a : α), ENNReal.ofReal (-f a) ∂μ).toReal =\n (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal",
"ppTerm... | [
"case pos\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : 0 ≤ᵐ[μ] f\nhfm : AEStronglyMeasurable f μ\nhfi : Integrable f μ\nh_min : ∫⁻ (a : α), ENNReal.ofReal (-f a) ∂μ = 0\n⊢ (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal - (∫⁻ (a : α), ENNReal.ofReal (-f a) ∂μ).toReal =\n (∫⁻ (a : α), ENNReal... | have h_min : ∫⁻ a, ENNReal.ofReal (-f a) ∂μ = 0 := by
rw [lintegral_eq_zero_iff']
· refine hf.mono ?_
simp only [Pi.zero_apply]
intro a h
simp only [h, neg_nonpos, ofReal_eq_zero]
· exact measurable_ofReal.comp_aemeasurable hfm.aemeasurable.neg | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 198,
"column": 2
} | {
"line": 203,
"column": 84
} | {
"line": 205,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : NormedField β\nu v : α → β\nl : Filter α\nhuv : Tendsto (u / v) l (𝓝 1)\n⊢ u ~[l] v",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NormedCommRing.toSeminormedCommRing",
"Fals... | [] | suffices ∀ᶠ x in l, v x = 0 → u x = 0 by
rw [isEquivalent_iff_exists_eq_mul]
exact ⟨u / v, huv, this.mono fun x hz' ↦ (div_mul_cancel_of_imp hz').symm⟩
by_contra! h
replace h : ∃ᶠ t in l, (u / v) t = 0 := h.mono fun x ⟨hv, hu⟩ ↦ by simp [hv]
simpa using tendsto_nhds_unique_of_frequently_eq (b := 0) huv te... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 198,
"column": 2
} | {
"line": 203,
"column": 84
} | {
"line": 205,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : NormedField β\nu v : α → β\nl : Filter α\nhuv : Tendsto (u / v) l (𝓝 1)\n⊢ u ~[l] v",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NormedCommRing.toSeminormedCommRing",
"Fals... | [] | suffices ∀ᶠ x in l, v x = 0 → u x = 0 by
rw [isEquivalent_iff_exists_eq_mul]
exact ⟨u / v, huv, this.mono fun x hz' ↦ (div_mul_cancel_of_imp hz').symm⟩
by_contra! h
replace h : ∃ᶠ t in l, (u / v) t = 0 := h.mono fun x ⟨hv, hu⟩ ↦ by simp [hv]
simpa using tendsto_nhds_unique_of_frequently_eq (b := 0) huv te... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.AffineSubspace | {
"line": 106,
"column": 2
} | {
"line": 107,
"column": 54
} | {
"line": 108,
"column": 2
} | [
{
"pp": "case inr\nR : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module R V\ninst✝⁴ : TopologicalSpace P\ninst✝³ : AddTorsor V P\ninst✝² : TopologicalSpace V\ninst✝¹ : IsTopologicalAddTorsor P\ninst✝ : T1Space V\ns : AffineSubspace R P\nx : P\nhx : x ∈ ↑s\n⊢ IsClos... | [
"case inr\nR : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module R V\ninst✝⁴ : TopologicalSpace P\ninst✝³ : AddTorsor V P\ninst✝² : TopologicalSpace V\ninst✝¹ : IsTopologicalAddTorsor P\ninst✝ : T1Space V\ns : AffineSubspace R P\nx : P\nhx : x ∈ ↑s\n⊢ IsClosed ((fun x_1... | rw [← (Homeomorph.vaddConst x).symm.isClosed_image,
AffineSubspace.coe_direction_eq_vsub_set_right hx] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 827,
"column": 2
} | {
"line": 873,
"column": 69
} | {
"line": 875,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : ℕ → α → ℝ\nF : α → ℝ\nhf_int : ∀ (n : ℕ), Integrable (f n) μ\nhF_int : Integrable F μ\nhf_tendsto : Tendsto (fun i ↦ ∫ (a : α), f i a ∂μ) atTop (𝓝 (∫ (a : α), F a ∂μ))\nhf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a\nhf_bound : ∀ᵐ (a : α) ∂μ, ∀ ... | [] | let f' : ℕ → α → ℝ≥0∞ := fun n a ↦ ENNReal.ofReal (f n a - f 0 a)
let F' : α → ℝ≥0∞ := fun a ↦ ENNReal.ofReal (F a - f 0 a)
have hf'_int_eq : ∀ i, ∫⁻ a, f' i a ∂μ = ENNReal.ofReal (∫ a, f i a ∂μ - ∫ a, f 0 a ∂μ) := by
intro i
unfold f'
