module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.Seminorm | {
"line": 544,
"column": 6
} | {
"line": 544,
"column": 28
} | {
"line": 544,
"column": 29
} | [
{
"pp": "𝕜 : Type u_3\nE : Type u_7\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nι : Sort u_12\np : ι → Seminorm 𝕜 E\n⊢ BddAbove (range p) ↔ ∀ (x : E), BddAbove (range fun i ↦ (p i) x)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Norme... | [
"𝕜 : Type u_3\nE : Type u_7\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nι : Sort u_12\np : ι → Seminorm 𝕜 E\n⊢ BddAbove (DFunLike.coe '' range p) ↔ ∀ (x : E), BddAbove (range fun i ↦ (p i) x)"
] | Seminorm.bddAbove_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Seminorm | {
"line": 573,
"column": 4
} | {
"line": 573,
"column": 27
} | {
"line": 574,
"column": 4
} | [
{
"pp": "case refine_1\n𝕜 : Type u_3\nE : Type u_7\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set (Seminorm 𝕜 E)\nhs₁ : BddAbove s\nhs₂ : s.Nonempty\np : Seminorm 𝕜 E\nhp : p ∈ s\nx : E\nthis : Nonempty ↑s\n⊢ p x ≤ ⨆ i, ↑i x",
"ppTerm": "?refine_1",
"assigned": true,
... | [
"case refine_1\n𝕜 : Type u_3\nE : Type u_7\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set (Seminorm 𝕜 E)\nhs₂ : s.Nonempty\np : Seminorm 𝕜 E\nhp : p ∈ s\nx : E\nthis : Nonempty ↑s\nq : Seminorm 𝕜 E\nhq : q ∈ upperBounds s\n⊢ p x ≤ ⨆ i, ↑i x"
] | rcases hs₁ with ⟨q, hq⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Seminorm | {
"line": 1002,
"column": 2
} | {
"line": 1008,
"column": 31
} | {
"line": 1010,
"column": 0
} | [
{
"pp": "𝕜 : Type u_3\nE : Type u_7\ninst✝⁵ : NormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : NormedSpace ℝ 𝕜\ninst✝² : Module 𝕜 E\ninst✝¹ : SMul ℝ E\ninst✝ : IsScalarTower ℝ 𝕜 E\np : Seminorm 𝕜 E\nx : E\nx✝² : x ∈ univ\ny : E\nx✝¹ : y ∈ univ\na b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nx✝ : a + b = 1\n⊢ p (a • x +... | [] | calc
p (a • x + b • y) ≤ p (a • x) + p (b • y) := map_add_le_add p _ _
_ = ‖a • (1 : 𝕜)‖ * p x + ‖b • (1 : 𝕜)‖ * p y := by
rw [← map_smul_eq_mul p, ← map_smul_eq_mul p, smul_one_smul, smul_one_smul]
_ = a * p x + b * p y := by
rw [norm_smul, norm_smul, norm_one, mul_one, mul_one, Real.norm_of_... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic | {
"line": 197,
"column": 4
} | {
"line": 197,
"column": 34
} | {
"line": 198,
"column": 4
} | [
{
"pp": "case refine_2\nk : Type u_1\nV : Type u_2\nP : Type u_5\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex k P n\nw : Fin (n + 1) → k\nhw : ∑ i, w i = 1\ni : Fin (n + 1)\nh : w i = 0\n⊢ (affineCombination k univ s.points) w ∈ a... | [
"case refine_2\nk : Type u_1\nV : Type u_2\nP : Type u_5\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex k P n\nw : Fin (n + 1) → k\nhw : ∑ i, w i = 1\ni : Fin (n + 1)\nh : w i = 0\n⊢ (affineCombination k univ s.points) w ∈ affineSpan k ... | rw [range_faceOpposite_points] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.NhdsKer | {
"line": 106,
"column": 46
} | {
"line": 106,
"column": 65
} | {
"line": 106,
"column": 65
} | [
{
"pp": "X : Type u_2\ninst✝ : TopologicalSpace X\n⊢ nhdsKer ∅ = ∅",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"nhdsKer_empty"
],
"usedFVars": [
"X",
"inst✝"
],
"usedGoals": []
}
] | [] | exact nhdsKer_empty | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.AlexandrovDiscrete | {
"line": 198,
"column": 4
} | {
"line": 198,
"column": 28
} | {
"line": 199,
"column": 4
} | [
{
"pp": "α : Type u_3\ninst✝ : TopologicalSpace α\nhα : ∀ (a : α), 𝓝 a = 𝓟 (nhdsKer {a})\nS : Set (Set α)\nhS : ∀ s ∈ S, ∀ (a : α), (nhdsKer {a} ∩ s).Nonempty → a ∈ s\na : α\ns : Set α\nhs : s ∈ S\nhas : (nhdsKer {a} ∩ s).Nonempty\n⊢ a ∈ ⋃₀ S",
"ppTerm": "?m.71",
"assigned": true,
"usedConstants":... | [
"α : Type u_3\ninst✝ : TopologicalSpace α\nhα : ∀ (a : α), 𝓝 a = 𝓟 (nhdsKer {a})\nS : Set (Set α)\na : α\ns : Set α\nhS : a ∈ s\nhs : s ∈ S\nhas : (nhdsKer {a} ∩ s).Nonempty\n⊢ a ∈ ⋃₀ S"
] | specialize hS s hs a has | Lean.Elab.Tactic.evalSpecialize | Lean.Parser.Tactic.specialize |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 943,
"column": 4
} | {
"line": 944,
"column": 61
} | {
"line": 945,
"column": 4
} | [
{
"pp": "case refine_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : Ring k\ninst✝⁴ : LinearOrder k\ninst✝³ : IsStrictOrderedRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw : ι → k\ns : Finset ι\nhw : ∑ i ∈ s, w i = 1\ni₁... | [
"case refine_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : Ring k\ninst✝⁴ : LinearOrder k\ninst✝³ : IsStrictOrderedRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw : ι → k\ns : Finset ι\nhw : ∑ i ∈ s, w i = 1\ni₁ i₂ i₃ : ι\n... | rw [Finset.affineCombinationLineMapWeights_apply_left h₂₃,
Finset.affineCombinationLineMapWeights_apply_right h₂₃] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.Module.LocallyConvex | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 68
} | {
"line": 151,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁸ : Field 𝕜\ninst✝⁷ : PartialOrder 𝕜\ninst✝⁶ : ZeroLEOneClass 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : TopologicalSpace E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousConstSMul 𝕜 E\ninst✝ : LocallyConvexSpace 𝕜 E\ns t : Set E\ndisj : Dis... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁸ : Field 𝕜\ninst✝⁷ : PartialOrder 𝕜\ninst✝⁶ : ZeroLEOneClass 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : TopologicalSpace E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousConstSMul 𝕜 E\ninst✝ : LocallyConvexSpace 𝕜 E\ns t : Set E\ndisj : Disjoint s t\nh... | simp_rw [UniformSpace.ball, ← preimage_comp, sub_eq_neg_add] at hV | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 155,
"column": 45
} | {
"line": 155,
"column": 54
} | {
"line": 155,
"column": 55
} | [
{
"pp": "case neg\nR : Type u_1\nE : Type u_6\nι : Type u_9\ninst✝² : SeminormedRing R\ninst✝¹ : AddCommGroup E\ninst✝ : Module R E\np : SeminormFamily R E ι\nv : E\nU : Set E\nhU✝ : U ∈ p.basisSets\ns : Finset ι\nr : ℝ\nhr : 0 < r\nhU : U = (s.sup p).ball 0 r\nh : (s.sup p) v ≤ 0\n⊢ {x | ‖x‖ * 0 < r} ∈ 𝓝 0",
... | [
"case neg\nR : Type u_1\nE : Type u_6\nι : Type u_9\ninst✝² : SeminormedRing R\ninst✝¹ : AddCommGroup E\ninst✝ : Module R E\np : SeminormFamily R E ι\nv : E\nU : Set E\nhU✝ : U ∈ p.basisSets\ns : Finset ι\nr : ℝ\nhr : 0 < r\nhU : U = (s.sup p).ball 0 r\nh : (s.sup p) v ≤ 0\n⊢ {x | 0 < r} ∈ 𝓝 0"
] | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 213,
"column": 2
} | {
"line": 218,
"column": 34
} | {
"line": 220,
"column": 0
} | [
{
"pp": "𝕜 : Type u_11\nE : Type u_12\nι : Type u_13\ninst✝³ : NormedField 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : TopologicalSpace E\np : SeminormFamily 𝕜 E ι\nhp : ∀ (i : ι), Continuous[inst✝, _] ⇑(p i)\ns : Finset ι\nr : ℝ\nhr : 0 < r\n⊢ Continuous[inst✝, _] ⇑(s.sup p)",
"ppTerm": "?... | [] | classical
induction s using Finset.induction_on with
| empty => simpa using continuous_zero
| insert a s _ hs =>
simp only [Finset.sup_insert, coe_sup]
exact Continuous.max (hp a) hs | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Analysis.Convex.Jensen | {
"line": 404,
"column": 76
} | {
"line": 405,
"column": 88
} | {
"line": 407,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nβ : Type u_4\ninst✝⁷ : Field 𝕜\ninst✝⁶ : LinearOrder 𝕜\ninst✝⁵ : IsStrictOrderedRing 𝕜\ninst✝⁴ : AddCommGroup β\ninst✝³ : LinearOrder β\ninst✝² : IsOrderedAddMonoid β\ninst✝¹ : Module 𝕜 β\ninst✝ : IsStrictOrderedModule 𝕜 β\ns : Set 𝕜\nf : 𝕜 → β\nx y z : 𝕜\nhf : ConvexOn 𝕜 s f\nh... | [] | by
rw [← segment_eq_Icc (hz.1.trans hz.2)] at hz; exact hf.le_max_of_mem_segment hx hy hz | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 338,
"column": 2
} | {
"line": 338,
"column": 37
} | {
"line": 340,
"column": 0
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_6\nι : Type u_9\ninst✝³ : NormedField 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : TopologicalSpace E\np : SeminormFamily 𝕜 E ι\nhp : WithSeminorms p\nx : E\nthis✝ : IsTopologicalAddGroup E\nsr : Finset ι × ℝ\nthis : (sr.1.sup p).ball (x +ᵥ 0) sr.2 = x +ᵥ (sr.1.... | [] | rwa [vadd_eq_add, add_zero] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 570,
"column": 50
} | {
"line": 570,
