module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Group.FoelnerFilter | {
"line": 107,
"column": 32
} | {
"line": 107,
"column": 48
} | {
"line": 108,
"column": 2
} | [
{
"pp": "G : Type u_1\nX : Type u_2\ninst✝⁴ : MeasurableSpace X\nμ : Measure X\ninst✝³ : Group G\ninst✝² : MulAction G X\nι : Type u_3\nl : Filter ι\ninst✝¹ : NeZero μ\ninst✝ : IsFiniteMeasure μ\n⊢ ∀ᶠ (i : ι) in l, μ univ ≠ 0",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"_private.... | [] | simp [NeZero.ne] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.ContinuousMap.CompactlySupported | {
"line": 360,
"column": 8
} | {
"line": 360,
"column": 22
} | {
"line": 360,
"column": 23
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : TopologicalSpace β\nx : α\nR : Type u_5\ninst✝⁵ : Semiring R\ninst✝⁴ : NonUnitalNonAssocSemiring β\ninst✝³ : IsTopologicalSemiring β\ninst✝² : Module R β\ninst✝¹ : ContinuousConstSMul R β\ninst✝ : IsScalarTowe... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : TopologicalSpace β\nx : α\nR : Type u_5\ninst✝⁵ : Semiring R\ninst✝⁴ : NonUnitalNonAssocSemiring β\ninst✝³ : IsTopologicalSemiring β\ninst✝² : Module R β\ninst✝¹ : ContinuousConstSMul R β\ninst✝ : IsScalarTower R β β\nr :... | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.ContinuousMap.CompactlySupported | {
"line": 360,
"column": 23
} | {
"line": 360,
"column": 37
} | {
"line": 360,
"column": 38
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : TopologicalSpace β\nx : α\nR : Type u_5\ninst✝⁵ : Semiring R\ninst✝⁴ : NonUnitalNonAssocSemiring β\ninst✝³ : IsTopologicalSemiring β\ninst✝² : Module R β\ninst✝¹ : ContinuousConstSMul R β\ninst✝ : IsScalarTowe... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : TopologicalSpace β\nx : α\nR : Type u_5\ninst✝⁵ : Semiring R\ninst✝⁴ : NonUnitalNonAssocSemiring β\ninst✝³ : IsTopologicalSemiring β\ninst✝² : Module R β\ninst✝¹ : ContinuousConstSMul R β\ninst✝ : IsScalarTower R β β\nr :... | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.ContinuousMap.CompactlySupported | {
"line": 368,
"column": 8
} | {
"line": 368,
"column": 22
} | {
"line": 368,
"column": 23
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : TopologicalSpace β\nx : α\nR : Type u_5\ninst✝⁵ : Semiring R\ninst✝⁴ : NonUnitalNonAssocSemiring β\ninst✝³ : IsTopologicalSemiring β\ninst✝² : Module R β\ninst✝¹ : ContinuousConstSMul R β\ninst✝ : SMulCommClas... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : TopologicalSpace β\nx : α\nR : Type u_5\ninst✝⁵ : Semiring R\ninst✝⁴ : NonUnitalNonAssocSemiring β\ninst✝³ : IsTopologicalSemiring β\ninst✝² : Module R β\ninst✝¹ : ContinuousConstSMul R β\ninst✝ : SMulCommClass R β β\nr :... | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.ContinuousMap.CompactlySupported | {
"line": 368,
"column": 23
} | {
"line": 368,
"column": 37
} | {
"line": 368,
"column": 38
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : TopologicalSpace β\nx : α\nR : Type u_5\ninst✝⁵ : Semiring R\ninst✝⁴ : NonUnitalNonAssocSemiring β\ninst✝³ : IsTopologicalSemiring β\ninst✝² : Module R β\ninst✝¹ : ContinuousConstSMul R β\ninst✝ : SMulCommClas... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : TopologicalSpace β\nx : α\nR : Type u_5\ninst✝⁵ : Semiring R\ninst✝⁴ : NonUnitalNonAssocSemiring β\ninst✝³ : IsTopologicalSemiring β\ninst✝² : Module R β\ninst✝¹ : ContinuousConstSMul R β\ninst✝ : SMulCommClass R β β\nr :... | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral | {
"line": 91,
"column": 2
} | {
"line": 107,
"column": 51
} | {
"line": 109,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\na b C : ℝ\nhab : a ≤ b\nhfc : ContinuousOn f (Icc a b)\nhfd : DifferentiableOn ℝ f (Ioo a b)\nhnorm : ∀ᵐ (t : ℝ), t ∈ Ioo a b → ‖deriv f t‖ ≤ C\ns : Set ℝ := toMeasurable volume {x | deriv f x ≠ 0}\nhsm : MeasurableSet s\n... | [] | calc
‖f b - f a‖ ≤ ∫ t in a..b, indicator s (fun _ ↦ C) t := by
apply norm_sub_le_integral_of_norm_deriv_le_of_le hab hfc hfd
· refine hnorm.mono fun t ht ht_mem ↦ ?_
apply le_indicator_apply
· exact fun ht' ↦ ht ht_mem
· simp only [s, norm_le_zero_iff]
exact not_imp_co... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule | {
"line": 83,
"column": 11
} | {
"line": 83,
"column": 36
} | {
"line": 83,
"column": 37
} | [
{
"pp": "f : ℝ → ℝ\nN : ℕ\na h : ℝ\nN_nonzero : 0 < N\n⊢ ∑ i ∈ Finset.range N, trapezoidal_integral f 1 (a + ↑i * h) (a + (↑i + 1) * h) =\n trapezoidal_integral f N a (a + ↑N * h)",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"HMul... | [
"f : ℝ → ℝ\nN : ℕ\na h : ℝ\nN_nonzero : 0 < N\n⊢ ∑ x ∈ Finset.range N, (a + (↑x + 1) * h - (a + ↑x * h)) / 2 * (f (a + ↑x * h) + f (a + (↑x + 1) * h)) =\n trapezoidal_integral f N a (a + ↑N * h)"
] | trapezoidal_integral_one, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule | {
"line": 125,
"column": 6
} | {
"line": 125,
"column": 20
} | {
"line": 125,
"column": 21
} | [
{
"pp": "f : ℝ → ℝ\nζ a b : ℝ\na_lt_b : a < b\nh_df : DifferentiableOn ℝ f (Set.Icc a b)\nh_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)\nfpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ\n⊢ |trapezoidal_error f 1 a b| ≤ (b - a) ^ 3 * ζ / 12",
"ppTerm": "?m.... | [
"f : ℝ → ℝ\nζ a b : ℝ\na_lt_b : a < b\nh_df : DifferentiableOn ℝ f (Set.Icc a b)\nh_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)\nfpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ\n⊢ |trapezoidal_error f 1 a b| ≤ (b - a) ^ 3 * (ζ / 12)"
] | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule | {
"line": 134,
"column": 4
} | {
"line": 134,
"column": 28
} | {
"line": 135,
"column": 4
} | [
{
"pp": "f : ℝ → ℝ\nζ a b : ℝ\na_lt_b : a < b\nh_df : DifferentiableOn ℝ f (Set.Icc a b)\nh_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)\nfpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ\ng : ℝ → ℝ := fun t ↦ trapezoidal_error f 1 a t\ndg : ℝ → ℝ := fun t ↦ 1 /... | [
"f : ℝ → ℝ\nζ a b : ℝ\na_lt_b : a < b\nh_df : DifferentiableOn ℝ f (Set.Icc a b)\nh_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)\nfpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ\ng : ℝ → ℝ := fun t ↦ trapezoidal_error f 1 a t\ndg : ℝ → ℝ := fun t ↦ 1 / 2 * (f a + ... | simp_rw [← mul_comm_div] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule | {
"line": 223,
"column": 12
} | {
"line": 223,
"column": 26
} | {
"line": 223,
"column": 27
} | [
{
"pp": "case refine_2\nf : ℝ → ℝ\nζ a b : ℝ\na_lt_b : a < b\nh_df : DifferentiableOn ℝ f (Set.Icc a b)\nh_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)\nfpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ\nN : ℕ\nN_nonzero : 0 < N\nh : ℝ := (b - a) / ↑N\nak : ℕ → ... | [
"case refine_2\nf : ℝ → ℝ\nζ a b : ℝ\na_lt_b : a < b\nh_df : DifferentiableOn ℝ f (Set.Icc a b)\nh_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)\nfpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ\nN : ℕ\nN_nonzero : 0 < N\nh : ℝ := (b - a) / ↑N\nak : ℕ → ℝ := fun k ↦... | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic | {
"line": 285,
"column": 2
} | {
"line": 285,
"column": 94
} | {
"line": 286,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\nΛ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0\ninst✝¹ : T2Space X\ninst✝ : LocallyCompactSpace X\nK : Compacts X\n⊢ (rieszContent Λ) K = ⨅ K', ⨅ (_ : ↑K ⊆ interior ↑K'), (rieszContent Λ) K'",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"_private.M... | [
"X : Type u_1\ninst✝² : TopologicalSpace X\nΛ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0\ninst✝¹ : T2Space X\ninst✝ : LocallyCompactSpace X\nK : Compacts X\n⊢ (∀ (i : Compacts X), ↑K ⊆ interior ↑i → rieszContentAux Λ K ≤ rieszContentAux Λ i) ∧\n ⨅ K', ⨅ (_ : ↑K ⊆ interior ↑K'), ↑(rieszContentAux Λ K') ≤ ↑(rieszContentAux Λ K)"
... | simp only [rieszContent, le_antisymm_iff, le_iInf_iff, ENNReal.coe_le_coe, Content.mk_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic | {
"line": 300,
"column": 16
} | {