rw [← ofReal_integral_eq_lintegral_ofReal, integral_sub (hf_int i) (... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 827,
"column": 2
} | {
"line": 873,
"column": 69
} | {
"line": 875,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : ℕ → α → ℝ\nF : α → ℝ\nhf_int : ∀ (n : ℕ), Integrable (f n) μ\nhF_int : Integrable F μ\nhf_tendsto : Tendsto (fun i ↦ ∫ (a : α), f i a ∂μ) atTop (𝓝 (∫ (a : α), F a ∂μ))\nhf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a\nhf_bound : ∀ᵐ (a : α) ∂μ, ∀ ... | [] | let f' : ℕ → α → ℝ≥0∞ := fun n a ↦ ENNReal.ofReal (f n a - f 0 a)
let F' : α → ℝ≥0∞ := fun a ↦ ENNReal.ofReal (F a - f 0 a)
have hf'_int_eq : ∀ i, ∫⁻ a, f' i a ∂μ = ENNReal.ofReal (∫ a, f i a ∂μ - ∫ a, f 0 a ∂μ) := by
intro i
unfold f'
rw [← ofReal_integral_eq_lintegral_ofReal, integral_sub (hf_int i) (... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.SumMeasure | {
"line": 134,
"column": 56
} | {
"line": 134,
"column": 77
} | {
"line": 134,
"column": 78
} | [
{
"pp": "ι : Type u_1\nX : Type u_2\nE : Type u_3\nmX : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\nμ : ι → Measure X\nf : X → E\ninst✝ : NormedSpace ℝ E\nhf : Integrable f (Measure.sum μ)\nhfi : ∀ (i : ι), Integrable f (μ i)\nε : ℝ≥0\nε0 : 0 < ↑ε\nhf_lt : ∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ < ∞\nhmem : ∀ᶠ ... | [
"ι : Type u_1\nX : Type u_2\nE : Type u_3\nmX : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\nμ : ι → Measure X\nf : X → E\ninst✝ : NormedSpace ℝ E\nhf : Integrable f (Measure.sum μ)\nhfi : ∀ (i : ι), Integrable f (μ i)\nε : ℝ≥0\nε0 : 0 < ↑ε\nhf_lt : ∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ < ∞\nhmem : ∀ᶠ (y : ℝ≥0∞) i... | ← ENNReal.coe_lt_coe, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 1198,
"column": 4
} | {
"line": 1198,
"column": 57
} | {
"line": 1199,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\np q : ℝ\nhpq : p.HolderConjugate q\nf g : α → ℝ\nhf_nonneg : 0 ≤ᵐ[μ] f\nhg_nonneg : 0 ≤ᵐ[μ] g\nhf : MemLp f (ENNReal.ofReal p) μ\nhg : MemLp g (ENNReal.ofReal q) μ\nx : α\nhxf : 0 x ≤ f x\nhxg : 0 x ≤ g x\n⊢ f x * g x = ‖f x‖ * ‖g x‖",
"ppTerm": "... | [] | rw [Real.norm_of_nonneg hxf, Real.norm_of_nonneg hxg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.ContinuousMap.Bounded.Basic | {
"line": 257,
"column": 42
} | {
"line": 257,
"column": 71
} | {
"line": 258,
"column": 4
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : PseudoMetricSpace β\nf : α →ᵇ β\nl : Filter (α →ᵇ β)\n⊢ Tendsto id l (𝓝 f) ↔ Tendsto (⇑UniformFun.ofFun ∘ DFunLike.coe) l (𝓝 ((⇑UniformFun.ofFun ∘ DFunLike.coe) f))",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
... | [
"α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : PseudoMetricSpace β\nf : α →ᵇ β\nl : Filter (α →ᵇ β)\n⊢ TendstoUniformly (fun i ↦ ⇑(id i)) (⇑f) l ↔\n Tendsto (⇑UniformFun.ofFun ∘ DFunLike.coe) l (𝓝 ((⇑UniformFun.ofFun ∘ DFunLike.coe) f))"
] | tendsto_iff_tendstoUniformly, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.FiniteDimension | {
"line": 618,
"column": 2
} | {
"line": 619,
"column": 29
} | {
"line": 620,
"column": 2
} | [
{
"pp": "α : Type u_1\nN : ℕ\ng : α → Fin N → ℝ\nhg : Summable g\nthis : ∀ (i : Fin N), Summable fun x ↦ ‖g x i‖\nx : α\n⊢ ∀ (i : Fin N), ‖g x i‖ ≤ ∑ i, ‖g x i‖",
"ppTerm": "?m.280",
"assigned": true,
"usedConstants": [
"Finset.mem_univ",
"Norm.norm",
"NormedCommRing.toSeminormedCo... | [
"α : Type u_1\nN : ℕ\ng : α → Fin N → ℝ\nhg : Summable g\nthis : ∀ (i : Fin N), Summable fun x ↦ ‖g x i‖\nx : α\n⊢ 0 ≤ ∑ i, ‖g x i‖"