"column": 63
} | {
"line": 570,
"column": 64
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_6\nι : Type u_9\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\ns : Set E\nhp : WithSeminorms p\n⊢ (∀ (i : ι), ∃ r > 0, ∀ x ∈ s, (p i) x < r) ↔ ∀ (i : ι), BddAbove (⇑(p i) '' s)",
"... | [
"𝕜 : Type u_2\nE : Type u_6\nι : Type u_9\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\ns : Set E\nhp : WithSeminorms p\n⊢ (∀ (i : ι), ∃ r > 0, ∀ x ∈ s, (p i) x < r) ↔ ∀ (i : ι), ∃ x, ∀ y ∈ ⇑(p i) '' s, y ≤ x"
] | bddAbove_def, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Algebra.Module.Spaces.UniformConvergenceCLM | {
"line": 334,
"column": 2
} | {
"line": 334,
"column": 87
} | {
"line": 335,
"column": 2
} | [
{
"pp": "𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝¹² : NormedField 𝕜₁\ninst✝¹¹ : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_3\nF : Type u_4\ninst✝¹⁰ : AddCommGroup E\ninst✝⁹ : Module 𝕜₁ E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\nR : Type u_6\ninst✝⁵ : NormedDivisionRing R\ni... | [
"𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝¹² : NormedField 𝕜₁\ninst✝¹¹ : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_3\nF : Type u_4\ninst✝¹⁰ : AddCommGroup E\ninst✝⁹ : Module 𝕜₁ E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\nR : Type u_6\ninst✝⁵ : NormedDivisionRing R\ninst✝⁴ : Topo... | simp_rw [isVonNBounded_iff_absorbing_le, nhds_zero_eq, le_iInf_iff, le_principal_iff] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Topology.Algebra.Module.Spaces.ContinuousLinearMap | {
"line": 299,
"column": 56
} | {
"line": 299,
"column": 66
} | {
"line": 299,
"column": 66
} | [
{
"pp": "R : Type u_1\n𝕜₂ : Type u_3\n𝕜₃ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝¹³ : Semiring R\ninst✝¹² : NormedField 𝕜₂\ninst✝¹¹ : NormedField 𝕜₃\ninst✝¹⁰ : AddCommMonoid E\ninst✝⁹ : Module R E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\ninst✝⁵ : Topol... | [
"R : Type u_1\n𝕜₂ : Type u_3\n𝕜₃ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝¹³ : Semiring R\ninst✝¹² : NormedField 𝕜₂\ninst✝¹¹ : NormedField 𝕜₃\ninst✝¹⁰ : AddCommMonoid E\ninst✝⁹ : Module R E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\ninst✝⁵ : TopologicalSpace ... | smul_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Module.Spaces.ContinuousLinearMap | {
"line": 329,
"column": 18
} | {
"line": 329,
"column": 28
} | {
"line": 329,
"column": 28
} | [
{
"pp": "𝕜₂ : Type u_3\n𝕜₃ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝¹² : NormedField 𝕜₂\ninst✝¹¹ : NormedField 𝕜₃\ninst✝¹⁰ : AddCommMonoid E\ninst✝⁹ : Module 𝕜₃ E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\ninst✝⁵ : TopologicalSpace F\ninst✝⁴ : AddCommGro... | [
"𝕜₂ : Type u_3\n𝕜₃ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝¹² : NormedField 𝕜₂\ninst✝¹¹ : NormedField 𝕜₃\ninst✝¹⁰ : AddCommMonoid E\ninst✝⁹ : Module 𝕜₃ E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\ninst✝⁵ : TopologicalSpace F\ninst✝⁴ : AddCommGroup G\ninst✝³... | smul_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Filter.Germ.Basic | {
"line": 630,
"column": 6
} | {
"line": 630,
"column": 21
} | {
"line": 632,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nl : Filter α\nf✝¹ g h : α → β\nM : Type u_5\nN : Type u_6\nR : Type u_7\ninst✝¹ : Monoid M\ninst✝ : MulAction M β\nc₁ c₂ : M\nf✝ : l.Germ β\nf : α → β\n⊢ (c₁ * c₂) • f =ᶠ[l] c₁ • c₂ • f",
"ppTerm": "?m.30",
"assigned": true,
"usedConst... | [] | simp [mul_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.Filter.Germ.Basic | {
"line": 638,
"column": 6
} | {
"line": 638,
"column": 21
} | {
"line": 640,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nl : Filter α\nf✝¹ g h : α → β\nM : Type u_5\nN : Type u_6\nR : Type u_7\ninst✝¹ : Monoid M\ninst✝ : MulAction M β\nc₁✝ c₂✝ : l.Germ M\nf✝ : l.Germ β\nc₁ c₂ : α → M\nf : α → β\n⊢ (c₁ * c₂) • f =ᶠ[l] c₁ • c₂ • f",
"ppTerm": "?m.35",
"assigne... | [] | simp [mul_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Function.AEEqFun | {
"line": 806,
"column": 2
} | {
"line": 806,
"column": 24
} | {
"line": 807,
"column": 2
} | [
{
"pp": "α : Type u_1\nγ : Type u_3\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace γ\ninst✝¹ : CommMonoid γ\ninst✝ : ContinuousMul γ\nι : Type u_5\ns : Finset ι\nf : ι → α →ₘ[μ] γ\n⊢ ↑(∏ i ∈ s, f i) =ᵐ[μ] fun x ↦ ∏ i ∈ s, ↑(f i) x",
"ppTerm": "?m.26",
"assigned": true,
"usedCo... | [
"α : Type u_1\nγ : Type u_3\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace γ\ninst✝¹ : CommMonoid γ\ninst✝ : ContinuousMul γ\nι : Type u_5\ns : Finset ι\nf : ι → α →ₘ[μ] γ\n⊢ ∏ i ∈ s, ↑(f i) =ᵐ[μ] fun x ↦ ∏ i ∈ s, ↑(f i) x"
] | grw [coeFn_finsetProd] | Mathlib.Tactic.GRewrite._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_GRewrite_grwSeq_1 | Mathlib.Tactic.GRewrite.grwSeq |
Mathlib.Probability.UniformOn | {
"line": 178,
"column": 6
} | {
"line": 178,
"column": 22
} | {
"line": 178,
"column": 22
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nhs : s.Finite\n⊢ (uniformOn s) (t ∩ u) = (uniformOn (s ∩ u)) t * (uniformOn s) u",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"HMul.... | [
"Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nhs : s.Finite\n⊢ (uniformOn s) (u ∩ t) = (uniformOn (s ∩ u)) t * (uniformOn s) u"
] | ← Set.inter_comm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.ConditionalProbability | {
"line": 306,
"column": 15
} | {
"line": 306,
"column": 30
} | {
"line": 306,
"column": 31
} | [
{
"pp": "Ω : Type u_1\nα : Type u_3\nm : MeasurableSpace Ω\ninst✝³ : Fintype α\ninst✝² : MeasurableSpace α\ninst✝¹ : DiscreteMeasurableSpace α\nX : Ω → α\nhX : Measurable X\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nE : Set Ω\nhE : MeasurableSet E\n⊢ ∑ x, μ (X ⁻¹' {x}) * μ[E | X ⁻¹' {x}] = ∑ x, μ (X ⁻¹' {x} ∩ E... | [
"Ω : Type u_1\nα : Type u_3\nm : MeasurableSpace Ω\ninst✝³ : Fintype α\ninst✝² : MeasurableSpace α\ninst✝¹ : DiscreteMeasurableSpace α\nX : Ω → α\nhX : Measurable X\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nE : Set Ω\nhE : MeasurableSet E\n⊢ ∑ x, μ[E | X ⁻¹' {x}] * μ (X ⁻¹' {x}) = ∑ x, μ (X ⁻¹' {x} ∩ E)"
] | mul_comm (μ _), | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Function.EssSup | {
"line": 158,
"column": 2
} | {
"line": 158,
"column": 43
} | {
"line": 159,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : ConditionallyCompleteLattice β\nγ : Type u_3\nmγ : MeasurableSpace γ\nf : α → γ\ng : γ → β\nhf : AEMeasurable f μ\nhgf : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) (ae μ) (g ∘ f)\nhg : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (ae (Measur... | [
"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : ConditionallyCompleteLattice β\nγ : Type u_3\nmγ : MeasurableSpace γ\nf : α → γ\ng : γ → β\nhf : AEMeasurable f μ\nhgf : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) (ae μ) (g ∘ f)\nhg : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (ae (Measure.map f μ)) ... | refine limsSup_le_limsSup_of_le ?_ hgf hg | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.LpSeminorm.Defs | {
"line": 101,
"column": 48
} | {
"line": 101,
"column": 75
} | {
"line": 101,
"column": 75
} | [
{
"pp": "α : Type u_1\nε : Type u_2\nm0 : MeasurableSpace α\np : ℝ≥0∞\ninst✝ : ENorm ε\nμ : Measure α\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ∞\nf : α → ε\n⊢ eLpNorm' f p.toReal μ = (∫⁻ (x : α), ‖f x‖ₑ ^ p.toReal ∂μ) ^ (1 / p.toReal)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.m... | [
"α : Type u_1\nε : Type u_2\nm0 : MeasurableSpace α\np : ℝ≥0∞\ninst✝ : ENorm ε\nμ : Measure α\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ∞\nf : α → ε\n⊢ (∫⁻ (a : α), ‖f a‖ₑ ^ p.toReal ∂μ) ^ (1 / p.toReal) = (∫⁻ (x : α), ‖f x‖ₑ ^ p.toReal ∂μ) ^ (1 / p.toReal)"
] | eLpNorm'_eq_lintegral_enorm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Indicator | {
"line": 74,
"column": 59
} | {
"line": 77,
"column": 88
} | {
"line": 79,
"column": 0
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\ns : Set α\nc : ε\n⊢ eLpNormEssSup (s.indicator fun x ↦ c) μ ≤ ‖c‖ₑ",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedA... | [] | by
obtain rfl | hμ0 := eq_or_ne μ 0
· simp