"line": 300,
"column": 30
} | {
"line": 300,
"column": 31
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\nΛ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0\ninst✝¹ : T2Space X\ninst✝ : LocallyCompactSpace X\nK : Compacts X\nb : ℝ≥0∞\nhb : ∀ (i : Compacts X), b ≤ ⨅ (_ : ↑K ⊆ interior ↑i), ↑(rieszContentAux Λ i)\nthis : b < ∞\nε : ℝ≥0\nhε : 0 < ↑ε\nf : X →C_c ℝ≥0\nhfleoneonK : ∀ x ∈ K, ... | [
"X : Type u_1\ninst✝² : TopologicalSpace X\nΛ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0\ninst✝¹ : T2Space X\ninst✝ : LocallyCompactSpace X\nK : Compacts X\nb : ℝ≥0∞\nhb : ∀ (i : Compacts X), b ≤ ⨅ (_ : ↑K ⊆ interior ↑i), ↑(rieszContentAux Λ i)\nthis : b < ∞\nε : ℝ≥0\nhε : 0 < ↑ε\nf : X →C_c ℝ≥0\nhfleoneonK : ∀ x ∈ K, 1 ≤ f x\nhfl... | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.TorusIntegral | {
"line": 207,
"column": 4
} | {
"line": 208,
"column": 46
} | {
"line": 208,
"column": 47
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : (Fin 1 → ℂ) → E\nc : Fin 1 → ℂ\nR : Fin 1 → ℝ\nH₁ : (⇑(MeasurableEquiv.funUnique (Fin 1) ℝ).symm ⁻¹' Icc 0 fun x ↦ 2 * π) = Icc 0 (2 * π)\nH₂ : torusMap c R = fun θ x ↦ circleMap (c 0) (R 0) (θ 0)\n⊢ ∫ (θ : Fin 1 → ℝ) in Icc 0 fu... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : (Fin 1 → ℂ) → E\nc : Fin 1 → ℂ\nR : Fin 1 → ℝ\nH₁ : (⇑(MeasurableEquiv.funUnique (Fin 1) ℝ).symm ⁻¹' Icc 0 fun x ↦ 2 * π) = Icc 0 (2 * π)\nH₂ : torusMap c R = fun θ x ↦ circleMap (c 0) (R 0) (θ 0)\n⊢ ∫ (x : ℝ) in ⇑(MeasurableEquiv.funUnique ... | ← ((volume_preserving_funUnique (Fin 1) ℝ).symm _).setIntegral_preimage_emb
(MeasurableEquiv.measurableEmbedding _), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | {
"line": 140,
"column": 22
} | {
"line": 140,
"column": 31
} | {
"line": 140,
"column": 31
} | [
{
"pp": "case h.refine_2\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nf : X →C_c ℝ\na ε : ℝ\nhε : 0 < ε\nN : ℕ\nhf : range ⇑f ⊆ Ioo a (a + ↑N * ε)\nb : ℝ := a + ↑N * ε\ny : Fin N → ℝ := fun n ↦ a + ε * (↑↑n + 1)\nhy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m\nE : F... | [
"case h.refine_2\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nf : X →C_c ℝ\na ε : ℝ\nhε : 0 < ε\nN : ℕ\nhf : range ⇑f ⊆ Ioo a (a + ↑N * ε)\nb : ℝ := a + ↑N * ε\ny : Fin N → ℝ := fun n ↦ a + ε * (↑↑n + 1)\nhy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m\nE : Fin N → Set X... | and_assoc | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 73
} | {
"line": 161,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝³ : TopologicalSpace X\ninst✝² : T2Space X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nf : X →C_c ℝ\nε : ℝ\nhε : 0 < ε\nE : Set X\nμ : Content X\nhμ : μ.outerMeasure E ≠ ∞\nhμ' : MeasurableSet E\nc : ℝ\nhfE : ∀ x ∈ E, f x < c\nhε' : ε.toNNReal ≠ 0\n⊢ ∃ V, E ⊆ ↑V ∧ (∀ x ∈ V, f ... | [
"X : Type u_1\ninst✝³ : TopologicalSpace X\ninst✝² : T2Space X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nf : X →C_c ℝ\nε : ℝ\nhε : 0 < ε\nE : Set X\nμ : Content X\nhμ : μ.outerMeasure E ≠ ∞\nhμ' : MeasurableSet E\nc : ℝ\nhfE : ∀ x ∈ E, f x < c\nhε' : ε.toNNReal ≠ 0\nV₁ : Opens X\nhV₁ : E ⊆ ↑V₁ ∧ μ.outerMea... | obtain ⟨V₁ : Opens X, hV₁⟩ := Content.outerMeasure_exists_open μ hμ hε' | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | {
"line": 171,
"column": 51
} | {
"line": 171,
"column": 76
} | {
"line": 171,
"column": 76
} | [
{
"pp": "X : Type u_1\ninst✝³ : TopologicalSpace X\ninst✝² : T2Space X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nf : X →C_c ℝ\nε : ℝ\nhε : 0 < ε\nE : Set X\nμ : Content X\nhμ : μ.outerMeasure E ≠ ∞\nhμ' : MeasurableSet E\nc : ℝ\nhfE : ∀ x ∈ E, f x < c\nhε' : ε.toNNReal ≠ 0\nV₁ : Opens X\nhV₁ : E ⊆ ↑V₁ ... | [
"X : Type u_1\ninst✝³ : TopologicalSpace X\ninst✝² : T2Space X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nf : X →C_c ℝ\nε : ℝ\nhε : 0 < ε\nE : Set X\nμ : Content X\nhμ : μ.outerMeasure E ≠ ∞\nhμ' : MeasurableSet E\nc : ℝ\nhfE : ∀ x ∈ E, f x < c\nhε' : ε.toNNReal ≠ 0\nV₁ : Opens X\nhV₁ : E ⊆ ↑V₁ ∧ μ.outerMea... | ENNReal.ofNNReal_toNNReal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | {
"line": 216,
"column": 4
} | {
"line": 216,
"column": 13
} | {
"line": 217,
"column": 4
} | [
{
"pp": "X : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : T2Space X\ninst✝² : MeasurableSpace X\ninst✝¹ : BorelSpace X\nΛ : (X →C_c ℝ) →ₚ[ℝ] ℝ\ninst✝ : LocallyCompactSpace X\nf : X →C_c ℝ\nμ : Measure X := ⋯\nK : Set X := ⋯\nε : ℝ\nhε : 0 < ε\na b : ℝ\nhab : a < b ∧ range ⇑f ⊆ Ioo a b\nN : ℕ\nhN : 0 < N\nε' ... | [
"X : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : T2Space X\ninst✝² : MeasurableSpace X\ninst✝¹ : BorelSpace X\nΛ : (X →C_c ℝ) →ₚ[ℝ] ℝ\ninst✝ : LocallyCompactSpace X\nf : X →C_c ℝ\nμ : Measure X := rieszMeasure Λ\nK : Set X := tsupport ⇑f\nε : ℝ\nhε : 0 < ε\na b : ℝ\nhab : a < b ∧ range ⇑f ⊆ Ioo a b\nN : ℕ\nhN :... | dsimp [μ] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic | {
"line": 83,
"column": 70
} | {
"line": 83,
"column": 89
} | {
"line": 85,
"column": 0
} | [
{
"pp": "F : Type u_2\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\n⊢ probCharDual 0 = 1",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"Real",
"BoundedContinuousFunction.char_... | [] | simp [probCharDual] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic | {
"line": 83,
"column": 70
} | {
"line": 83,
"column": 89
} | {
"line": 85,
"column": 0
} | [
{
"pp": "F : Type u_2\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\n⊢ probCharDual 0 = 1",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"Real",
"BoundedContinuousFunction.char_... | [] | simp [probCharDual] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic | {
"line": 83,
"column": 70
} | {
"line": 83,
"column": 89
} | {
"line": 85,
"column": 0
} | [
{
"pp": "F : Type u_2\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\n⊢ probCharDual 0 = 1",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"Real",
"BoundedContinuousFunction.char_... | [] | simp [probCharDual] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Separation.CompletelyRegular | {
"line": 177,
"column": 66
} | {
"line": 177,
"column": 75
} | {
"line": 177,
"column": 75
} | [
{
"pp": "case a\nX : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompletelyRegularSpace X\nx : X\n⊢ ∀ (s : Set X), x ∈ s → IsOpen[inst✝¹] s → ∃ i, (stoneCechUnit x ∈ i ∧ IsOpen i) ∧ stoneCechUnit ⁻¹' i ⊆ s",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [
"case a\nX : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompletelyRegularSpace X\nx : X\n⊢ ∀ (s : Set X), x ∈ s → IsOpen[inst✝¹] s → ∃ i, stoneCechUnit x ∈ i ∧ IsOpen i ∧ stoneCechUnit ⁻¹' i ⊆ s"
] | and_assoc | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 38
} | {
"line": 66,
"column": 2
} | [
{
"pp": "case neg\nΩ : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : PseudoEMetricSpace Ω\nε₁ ε₂ : ℝ≥0∞\nμ ν : Measure Ω\nh_le : ε₁ ≤ ε₂\nB : Set Ω\nhε₁ : μ B ≤ ν (thickening ε₁.toReal B) + ε₁\nε_top : ¬ε₂ = ∞\n⊢ μ B ≤ ν (thickening ε₂.toReal B) + ε₂",
"ppTerm": "?neg✝",
"assigned": true,
"usedConst... | [
"Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : PseudoEMetricSpace Ω\nε₁ ε₂ : ℝ≥0∞\nμ ν : Measure Ω\nh_le : ε₁ ≤ ε₂\nB : Set Ω\nhε₁ : μ B ≤ ν (thickening ε₁.toReal B) + ε₁\nε_top : ¬ε₂ = ∞\n⊢ ν (thickening ε₁.toReal B) ≤ ν (thickening ε₂.toReal B)"
] | apply hε₁.trans (add_le_add ?_ h_le) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Measure.FiniteMeasureProd | {
"line": 126,
"column": 79
} | {
"line": 129,
"column": 36
} | {
"line": 131,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nβ : Type u_2\ninst✝ : MeasurableSpace β\nμ : ProbabilityMeasure α\nν : ProbabilityMeasure β\n⊢ (μ.prod ν).map ⋯ = ν",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"instHSMul",
"MeasureTheory.Measure",
"instSMulOfMul",
... | [] | by
apply Subtype.ext
simp only [val_eq_to_measure, toMeasure_map, toMeasure_prod, Measure.map_snd_prod,