] | · refine fun i => Finset.single_le_sum (f := fun i => ‖g x i‖) (fun i _ => ?_) (Finset.mem_univ i)
exact norm_nonneg (g x i) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.ContinuousMap.Compact | {
"line": 248,
"column": 4
} | {
"line": 248,
"column": 48
} | {
"line": 249,
"column": 4
} | [
{
"pp": "case a\nα : Type u_1\nE : Type u_3\ninst✝³ : TopologicalSpace α\ninst✝² : CompactSpace α\ninst✝¹ : SeminormedAddCommGroup E\nf : C(α, E)\ninst✝ : Fintype α\n⊢ ‖f‖ ≤ ‖⇑f‖",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.partialOrder",
... | [
"case a\nα : Type u_1\nE : Type u_3\ninst✝³ : TopologicalSpace α\ninst✝² : CompactSpace α\ninst✝¹ : SeminormedAddCommGroup E\nf : C(α, E)\ninst✝ : Fintype α\n⊢ ∀ (x : α), ‖f x‖ ≤ ‖⇑f‖"
] | rw [ContinuousMap.norm_le _ (by positivity)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 1226,
"column": 2
} | {
"line": 1226,
"column": 65
} | {
"line": 1227,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC : ℝ\nf : α → E\nhT : DominatedFinMeasAdditive μ T C\nhC : 0 ≤ C\nhF : Compl... | [
"case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC : ℝ\nf : α → E\nhT : DominatedFinMeasAdditive μ T C\nhC : 0 ≤ C\nhF : CompleteSpace F\n... | simp only [toReal_enorm, toReal_mul, coe_toReal, NNReal.coe_mk] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 21
} | {
"line": 79,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝² : LinearOrder R\ninst✝¹ : TopologicalSpace R\ninst✝ : OrderTopology R\na b : R\nh : ¬IsTop b\n⊢ IsOpen[inst✝¹] (Ioo a b)",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"_private.Mathlib.MeasureTheory.Measure.Stieltjes.0.isOpen_Iotop._simp_1_3"... | [] | · simp [isOpen_Ioo] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 407,
"column": 4
} | {
"line": 407,
"column": 46
} | {
"line": 408,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : LinearOrder R\ninst✝³ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝² : OrderTopology R\ninst✝¹ : CompactIccSpace R\ninst✝ : DenselyOrdered R\na b : R\nhab : a < b\ns : ℕ → Set R\nhs : Ioc a b ⊆ ⋃ i, s i\nε : ℝ≥0\nεpos : 0 < ε\nh : ∑' (i : ℕ), f.length (s i) < ∞\nδ : ℝ≥0 :=... | [
"R : Type u_1\ninst✝⁴ : LinearOrder R\ninst✝³ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝² : OrderTopology R\ninst✝¹ : CompactIccSpace R\ninst✝ : DenselyOrdered R\na b : R\nhab : a < b\ns : ℕ → Set R\nhs : Ioc a b ⊆ ⋃ i, s i\nε : ℝ≥0\nεpos : 0 < ε\nh : ∑' (i : ℕ), f.length (s i) < ∞\nδ : ℝ≥0 := ε / 2\nδpos... | simp only [iInf_lt_iff, exists_prop] at hl | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 471,
"column": 2
} | {
"line": 471,
"column": 59
} | {
"line": 472,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : LinearOrder R\ninst✝⁵ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁴ : OrderTopology R\ninst✝³ : CompactIccSpace R\ninst✝² : MeasurableSpace R\ninst✝¹ : BorelSpace R\ninst✝ : DenselyOrdered R\n⊢ f.outer.trim = f.outer",
"ppTerm": "?m.19",
"assigned": true,
"use... | [
"R : Type u_1\ninst✝⁶ : LinearOrder R\ninst✝⁵ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁴ : OrderTopology R\ninst✝³ : CompactIccSpace R\ninst✝² : MeasurableSpace R\ninst✝¹ : BorelSpace R\ninst✝ : DenselyOrdered R\ns : Set R\n⊢ f.outer.trim s ≤ f.outer s"
] | refine le_antisymm (fun s => ?_) (OuterMeasure.le_trim _) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 609,
"column": 4
} | {
"line": 609,
"column": 90
} | {
"line": 610,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\nhx : ∃ x, IsBot x\n⊢ f.measure bot... | [] | simp [botSet_eq_singleton_of_isBot hx.choose_spec, leftLim_eq_of_isBot hx.choose_spec] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 609,