· exact (eLpNormEssSup_indicator_le s fun _ => c).trans (eLpNormEssSup_const c hμ0).le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity | {
"line": 33,
"column": 11
} | {
"line": 33,
"column": 38
} | {
"line": 33,
"column": 38
} | [
{
"pp": "α : Type u_1\nF : Type u_3\nG : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → F\ng : α → G\nc : ℝ≥0\nh : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊\np : ℝ\nhp : 0 < p\n⊢ eLpNorm' f p μ ≤ c • eLpNorm' g p μ",
"ppTerm": "?m.38",
"assi... | [
"α : Type u_1\nF : Type u_3\nG : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → F\ng : α → G\nc : ℝ≥0\nh : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊\np : ℝ\nhp : 0 < p\n⊢ (∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ) ^ (1 / p) ≤ c • (∫⁻ (a : α), ‖g a‖ₑ ^ p ∂μ) ^ (1 / p)"... | eLpNorm'_eq_lintegral_enorm | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity | {
"line": 48,
"column": 11
} | {
"line": 48,
"column": 38
} | {
"line": 48,
"column": 38
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nε : Type u_5\nε' : Type u_6\ninst✝³ : TopologicalSpace ε\ninst✝² : ContinuousENorm ε\ninst✝¹ : TopologicalSpace ε'\ninst✝ : ContinuousENorm ε'\nf : α → ε\ng : α → ε'\nc : ℝ≥0\nh : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ ↑c * ‖g x‖ₑ\np : ℝ\nhp : 0 < p\n⊢ eLpNorm' f p ... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nε : Type u_5\nε' : Type u_6\ninst✝³ : TopologicalSpace ε\ninst✝² : ContinuousENorm ε\ninst✝¹ : TopologicalSpace ε'\ninst✝ : ContinuousENorm ε'\nf : α → ε\ng : α → ε'\nc : ℝ≥0\nh : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ ↑c * ‖g x‖ₑ\np : ℝ\nhp : 0 < p\n⊢ (∫⁻ (a : α), ‖f a‖ₑ ^ p ∂... | eLpNorm'_eq_lintegral_enorm | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity | {
"line": 79,
"column": 11
} | {
"line": 79,
"column": 38
} | {
"line": 79,
"column": 38
} | [
{
"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 * ‖... | eLpNorm'_eq_lintegral_enorm | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 195,
"column": 4
} | {
"line": 198,
"column": 44
} | {
"line": 199,
"column": 2
} | [
{
"pp": "α : Type u_1\nF : Type u_5\nm0 : MeasurableSpace α\nq : ℝ\nμ : Measure α\ninst✝¹ : NormedAddCommGroup F\ninst✝ : IsFiniteMeasure μ\nc : F\nhc_ne_zero : c ≠ 0\nhq_ne_zero : q ≠ 0\n⊢ (‖c‖ₑ ^ q) ^ (1 / q) * μ Set.univ ^ (1 / q) = ‖c‖ₑ * μ Set.univ ^ (1 / q)",
"ppTerm": "?m.56",
"assigned": true,
... | [] | congr
rw [← ENNReal.rpow_mul]
suffices hp_cancel : q * (1 / q) = 1 by rw [hp_cancel, ENNReal.rpow_one]
rw [one_div, mul_inv_cancel₀ hq_ne_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 195,
"column": 4
} | {
"line": 198,
"column": 44
} | {
"line": 199,
"column": 2
} | [
{
"pp": "α : Type u_1\nF : Type u_5\nm0 : MeasurableSpace α\nq : ℝ\nμ : Measure α\ninst✝¹ : NormedAddCommGroup F\ninst✝ : IsFiniteMeasure μ\nc : F\nhc_ne_zero : c ≠ 0\nhq_ne_zero : q ≠ 0\n⊢ (‖c‖ₑ ^ q) ^ (1 / q) * μ Set.univ ^ (1 / q) = ‖c‖ₑ * μ Set.univ ^ (1 / q)",
"ppTerm": "?m.56",
"assigned": true,
... | [] | congr
rw [← ENNReal.rpow_mul]
suffices hp_cancel : q * (1 / q) = 1 by rw [hp_cancel, ENNReal.rpow_one]
rw [one_div, mul_inv_cancel₀ hq_ne_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Constructions.Pi | {
"line": 827,
"column": 15
} | {
"line": 827,
"column": 55
} | {
"line": 828,
"column": 4
} | [
{
"pp": "case e_6\nι : Type u_1\ninst✝¹ : Fintype ι\nX : ι → Type u_4\ninst✝ : Unique ι\nm : (i : ι) → MeasurableSpace (X i)\nμ : (i : ι) → Measure (X i)\ne : ((i : ι) → X i) ≃ᵐ X default := ⋯\ns : Set ((i : ι) → X i)\n⊢ eval default '' s = ⇑e.symm ⁻¹' s",
"ppTerm": "?e_6",
"assigned": true,
"usedCo... | [] | exact e.toEquiv.image_eq_preimage_symm s | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 604,
"column": 45
} | {
"line": 604,
"column": 72
} | {
"line": 604,
"column": 72
} | [
{
"pp": "case neg\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 = ∞\n⊢ eLpNorm' f p.toReal (μ.restrict s) = eLpNorm' f p.toReal μ",
"ppTerm":... | [
"case neg\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 = ∞\n⊢ (∫⁻ (a : α) in s, ‖f a‖ₑ ^ p.toReal ∂μ) ^ (1 / p.toReal) = (∫⁻ (a : α), ‖f a‖ₑ ^ p.toR... | eLpNorm'_eq_lintegral_enorm | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Convex.SpecificFunctions.Basic | {
"line": 189,
"column": 51
} | {
"line": 189,
"column": 61
} | {
"line": 189,
"column": 62
} | [
{
"pp": "p : ℝ\nhp : 1 < p\nx y z : ℝ\nhx : 0 ≤ x\nhz : 0 ≤ z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nq : 0 < y - x\n⊢ 1 + -(p * ((y - x) / y)) < (1 + (x / y - 1)) ^ p",
"ppTerm": "?m.226",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real.partial... | [
"p : ℝ\nhp : 1 < p\nx y z : ℝ\nhx : 0 ≤ x\nhz : 0 ≤ z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nq : 0 < y - x\n⊢ 1 + p * -((y - x) / y) < (1 + (x / y - 1)) ^ p"
] | ← mul_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 915,
"column": 4
} | {
"line": 915,
"column": 28
} | {
"line": 916,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nE : Type u_4\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nR : ℝ≥0\np : ℝ≥0∞\nf : ℕ → α → E\nhfmeas : ∀ (n : ℕ), Measurable (f n)\nhbdd : ∀ (n : ℕ), eLpNorm (f n) p μ ≤ ↑R\nhp0 : p.toReal = 0\na... | [] | exact ENNReal.one_lt_top | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 155,
"column": 39
} | {
"line": 155,
"column": 47
} | {
"line": 155,
"column": 48
} | [
{
"pp": "case neg\np : ℝ\na b : ℝ≥0\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nh_zero : ¬a + b = 0\nh_nonzero : ¬(a = 0 ∧ b = 0)\nh_add : a / (a + b) + b / (a + b) = 1\nh : a ^ p / (a + b) ^ p + b ^ p / (a + b) ^ p ≤ 1\nhab_0 : (a + b) ^ p ≠ 0\nh_mul : (a + b) ^ p * (a ^ p * ((a + b) ^ p)⁻¹ + b ^ p * ((a + b) ^ p)⁻¹) ≤ (a +... | [
"case neg\np : ℝ\na b : ℝ≥0\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nh_zero : ¬a + b = 0\nh_nonzero : ¬(a = 0 ∧ b = 0)\nh_add : a / (a + b) + b / (a + b) = 1\nh : a ^ p / (a + b) ^ p + b ^ p / (a + b) ^ p ≤ 1\nhab_0 : (a + b) ^ p ≠ 0\nh_mul : (a + b) ^ p * (a ^ p * ((a + b) ^ p)⁻¹) + (a + b) ^ p * (b ^ p * ((a + b) ^ p)⁻¹) ≤ ... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 121,
"column": 14
} | {
"line": 121,
"column": 95
} | {
"line": 121,
"column": 95
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np q : ℝ\nhpq : p.HolderConjugate q\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhf_nontop : ∫⁻ (a : α), f a ^ p ∂μ ≠ ∞\nhg_nontop : ∫⁻ (a : α), g a ^ q ∂μ ≠ ∞\nhf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0\nhg_nonzero : ∫⁻ (a : α), g a ^ q ∂μ ≠ 0\nnpf : ℝ≥0∞... | [] | simp [npf, nqg, hf_nontop, hg_nontop, hf_nonzero, hg_nonzero, ENNReal.mul_eq_top] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 121,
"column": 14
} | {
"line": 121,
"column": 95
} | {
"line": 121,
"column": 95
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np q : ℝ\nhpq : p.HolderConjugate q\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhf_nontop : ∫⁻ (a : α), f a ^ p ∂μ ≠ ∞\nhg_nontop : ∫⁻ (a : α), g a ^ q ∂μ ≠ ∞\nhf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0\nhg_nonzero : ∫⁻ (a : α), g a ^ q ∂μ ≠ 0\nnpf : ℝ≥0∞... | [] | simp [npf, nqg, hf_nontop, hg_nontop, hf_nonzero, hg_nonzero, ENNReal.mul_eq_top] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 121,
"column": 14
} | {
"line": 121,
"column": 95
} | {
"line": 121,
"column": 95
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np q : ℝ\nhpq : p.HolderConjugate q\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhf_nontop : ∫⁻ (a : α), f a ^ p ∂μ ≠ ∞\nhg_nontop : ∫⁻ (a : α), g a ^ q ∂μ ≠ ∞\nhf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0\nhg_nonzero : ∫⁻ (a : α), g a ^ q ∂μ ≠ 0\nnpf : ℝ≥0∞... | [] | simp [npf, nqg, hf_nontop, hg_nontop, hf_nonzero, hg_nonzero, ENNReal.mul_eq_top] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 236,
"column": 4
} | {
"line": 261,
"column": 35
} | {
"line": 263,
"column": 0
} | [
{
"pp": "case refine_2\nι : Type u\ns : Finset ι\nw z : ι → ℝ≥0∞\nhw' : ∑ i ∈ s, w i = 1\np : ℝ\nhp : 1 ≤ p\nhp_pos : 0 < p\nhp_nonneg : 0 ≤ p\nhp_not_neg : ¬p < 0\nh_top_iff_rpow_top : ∀ i ∈ s, w i * z i = ∞ ↔ w i * z i ^ p = ∞\n⊢ (∑ i ∈ s, w i * z i) ^ p ≠ ∞ →\n ∑ i ∈ s, w i * z i ^ p ≠ ∞ → ((∑ i ∈ s, w i ... | [] | intro h_top_rpow_sum _
-- show hypotheses needed to put the `.toNNReal` inside the sums.