measure_univ, one_smul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 230,
"column": 2
} | {
"line": 230,
"column": 60
} | {
"line": 231,
"column": 2
} | [
{
"pp": "Ω : Type u_1\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : PseudoEMetricSpace Ω\ninst✝² : OpensMeasurableSpace Ω\nμ ν : Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : IsProbabilityMeasure ν\nδ : ℝ≥0∞\nh : ∀ (ε : ℝ≥0∞) (B : Set Ω), δ < ε → ε < ∞ → MeasurableSet B → μ B ≤ ν (thickening ε.toReal B) + ε\nε : ... | [
"Ω : Type u_1\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : PseudoEMetricSpace Ω\ninst✝² : OpensMeasurableSpace Ω\nμ ν : Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : IsProbabilityMeasure ν\nδ : ℝ≥0∞\nh : ∀ (ε : ℝ≥0∞) (B : Set Ω), δ < ε → ε < ∞ → MeasurableSet B → μ B ≤ ν (thickening ε.toReal B) + ε\nε : ℝ≥0∞\nB : Se... | rw [prob_compl_eq_one_sub isOpen_thickening.measurableSet] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | {
"line": 337,
"column": 4
} | {
"line": 338,
"column": 100
} | {
"line": 340,
"column": 0
} | [
{
"pp": "case calc_7\nX : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : T2Space X\ninst✝² : MeasurableSpace X\ninst✝¹ : BorelSpace X\nΛ : (X →C_c ℝ) →ₚ[ℝ] ℝ\ninst✝ : LocallyCompactSpace X\nf : X →C_c ℝ\nμ : Measure X := rieszMeasure Λ\nK : Set X := tsupport ⇑f\nε : ℝ\nhε : 0 < ε\na b : ℝ\nhab : a < b ∧ range ... | [] | simpa [show (N : ℝ) ≠ 0 by simp [hN.ne.symm], mul_comm _ ε', div_eq_mul_inv, mul_assoc]
using (mul_le_mul_iff_of_pos_left hε'.1).mpr <| (inv_mul_le_iff₀ (Nat.cast_pos'.mpr hN)).mpr h | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Measure.Haar.MulEquivHaarChar | {
"line": 77,
"column": 6
} | {
"line": 77,
"column": 51
} | {
"line": 77,
"column": 52
} | [
{
"pp": "G : Type u_1\ninst✝⁷ : Group G\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : BorelSpace G\ninst✝³ : IsTopologicalGroup G\ninst✝² : LocallyCompactSpace G\nμ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ.Regular\nφ : G ≃ₜ* G\ne : G ≃ᵐ G := φ.toHomeomorph.toMeasurableEquiv\nthis : ... | [
"G : Type u_1\ninst✝⁷ : Group G\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : BorelSpace G\ninst✝³ : IsTopologicalGroup G\ninst✝² : LocallyCompactSpace G\nμ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ.Regular\nφ : G ≃ₜ* G\ne : G ≃ᵐ G := φ.toHomeomorph.toMeasurableEquiv\nthis : (map (⇑φ.sym... | ← mulEquivHaarChar_smul_map (map φ.symm μ) φ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Haar.Extension | {
"line": 256,
"column": 28
} | {
"line": 256,
"column": 42
} | {
"line": 256,
"column": 43
} | [
{
"pp": "case inr\nA : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝¹⁷ : Group A\ninst✝¹⁶ : Group B\ninst✝¹⁵ : Group C\ninst✝¹⁴ : TopologicalSpace A\ninst✝¹³ : TopologicalSpace B\ninst✝¹² : TopologicalSpace C\nφ : A →* B\nψ : B →* C\nH : IsSES φ ψ\ninst✝¹¹ : IsTopologicalGroup A\ninst✝¹⁰ : IsTopologicalGroup B\ni... | [
"case inr\nA : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝¹⁷ : Group A\ninst✝¹⁶ : Group B\ninst✝¹⁵ : Group C\ninst✝¹⁴ : TopologicalSpace A\ninst✝¹³ : TopologicalSpace B\ninst✝¹² : TopologicalSpace C\nφ : A →* B\nψ : B →* C\nH : IsSES φ ψ\ninst✝¹¹ : IsTopologicalGroup A\ninst✝¹⁰ : IsTopologicalGroup B\ninst✝⁹ : Meas... | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls | {
"line": 69,
"column": 6
} | {
"line": 69,
"column": 20
} | {
"line": 69,
"column": 21
} | [
{
"pp": "case inr\nE : Type u_1\np : ℝ\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nhp : 0 < p\nhE : Nontrivial E\nthis✝ : 0 < ↑(finrank ℝ E)\nthis :\n (∫ (y : ℝ) in Set.Io... | [
"case inr\nE : Type u_1\np : ℝ\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nhp : 0 < p\nhE : Nontrivial E\nthis✝ : 0 < ↑(finrank ℝ E)\nthis :\n (∫ (y : ℝ) in Set.Ioi 0, y ^ (fi... | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls | {
"line": 69,
"column": 21
} | {
"line": 69,
"column": 35
} | {
"line": 69,
"column": 36
} | [
{
"pp": "case inr\nE : Type u_1\np : ℝ\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nhp : 0 < p\nhE : Nontrivial E\nthis✝ : 0 < ↑(finrank ℝ E)\nthis :\n (∫ (y : ℝ) in Set.Io... | [
"case inr\nE : Type u_1\np : ℝ\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nhp : 0 < p\nhE : Nontrivial E\nthis✝ : 0 < ↑(finrank ℝ E)\nthis :\n (∫ (y : ℝ) in Set.Ioi 0, y ^ (fi... | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.HasOuterApproxClosedProd | {
"line": 101,
"column": 6
} | {
"line": 101,
"column": 63
} | {
"line": 102,
"column": 6
} | [
{
"pp": "case hC\nι : Type u_1\nκ : Type u_2\nX : ι → Type u_5\nY : κ → Type u_6\nmX : (i : ι) → MeasurableSpace (X i)\ninst✝⁸ : (i : ι) → TopologicalSpace (X i)\ninst✝⁷ : ∀ (i : ι), BorelSpace (X i)\ninst✝⁶ : ∀ (i : ι), HasOuterApproxClosed (X i)\nmY : (j : κ) → MeasurableSpace (Y j)\ninst✝⁵ : (j : κ) → Topolo... | [
"case hC.hC\nι : Type u_1\nκ : Type u_2\nX : ι → Type u_5\nY : κ → Type u_6\nmX : (i : ι) → MeasurableSpace (X i)\ninst✝⁸ : (i : ι) → TopologicalSpace (X i)\ninst✝⁷ : ∀ (i : ι), BorelSpace (X i)\ninst✝⁶ : ∀ (i : ι), HasOuterApproxClosed (X i)\nmY : (j : κ) → MeasurableSpace (Y j)\ninst✝⁵ : (j : κ) → TopologicalSpac... | rw [← generateFrom_eq_pi (C := fun _ ↦ {s | IsClosed s})] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls | {
"line": 188,
"column": 77
} | {
"line": 204,
"column": 80
} | {
"line": 206,
"column": 0
} | [
{
"pp": "ι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : Nonempty ι\np : ℝ\nhp : 1 ≤ p\nr : ℝ\n⊢ volume {x | (∑ i, |x i| ^ p) ^ (1 / p) < r} =\n ENNReal.ofReal r ^ card ι * ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma (↑(card ι) / p + 1))",
"ppTerm": "?m.86",
"assigned": true,
"usedConstant... | [] | by
have h₁ (x : ι → ℝ) : 0 ≤ ∑ i, |x i| ^ p := by positivity
have h₂ : ∀ x : ι → ℝ, 0 ≤ (∑ i, |x i| ^ p) ^ (1 / p) := fun x => rpow_nonneg (h₁ x) _
obtain hr | hr := le_or_gt r 0
· have : {x : ι → ℝ | (∑ i, |x i| ^ p) ^ (1 / p) < r} = ∅ := by
ext x
refine ⟨fun hx => ?_, fun hx => hx.elim⟩
exac... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls | {
"line": 379,
"column": 66
} | {
"line": 379,
"column": 80
} | {
"line": 379,
"column": 81
} | [
{
"pp": "E : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace ℝ E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nk : ℕ\nhk : finrank ℝ E = 2 * k + 1\nx : E\nr : ℝ\nthis : Nontrivial E\n⊢ ENNReal.ofReal r ^ (2 * k + 1) * ENNReal.ofReal (π ^ k * √π / Gamma (↑... | [
"E : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace ℝ E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nk : ℕ\nhk : finrank ℝ E = 2 * k + 1\nx : E\nr : ℝ\nthis : Nontrivial E\n⊢ ENNReal.ofReal r ^ (2 * k + 1) * ENNReal.ofReal (π ^ k * (√π / Gamma (↑(2 * k + 1)... | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.LevyConvergence | {
"line": 139,
"column": 6
} | {
"line": 139,
"column": 95
} | {
"line": 140,
"column": 6
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : ℕ → Measure E\ninst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μ i)\nf : E → ℂ\nhf : ContinuousAt f 0\nh : ∀ (t : E), Tendsto (fun n ↦ charFun (μ ... | [
"E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : ℕ → Measure E\ninst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μ i)\nf : E → ℂ\nhf : ContinuousAt f 0\nh : ∀ (t : E), Tendsto (fun n ↦ charFun (μ n) t) atTop ... | refine measurable_of_tendsto_metrizable (f := fun n t ↦ charFun (μ n) t) (by fun_prop) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.PreVariation | {
"line": 218,
"column": 10
} | {
"line": 218,
"column": 48
} | {
"line": 219,
"column": 10
} | [
{
"pp": "X : Type u_1\ninst✝ : MeasurableSpace X\nf : Set X → ℝ≥0∞\nhf : IsSigmaSubadditiveSetFun f\nhf' : f ∅ = 0\ns : ℕ → Set X\nhs : ∀ (i : ℕ), MeasurableSet (s i)\nhs' : Pairwise (Disjoint on s)\nb : ℝ≥0∞\nQ : Finpartition ⟨⋃ i, s i, ⋯⟩\nhQ : b < ∑ p ∈ Q.parts, f ↑p\ns' : ℕ → Subtype MeasurableSet := fun i ... | [