"column": 4
} | {
"line": 609,
"column": 90
} | {
"line": 610,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\nhx : ∃ x, IsBot x\n⊢ f.measure bot... | [] | simp [botSet_eq_singleton_of_isBot hx.choose_spec, leftLim_eq_of_isBot hx.choose_spec] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 609,
"column": 4
} | {
"line": 609,
"column": 90
} | {
"line": 610,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\nhx : ∃ x, IsBot x\n⊢ f.measure bot... | [] | simp [botSet_eq_singleton_of_isBot hx.choose_spec, leftLim_eq_of_isBot hx.choose_spec] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Content | {
"line": 348,
"column": 4
} | {
"line": 348,
"column": 67
} | {
"line": 349,
"column": 4
} | [
{
"pp": "G : Type w\ninst✝² : TopologicalSpace G\nμ : Content G\ninst✝¹ : R1Space G\nS : MeasurableSpace G\ninst✝ : BorelSpace G\nU : Set G\nhU : U ∈ {s | IsOpen s}\nU' : Opens G\nthis✝¹ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }\nL : Compacts G\nL' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }\nhL : ↑L ⊆ ↑U... | [
"G : Type w\ninst✝² : TopologicalSpace G\nμ : Content G\ninst✝¹ : R1Space G\nS : MeasurableSpace G\ninst✝ : BorelSpace G\nU : Set G\nhU : U ∈ {s | IsOpen s}\nU' : Opens G\nthis✝¹ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }\nL : Compacts G\nL' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }\nhL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U\n... | simp only [Compacts.coe_sup, union_subset_iff, hL'U', true_and] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 787,
"column": 4
} | {
"line": 788,
"column": 67
} | {
"line": 789,
"column": 2
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\nf g : StieltjesFunction R\na b : R\nh : a ≤ b\nha : (𝓝[<] ... | [] | · exact tendsto_nhds_unique ((f + g).mono.tendsto_leftLim a)
((f.mono.tendsto_leftLim a).add (g.mono.tendsto_leftLim a)) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 792,
"column": 2
} | {
"line": 792,
"column": 9
} | {
"line": 794,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\nf g : StieltjesFunction R\na b : R\nh : a ≤ b\nthis : leftLim (↑(f + ... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 764,
"column": 6
} | {
"line": 764,
"column": 54
} | {
"line": 764,
"column": 55
} | [
{
"pp": "G : Type u_1\nα : Type u_3\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : MeasurableSpace α\nν : Measure α\nμ : Measure (Quotient α_mod_G)\ni : QuotientMeasureEqMeasurePreimage ν μ\nt : Set α\nfund_dom_t : IsFundamentalDomain G t ν\nU : Set (Quotient α_mod_G)\nmeas_U : MeasurableSet U\n⊢ μ U = ν (Q... | [
"G : Type u_1\nα : Type u_3\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : MeasurableSpace α\nν : Measure α\nμ : Measure (Quotient α_mod_G)\ni : QuotientMeasureEqMeasurePreimage ν μ\nt : Set α\nfund_dom_t : IsFundamentalDomain G t ν\nU : Set (Quotient α_mod_G)\nmeas_U : MeasurableSet U\n⊢ (Measure.map (Quotient... | fund_dom_t.projection_respects_measure (μ := μ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 844,
"column": 37
} | {
"line": 844,
"column": 72
} | {
"line": 844,
"column": 72
} | [
{
"pp": "G : Type u_1\nα : Type u_3\ninst✝⁶ : Group G\ninst✝⁵ : MulAction G α\ninst✝⁴ : MeasurableSpace α\nν : Measure α\ninst✝³ : SMulInvariantMeasure G α ν\ninst✝² : Countable G\ninst✝¹ : MeasurableConstSMul G α\ni : SigmaFinite ν\ni' : HasFundamentalDomain G α ν\nμ : Measure (Quotient α_mod_G)\ninst✝ : Quoti... | [] | by convert! Quotient.mk'_surjective | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 375,