have h_top : ∀ a : ι, a ∈ s → w a * z a ≠ ⊤ :=
haveI h_top_sum : ∑ i ∈ s, w i * z i ≠ ⊤ := by
intro h
rw [h, top_rpow_of_pos hp_pos] at h_top_rpow_sum
exact h_top_rpow_sum rfl
fun a ha =>... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 236,
"column": 4
} | {
"line": 261,
"column": 35
} | {
"line": 263,
"column": 0
} | [
{
"pp": "case refine_2\nι : Type u\ns : Finset ι\nw z : ι → ℝ≥0∞\nhw' : ∑ i ∈ s, w i = 1\np : ℝ\nhp : 1 ≤ p\nhp_pos : 0 < p\nhp_nonneg : 0 ≤ p\nhp_not_neg : ¬p < 0\nh_top_iff_rpow_top : ∀ i ∈ s, w i * z i = ∞ ↔ w i * z i ^ p = ∞\n⊢ (∑ i ∈ s, w i * z i) ^ p ≠ ∞ →\n ∑ i ∈ s, w i * z i ^ p ≠ ∞ → ((∑ i ∈ s, w i ... | [] | intro h_top_rpow_sum _
-- show hypotheses needed to put the `.toNNReal` inside the sums.
have h_top : ∀ a : ι, a ∈ s → w a * z a ≠ ⊤ :=
haveI h_top_sum : ∑ i ∈ s, w i * z i ≠ ⊤ := by
intro h
rw [h, top_rpow_of_pos hp_pos] at h_top_rpow_sum
exact h_top_rpow_sum rfl
fun a ha =>... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 43,
"column": 14
} | {
"line": 43,
"column": 41
} | {
"line": 43,
"column": 41
} | [
{
"pp": "case neg\nα : Type u_1\nε : Type u_2\nm : MeasurableSpace α\nμ : Measure α\nf : α → ε\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\np q : ℝ\nhp0_lt : 0 < p\nhpq✝ : p ≤ q\nhf : AEStronglyMeasurable f μ\nhq0_lt : 0 < q\nhpq_eq : ¬p = q\nhpq : p < q\ng : α → ℝ≥0∞ := fun x ↦ 1\nh_rw : ∫⁻ (a : α)... | [
"case neg\nα : Type u_1\nε : Type u_2\nm : MeasurableSpace α\nμ : Measure α\nf : α → ε\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\np q : ℝ\nhp0_lt : 0 < p\nhpq✝ : p ≤ q\nhf : AEStronglyMeasurable f μ\nhq0_lt : 0 < q\nhpq_eq : ¬p = q\nhpq : p < q\ng : α → ℝ≥0∞ := fun x ↦ 1\nh_rw : ∫⁻ (a : α), ‖f a‖ₑ ^ p... | eLpNorm'_eq_lintegral_enorm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 43,
"column": 14
} | {
"line": 43,
"column": 41
} | {
"line": 43,
"column": 41
} | [
{
"pp": "case neg\nα : Type u_1\nε : Type u_2\nm : MeasurableSpace α\nμ : Measure α\nf : α → ε\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\np q : ℝ\nhp0_lt : 0 < p\nhpq✝ : p ≤ q\nhf : AEStronglyMeasurable f μ\nhq0_lt : 0 < q\nhpq_eq : ¬p = q\nhpq : p < q\ng : α → ℝ≥0∞ := fun x ↦ 1\nh_rw : ∫⁻ (a : α)... | [
"case neg\nα : Type u_1\nε : Type u_2\nm : MeasurableSpace α\nμ : Measure α\nf : α → ε\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\np q : ℝ\nhp0_lt : 0 < p\nhpq✝ : p ≤ q\nhf : AEStronglyMeasurable f μ\nhq0_lt : 0 < q\nhpq_eq : ¬p = q\nhpq : p < q\ng : α → ℝ≥0∞ := fun x ↦ 1\nh_rw : ∫⁻ (a : α), ‖f a‖ₑ ^ p... | eLpNorm'_eq_lintegral_enorm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 43,
"column": 14
} | {
"line": 43,
"column": 41
} | {
"line": 43,
"column": 41
} | [
{
"pp": "case neg\nα : Type u_1\nε : Type u_2\nm : MeasurableSpace α\nμ : Measure α\nf : α → ε\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\np q : ℝ\nhp0_lt : 0 < p\nhpq✝ : p ≤ q\nhf : AEStronglyMeasurable f μ\nhq0_lt : 0 < q\nhpq_eq : ¬p = q\nhpq : p < q\ng : α → ℝ≥0∞ := fun x ↦ 1\nh_rw : ∫⁻ (a : α)... | [
"case neg\nα : Type u_1\nε : Type u_2\nm : MeasurableSpace α\nμ : Measure α\nf : α → ε\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\np q : ℝ\nhp0_lt : 0 < p\nhpq✝ : p ≤ q\nhf : AEStronglyMeasurable f μ\nhq0_lt : 0 < q\nhpq_eq : ¬p = q\nhpq : p < q\ng : α → ℝ≥0∞ := fun x ↦ 1\nh_rw : ∫⁻ (a : α), ‖f a‖ₑ ^ p... | eLpNorm'_eq_lintegral_enorm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 342,
"column": 4
} | {
"line": 342,
"column": 28
} | {
"line": 344,
"column": 0
} | [
{
"pp": "case neg\np : ℝ≥0∞\nh : p ∉ Set.Ioo 0 1\n⊢ 1 < ∞",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"ENNReal.one_lt_top"
],
"usedFVars": [],
"usedGoals": []
}
] | [] | exact ENNReal.one_lt_top | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 342,
"column": 4
} | {
"line": 342,
"column": 28
} | {
"line": 344,
"column": 0
} | [
{
"pp": "case neg\np : ℝ≥0∞\nh : p ∉ Set.Ioo 0 1\n⊢ 1 < ∞",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"ENNReal.one_lt_top"
],
"usedFVars": [],
"usedGoals": []
}
] | [] | exact ENNReal.one_lt_top | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 342,
"column": 4
} | {
"line": 342,
"column": 28
} | {
"line": 344,
"column": 0
} | [
{
"pp": "case neg\np : ℝ≥0∞\nh : p ∉ Set.Ioo 0 1\n⊢ 1 < ∞",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"ENNReal.one_lt_top"
],
"usedFVars": [],
"usedGoals": []
}
] | [] | exact ENNReal.one_lt_top | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 153,
"column": 14
} | {
"line": 153,
"column": 75
} | {
"line": 153,
"column": 75
} | [
{
"pp": "α : Type u_1\nε' : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : TopologicalSpace ε'\ninst✝ : ESeminormedAddMonoid ε'\np q : ℝ≥0∞\nf : α → ε'\ns : Set α\nhfq : MemLp ((toMeasurable μ s).indicator f) q μ\nhf : ∀ x ∉ s, f x = 0\nhs : μ s ≠ ∞\nhpq : p ≤ q\nthis : (toMeasurable μ s).indicator f ... | [
"α : Type u_1\nε' : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : TopologicalSpace ε'\ninst✝ : ESeminormedAddMonoid ε'\np q : ℝ≥0∞\nf : α → ε'\ns : Set α\nhfq : MemLp f q (μ.restrict (toMeasurable μ s))\nhf : ∀ x ∉ s, f x = 0\nhs : μ s ≠ ∞\nhpq : p ≤ q\nthis : (toMeasurable μ s).indicator f = f\n⊢ MemLp ... | memLp_indicator_iff_restrict (measurableSet_toMeasurable μ s) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.MeanInequalities | {
"line": 435,
"column": 2
} | {
"line": 435,
"column": 81
} | {
"line": 436,
"column": 2
} | [
{
"pp": "w₁ w₂ p₁ p₂ : ℝ\nhw₁ : 0 ≤ w₁\nhw₂ : 0 ≤ w₂\nhp₁ : 0 ≤ p₁\nhp₂ : 0 ≤ p₂\nhw : w₁ + w₂ = 1\n⊢ p₁ ^ w₁ * p₂ ^ w₂ < w₁ * p₁ + w₂ * p₂ ↔ w₁ ≠ 0 ∧ w₂ ≠ 0 ∧ p₁ ≠ p₂",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Real.instPow",
"Real.instLE",
"Real",
"HMul.hMul"... | [
"w₁ w₂ p₁ p₂ : ℝ\nhw₁ : 0 ≤ w₁\nhw₂ : 0 ≤ w₂\nhp₁ : 0 ≤ p₁\nhp₂ : 0 ≤ p₂\nhw : w₁ + w₂ = 1\nthis :\n (∀ i ∈ univ, 0 ≤ ![w₁, w₂] i) →\n ∑ i, ![w₁, w₂] i = 1 →\n (∀ i ∈ univ, 0 ≤ ![p₁, p₂] i) →\n (∏ i, ![p₁, p₂] i ^ ![w₁, w₂] i < ∑ i, ![w₁, w₂] i * ![p₁, p₂] i ↔\n ∃ j ∈ univ, ∃ k ∈ univ, ![w₁... | have := geom_mean_lt_arith_mean_weighted_iff_of_nonneg univ ![w₁, w₂] ![p₁, p₂] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 364,
"column": 2
} | {
"line": 368,
"column": 70
} | {
"line": 369,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np q : ℝ\nhpq : p.HolderConjugate q\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\nh_add_zero : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ 0\nh_add_top : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ ∞\nh0_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ 0\n... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np q : ℝ\nhpq : p.HolderConjugate q\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\nh_add_zero : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ 0\nh_add_top : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ ∞\nh0_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ 0\nh :\n ∫⁻ (a... | have h :
(∫⁻ a : α, (f + g) a ^ p ∂μ) ≤
((∫⁻ a : α, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a : α, g a ^ p ∂μ) ^ (1 / p)) *
(∫⁻ a : α, (f + g) a ^ p ∂μ) ^ (1 / q) :=
lintegral_rpow_add_le_add_eLpNorm_mul_lintegral_rpow_add hpq hf hg | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Function.ConvergenceInMeasure | {
"line": 148,
"column": 4
} | {
"line": 148,
"column": 68
} | {
"line": 149,
"column": 4
} | [
{
"pp": "case refine_2\nα : Type u_1\nι : Type u_2\nE : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : EDist E\ninst✝ : IsFiniteMeasure μ\nf : ι → α → E\nl : Filter ι\ng : α → E\nhfin : ∀ (ε : ℝ≥0∞) (i : ι), μ {x | ε ≤ edist (f i x) (g x)} ≠ ∞\nh : ∀ (ε : ℝ≥0∞), 0 < ε → Tendsto (fun i ↦ (μ {x | ε ≤ ed... | [