"X : Type u_1\ninst✝ : MeasurableSpace X\nf : Set X → ℝ≥0∞\nhf : IsSigmaSubadditiveSetFun f\nhf' : f ∅ = 0\ns : ℕ → Set X\nhs : ∀ (i : ℕ), MeasurableSet (s i)\nhs' : Pairwise (Disjoint on s)\nb : ℝ≥0∞\nQ : Finpartition ⟨⋃ i, s i, ⋯⟩\nhQ : b < ∑ p ∈ Q.parts, f ↑p\ns' : ℕ → Subtype MeasurableSet := fun i ↦ ⟨s i, ⋯⟩\n... | apply Finset.sum_le_sum fun q hq => ?_ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 567,
"column": 8
} | {
"line": 567,
"column": 27
} | {
"line": 568,
"column": 6
} | [
{
"pp": "𝓧 : Type u_1\nm𝓧 : MeasurableSpace 𝓧\ninst✝² : PseudoMetricSpace 𝓧\ninst✝¹ : OpensMeasurableSpace 𝓧\ninst✝ : SecondCountableTopology 𝓧\nS : Set (ProbabilityMeasure 𝓧)\nU : ℕ → Set 𝓧\nO : ∀ (i : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)\nCov : ⋃ i, U i = univ\nhcomp :... | [] | exact mod_cast this | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 567,
"column": 8
} | {
"line": 567,
"column": 27
} | {
"line": 568,
"column": 6
} | [
{
"pp": "𝓧 : Type u_1\nm𝓧 : MeasurableSpace 𝓧\ninst✝² : PseudoMetricSpace 𝓧\ninst✝¹ : OpensMeasurableSpace 𝓧\ninst✝ : SecondCountableTopology 𝓧\nS : Set (ProbabilityMeasure 𝓧)\nU : ℕ → Set 𝓧\nO : ∀ (i : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)\nCov : ⋃ i, U i = univ\nhcomp :... | [] | exact mod_cast this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 567,
"column": 8
} | {
"line": 567,
"column": 27
} | {
"line": 568,
"column": 6
} | [
{
"pp": "𝓧 : Type u_1\nm𝓧 : MeasurableSpace 𝓧\ninst✝² : PseudoMetricSpace 𝓧\ninst✝¹ : OpensMeasurableSpace 𝓧\ninst✝ : SecondCountableTopology 𝓧\nS : Set (ProbabilityMeasure 𝓧)\nU : ℕ → Set 𝓧\nO : ∀ (i : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)\nCov : ⋃ i, U i = univ\nhcomp :... | [] | exact mod_cast this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.OuterMeasure.OfAddContent | {
"line": 64,
"column": 6
} | {
"line": 64,
"column": 66
} | {
"line": 65,
"column": 4
} | [
{
"pp": "α : Type u_1\nC : Set (Set α)\ns : Set α\nhC : IsSetSemiring C\nm : AddContent ℝ≥0∞ C\nm_sigma_subadd : m.IsSigmaSubadditive\nm_top : ∀ s ∉ C, m s = ∞\nhs : s ∈ C\nf : ℕ → Set α\nhf : ∀ (i : ℕ), f i ∈ C\nhs_subset : s ⊆ ⋃ i, f i\n⊢ ⋃ i, s ∩ f i ∈ C",
"ppTerm": "?m.127",
"assigned": true,
"u... | [] | rwa [← inter_iUnion, inter_eq_self_of_subset_left hs_subset] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.MeasureTheory.SpecificCodomains.WithLp | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 34
} | {
"line": 50,
"column": 0
} | [
{
"pp": "case φ_int\nX : Type u_1\nmX : MeasurableSpace X\nμ : Measure X\nq : ℝ≥0∞\ninst✝⁴ : Fact (1 ≤ q)\nι : Type u_2\ninst✝³ : Fintype ι\nE : ι → Type u_3\ninst✝² : (i : ι) → NormedAddCommGroup (E i)\nf : X → PiLp q E\ninst✝¹ : (i : ι) → NormedSpace ℝ (E i)\ninst✝ : ∀ (i : ι), CompleteSpace (E i)\nhf : ∀ (i ... | [] | exact Integrable.of_eval_piLp hf | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.VectorMeasure.BoundedVariation | {
"line": 239,
"column": 28
} | {
"line": 239,
"column": 57
} | {
"line": 239,
"column": 57
} | [
{
"pp": "α : Type u_1\ninst✝⁸ : LinearOrder α\ninst✝⁷ : DenselyOrdered α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : OrderTopology α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : CompactIccSpace α\nhα : MeasurableSpace α\ninst✝² : BorelSpace α\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : CompleteSpace E\nf ... | [] | by simp [Icc_subset_Iic_self] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.VectorMeasure.BoundedVariation | {
"line": 261,
"column": 2
} | {
"line": 262,
"column": 59
} | {
"line": 263,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝⁸ : LinearOrder α\ninst✝⁷ : DenselyOrdered α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : OrderTopology α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : CompactIccSpace α\nhα : MeasurableSpace α\ninst✝² : BorelSpace α\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : CompleteSpace E\nf ... | [
"α : Type u_1\ninst✝⁸ : LinearOrder α\ninst✝⁷ : DenselyOrdered α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : OrderTopology α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : CompactIccSpace α\nhα : MeasurableSpace α\ninst✝² : BorelSpace α\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : CompleteSpace E\nf : α → E\nhf ... | rw [← Iio_union_Icc_eq_Iic le_rfl, VectorMeasure.of_union (by simp) measurableSet_Iio
measurableSet_Icc, hf.vectorMeasure_Icc le_rfl] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 71,
"column": 4
} | {
"line": 84,
"column": 68
} | {
"line": 86,
"column": 0
} | [] | [] | ∑ p ∈ P, ‖μ p‖ₑ = ∑ p ∈ Q.parts, ‖μ p‖ₑ :=
(Finpartition.sum_ofPairwiseDisjoint_eq_sum hP₂ (by simp)).symm
_ = ∑ p ∈ Q'.parts, ‖μ p‖ₑ := (Q.sum_ofSubset_eq_sum _ _ _ (by simp_all)).symm
_ ≤ ∑ p ∈ (Q'.extendOfLE (Finset.sup_le hQ')).parts, ‖μ p‖ₑ :=
sum_le_sum_of_subset (Q'.parts_subset_extendOfLE (F... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 191,
"column": 4
} | {
"line": 193,
"column": 53
} | {
"line": 195,
"column": 0
} | [] | [] | _ ≤ ‖μ E‖ₑ + ‖ν E‖ₑ := enorm_add_le _ _
_ ≤ μ.variation E + ν.variation E := by
gcongr <;> exact enorm_measure_le_variation _ E | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 196,
"column": 59
} | {
"line": 202,
"column": 68
} | {
"line": 204,
"column": 0
} | [
{
"pp": "X : Type u_1\nV : Type u_2\nmX : MeasurableSpace X\ninst✝³ : TopologicalSpace V\ninst✝² : ENormedAddCommMonoid V\ninst✝¹ : T2Space V\ninst✝ : ContinuousAdd V\nι : Type u_3\ns : Finset ι\nμ : ι → VectorMeasure X V\n⊢ (∑ i ∈ s, μ i).variation ≤ ∑ i ∈ s, (μ i).variation",
"ppTerm": "?m.39",
"assig... | [] | by
classical
induction s using Finset.induction_on with
| empty => simp
| insert i s his ih =>
simpa [Finset.sum_insert his] using
variation_add_le.trans (add_le_add_right ih ((μ i).variation)) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.VectorMeasure.Variation.Semivariation | {
"line": 58,
"column": 2
} | {
"line": 61,
"column": 32
} | {
"line": 63,
"column": 0
} | [
{
"pp": "X : Type u_1\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nmX : MeasurableSpace X\nμ : VectorMeasure X E\ns t : Set X\n⊢ μ.semivariation (s ∪ t) ≤ μ.semivariation s + μ.semivariation t",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [] | simp only [semivariation, iSup_le_iff]
intro ℓ hℓ
apply (measure_union_le _ _).trans
gcongr <;> apply le_biSup _ hℓ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Variation.Semivariation | {
"line": 58,
"column": 2
} | {
"line": 61,
"column": 32
} | {
"line": 63,
"column": 0
} | [
{
"pp": "X : Type u_1\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nmX : MeasurableSpace X\nμ : VectorMeasure X E\ns t : Set X\n⊢ μ.semivariation (s ∪ t) ≤ μ.semivariation s + μ.semivariation t",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [] | simp only [semivariation, iSup_le_iff]
intro ℓ hℓ
apply (measure_union_le _ _).trans
gcongr <;> apply le_biSup _ hℓ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | {
"line": 200,
"column": 11
} | {
"line": 200,
"column": 29
} | {
"line": 200,
"column": 30
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nF : Type u_5\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid N\ninst✝¹ : FunLike F M N\ninst✝ : AddMonoidHomClass F M N\nf : F\nS : Set (Set M)\nhS : S.Finite\nhS' : ∀ t ∈ S, IsLinearSet t\n⊢ IsSemilinearSet (⇑f '' ⋃₀ S)",
"ppTerm": "?m.40",
"assigned": true,
"... | [
"M : Type u_1\nN : Type u_2\nF : Type u_5\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid N\ninst✝¹ : FunLike F M N\ninst✝ : AddMonoidHomClass F M N\nf : F\nS : Set (Set M)\nhS : S.Finite\nhS' : ∀ t ∈ S, IsLinearSet t\n⊢ IsSemilinearSet (⇑f '' ⋃ i ∈ S, i)"
] | sUnion_eq_biUnion, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | {
"line": 434,
"column": 6
} | {
"line": 434,
"column": 48
} | {
"line": 435,
"column": 2
} | [
{
"pp": "case mp.refine_2\nS : Finset (Set ℕ)\nk p : Set ℕ → ℕ\nhS : ∀ t ∈ S, p t > 0\nhS' : ∀ t ∈ S, ∀ x ≥ k t, x ∈ t ↔ x + p t ∈ t\nx : ℕ\nhx : x ≥ S.sup k\nt : Set ℕ\nht : t ∈ S\nm : ℕ\n⊢ x ∈ t ↔ x + m * p t ∈ t",