"column": 4
} | {
"line": 375,
"column": 41
} | {
"line": 375,
"column": 41
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\neval : (Compacts G → ℝ) → ℝ := fun f ↦ f ⊥\nthis : Continuous eval\n⊢ IsClosed (eval ⁻¹' {0})",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Real",
"Co... | [
"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\neval : (Compacts G → ℝ) → ℝ := ⋯\nthis : Continuous eval\n⊢ IsClosed {0}"
] | apply continuous_iff_isClosed.mp this | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 384,
"column": 38
} | {
"line": 384,
"column": 56
} | {
"line": 385,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\neval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₀.toCompacts\nthis : Continuous eval\nU : Set G\nleft✝ : U ⊆ ↑⊤.toOpens\nh2U : IsOpen[inst✝¹] U\nh3U : 1 ∈ U\n⊢ prehaar (↑K₀) U ∈ eval ⁻¹' {1}",
... | [
"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\neval : (Compacts G → ℝ) → ℝ := ⋯\nthis : Continuous eval\nU : Set G\nleft✝ : U ⊆ ↑⊤.toOpens\nh2U : IsOpen[inst✝¹] U\nh3U : 1 ∈ U\n⊢ (interior U).Nonempty"
] | apply prehaar_self | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 386,
"column": 4
} | {
"line": 386,
"column": 41
} | {
"line": 386,
"column": 41
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\neval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₀.toCompacts\nthis : Continuous eval\n⊢ IsClosed (eval ⁻¹' {1})",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"Real... | [
"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\neval : (Compacts G → ℝ) → ℝ := ⋯\nthis : Continuous eval\n⊢ IsClosed {1}"
] | apply continuous_iff_isClosed.mp this | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 398,
"column": 4
} | {
"line": 398,
"column": 41
} | {
"line": 398,
"column": 41
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\nK₁ K₂ : Compacts G\nh : ↑K₁ ⊆ ↑K₂\neval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₂ - f K₁\nthis : Continuous eval\n⊢ IsClosed (eval ⁻¹' Ici 0)",
"ppTerm": "?m.84",
"assigned": true,
... | [
"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\nK₁ K₂ : Compacts G\nh : ↑K₁ ⊆ ↑K₂\neval : (Compacts G → ℝ) → ℝ := ⋯\nthis : Continuous eval\n⊢ IsClosed (Ici 0)"
] | apply continuous_iff_isClosed.mp this | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 411,
"column": 4
} | {
"line": 411,
"column": 41
} | {
"line": 411,
"column": 41
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\nK₁ K₂ : Compacts G\neval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₁ + f K₂ - f (K₁ ⊔ K₂)\nthis : Continuous eval\n⊢ IsClosed (eval ⁻¹' Ici 0)",
"ppTerm": "?m.113",
"assigned": true,
... | [
"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\nK₁ K₂ : Compacts G\neval : (Compacts G → ℝ) → ℝ := ⋯\nthis : Continuous eval\n⊢ IsClosed (Ici 0)"
] | apply continuous_iff_isClosed.mp this | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 445,
"column": 4
} | {
"line": 445,
"column": 41
} | {
"line": 445,
"column": 41
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\nK₁ K₂ : Compacts G\nh : Disjoint K₁.carrier K₂.carrier\nh₂ : IsClosed[inst✝¹] K₂.carrier\nU₁ U₂ : Set G\nh1U₁ : IsOpen[inst✝¹] U₁\nh1U₂ : IsOpen[inst✝¹] U₂\nh2U₁ : K₁.carrier ⊆ U₁\nh2U₂ :... | [
"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\nK₁ K₂ : Compacts G\nh : Disjoint K₁.carrier K₂.carrier\nh₂ : IsClosed[inst✝¹] K₂.carrier\nU₁ U₂ : Set G\nh1U₁ : IsOpen[inst✝¹] U₁\nh1U₂ : IsOpen[inst✝¹] U₂\nh2U₁ : K₁.carrier ⊆ U₁\nh2U₂ : K₂.carrier ... | apply continuous_iff_isClosed.mp this | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 458,