"case refine_2\nα : Type u_1\nι : Type u_2\nE : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : EDist E\ninst✝ : IsFiniteMeasure μ\nf : ι → α → E\nl : Filter ι\ng : α → E\nhfin : ∀ (ε : ℝ≥0∞) (i : ι), μ {x | ε ≤ edist (f i x) (g x)} ≠ ∞\nh : ∀ (ε : ℝ≥0∞), 0 < ε → Tendsto (fun i ↦ (μ {x | ε ≤ edist (f i x) ... | rw [← ENNReal.tendsto_toNNReal_iff ENNReal.zero_ne_top (hfin ε)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.MeanInequalities | {
"line": 933,
"column": 2
} | {
"line": 938,
"column": 72
} | {
"line": 940,
"column": 0
} | [
{
"pp": "ι : Type u\nf g : ι → ℝ\np q r : ℝ\nhpqr : p.HolderTriple q r\nhf : ∀ (i : ι), 0 ≤ f i\nhg : ∀ (i : ι), 0 ≤ g i\nhf_sum : Summable fun i ↦ f i ^ p\nhg_sum : Summable fun i ↦ g i ^ q\n⊢ (Summable fun i ↦ (f i * g i) ^ r) ∧\n ∑' (i : ι), (f i * g i) ^ r ≤ (∑' (i : ι), f i ^ p) ^ (r / p) * (∑' (i : ι),... | [] | lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After https://github.com/leanprover/lean4/pull/2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.summable_and_Lr_rpow_le_Lp_mul_Lq_tsum hpqr hf_sum hg_sum | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.MeanInequalities | {
"line": 933,
"column": 2
} | {
"line": 938,
"column": 72
} | {
"line": 940,
"column": 0
} | [
{
"pp": "ι : Type u\nf g : ι → ℝ\np q r : ℝ\nhpqr : p.HolderTriple q r\nhf : ∀ (i : ι), 0 ≤ f i\nhg : ∀ (i : ι), 0 ≤ g i\nhf_sum : Summable fun i ↦ f i ^ p\nhg_sum : Summable fun i ↦ g i ^ q\n⊢ (Summable fun i ↦ (f i * g i) ^ r) ∧\n ∑' (i : ι), (f i * g i) ^ r ≤ (∑' (i : ι), f i ^ p) ^ (r / p) * (∑' (i : ι),... | [] | lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After https://github.com/leanprover/lean4/pull/2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.summable_and_Lr_rpow_le_Lp_mul_Lq_tsum hpqr hf_sum hg_sum | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 707,
"column": 79
} | {
"line": 707,
"column": 82
} | {
"line": 707,
"column": 83
} | [
{
"pp": "α : Type u_1\nE : Type u_4\nF : Type u_5\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedAddCommGroup F\ng : E → F\nc : ℝ≥0\nhg : LipschitzWith c g\ng0 : g 0 = 0\nf f' : ↥(Lp E p μ)\na : α\n⊢ ↑↑(hg.compLp g0 f) a = (g ∘ ↑↑f) a →\n ↑↑(hg.compLp g0 f') a =... | [
"α : Type u_1\nE : Type u_4\nF : Type u_5\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedAddCommGroup F\ng : E → F\nc : ℝ≥0\nhg : LipschitzWith c g\ng0 : g 0 = 0\nf f' : ↥(Lp E p μ)\na : α\nha1 : ↑↑(hg.compLp g0 f) a = (g ∘ ↑↑f) a\n⊢ ↑↑(hg.compLp g0 f') a = (g ∘ ↑↑f') ... | ha1 | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 793,
"column": 12
} | {
"line": 793,
"column": 15
} | {
"line": 793,
"column": 16
} | [
{
"pp": "α : Type u_1\n𝕜✝ : Type u_2\n𝕜'✝ : Type u_3\nE : Type u_4\nF : Type u_5\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ng✝ : E → F\nc : ℝ≥0\n𝕜 : Type u_6\n𝕜' : Type u_7\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NontriviallyNormedFi... | [
"α : Type u_1\n𝕜✝ : Type u_2\n𝕜'✝ : Type u_3\nE : Type u_4\nF : Type u_5\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ng✝ : E → F\nc : ℝ≥0\n𝕜 : Type u_6\n𝕜' : Type u_7\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NontriviallyNormedField 𝕜'\nins... | ha1 | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 798,
"column": 31
} | {
"line": 798,
"column": 34
} | {
"line": 798,
"column": 35
} | [
{
"pp": "α : Type u_1\n𝕜✝ : Type u_2\n𝕜'✝ : Type u_3\nE : Type u_4\nF : Type u_5\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ng : E → F\nc✝ : ℝ≥0\n𝕜 : Type u_6\n𝕜' : Type u_7\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NontriviallyNormedFi... | [
"α : Type u_1\n𝕜✝ : Type u_2\n𝕜'✝ : Type u_3\nE : Type u_4\nF : Type u_5\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ng : E → F\nc✝ : ℝ≥0\n𝕜 : Type u_6\n𝕜' : Type u_7\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NontriviallyNormedField 𝕜'\nins... | ha1 | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Logic.Equiv.Embedding | {
"line": 45,
"column": 8
} | {
"line": 45,
"column": 49
} | {
"line": 46,
"column": 6
} | [
{
"pp": "case inr.inl\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nx✝ : { f // Disjoint (Set.range ⇑f.1) (Set.range ⇑f.2) }\nf : α ↪ γ\ng : β ↪ γ\ndisj : Disjoint (Set.range ⇑(f, g).1) (Set.range ⇑(f, g).2)\nb₁ : β\na₂ : α\nf_eq : g b₁ = f a₂\n⊢ False",
"ppTerm": "?inr.inl",
"assigned": true,
"usedCon... | [] | exact disj.le_bot ⟨⟨a₂, rfl⟩, ⟨b₁, f_eq⟩⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.LpSpace.Complete | {
"line": 339,
"column": 2
} | {
"line": 339,
"column": 23
} | {
"line": 340,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nE : Type u_3\ninst✝ : NormedAddCommGroup E\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : ∑' (i : ℕ), B i ≠ ∞\nh_cau : ∀ (N n m_1 : ℕ), N ≤ n → N ≤ m_1 → eLpNorm (f n - f m_1) p μ < B N\nh_lim... | [
"α : Type u_1\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nE : Type u_3\ninst✝ : NormedAddCommGroup E\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : ∑' (i : ℕ), B i ≠ ∞\nh_cau : ∀ (N n m_1 : ℕ), N ≤ n → N ≤ m_1 → eLpNorm (f n - f m_1) p μ < B N\nh_lim : ∀ᵐ (x : α... | refine h_sub.trans ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Normed.Operator.NormedSpace | {
"line": 214,
"column": 30
} | {
"line": 214,
"column": 55
} | {
"line": 216,
"column": 0
} | [
{
"pp": "𝕜₁ : Type u_2\n𝕜₂ : Type u_3\n𝕜₃ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_8\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup F\ninst✝¹¹ : NormedAddCommGroup G\ninst✝¹⁰ : NontriviallyNormedField 𝕜₁\ninst✝⁹ : NormedSpace 𝕜₁ E\ninst✝⁸ : NontriviallyNormedField 𝕜₂\ninst✝⁷ : Nor... | [] | by simp [opNNNorm_le_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Operator.NormedSpace | {
"line": 356,
"column": 13
} | {
"line": 356,
"column": 26
} | {
"line": 356,
"column": 27
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_3\nE : Type u_5\nF : Type u_6\nι : Type u_9\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝⁴ : RingHomIsometric σ₁₂\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : N... | [
"𝕜 : Type u_1\n𝕜₂ : Type u_3\nE : Type u_5\nF : Type u_6\nι : Type u_9\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝⁴ : RingHomIsometric σ₁₂\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace �... | bddAbove_def, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Measure.Real | {
"line": 483,
"column": 28
} | {
"line": 483,
"column": 41
} | {
"line": 483,
"column": 41
} | [
{
"pp": "α : Type u_1\nx✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\ninst✝ : IsProbabilityMeasure μ\nh : NullMeasurableSet s μ\n⊢ μ.real univ - μ.real s = 1 - μ.real s",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Real.instSub",
... | [
"α : Type u_1\nx✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\ninst✝ : IsProbabilityMeasure μ\nh : NullMeasurableSet s μ\n⊢ 1 - μ.real s = 1 - μ.real s"
] | probReal_univ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 141,
"column": 2
} | {
"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... | [] | 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 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 141,
"column": 2
} | {
"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... | [] | 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 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 215,
"column": 4
} | {
"line": 215,
"column": 74
} | {
"line": 216,
"column": 4
} | [
{