"ppTerm": "?mp.refine_2",
"assigned": true,
"usedConstants": [
"Nat.recAux",
... | [] | induction m with grind [Finset.sup_le_iff] | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.MeasureTheory.VectorMeasure.SetIntegral | {
"line": 213,
"column": 2
} | {
"line": 213,
"column": 63
} | {
"line": 213,
"column": 63
} | [
{
"pp": "case pos\nX : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nmX : MeasurableSpace X\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedAddCommGroup G\nμ : VectorMeasure X F\nf : X → E\nt : Set X\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedSpace ℝ F\ninst✝ : NormedSpace ℝ G... | [
"case pos\nX : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nmX : MeasurableSpace X\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedAddCommGroup G\nμ : VectorMeasure X F\nf : X → E\nt : Set X\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedSpace ℝ F\ninst✝ : NormedSpace ℝ G\nB : E →L[ℝ... | by_cases hf : AEStronglyMeasurable f (μ.restrict t).variation | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 221,
"column": 11
} | {
"line": 221,
"column": 29
} | {
"line": 221,
"column": 30
} | [
{
"pp": "M : Type u_1\nι : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : Finite ι\nF : Type u_5\ninst✝¹ : FunLike F (ι → ℕ) M\ninst✝ : AddMonoidHomClass F (ι → ℕ) M\nf : F\nS : Set (Set M)\nhS : S.Finite\nhS' : ∀ t ∈ S, IsLinearSet t\n⊢ IsSemilinearSet (⇑f ⁻¹' ⋃₀ S)",
"ppTerm": "?m.39",
"assigned": true,... | [
"M : Type u_1\nι : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : Finite ι\nF : Type u_5\ninst✝¹ : FunLike F (ι → ℕ) M\ninst✝ : AddMonoidHomClass F (ι → ℕ) M\nf : F\nS : Set (Set M)\nhS : S.Finite\nhS' : ∀ t ∈ S, IsLinearSet t\n⊢ IsSemilinearSet (⇑f ⁻¹' ⋃ i ∈ S, i)"
] | sUnion_eq_biUnion, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 275,
"column": 2
} | {
"line": 275,
"column": 32
} | {
"line": 276,
"column": 2
} | [
{
"pp": "ι : Type u_3\ninst✝ : Finite ι\ns₁ s₂ : Set (ι → ℕ)\nhs₁ : ∃ v n A, s₁ = {x | ∃ x_1, v + A *ᵥ x_1 = x}\nhs₂ : ∃ v n A, s₂ = {x | ∃ x_1, v + A *ᵥ x_1 = x}\nthis : Fintype ι\n⊢ IsSemilinearSet (s₁ ∩ s₂)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Pi.addCommMonoid",
... | [
"ι : Type u_3\ninst✝ : Finite ι\ns₂ : Set (ι → ℕ)\nhs₂ : ∃ v n A, s₂ = {x | ∃ x_1, v + A *ᵥ x_1 = x}\nthis : Fintype ι\nu : ι → ℕ\nn : ℕ\nA : Matrix ι (Fin n) ℕ\n⊢ IsSemilinearSet ({x | ∃ x_1, u + A *ᵥ x_1 = x} ∩ s₂)"
] | rcases hs₁ with ⟨u, n, A, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 300,
"column": 2
} | {
"line": 300,
"column": 39
} | {
"line": 301,
"column": 2
} | [
{
"pp": "M : Type u_1\ninst✝ : AddCommMonoid M\nP : AddSubmonoid M\ns₁' s₂' : Set ↥P\nhs₁' : IsSemilinearSet s₁'\nhs₁ : IsSemilinearSet (Subtype.val '' s₁')\nhs₂' : IsSemilinearSet s₂'\nhs₂ : IsSemilinearSet (Subtype.val '' s₂')\nn : ℕ\nf : (Fin n → ℕ) →+ ↥P\nhf : AddMonoidHom.mrange f = ⊤\n⊢ IsSemilinearSet (s... | [
"M : Type u_1\ninst✝ : AddCommMonoid M\nP : AddSubmonoid M\ns₁' s₂' : Set ↥P\nhs₁' : IsSemilinearSet s₁'\nhs₁ : IsSemilinearSet (Subtype.val '' s₁')\nhs₂' : IsSemilinearSet s₂'\nhs₂ : IsSemilinearSet (Subtype.val '' s₂')\nn : ℕ\nf : (Fin n → ℕ) →+ ↥P\nhf : Function.Surjective ⇑f\n⊢ IsSemilinearSet (s₁' ∩ s₂')"
] | rw [AddMonoidHom.mrange_eq_top] at hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 368,
"column": 6
} | {
"line": 368,
"column": 44
} | {
"line": 368,
"column": 44
} | [
{
"pp": "ι : Type u_3\ns : Set (ι → ℕ)\nhs : LinearIndepOn ℕ id s\nt : Finset (ι → ℕ)\nf : (ι → ℕ) → ℤ\nht : ↑t ⊆ s\nhf : ∀ i ∉ t, f i = 0\nheq : ∑ i ∈ t, f i • toRatVec i = 0\ni : ι → ℕ\nhi : i ∈ t\n⊢ f i = 0",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"instHSMul",
"Pi.add... | [
"ι : Type u_3\ns : Set (ι → ℕ)\nhs : ∀ (t : Finset (ι → ℕ)), ↑t ⊆ s → LinearIndepOn ℕ id ↑t\nt : Finset (ι → ℕ)\nf : (ι → ℕ) → ℤ\nht : ↑t ⊆ s\nhf : ∀ i ∉ t, f i = 0\nheq : ∑ i ∈ t, f i • toRatVec i = 0\ni : ι → ℕ\nhi : i ∈ t\n⊢ f i = 0"
] | linearIndepOn_iff_linearIndepOn_finset | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 520,
"column": 2
} | {
"line": 521,
"column": 54
} | {
"line": 523,
"column": 0
} | [
{
"pp": "ι : Type u_3\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\ninst✝ : Finite ι\nx : ι → ℕ\n⊢ x + ∑ i, (-hs.floor x i).toNat • ↑i = hs.fract x + ∑ i, (hs.floor x i).toNat • ↑i",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"N... | [] | simp only [fract]
rw [tsub_add_cancel_of_le (hs.floor_toNat_sum_le x)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 520,
"column": 2
} | {
"line": 521,
"column": 54
} | {
"line": 523,
"column": 0
} | [
{
"pp": "ι : Type u_3\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\ninst✝ : Finite ι\nx : ι → ℕ\n⊢ x + ∑ i, (-hs.floor x i).toNat • ↑i = hs.fract x + ∑ i, (hs.floor x i).toNat • ↑i",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"N... | [] | simp only [fract]
rw [tsub_add_cancel_of_le (hs.floor_toNat_sum_le x)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 768,
"column": 11
} | {
"line": 768,
"column": 29
} | {
"line": 768,
"column": 30
} | [
{
"pp": "ι : Type u_3\ninst✝ : Finite ι\nS : Set (Set (ι → ℕ))\nhS : S.Finite\nhS' : ∀ t ∈ S, IsProperLinearSet t\nhs : IsSemilinearSet (⋃₀ S)\n⊢ IsSemilinearSet (⋃₀ S)ᶜ",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.addCommMonoid",
"congrArg",
"Compl... | [
"ι : Type u_3\ninst✝ : Finite ι\nS : Set (Set (ι → ℕ))\nhS : S.Finite\nhS' : ∀ t ∈ S, IsProperLinearSet t\nhs : IsSemilinearSet (⋃₀ S)\n⊢ IsSemilinearSet (⋃ i ∈ S, i)ᶜ"
] | sUnion_eq_biUnion, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 785,
"column": 2
} | {
"line": 785,
"column": 39
} | {
"line": 786,
"column": 2
} | [
{
"pp": "M : Type u_1\ninst✝ : AddCommMonoid M\nP : AddSubmonoid M\ns₁' s₂' : Set ↥P\nhs₁' : IsSemilinearSet s₁'\nhs₁ : IsSemilinearSet (Subtype.val '' s₁')\nhs₂' : IsSemilinearSet s₂'\nhs₂ : IsSemilinearSet (Subtype.val '' s₂')\nn : ℕ\nf : (Fin n → ℕ) →+ ↥P\nhf : AddMonoidHom.mrange f = ⊤\n⊢ IsSemilinearSet (s... | [
"M : Type u_1\ninst✝ : AddCommMonoid M\nP : AddSubmonoid M\ns₁' s₂' : Set ↥P\nhs₁' : IsSemilinearSet s₁'\nhs₁ : IsSemilinearSet (Subtype.val '' s₁')\nhs₂' : IsSemilinearSet s₂'\nhs₂ : IsSemilinearSet (Subtype.val '' s₂')\nn : ℕ\nf : (Fin n → ℕ) →+ ↥P\nhf : Function.Surjective ⇑f\n⊢ IsSemilinearSet (s₁' \\ s₂')"
] | rw [AddMonoidHom.mrange_eq_top] at hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 795,
"column": 2
} | {
"line": 796,
"column": 22
} | {
"line": 797,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝¹ : AddCommMonoid M\ns : Set M\ninst✝ : AddMonoid.FG M\nhs : IsSemilinearSet s\n⊢ IsSemilinearSet sᶜ",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"Set.univ",
"Set.compl_eq_univ_sdiff",
"i... | [] | rw [compl_eq_univ_sdiff]
exact sdiff .univ hs | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 795,
"column": 2
} | {
"line": 796,
"column": 22
} | {
"line": 797,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝¹ : AddCommMonoid M\ns : Set M\ninst✝ : AddMonoid.FG M\nhs : IsSemilinearSet s\n⊢ IsSemilinearSet sᶜ",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"Set.univ",
"Set.compl_eq_univ_sdiff",
"i... | [] | rw [compl_eq_univ_sdiff]
exact sdiff .univ hs | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.WithDensityVec | {
"line": 267,
"column": 6
} | {
"line": 269,
"column": 29
} | {
"line": 270,
"column": 4
} | [
{
"pp": "case hm\nX : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nmX : MeasurableSpace X\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\nμ : VectorMeasure X F\nf : X → E\nB : E →L[... | [] | · rintro ⟨i, p⟩ hip
simp only [Finset.mem_sigma, SimpleFunc.mem_range, mem_range] at hip
exact Pmeas i p hip.2 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.ModelTheory.Equivalence | {
"line": 97,
"column": 34
} | {
"line": 99,
"column": 43
} | {
"line": 101,
"column": 0
} | [
{