"column": 4
} | {
"line": 458,
"column": 41
} | {
"line": 458,
"column": 41
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G → ℝ) → ℝ := fun f ↦ f (Compacts.map (fun x ↦ g * x) ⋯ K) - f K\nthis : Continuous eval\n⊢ IsClosed (eval ⁻¹' {0})",
"ppTerm": "?m.103",
"... | [
"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G → ℝ) → ℝ := ⋯\nthis : Continuous eval\n⊢ IsClosed {0}"
] | apply continuous_iff_isClosed.mp this | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 674,
"column": 57
} | {
"line": 679,
"column": 46
} | {
"line": 681,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nx : E\nr : 𝕜\nhx : x ≠ 0\nhr : r ≠ 0\n⊢ ‖⟪x, r • x⟫‖ / (‖x‖ * ‖r • x‖) = 1",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"... | [] | by
have hx' : ‖x‖ ≠ 0 := by simp [hx]
have hr' : ‖r‖ ≠ 0 := by simp [hr]
rw [inner_smul_right, norm_mul, ← inner_self_re_eq_norm, inner_self_eq_norm_mul_norm, norm_smul]
rw [← mul_assoc, ← div_div, mul_div_cancel_right₀ _ hx', ← div_div, mul_comm,
mul_div_cancel_right₀ _ hr', div_self hx'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 869,
"column": 6
} | {
"line": 869,
"column": 100
} | {
"line": 870,
"column": 4
} | [
{
"pp": "F : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx y : F\n⊢ ‖x + y‖ = ‖x‖ + ‖y‖ ↔ ‖y‖ • x = ‖x‖ • y",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
... | [
"F : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx y : F\n⊢ ‖x + y‖ ^ 2 = (‖x‖ + ‖y‖) ^ 2 ↔ ‖y‖ • x = ‖x‖ • y"
] | ← pow_left_inj₀ (norm_nonneg _) (Left.add_nonneg (norm_nonneg _) (norm_nonneg _)) two_ne_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.LinearMap | {
"line": 385,
"column": 69
} | {
"line": 385,
"column": 76
} | {
"line": 386,
"column": 4
} | [
{
"pp": "𝕜 : Type u_4\ninst✝⁴ : RCLike 𝕜\nF : Type u_8\nH : Type u_9\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace 𝕜 H\na c : F\nb d : H\nha : a ≠ 0\nhb : b ≠ 0\nh : ∀ (x : H), ⟪b, x⟫_𝕜 • a = ⟪d, x⟫_𝕜 • c\nh₂ : ∀ (x : F), ⟪a, x⟫_𝕜... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.InnerProductSpace.LinearMap | {
"line": 386,
"column": 66
} | {
"line": 386,
"column": 73
} | {
"line": 387,
"column": 2
} | [
{
"pp": "𝕜 : Type u_4\ninst✝⁴ : RCLike 𝕜\nF : Type u_8\nH : Type u_9\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace 𝕜 H\na c : F\nb d : H\nha : a ≠ 0\nhb : b ≠ 0\nh : ∀ (x : H), ⟪b, x⟫_𝕜 • a = ⟪d, x⟫_𝕜 • c\nh₂ : ∀ (x : F), ⟪a, x⟫_𝕜... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.InnerProductSpace.LinearMap | {
"line": 389,
"column": 69
} | {
"line": 389,
"column": 76
} | {
"line": 390,
"column": 4
} | [
{
"pp": "𝕜 : Type u_4\ninst✝⁴ : RCLike 𝕜\nF : Type u_8\nH : Type u_9\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace 𝕜 H\na c : F\nb d : H\nha : a ≠ 0\nhb : b ≠ 0\nh : ∀ (x : H), ⟪b, x⟫_𝕜 • a = ⟪d, x⟫_𝕜 • c\nh₂ : ∀ (x : F), ⟪a, x⟫_𝕜... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.InnerProductSpace.LinearMap | {
"line": 391,
"column": 53
} | {
"line": 391,
"column": 60
} | {
"line": 392,
"column": 2
} | [
{
"pp": "𝕜 : Type u_4\ninst✝⁴ : RCLike 𝕜\nF : Type u_8\nH : Type u_9\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace 𝕜 H\na c : F\nb d : H\nha : a ≠ 0\nhb : b ≠ 0\nh : ∀ (x : H), ⟪b, x⟫_𝕜 • a = ⟪d, x⟫_𝕜 • c\nh₂ : ∀ (x : F), ⟪a, x⟫_𝕜... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Topology.Baire.CompleteMetrizable | {
"line": 60,
"column": 2
} | {
"line": 60,
"column": 88
} | {