"pp": "α : Type u_1\nε' : Type u_4\nmα : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace ε'\ninst✝¹ : ESeminormedAddMonoid ε'\nf : α → ε'\nx : α\ninst✝ : MeasurableSingletonClass α\nhfx : ‖f x‖ₑ ≠ ∞\n⊢ f =ᵐ[μ.restrict {x}] fun x_1 ↦ f x",
"ppTerm": "?m.45",
"assigned": true,
"usedConst... | [
"α : Type u_1\nε' : Type u_4\nmα : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace ε'\ninst✝¹ : ESeminormedAddMonoid ε'\nf : α → ε'\nx : α\ninst✝ : MeasurableSingletonClass α\nhfx : ‖f x‖ₑ ≠ ∞\na✝ : α\nha : a✝ ∈ {x}\n⊢ f a✝ = f x"
] | filter_upwards [ae_restrict_mem (measurableSet_singleton x)] with _ ha | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 618,
"column": 41
} | {
"line": 618,
"column": 63
} | {
"line": 618,
"column": 64
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf : α → β\nhf : AEStronglyMeasurable f μ\n⊢ AEStronglyMeasurable (fun a ↦ ‖f a‖) μ ∧ HasFiniteIntegral (fun a ↦ ‖f a‖) μ ↔ HasFiniteIntegral f μ",
"ppTerm": "?m.25",
"assigned": true,
"usedConsta... | [
"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf : α → β\nhf : AEStronglyMeasurable f μ\n⊢ HasFiniteIntegral (fun a ↦ ‖f a‖) μ ↔ HasFiniteIntegral f μ"
] | and_iff_right hf.norm, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 313,
"column": 6
} | {
"line": 313,
"column": 74
} | {
"line": 313,
"column": 75
} | [
{
"pp": "α : Type u_1\nε : Type u_3\nmα : MeasurableSpace α\nf : α → ε\ns : Set α\nμ : Measure α\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\nhs : MeasurableSet s\n⊢ IntegrableOn f s μ ↔ Integrable (f ∘ Subtype.val) (Measure.comap Subtype.val μ)",
"ppTerm": "?m.26",
"assigned": true,
"us... | [
"α : Type u_1\nε : Type u_3\nmα : MeasurableSpace α\nf : α → ε\ns : Set α\nμ : Measure α\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\nhs : MeasurableSet s\n⊢ IntegrableOn f s μ ↔ IntegrableOn f (range Subtype.val) μ"
] | ← (MeasurableEmbedding.subtype_coe hs).integrableOn_range_iff_comap, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 505,
"column": 4
} | {
"line": 505,
"column": 17
} | {
"line": 506,
"column": 2
} | [
{
"pp": "case mp\nα : Type u_1\nβ : Type u_2\nε : Type u_3\nmα : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\nl : Filter α\ninst✝ : MeasurableSpace β\ne : α → β\nhe : MeasurableEmbedding e\nf : β → ε\ns : Set β\nhs : s ∈ map e l ∧ IntegrableOn (f ∘ e) (e ⁻¹' s) μ\n⊢... | [] | exact ⟨_, hs⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 505,
"column": 4
} | {
"line": 505,
"column": 17
} | {
"line": 506,
"column": 2
} | [
{
"pp": "case mp\nα : Type u_1\nβ : Type u_2\nε : Type u_3\nmα : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\nl : Filter α\ninst✝ : MeasurableSpace β\ne : α → β\nhe : MeasurableEmbedding e\nf : β → ε\ns : Set β\nhs : s ∈ map e l ∧ IntegrableOn (f ∘ e) (e ⁻¹' s) μ\n⊢... | [] | exact ⟨_, hs⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 505,
"column": 4
} | {
"line": 505,
"column": 17
} | {
"line": 506,
"column": 2
} | [
{
"pp": "case mp\nα : Type u_1\nβ : Type u_2\nε : Type u_3\nmα : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\nl : Filter α\ninst✝ : MeasurableSpace β\ne : α → β\nhe : MeasurableEmbedding e\nf : β → ε\ns : Set β\nhs : s ∈ map e l ∧ IntegrableOn (f ∘ e) (e ⁻¹' s) μ\n⊢... | [] | exact ⟨_, hs⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 874,
"column": 25
} | {
"line": 874,
"column": 28
} | {
"line": 875,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nε : Type u_5\nε' : Type u_6\nε'' : Type u_7\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁹ : MeasurableSpace δ\ninst✝⁸ : NormedAddCommGroup β\ninst✝⁷ : NormedAddCommGroup γ\ninst✝⁶ : TopologicalSpace ε\ninst✝⁵ : ContinuousENorm ε\ninst✝⁴ : Topolo... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nε : Type u_5\nε' : Type u_6\nε'' : Type u_7\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁹ : MeasurableSpace δ\ninst✝⁸ : NormedAddCommGroup β\ninst✝⁷ : NormedAddCommGroup γ\ninst✝⁶ : TopologicalSpace ε\ninst✝⁵ : ContinuousENorm ε\ninst✝⁴ : TopologicalSpace ε... | h'' | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 887,
"column": 17
} | {
"line": 887,
"column": 20
} | {
"line": 887,
"column": 21
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nε : Type u_5\nε' : Type u_6\nε'' : Type u_7\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁹ : MeasurableSpace δ\ninst✝⁸ : NormedAddCommGroup β\ninst✝⁷ : NormedAddCommGroup γ\ninst✝⁶ : TopologicalSpace ε\ninst✝⁵ : ContinuousENorm ε\ninst✝⁴ : Topolo... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nε : Type u_5\nε' : Type u_6\nε'' : Type u_7\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁹ : MeasurableSpace δ\ninst✝⁸ : NormedAddCommGroup β\ninst✝⁷ : NormedAddCommGroup γ\ninst✝⁶ : TopologicalSpace ε\ninst✝⁵ : ContinuousENorm ε\ninst✝⁴ : TopologicalSpace ε... | h'' | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 470,
"column": 4
} | {
"line": 474,
"column": 44
} | {
"line": 475,
"column": 2
} | [
{
"pp": "case h\n𝕜 : Type u\nι : Type v\nE : ι → Type wE\nG : Type wG\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝³ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝² : SeminormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : Fintype ι\nA : ∀ (f : ContinuousMultilinear... | [] | calc
‖f x‖ ≤ 1 := hf _ <| (pi_norm_le_iff_of_nonneg (norm_nonneg c)).2 fun i ↦ (hx i).le
_ = ∏ i : ι, 1 := by simp
_ ≤ ∏ i, ‖x i‖ := by gcongr with i; simpa only [div_self hc₀.ne'] using hcx i
_ = 1 * ∏ i, ‖x i‖ := (one_mul _).symm | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 539,
"column": 30
} | {
"line": 540,
"column": 84
} | {
"line": 542,
"column": 0
} | [
{
"pp": "𝕜 : Type u\nι : Type v\nE : ι → Type wE\nG : Type wG\nG' : Type wG'\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝⁵ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁴ : SeminormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\ninst✝² : SeminormedAddCommGroup G'\ninst✝¹ :... | [] | by
simp only [opNNNorm_le_iff, prod_apply, Prod.nnnorm_def, max_le_iff, forall_and] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 720,
"column": 2
} | {
"line": 720,
"column": 34
} | {
"line": 721,
"column": 2
} | [
{
"pp": "case a\n𝕜 : Type u\nι : Type v\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : Fintype ι\nA : Type u_1\ninst✝² : NormedCommRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : IsEmpty ι\n⊢ ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ ≤ ‖1‖",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
... | [
"case a\n𝕜 : Type u\nι : Type v\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : Fintype ι\nA : Type u_1\ninst✝² : NormedCommRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : IsEmpty ι\n⊢ ‖1‖ ≤ ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖"
] | · apply opNorm_le_bound <;> simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Constructions.Polish.StronglyMeasurable | {
"line": 122,
"column": 45
} | {
"line": 131,
"column": 77
} | {
"line": 133,
"column": 0
} | [
{
"pp": "X : Type u_4\nE : Type u_5\nι : Type u_6\ninst✝⁶ : MeasurableSpace X\ninst✝⁵ : CommMonoid E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : PseudoMetrizableSpace E\ninst✝² : ContinuousMul E\nL : SummationFilter ι\ninst✝¹ : L.NeBot\ninst✝ : L.filter.IsCountablyGenerated\nf : ι → X → E\nh : ∀ (i : ι), StronglyMea... | [] | by
rw [tprod_def, finprod_def']
split_ifs with hm
any_goals exact stronglyMeasurable_one
· refine Finset.stronglyMeasurable_prod _ (fun _ _ ↦ ?_)
rw [Set.mulIndicator]
split_ifs
· fun_prop
· exact stronglyMeasurable_one
· exact stronglyMeasurable_of_tendsto L.filter (by fun_prop) hm.choose_spe... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 1196,