"pp": "L : Language\nT : L.Theory\nα : Type w\nn : ℕ\nφ ψ θ : L.BoundedFormula α n\nh₁ : φ ⟹[T] ψ\nh₂ : φ ⟹[T] θ\nM : T.ModelType\nv : α → ↑M\nxs : Fin n → ↑M\n⊢ (φ ⟹ ψ ⊓ θ).Realize v xs",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"FirstOrder.Language.BoundedFormula.imp",
... | [] | by
simp only [BoundedFormula.realize_imp, BoundedFormula.realize_inf]
exact fun h => ⟨h₁ M v xs h, h₂ M v xs h⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.ModelTheory.Equivalence | {
"line": 145,
"column": 34
} | {
"line": 145,
"column": 43
} | {
"line": 145,
"column": 44
} | [
{
"pp": "L : Language\nT : L.Theory\nα : Type w\nn : ℕ\nφ ψ : L.BoundedFormula α n\nh : φ ⇔[T] ψ\nM : T.ModelType\nv : α → ↑M\nxs : Fin n → ↑M\n⊢ ψ.Realize v xs ↔ φ.Realize v xs",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"FirstOrder.Language.Theory.ModelType.stru... | [
"L : Language\nT : L.Theory\nα : Type w\nn : ℕ\nφ ψ : L.BoundedFormula α n\nh : φ ⇔[T] ψ\nM : T.ModelType\nv : α → ↑M\nxs : Fin n → ↑M\n⊢ φ.Realize v xs ↔ ψ.Realize v xs"
] | Iff.comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.DirectLimit | {
"line": 234,
"column": 2
} | {
"line": 239,
"column": 5
} | {
"line": 241,
"column": 0
} | [
{
"pp": "L : Language\nι : Type v\ninst✝⁴ : Preorder ι\nG : ι → Type w\ninst✝³ : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝² : IsDirectedOrder ι\ninst✝¹ : DirectedSystem G fun i j h ↦ ⇑(f i j h)\ninst✝ : Nonempty ι\nn : ℕ\nF : L.Functions n\ni : ι\nx : Fin n → G i\n⊢ (funMap F fun ... | [] | simp only [funMap_quotient_mk', Quotient.eq]
obtain ⟨k, ik, jk⟩ :=
directed_of (· ≤ ·) i (Classical.choose (Finite.bddAbove_range fun _ : Fin n => i))
refine ⟨k, jk, ik, ?_⟩
simp only [Embedding.map_fun, comp_unify]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.DirectLimit | {
"line": 234,
"column": 2
} | {
"line": 239,
"column": 5
} | {
"line": 241,
"column": 0
} | [
{
"pp": "L : Language\nι : Type v\ninst✝⁴ : Preorder ι\nG : ι → Type w\ninst✝³ : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝² : IsDirectedOrder ι\ninst✝¹ : DirectedSystem G fun i j h ↦ ⇑(f i j h)\ninst✝ : Nonempty ι\nn : ℕ\nF : L.Functions n\ni : ι\nx : Fin n → G i\n⊢ (funMap F fun ... | [] | simp only [funMap_quotient_mk', Quotient.eq]
obtain ⟨k, ik, jk⟩ :=
directed_of (· ≤ ·) i (Classical.choose (Finite.bddAbove_range fun _ : Fin n => i))
refine ⟨k, jk, ik, ?_⟩
simp only [Embedding.map_fun, comp_unify]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.DirectLimit | {
"line": 233,
"column": 88
} | {
"line": 239,
"column": 5
} | {
"line": 241,
"column": 0
} | [
{
"pp": "L : Language\nι : Type v\ninst✝⁴ : Preorder ι\nG : ι → Type w\ninst✝³ : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝² : IsDirectedOrder ι\ninst✝¹ : DirectedSystem G fun i j h ↦ ⇑(f i j h)\ninst✝ : Nonempty ι\nn : ℕ\nF : L.Functions n\ni : ι\nx : Fin n → G i\n⊢ (funMap F fun ... | [] | by
simp only [funMap_quotient_mk', Quotient.eq]
obtain ⟨k, ik, jk⟩ :=
directed_of (· ≤ ·) i (Classical.choose (Finite.bddAbove_range fun _ : Fin n => i))
refine ⟨k, jk, ik, ?_⟩
simp only [Embedding.map_fun, comp_unify]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.ModelTheory.Graph | {
"line": 80,
"column": 2
} | {
"line": 82,
"column": 28
} | {
"line": 84,
"column": 0
} | [
{
"pp": "V : Type u\nn : ℕ\nG : SimpleGraph V\n⊢ V ⊨ Theory.simpleGraph",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Std.Irrefl",
"Eq.mpr",
"Std.Symm",
"congrArg",
"id",
"instOfNatNat",
"FirstOrder.Language.Structure.RelMap",
"FirstOrder.L... | [] | letI := G.structure
rw [Theory.simpleGraph_model_iff]
exact ⟨G.loopless, G.symm⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Graph | {
"line": 80,
"column": 2
} | {
"line": 82,
"column": 28
} | {
"line": 84,
"column": 0
} | [
{
"pp": "V : Type u\nn : ℕ\nG : SimpleGraph V\n⊢ V ⊨ Theory.simpleGraph",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Std.Irrefl",
"Eq.mpr",
"Std.Symm",
"congrArg",
"id",
"instOfNatNat",
"FirstOrder.Language.Structure.RelMap",
"FirstOrder.L... | [] | letI := G.structure
rw [Theory.simpleGraph_model_iff]
exact ⟨G.loopless, G.symm⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.Integral | {
"line": 680,
"column": 59
} | {
"line": 691,
"column": 28
} | {
"line": 693,
"column": 0
} | [
{
"pp": "X : Type u_2\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\nf : X → E\nB : E →L[ℝ] F →L[ℝ] G\ninst✝¹ : MeasurableSpace X\ninst✝ : Comple... | [] | by
borelize E
have : IsFiniteMeasure ((dirac a v).transpose B).variation := by
have : ‖B.flip v‖ₑ • Measure.dirac a = ‖B.flip v‖₊ • Measure.dirac a := rfl
simp only [transpose_dirac, variation_dirac, this]
infer_instance
calc
∫ᵛ x, f x ∂[B; VectorMeasure.dirac a v] = ∫ᵛ _, f a ∂[B; VectorMeasure.d... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.VectorMeasure.Integral | {
"line": 711,
"column": 51
} | {
"line": 711,
"column": 96
} | {
"line": 712,
"column": 4
} | [
{
"pp": "X : Type u_2\nE : Type u_4\nF : Type u_5\nG : Type u_6\nmX : MeasurableSpace X\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\nf : X → E\nμ : VectorMeasure X F\nB : E →L[ℝ] F →L[ℝ... | [] | by congr with x; congr; exact Unique.uniq _ x | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.VectorMeasure.Integral | {
"line": 727,
"column": 71
} | {
"line": 730,
"column": 10
} | {
"line": 732,
"column": 0
} | [
{
"pp": "X : Type u_2\nE : Type u_4\nF✝ : Type u_5\nG : Type u_6\nmX : MeasurableSpace X\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F✝\ninst✝² : NormedSpace ℝ F✝\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nμ : VectorMeasure X F✝\nB : E →L[ℝ] F✝ →L[ℝ] G\nι ... | [] | by
refine tendsto_integral_of_L1 f hfi hFi ?_
simp_rw [eLpNorm_one_eq_lintegral_enorm, Pi.sub_apply] at hF
exact hF | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.AlmostPrime | {
"line": 63,
"column": 4
} | {
"line": 63,
"column": 84
} | {
"line": 65,
"column": 0
} | [
{
"pp": "case mpr\nn : ℕ\n⊢ Prime n → IsAlmostPrime 1 n",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Nat.instMulZeroClass",
"Nat.Prime",
"ArithmeticFunction.instFunLikeNat",
"ArithmeticFunction.cardFactors",
"Ne",
"instOfNatNat",
... | [] | exact fun h ↦ ⟨h.ne_zero, ArithmeticFunction.cardFactors_eq_one_iff_prime.mpr h⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.AlmostPrime | {
"line": 63,
"column": 4
} | {
"line": 63,
"column": 84
} | {
"line": 65,
"column": 0
} | [
{
"pp": "case mpr\nn : ℕ\n⊢ Prime n → IsAlmostPrime 1 n",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Nat.instMulZeroClass",
"Nat.Prime",
"ArithmeticFunction.instFunLikeNat",
"ArithmeticFunction.cardFactors",
"Ne",
"instOfNatNat",
... | [] | exact fun h ↦ ⟨h.ne_zero, ArithmeticFunction.cardFactors_eq_one_iff_prime.mpr h⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.AlmostPrime | {
"line": 63,
"column": 4
} | {
"line": 63,
"column": 84
} | {
"line": 65,
"column": 0
} | [
{
"pp": "case mpr\nn : ℕ\n⊢ Prime n → IsAlmostPrime 1 n",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Nat.instMulZeroClass",
"Nat.Prime",
"ArithmeticFunction.instFunLikeNat",
"ArithmeticFunction.cardFactors",
"Ne",
"instOfNatNat",
... | [] | exact fun h ↦ ⟨h.ne_zero, ArithmeticFunction.cardFactors_eq_one_iff_prime.mpr h⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 101,
"column": 6
} | {
"line": 104,
"column": 23
} | {
"line": 104,
"column": 23
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nu v : R\np : ℕ\nhp : Nat.Prime p\nhvu : v ∣ u\nhpuv : ↑p * u * v ∣ u ^ p\nx : R\n⊢ ∃ y,\n 1 +\n ((u * x) ^ 1 * ↑(p.choose 1) +\n ∑ x_1 ∈ ((Finset.range (p + 1)).erase 0).erase 1, (u * x) ^ x_1 * ↑(p.choose x_1)) =\n 1 + ↑p * u * (x + v * y)"... | [
"R : Type u_1\ninst✝ : CommSemiring R\nu v : R\np : ℕ\nhp : Nat.Prime p\nhvu : v ∣ u\nhpuv : ↑p * u * v ∣ u ^ p\nx : R\n⊢ ∃ y,\n 1 +\n ((u * x) ^ 1 * ↑(p.choose 1) +\n (∑ x_1 ∈ (((Finset.range (p + 1)).erase 0).erase 1).erase p, (u * x) ^ x_1 * ↑(p.choose x_1) +\n (u * x) ^ p * ↑(p.cho... | ← Finset.sum_erase_add (a := p) _ _ (by -- aesop works but is slow