"line": 66,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : IsCompletelyPseudoMetrizableSpace X\nx✝ : UpgradedIsCompletelyPseudoMetrizableSpace X := upgradeIsCompletelyPseudoMetrizable X\nf : ℕ → Set X\nho : ∀ (n : ℕ), IsOpen (f n)\nhd : ∀ (n : ℕ), Dense (f n)\nB : ℕ → ℝ≥0∞ := fun n ↦ 1 / 2 ^ n\nBpos : ∀ (n : ℕ... | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : IsCompletelyPseudoMetrizableSpace X\nx✝ : UpgradedIsCompletelyPseudoMetrizableSpace X := upgradeIsCompletelyPseudoMetrizable X\nf : ℕ → Set X\nho : ∀ (n : ℕ), IsOpen (f n)\nhd : ∀ (n : ℕ), Dense (f n)\nB : ℕ → ℝ≥0∞ := fun n ↦ 1 / 2 ^ n\nBpos : ∀ (n : ℕ), 0 < B n\n... | refine fun x => (mem_closure_iff_nhds_basis nhds_basis_closedEBall).2 fun ε εpos => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Algebra.Module.ClosedSubmodule | {
"line": 246,
"column": 2
} | {
"line": 246,
"column": 48
} | {
"line": 247,
"column": 2
} | [
{
"pp": "R : Type u_2\nN : Type u_4\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid N\ninst✝³ : TopologicalSpace N\ninst✝² : Module R N\ninst✝¹ : ContinuousAdd N\ninst✝ : ContinuousConstSMul R N\ns t : ClosedSubmodule R N\n⊢ ↑(s ⊔ t) = closure (↑s ⊔ ↑t).carrier",
"ppTerm": "?m.53",
"assigned": true,
"u... | [
"R : Type u_2\nN : Type u_4\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid N\ninst✝³ : TopologicalSpace N\ninst✝² : Module R N\ninst✝¹ : ContinuousAdd N\ninst✝ : ContinuousConstSMul R N\ns t : ClosedSubmodule R N\n⊢ ↑↑(↑s ⊔ ↑t).closure = closure (↑s ⊔ ↑t).carrier"
] | simp only [← coe_toSubmodule, toSubmodule_sup] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 123,
"column": 6
} | {
"line": 123,
"column": 30
} | {
"line": 123,
"column": 31
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nn k : ℕ\nx✝ : 0 ≤ k\nih : (T ^ k).IsSymmetric\n⊢ (T ^ (k + 1)).IsSymmetric",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nn k : ℕ\nx✝ : 0 ≤ k\nih : (T ^ k).IsSymmetric\n⊢ ((T ^ k) ∘ₗ T).IsSymmetric"
] | Module.End.iterate_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Projection.Submodule | {
"line": 37,
"column": 57
} | {
"line": 45,
"column": 91
} | {
"line": 47,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nK₁ K₂ : Submodule 𝕜 E\nh : K₁ ≤ K₂\ninst✝ : K₁.HasOrthogonalProjection\n⊢ K₁ ⊔ K₁ᗮ ⊓ K₂ = K₂",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Inn... | [] | by
ext x
rw [Submodule.mem_sup]
let v : K₁ := orthogonalProjectionOnto K₁ x
have hvm : x - v ∈ K₁ᗮ := sub_starProjection_mem_orthogonal x
constructor
· rintro ⟨y, hy, z, hz, rfl⟩
exact K₂.add_mem (h hy) hz.2
· exact fun hx => ⟨v, v.prop, x - v, ⟨hvm, K₂.sub_mem hx (h v.prop)⟩, add_sub_cancel _ _⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional | {
"line": 74,
"column": 45
} | {
"line": 74,
"column": 62
} | {
"line": 74,
"column": 63
} | [
{
"pp": "case pos\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\ninst✝ : FiniteDimensional 𝕜 ↥K\nhK : FiniteDimensional 𝕜 ↥Kᗮ\ne : (↥K × ↥Kᗮ) ≃ₗ[𝕜] E := K.prodEquivOfIsCompl Kᗮ ⋯\nb : Basis (Fin (finrank 𝕜 ↥K) ⊕ Fin (finr... | [
"case pos\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\ninst✝ : FiniteDimensional 𝕜 ↥K\nhK : FiniteDimensional 𝕜 ↥Kᗮ\ne : (↥K × ↥Kᗮ) ≃ₗ[𝕜] E := K.prodEquivOfIsCompl Kᗮ ⋯\nb : Basis (Fin (finrank 𝕜 ↥K) ⊕ Fin (finrank 𝕜 ↥Kᗮ))... | Fintype.card_fin, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 637,
"column": 27
} | {
"line": 637,
"column": 71
} | {
"line": 637,
"column": 71
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nw✝¹ : Submodule 𝕜 E\nw✝ : w✝¹.HasOrthogonalProjection\n⊢ (↑w✝¹.starProjection).IsSymmetricProjection",
"ppTerm": "?m.85",
"assigned": true,
"usedConstants": [