"column": 60
} | {
"line": 1196,
"column": 71
} | {
"line": 1196,
"column": 72
} | [
{
"pp": "𝕜 : Type u\nι : Type v\nE₁ : ι → Type wE₁\nG : Type wG\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : (i : ι) → SeminormedAddCommGroup (E₁ i)\ninst✝⁵ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁴ : SeminormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\ninst✝² : Fintype ι\nα : Type u_1\ninst✝¹ : Fintype α\nf... | [
"𝕜 : Type u\nι : Type v\nE₁ : ι → Type wE₁\nG : Type wG\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : (i : ι) → SeminormedAddCommGroup (E₁ i)\ninst✝⁵ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁴ : SeminormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\ninst✝² : Fintype ι\nα : Type u_1\ninst✝¹ : Fintype α\nf : Continuou... | prod_const, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.LocallyIntegrable | {
"line": 390,
"column": 2
} | {
"line": 390,
"column": 39
} | {
"line": 391,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nε'' : Type u_5\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : TopologicalSpace ε''\ninst✝² : ESeminormedAddMonoid ε''\ninst✝¹ : BorelSpace X\ninst✝ : BorelSpace Y\ne : X ≃ₜ Y\nf : Y → ε''\nμ : Measu... | [
"case refine_1\nX : Type u_1\nY : Type u_2\nε'' : Type u_5\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : TopologicalSpace ε''\ninst✝² : ESeminormedAddMonoid ε''\ninst✝¹ : BorelSpace X\ninst✝ : BorelSpace Y\ne : X ≃ₜ Y\nf : Y → ε''\nμ : Me... | refine ⟨fun h x => ?_, fun h x => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 163,
"column": 4
} | {
"line": 163,
"column": 18
} | {
"line": 165,
"column": 0
} | [
{
"pp": "case refine_2\nβ : Type u_2\nE : Type u_4\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\np : ℝ≥0∞\ninst✝¹ : BorelSpace E\nf : β → E\nhp_ne_top : p ≠ ∞\nμ : Measure β\nfmeas : Measurable f\ninst✝ : SeparableSpace ↑(Set.range f ∪ {0})\nhf : eLpNorm f p μ < ∞\n⊢ eL... | [] | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 163,
"column": 4
} | {
"line": 163,
"column": 18
} | {
"line": 165,
"column": 0
} | [
{
"pp": "case refine_2\nβ : Type u_2\nE : Type u_4\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\np : ℝ≥0∞\ninst✝¹ : BorelSpace E\nf : β → E\nhp_ne_top : p ≠ ∞\nμ : Measure β\nfmeas : Measurable f\ninst✝ : SeparableSpace ↑(Set.range f ∪ {0})\nhf : eLpNorm f p μ < ∞\n⊢ eL... | [] | simpa using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 163,
"column": 4
} | {
"line": 163,
"column": 18
} | {
"line": 165,
"column": 0
} | [
{
"pp": "case refine_2\nβ : Type u_2\nE : Type u_4\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\np : ℝ≥0∞\ninst✝¹ : BorelSpace E\nf : β → E\nhp_ne_top : p ≠ ∞\nμ : Measure β\nfmeas : Measurable f\ninst✝ : SeparableSpace ↑(Set.range f ∪ {0})\nhf : eLpNorm f p μ < ∞\n⊢ eL... | [] | simpa using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 552,
"column": 96
} | {
"line": 556,
"column": 24
} | {
"line": 558,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_4\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nμ : Measure α\nf : ↥(simpleFunc E p μ)\n⊢ ⇑(toSimpleFunc (-f)) =ᵐ[μ] -⇑(toSimpleFunc f)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"AddGroup.toSubtr... | [] | by
filter_upwards [toSimpleFunc_eq_toFun (-f), toSimpleFunc_eq_toFun f,
Lp.coeFn_neg (f : Lp E p μ)] with _
simp only [Pi.neg_apply, AddSubgroup.coe_neg]
repeat intro h; rw [h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.LocallyIntegrable | {
"line": 806,
"column": 2
} | {
"line": 806,
"column": 87
} | {
"line": 808,
"column": 0
} | [
{
"pp": "X : Type u_1\nE : Type u_6\ninst✝⁹ : MeasurableSpace X\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : NormedAddCommGroup E\nμ : Measure X\ninst✝⁶ : OpensMeasurableSpace X\ninst✝⁵ : LocallyCompactSpace X\ninst✝⁴ : T2Space X\n𝕜 : Type u_9\ninst✝³ : NormedRing 𝕜\ninst✝² : SecondCountableTopologyEither X E\ninst... | [] | exact fun k hk_sub hk_c => (hf k hk_sub hk_c).smul_continuousOn (hg.mono hk_sub) hk_c | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Operator.Extend | {
"line": 258,
"column": 14
} | {
"line": 263,
"column": 59
} | {
"line": 264,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_3\nEₗ : Type u_4\nF : Type u_5\nFₗ : Type u_6\ninst✝¹⁵ : NormedDivisionRing 𝕜\ninst✝¹⁴ : NormedDivisionRing 𝕜₂\ninst✝¹³ : AddCommGroup E\ninst✝¹² : NormedAddCommGroup Eₗ\ninst✝¹¹ : AddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup Fₗ\ninst✝⁹ : Module 𝕜 E\ninst✝⁸ ... | [] | by
refine h_dense₁.induction ?_ ?_
· rintro _ ⟨_, rfl⟩
simp [LinearMap.extendOfNorm_eq, h_dense₁, h_norm₁, h_dense₂, h_norm₂]
· exact isClosed_eq (by simp only [AddHom.toFun_eq_coe, LinearMap.coe_toAddHom,
ContinuousLinearMap.coe_coe]; fun_prop) continuous_id | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 332,
"column": 2
} | {
"line": 332,
"column": 23
} | {
"line": 333,
"column": 2
} | [
{
"pp": "α : 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\n⊢ 0 ≤ μ.real (⇑f ⁻¹' {f y}) • f y",
"ppTerm": "?m.56",
... | [
"case pos\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\n⊢ 0 ≤ μ.real (⇑f ⁻¹' {f y}) • f y",
"case neg... | by_cases hy : 0 ≤ f y | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 115,
"column": 2
} | {
"line": 117,
"column": 72
} | {
"line": 119,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : NormedAddCommGroup β\nu : α → β\nl : Filter α\n⊢ u ~[l] 0 ↔ u =O[l] 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Asymptotics.isBigO_zero_right_iff",
"Eq.mpr",
"congrArg",
"Asymptotics.... | [] | refine ⟨IsEquivalent.isBigO, fun h ↦ ?_⟩
rw [isEquivalent_zero_iff_eventually_zero, eventuallyEq_iff_exists_mem]
exact ⟨{ x : α | u x = 0 }, isBigO_zero_right_iff.mp h, fun x hx ↦ hx⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 115,
"column": 2
} | {
"line": 117,
"column": 72
} | {
"line": 119,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : NormedAddCommGroup β\nu : α → β\nl : Filter α\n⊢ u ~[l] 0 ↔ u =O[l] 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Asymptotics.isBigO_zero_right_iff",
"Eq.mpr",
"congrArg",
"Asymptotics.... | [] | refine ⟨IsEquivalent.isBigO, fun h ↦ ?_⟩
rw [isEquivalent_zero_iff_eventually_zero, eventuallyEq_iff_exists_mem]
exact ⟨{ x : α | u x = 0 }, isBigO_zero_right_iff.mp h, fun x hx ↦ hx⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 482,
"column": 4
} | {
"line": 482,
"column": 58
} | {
"line": 483,
"column": 4
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : Integrable f μ\nf₁ : ↥(Lp ℝ 1 μ) := ⋯\neq₁ : (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal = ‖Lp.posPart f₁‖\na✝ : α\nh₁ : ↑↑(Lp.negPart f₁) a✝ = -min (↑↑f₁ a✝) 0\nh₂ : ↑↑(Integrable.toL1 f hf) a✝ = f a✝\n⊢ ↑(-f a✝).toNNReal = ↑‖-min (f... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : Integrable f μ\nf₁ : ↥(Lp ℝ 1 μ) := Integrable.toL1 f hf\neq₁ : (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal = ‖Lp.posPart f₁‖\na✝ : α\nh₁ : ↑↑(Lp.negPart f₁) a✝ = -min (↑↑f₁ a✝) 0\nh₂ : ↑↑(Integrable.toL1 f hf) a✝ = f a✝\n⊢ max (-f a✝) 0 = ‖min (... | simp only [Real.coe_toNNReal', coe_nnnorm, nnnorm_neg] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 862,
"column": 4
} | {
"line": 862,
"column": 50
} | {
"line": 863,
"column": 4
} | [
{
"pp": "α : Type u_1\nE : Type u_4\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ∞\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → μ s < ∞ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, eLpNorm (g - s.indicator fun x ↦ c) p μ ≤ ε ∧ P g\nh1P : ∀ (f... | [