simp only [Finset.mem_erase]
rw [← and_assoc, and_comm (a := ¬ _), ← Nat.two_le_iff]
simp [hp.two_le]) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PowerSeries.Derivative | {
"line": 92,
"column": 60
} | {
"line": 93,
"column": 43
} | {
"line": 95,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\n⊢ derivativeFun 1 = 0",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"MvPowerSeries.instZero",
"congrArg",
"CommSemiring.toSem... | [] | by
rw [← map_one C, derivativeFun_C (1 : R)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.AbelSummation | {
"line": 146,
"column": 11
} | {
"line": 146,
"column": 25
} | {
"line": 146,
"column": 26
} | [
{
"pp": "case inr\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\nf : ℝ → 𝕜\na b : ℝ\nha : 0 ≤ a\nhab : a ≤ b\nhf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t\nhf_int : IntegrableOn (deriv f) (Set.Icc a b) volume\naux1 : ↑⌊a⌋₊ ≤ a\naux2 : b ≤ ↑⌊b⌋₊ + 1\nhb : ⌊a⌋₊ < ⌊b⌋₊\naux3 : a ≤ ↑⌊a⌋₊ + 1\naux4 : ↑⌊a⌋₊ +... | [
"case inr\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\nf : ℝ → 𝕜\na b : ℝ\nha : 0 ≤ a\nhab : a ≤ b\nhf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t\nhf_int : IntegrableOn (deriv f) (Set.Icc a b) volume\naux1 : ↑⌊a⌋₊ ≤ a\naux2 : b ≤ ↑⌊b⌋₊ + 1\nhb : ⌊a⌋₊ < ⌊b⌋₊\naux3 : a ≤ ↑⌊a⌋₊ + 1\naux4 : ↑⌊a⌋₊ + 1 ≤ b\naux5... | ← smul_eq_mul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.BernoulliPolynomials | {
"line": 140,
"column": 31
} | {
"line": 140,
"column": 45
} | {
"line": 140,
"column": 46
} | [
{
"pp": "case a\nn x✝¹ : ℕ\na✝¹ : x✝¹ ∈ range (n + 1 - 0)\nx✝ : ℕ\na✝ : x✝ ∈ range (n + 1 - x✝¹)\n| (monomial x✝¹) (↑((n + 1 - x✝¹).choose x✝) * _root_.bernoulli x✝ * ↑((n + 1).choose x✝¹))",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCo... | [
"case a\nn x✝¹ : ℕ\na✝¹ : x✝¹ ∈ range (n + 1 - 0)\nx✝ : ℕ\na✝ : x✝ ∈ range (n + 1 - x✝¹)\n| (monomial x✝¹) ((↑((n + 1 - x✝¹).choose x✝) * _root_.bernoulli x✝) • ↑((n + 1).choose x✝¹))"
] | ← smul_eq_mul, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 351,
"column": 4
} | {
"line": 351,
"column": 91
} | {
"line": 352,
"column": 4
} | [
{
"pp": "case neg.inr.inl\nn : ℕ\nh0 : ¬2 * n = 0\nh1 : ¬2 * n = 1\nh2 : ¬2 * n = 2\nh4 : ¬2 * n = 4\nhn✝ : Even (2 * n)\nhn : Odd n\n⊢ IsCyclic (ZMod (2 * n))ˣ ↔\n (∃ x x_1, Nat.Prime x ∧ Odd x ∧ 1 ≤ x_1 ∧ 2 * n = x ^ x_1) ∨\n ∃ x x_1, Nat.Prime x ∧ Odd x ∧ 1 ≤ x_1 ∧ 2 * n = 2 * x ^ x_1",
"ppTerm":... | [
"case neg.inr.inl\nn : ℕ\nh0 : ¬2 * n = 0\nh1 : ¬2 * n = 1\nh2 : ¬2 * n = 2\nh4 : ¬2 * n = 4\nhn✝ : Even (2 * n)\nhn : Odd n\n⊢ (∃ p m, Nat.Prime p ∧ Odd p ∧ n = p ^ m) ↔ ∃ x x_1, Nat.Prime x ∧ Odd x ∧ 1 ≤ x_1 ∧ 2 * n = 2 * x ^ x_1",
"case neg.inr.inl\nn : ℕ\nh0 : ¬2 * n = 0\nh1 : ¬2 * n = 1\nh2 : ¬2 * n = 2\nh4 ... | rw [isCyclic_units_two_mul_iff_of_odd _ hn, isCyclic_units_iff_of_odd hn, or_iff_right] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.PrimeCounting | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 49
} | {
"line": 258,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ #({p ∈ range n | Prime p}) = count Prime n",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Nat.Prime",
"Nat.count_eq_card_filter_range",
"Finset.range",
"Nat",
"Nat.count",
"Finset.card",
"Eq.symm",
"Finset.filter",
"N... | [] | exact (count_eq_card_filter_range Prime n).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Primorial | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 33
} | {
"line": 48,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ n#.primeFactors = n.primesLE",
"ppTerm": "?m.2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"primorial_eq_prod_primesLE",
"Nat.primesLE",
"id",
"Finset.prod",
"Nat",
"Nat.instCommMonoid",
"primori... | [
"n : ℕ\n⊢ (∏ p ∈ n.primesLE, p).primeFactors = n.primesLE"
] | rw [primorial_eq_prod_primesLE] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Primorial | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 33
} | {
"line": 151,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ Squarefree (n#)",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Nat.instMonoid",
"primorial_eq_prod_primesLE",
"Nat.primesLE",
"id",
"Finset.prod",
"Nat",
"Nat.instCommMonoid",
"primorial",
... | [
"n : ℕ\n⊢ Squarefree (∏ p ∈ n.primesLE, p)"
] | rw [primorial_eq_prod_primesLE] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Bernoulli | {
"line": 224,
"column": 4
} | {
"line": 224,
"column": 28
} | {
"line": 225,
"column": 4
} | [
{
"pp": "case inr\nn : ℕ\nhn : n ≠ 1\nhgt : 1 < n\nx✝ : Even n ∨ Odd n\n⊢ bernoulli n = bernoulli' n",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"bernoulli.eq_1",
"Rat.instOfNat",
"Eq.mpr",
"MulOne.toOne",
"Rat.instMul",
"NonUnitalCommRing.toNonUnital... | [] | cases n.even_or_odd with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.NumberTheory.Bertrand | {
"line": 174,
"column": 67
} | {
"line": 175,
"column": 63
} | {
"line": 175,
"column": 63
} | [
{
"pp": "n : ℕ\nn_large : 2 < n\nno_prime : ∀ (p : ℕ), Nat.Prime p → n < p → 2 * n < p\nn_pos : 0 < n\nn2_pos : 1 ≤ 2 * n\nS : Finset ℕ := {p ∈ Finset.range (2 * n / 3 + 1) | Nat.Prime p}\nf : ℕ → ℕ := fun x ↦ x ^ n.centralBinom.factorization x\nthis : ∏ x ∈ S, f x = ∏ x ∈ Finset.range (2 * n / 3 + 1), f x\n⊢ ∏... | [
"n : ℕ\nn_large : 2 < n\nno_prime : ∀ (p : ℕ), Nat.Prime p → n < p → 2 * n < p\nn_pos : 0 < n\nn2_pos : 1 ≤ 2 * n\nS : Finset ℕ := {p ∈ Finset.range (2 * n / 3 + 1) | Nat.Prime p}\nf : ℕ → ℕ := fun x ↦ x ^ n.centralBinom.factorization x\nthis : ∏ x ∈ S, f x = ∏ x ∈ Finset.range (2 * n / 3 + 1), f x\n⊢ (∏ x ∈ S with... | ←
Finset.prod_filter_mul_prod_filter_not S (· ≤ sqrt (2 * n)) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Bernoulli | {
"line": 377,
"column": 18
} | {
"line": 377,
"column": 70
} | {
"line": 378,
"column": 2
} | [
{
"pp": "n p : ℕ\nf : ℕ → ℚ := fun i ↦ bernoulli i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ\nf' : ℕ → ℚ := fun i ↦ bernoulli' i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ\nhle : 1 ≤ n + 1\nhne : ↑p + 1 + 1 ≠ 0\nr : ℚ\n⊢ r * (↑p + 1 + 1) * ↑n ^ p.succ / (↑p + 1 + 1... | [] | rw [mul_div_right_comm, mul_div_cancel_right₀ _ hne] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Bernoulli | {
"line": 377,
"column": 18
} | {
"line": 377,
"column": 70
} | {
"line": 378,
"column": 2
} | [
{
"pp": "n p : ℕ\nf : ℕ → ℚ := fun i ↦ bernoulli i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ\nf' : ℕ → ℚ := fun i ↦ bernoulli' i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ\nhle : 1 ≤ n + 1\nhne : ↑p + 1 + 1 ≠ 0\nr : ℚ\n⊢ r * (↑p + 1 + 1) * ↑n ^ p.succ / (↑p + 1 + 1... | [] | rw [mul_div_right_comm, mul_div_cancel_right₀ _ hne] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Bernoulli | {
"line": 377,
"column": 18
} | {
"line": 377,
"column": 70
} | {
"line": 378,
"column": 2
} | [
{
"pp": "n p : ℕ\nf : ℕ → ℚ := fun i ↦ bernoulli i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ\nf' : ℕ → ℚ := fun i ↦ bernoulli' i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ\nhle : 1 ≤ n + 1\nhne : ↑p + 1 + 1 ≠ 0\nr : ℚ\n⊢ r * (↑p + 1 + 1) * ↑n ^ p.succ / (↑p + 1 + 1... | [] | rw [mul_div_right_comm, mul_div_cancel_right₀ _ hne] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Bernoulli | {
"line": 554,
"column": 2
} | {
"line": 554,
"column": 14
} | {
"line": 555,
"column": 2
} | [
{
"pp": "k m p : ℕ\nhm_lt : m < k\ninst✝ : Fact (Nat.Prime p)\nih : pIntegral p (bernoulli (2 * m) + vonStaudtIndicator (2 * m) p / ↑p)\nhp_ne : ↑p ≠ 0\nP : ℚ := ↑p ^ (2 * k - 2 * m - 1)\nhpow : ↑p ^ (2 * k - 2 * m) = P * ↑p\nhdecomp :\n bernoulli (2 * m) * ↑((2 * k + 1).choose (2 * m)) * ↑p ^ (2 * k - 2 * m) ... | [