"Submodule.is... | [] | exact isSymmetricProjection_starProjection _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 222,
"column": 4
} | {
"line": 222,
"column": 86
} | {
"line": 222,
"column": 87
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nE : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜' F\nf : E →SL[σ] F\nσ' : 𝕜' →+* 𝕜\nin... | [
"case refine_1\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nE : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜' F\nf : E →SL[σ] F\nσ' : 𝕜' →+* 𝕜\... | refine (_root_.tendsto_pow_atTop_nhds_zero_of_lt_one ?_ ?_).mul tendsto_const_nhds | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 627,
"column": 81
} | {
"line": 633,
"column": 94
} | {
"line": 635,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜'\nE : Type u_3\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nσ : 𝕜 →+* 𝕜'\nσ' : 𝕜' →+* 𝕜\ninst✝⁷ : RingHomInvPair σ σ'\nF : Type u_4\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : Normed... | [] | by
refine ⟨fun h ↦ ⟨?eq_top, ?anti⟩, fun ⟨hd, c, hf⟩ ↦ ⟨hf.injective, ?surj⟩⟩
case eq_top => simpa [SetLike.ext'_iff] using! h.2.denseRange.closure_eq
case anti =>
refine ⟨_, ContinuousLinearEquiv.ofBijective f ?_ ?_ |>.antilipschitz⟩ <;>
simp only [LinearMap.range_eq_top, LinearMap.ker_eq_bot, f.coe_coe,... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 353,
"column": 10
} | {
"line": 354,
"column": 55
} | {
"line": 355,
"column": 10
} | [
{
"pp": "p : ℝ≥0∞\n𝕜 : Type u_1\nα : Type u_2\nβ : Type u_3\nhp✝ : Fact (1 ≤ p)\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nf g h : WithLp p (α × β)\nhp : 1 ≤ p.toReal\nthis :\n (∑ i ∈ {0, 1},\n ((if i = 0 then edist f.fst g.fst else edist f.snd g.snd) +\n if i = 0 then edi... | [
"p : ℝ≥0∞\n𝕜 : Type u_1\nα : Type u_2\nβ : Type u_3\nhp✝ : Fact (1 ≤ p)\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nf g h : WithLp p (α × β)\nhp : 1 ≤ p.toReal\nthis :\n (((if True then edist f.fst g.fst else edist f.snd g.snd) + if True then edist g.fst h.fst else edist g.snd h.snd) ^\n ... | simp only [Finset.mem_singleton, not_false_eq_true, Finset.sum_insert,
Finset.sum_singleton, reduceCtorEq] at this | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional | {
"line": 382,
"column": 6
} | {
"line": 382,
"column": 53
} | {
"line": 383,
"column": 6
} | [
{
"pp": "case mp.refine_2.inr\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhe'' : e ∉ v\nh_end :... | [
"case mp.refine_2.inr.inl\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhe'' : e ∉ v\nh_end : ∀ a ∈ v... | rcases eq_or_mem_of_mem_insert hb' with hb | hb | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 417,
"column": 4
} | {
"line": 417,
"column": 53
} | {
"line": 418,
"column": 4
} | [
{
"pp": "case hdiag\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : Matrix ι ι ℝ\nhM : M.det ≠ 0\nD : ι → ℝ\nhD : (Matrix.diagonal D).det ≠ 0\n⊢ Measure.map (⇑(toLin' (Matrix.diagonal D))) volume = ofReal |(Matrix.diagonal D).det⁻¹| • volume",
"ppTerm": "?hdiag",
"assigned": true,
"use... | [
"case hdiag\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : Matrix ι ι ℝ\nhM : M.det ≠ 0\nD : ι → ℝ\nhD : (Matrix.diagonal D).det ≠ 0\n⊢ Measure.map (⇑(toLin' (Matrix.diagonal D))) volume =\n ofReal |(Matrix.diagonal D).det⁻¹| •\n ofReal |(Matrix.diagonal D).det| • Measure.map (⇑(toLin' (Matri... | conv_rhs => rw [← smul_map_diagonal_volume_pi hD] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convRHS_1 | Mathlib.Tactic.Conv.convRHS |
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