"α : Type u_1\nE : Type u_4\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ∞\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → μ s < ∞ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, eLpNorm (g - s.indicator fun x ↦ c) p μ ≤ ε ∧ P g\nh1P : ∀ (f g : α → E),... | rcases H f' η ηpos.ne' f'_mem with ⟨g, hg, Pg⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 655,
"column": 2
} | {
"line": 655,
"column": 17
} | {
"line": 656,
"column": 2
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nm : MeasurableSpace α\nμ : Measure α\ninst✝⁵ : PartialOrder E\ninst✝⁴ : IsOrderedAddMonoid E\ninst✝³ : IsOrderedModule ℝ E\ninst✝² : ClosedIciTopology E\nβ : Type u_6\ninst✝¹ : AddCommMonoid β\ninst✝ : Module ℝ β\nf : ... | [
"α : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nm : MeasurableSpace α\nμ : Measure α\ninst✝⁵ : PartialOrder E\ninst✝⁴ : IsOrderedAddMonoid E\ninst✝³ : IsOrderedModule ℝ E\ninst✝² : ClosedIciTopology E\nβ : Type u_6\ninst✝¹ : AddCommMonoid β\ninst✝ : Module ℝ β\nf : α → β → E\ns... | refine ⟨hs, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 805,
"column": 4
} | {
"line": 805,
"column": 53
} | {
"line": 807,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : ℕ → α → ℝ\nF : α → ℝ\nhf : ∀ (n : ℕ), Integrable (f n) μ\nhF : Integrable F μ\nh_mono : ∀ᵐ (x : α) ∂μ, Antitone fun n ↦ f n x\nh_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))\n⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ -f... | [] | filter_upwards [h_tendsto] with x hx using hx.neg | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 805,
"column": 4
} | {
"line": 805,
"column": 53
} | {
"line": 807,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : ℕ → α → ℝ\nF : α → ℝ\nhf : ∀ (n : ℕ), Integrable (f n) μ\nhF : Integrable F μ\nh_mono : ∀ᵐ (x : α) ∂μ, Antitone fun n ↦ f n x\nh_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))\n⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ -f... | [] | filter_upwards [h_tendsto] with x hx using hx.neg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 805,
"column": 4
} | {
"line": 805,
"column": 53
} | {
"line": 807,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : ℕ → α → ℝ\nF : α → ℝ\nhf : ∀ (n : ℕ), Integrable (f n) μ\nhF : Integrable F μ\nh_mono : ∀ᵐ (x : α) ∂μ, Antitone fun n ↦ f n x\nh_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))\n⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ -f... | [] | filter_upwards [h_tendsto] with x hx using hx.neg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.ContinuousAffineMap | {
"line": 229,
"column": 2
} | {
"line": 239,
"column": 55
} | {
"line": 241,
"column": 0
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nW : Type u_3\nP : Type u_4\nQ : Type u_5\ninst✝¹² : Ring R\ninst✝¹¹ : AddCommGroup V\ninst✝¹⁰ : Module R V\ninst✝⁹ : TopologicalSpace P\ninst✝⁸ : AddTorsor V P\ninst✝⁷ : AddCommGroup W\ninst✝⁶ : Module R W\ninst✝⁵ : TopologicalSpace Q\ninst✝⁴ : AddTorsor W Q\ninst✝³ : Topolo... | [] | have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine ⟨fun h => ?_, fun h => ?_⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine ⟨fun h =... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.ContinuousAffineMap | {
"line": 229,
"column": 2
} | {
"line": 239,
"column": 55
} | {
"line": 241,
"column": 0
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nW : Type u_3\nP : Type u_4\nQ : Type u_5\ninst✝¹² : Ring R\ninst✝¹¹ : AddCommGroup V\ninst✝¹⁰ : Module R V\ninst✝⁹ : TopologicalSpace P\ninst✝⁸ : AddTorsor V P\ninst✝⁷ : AddCommGroup W\ninst✝⁶ : Module R W\ninst✝⁵ : TopologicalSpace Q\ninst✝⁴ : AddTorsor W Q\ninst✝³ : Topolo... | [] | have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine ⟨fun h => ?_, fun h => ?_⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine ⟨fun h =... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 840,
"column": 2
} | {
"line": 843,
"column": 61
} | {
"line": 845,
"column": 2
} | [
{
"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, ∀ ... | [
"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, ∀ (i : ℕ), f i... | have h_bound : ∀ᵐ a ∂μ, ∀ i, f' i a ≤ F' a := by
filter_upwards [hf_bound] with a ha_bound i
refine ENNReal.ofReal_le_ofReal ?_
simp only [tsub_le_iff_right, sub_add_cancel, ha_bound i] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Module.RieszLemma | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 47
} | {
"line": 101,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nR : ℝ\nhR : ‖c‖ < R\nF : Subspace 𝕜 E\nhFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F\nhF : ∃ x, x ∉ F\nRpos : 0 < R\nthis : ‖c‖ / R < 1\nx : E\n... | [
"𝕜 : Type u_1\ninst✝² : NormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nR : ℝ\nhR : ‖c‖ < R\nF : Subspace 𝕜 E\nhFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F\nhF : ∃ x, x ∉ F\nRpos : 0 < R\nthis : ‖c‖ / R < 1\nx : E\nxF : x ∉ F\n... | have x0 : x ≠ 0 := fun H => by simp [H] at xF | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Module.RieszLemma | {
"line": 113,
"column": 53
} | {
"line": 113,
"column": 89
} | {
"line": 113,
"column": 89
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nR : ℝ\nhR : ‖c‖ < R\nF : Subspace 𝕜 E\nhFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F\nhF : ∃ x, x ∉ F\nRpos : 0 < R\nthis : ‖c‖ / R < 1\nx : E\n... | [] | simp [y', Submodule.smul_mem _ _ hy] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Module.RieszLemma | {
"line": 113,
"column": 53
} | {
"line": 113,
"column": 89
} | {
"line": 113,
"column": 89
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nR : ℝ\nhR : ‖c‖ < R\nF : Subspace 𝕜 E\nhFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F\nhF : ∃ x, x ∉ F\nRpos : 0 < R\nthis : ‖c‖ / R < 1\nx : E\n... | [] | simp [y', Submodule.smul_mem _ _ hy] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.RieszLemma | {
"line": 113,
"column": 53
} | {
"line": 113,
"column": 89
} | {
"line": 113,
"column": 89
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nR : ℝ\nhR : ‖c‖ < R\nF : Subspace 𝕜 E\nhFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F\nhF : ∃ x, x ∉ F\nRpos : 0 < R\nthis : ‖c‖ / R < 1\nx : E\n... | [] | simp [y', Submodule.smul_mem _ _ hy] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 775,
"column": 4
} | {
"line": 776,
"column": 36
} | {
"line": 778,
"column": 0
} | [
{
"pp": "case neg\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\nhT : DominatedFinMeasAdditive μ T C\nf : α → E\nhF : CompleteSpace F\n... | [] | rw [setToFun_undef hT hf, setToFun_undef hT, neg_zero]
rwa [← integrable_neg_iff] at hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 775,
"column": 4
} | {
"line": 776,
"column": 36
} | {
"line": 778,
"column": 0
} | [
{
"pp": "case neg\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\nhT : DominatedFinMeasAdditive μ T C\nf : α → E\nhF : CompleteSpace F\n... | [] | rw [setToFun_undef hT hf, setToFun_undef hT, neg_zero]
rwa [← integrable_neg_iff] at hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Indicator | {
"line": 30,
"column": 37
} | {
"line": 31,
"column": 50
} | {
"line": 33,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\nf : α → β\ns : Set α\ninst✝ : One β\nhs : ∀ a ∈ frontier s, f a = 1\nhf : Continuous[inst✝², inst✝¹] f\n⊢ Continuous[inst✝², inst✝¹] (s.mulIndicator f)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants... | [] | by
classical exact hf.piecewise hs continuous_const | [anonymous] | Lean.Parser.Term.byTactic |
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