"k m p : ℕ\nhm_lt : m < k\ninst✝ : Fact (Nat.Prime p)\nih : pIntegral p (bernoulli (2 * m) + vonStaudtIndicator (2 * m) p / ↑p)\nhp_ne : ↑p ≠ 0\nP : ℚ := ↑p ^ (2 * k - 2 * m - 1)\nhpow : ↑p ^ (2 * k - 2 * m) = P * ↑p\nhdecomp :\n bernoulli (2 * m) * ↑((2 * k + 1).choose (2 * m)) * ↑p ^ (2 * k - 2 * m) / (2 * ↑k + ... | rw [hdecomp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Chebyshev | {
"line": 374,
"column": 4
} | {
"line": 374,
"column": 40
} | {
"line": 376,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : AddCommMonoid R\nf : ℕ → R\nx : ℝ\nhx : 0 ≤ x\nN : ℕ\nhN : ⌊log x / log 2⌋₊ ≤ N\nk p : ℕ\nx✝ : 1 ≤ k ∧ k ≤ N\nh : (0 < p ∧ p ≤ ⌊x ^ (↑k)⁻¹⌋₊) ∧ Nat.Prime p\nthis : x < 1\n⊢ x ^ (↑k)⁻¹ < 1",
"ppTerm": "?m.536",
"assigned": true,
"usedConstants": [
"Real.instIsOrde... | [] | apply rpow_lt_one hx this (by bound) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.NumberTheory.Bernoulli | {
"line": 618,
"column": 2
} | {
"line": 618,
"column": 32
} | {
"line": 619,
"column": 2
} | [
{
"pp": "k p : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp_ne : ↑p ≠ 0\ni : ℕ\nhi : i ∈ range (2 * k)\n⊢ bernoulli i * ↑((2 * k + 1).choose i) * ↑p ^ (2 * k + 1 - i) / (2 * ↑k + 1) / ↑p =\n bernoulli i * ↑((2 * k + 1).choose i) * ↑p ^ (2 * k - i) / (2 * ↑k + 1)",
"ppTerm": "?m.163",
"assigned": true,
"used... | [
"k p : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp_ne : ↑p ≠ 0\ni : ℕ\nhi : i ∈ range (2 * k)\nthis : i < 2 * k\n⊢ bernoulli i * ↑((2 * k + 1).choose i) * ↑p ^ (2 * k + 1 - i) / (2 * ↑k + 1) / ↑p =\n bernoulli i * ↑((2 * k + 1).choose i) * ↑p ^ (2 * k - i) / (2 * ↑k + 1)"
] | have := Finset.mem_range.mp hi | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.Chebyshev | {
"line": 492,
"column": 4
} | {
"line": 492,
"column": 20
} | {
"line": 494,
"column": 0
} | [
{
"pp": "case h₂\nx : ℝ\nhx : 0 ≤ x\n⊢ 1 ≤ 2",
"ppTerm": "?h₂",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZeroLEOneClass",
"AddGroupWithOne.toAddMonoidWithOne",
"Preorder.toLE",
"Real.instRing",
"Real.instAddCommMonoid",
"one_le_two",
"Rea... | [] | exact one_le_two | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Chebyshev | {
"line": 492,
"column": 4
} | {
"line": 492,
"column": 20
} | {
"line": 494,
"column": 0
} | [
{
"pp": "case h₂\nx : ℝ\nhx : 0 ≤ x\n⊢ 1 ≤ 2",
"ppTerm": "?h₂",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZeroLEOneClass",
"AddGroupWithOne.toAddMonoidWithOne",
"Preorder.toLE",
"Real.instRing",
"Real.instAddCommMonoid",
"one_le_two",
"Rea... | [] | exact one_le_two | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Chebyshev | {
"line": 492,
"column": 4
} | {
"line": 492,
"column": 20
} | {
"line": 494,
"column": 0
} | [
{
"pp": "case h₂\nx : ℝ\nhx : 0 ≤ x\n⊢ 1 ≤ 2",
"ppTerm": "?h₂",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZeroLEOneClass",
"AddGroupWithOne.toAddMonoidWithOne",
"Preorder.toLE",
"Real.instRing",
"Real.instAddCommMonoid",
"one_le_two",
"Rea... | [] | exact one_le_two | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Chebyshev | {
"line": 495,
"column": 7
} | {
"line": 495,
"column": 45
} | {
"line": 495,
"column": 45
} | [
{
"pp": "x : ℝ\nhx : 1 ≤ x\n⊢ ψ x - θ x ≤ 2 * √x * log x",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"le_refl",
"Chebyshev.abs_psi_sub_theta_le_sqrt_mul_log",
"Real",
"HMul.hMul",
"Real.lattice",
"abs",
"Real.instSub",
"Nat.instAtLeastTwo... | [
"x : ℝ\nhx : 1 ≤ x\n⊢ ψ x - θ x ≤ |ψ x - θ x|"
] | ← abs_psi_sub_theta_le_sqrt_mul_log hx | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.NumberTheory.Chebyshev | {
"line": 506,
"column": 7
} | {
"line": 506,
"column": 28
} | {
"line": 506,
"column": 28
} | [
{
"pp": "x : ℝ\nhx : 1 ≤ x\n⊢ (x - 1) * log 2 - log (x + 2) - 2 * √x * log x ≤ θ x",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"sub_le_sub_right",
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAs... | [
"x : ℝ\nhx : 1 ≤ x\n⊢ ψ x - 2 * √x * log x ≤ θ x"
] | psi_ge' (by linarith) | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.NumberTheory.ClassNumber.Finite | {
"line": 219,
"column": 8
} | {
"line": 219,
"column": 15
} | {
"line": 219,
"column": 16
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : EuclideanDomain R\ninst✝⁶ : CommRing S\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R S\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nbS : Basis ι R S\nadm : abv.IsAdmissible\ninst✝¹ : Infinite R\ninst✝ : DecidableEq R\na : S\nb : R\n... | [
"R : Type u_1\nS : Type u_2\ninst✝⁷ : EuclideanDomain R\ninst✝⁶ : CommRing S\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R S\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nbS : Basis ι R S\nadm : abv.IsAdmissible\ninst✝¹ : Infinite R\ninst✝ : DecidableEq R\na : S\nb : R\nhb : b ≠ 0\n... | ← s_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ClassNumber.AdmissibleCardPowDegree | {
"line": 213,
"column": 6
} | {
"line": 215,
"column": 19
} | {
"line": 216,
"column": 4
} | [
{
"pp": "case refine_2.refine_1\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Field Fq\nε : ℝ\nhε : 0 < ε\nb : Fq[X]\nhb : b ≠ 0\nhbε : 0 < cardPowDegree b • ε\nn : ℕ\nih :\n ∀ (A : Fin n → Fq[X]),\n ∃ t, ∀ (i₀ i₁ : Fin n), t i₀ = t i₁ ↔ ↑(cardPowDegree (A i₁ % b - A i₀ % b)) < cardPowDegree b • ε\nA : Fin (... | [] | rw [Fin.cons_succ, Fin.cons_zero, ← not_le] at approx
have := (Classical.choose_spec (hg j₀)).2
contradiction | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ClassNumber.AdmissibleCardPowDegree | {
"line": 213,
"column": 6
} | {
"line": 215,
"column": 19
} | {
"line": 216,
"column": 4
} | [
{
"pp": "case refine_2.refine_1\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Field Fq\nε : ℝ\nhε : 0 < ε\nb : Fq[X]\nhb : b ≠ 0\nhbε : 0 < cardPowDegree b • ε\nn : ℕ\nih :\n ∀ (A : Fin n → Fq[X]),\n ∃ t, ∀ (i₀ i₁ : Fin n), t i₀ = t i₁ ↔ ↑(cardPowDegree (A i₁ % b - A i₀ % b)) < cardPowDegree b • ε\nA : Fin (... | [] | rw [Fin.cons_succ, Fin.cons_zero, ← not_le] at approx
have := (Classical.choose_spec (hg j₀)).2
contradiction | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Chebyshev | {
"line": 680,
"column": 4
} | {
"line": 680,
"column": 36
} | {
"line": 681,
"column": 2
} | [
{
"pp": "x : ℝ\nhx : 2 ≤ x\na : ℕ → ℝ := (setOf Nat.Prime).indicator fun n ↦ 1\nn : ℕ\nx✝ : n ∈ Icc 0 ⌊x⌋₊\n⊢ (if Nat.Prime n then log ↑n else 0) = log ↑n * a n",
"ppTerm": "?m.85",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Real",
"Nat.Prime",
"HMul.hMul"... | [] | split_ifs with h <;> simp [a, h] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.NumberTheory.ClassNumber.AdmissibleCardPowDegree | {
"line": 172,
"column": 75
} | {
"line": 233,
"column": 15
} | {
"line": 235,
"column": 0
} | [
{
"pp": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Field Fq\nn : ℕ\nε : ℝ\nhε : 0 < ε\nb : Fq[X]\nhb : b ≠ 0\nA : Fin n → Fq[X]\n⊢ ∃ t, ∀ (i₀ i₁ : Fin n), t i₀ = t i₁ ↔ ↑(cardPowDegree (A i₁ % b - A i₀ % b)) < cardPowDegree b • ε",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Mat... | [] | by
have hbε : 0 < cardPowDegree b • ε := by
rw [Algebra.smul_def, eq_intCast]
exact mul_pos (Int.cast_pos.mpr (AbsoluteValue.pos _ hb)) hε
-- We go by induction on the size `A`.
induction n with | zero => refine ⟨finZeroElim, finZeroElim⟩ | succ n ih =>
-- Show `anti_archimedean` also holds for real dis... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Localization.NormTrace | {
"line": 56,
"column": 2
} | {
"line": 58,
"column": 89
} | {
"line": 60,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹⁴ : CommRing R\ninst✝¹³ : CommRing S\ninst✝¹² : Algebra R S\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝¹¹ : CommRing Rₘ\ninst✝¹⁰ : Algebra R Rₘ\ninst✝⁹ : CommRing Sₘ\ninst✝⁸ : Algebra S Sₘ\nM : Submonoid R\ninst✝⁷ : IsLocalization M Rₘ\ninst✝⁶ : IsLocalization (algebraMapSubm... | [] | ext i j
simp only [Matrix.map_apply, RingHom.mapMatrix_apply, leftMulMatrix_eq_repr_mul, ← map_mul,
Basis.localizationLocalization_apply, Basis.localizationLocalization_repr_algebraMap] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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