module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 493,
"column": 8
} | {
"line": 493,
"column": 53
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)\nν : LevyProkhorov (ProbabilityMeasure Ω)\nhμs : Tendsto μs atTop (𝓝 ν)\nP : ProbabilityMeasure Ω := ν.toMeasure\nPs : ℕ → ProbabilityMeasure Ω := toMea... | ofReal_toReal (levyProkhorovEDist_ne_top _ _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 496,
"column": 2
} | {
"line": 498,
"column": 76
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)\nν : LevyProkhorov (ProbabilityMeasure Ω)\nhμs : Tendsto μs atTop (𝓝 ν)\nP : ProbabilityMeasure Ω := ν.toMeasure\nPs : ℕ → ProbabilityMeasure Ω := toMea... | · simp only [IsCoboundedUnder, IsCobounded, eventually_map, eventually_atTop,
forall_exists_index]
refine ⟨0, fun a i hia ↦ le_trans (integral_nonneg f_nn) (hia i le_rfl)⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 626,
"column": 2
} | {
"line": 626,
"column": 49
} | [
{
"pp": "Ω : Type u_1\ninst✝³ : PseudoMetricSpace Ω\ninst✝² : MeasurableSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\ninst✝ : SeparableSpace Ω\nP : ProbabilityMeasure Ω\nε : ℝ\nε_pos : ε > 0\nthird_ε_pos : 0 < ε / 3\nthird_ε_pos' : 0 < ENNReal.ofReal (ε / 3)\nEs : ℕ → Set Ω\nEs_mble : ∀ (n : ℕ), MeasurableSet (Es n... | set JB := {i | (B ∩ Es i).Nonempty ∧ i ∈ Iio N} | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 319,
"column": 4
} | {
"line": 319,
"column": 24
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\na b c : E\ns : Set E\nω : E → E →L[𝕜] F\ndω : E → E →L[ℝ]... | apply mem_range_self | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 321,
"column": 8
} | {
"line": 321,
"column": 65
} | [
{
"pp": "case refine_3\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\na b c : E\ns : Set E\nω : E → E →L[𝕜] F\ndω : E → E →L[ℝ]... | (isCompact_range <| map_continuous _).isClosed.closure_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls | {
"line": 258,
"column": 28
} | {
"line": 258,
"column": 57
} | [
{
"pp": "case h.e'_3.h.e'_1.h.e'_6.h.e'_1\nι : Type u_1\ninst✝ : Fintype ι\np : ℝ\nhp : 1 ≤ p\nh₁ : 0 < p\nthis✝ : (ENNReal.ofReal p).toReal = p\nh₂ : ∀ (x : ι → ℂ), 0 ≤ ∑ i, ‖x i‖ ^ p\neq_norm : ∀ (x : ι → ℂ), ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, ‖x i‖ ^ p) ^ (1 / p)\nthis : Fact (1 ≤ ENNReal.ofReal p)\neq_zero... | Complex.finrank_real_complex, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls | {
"line": 280,
"column": 37
} | {
"line": 280,
"column": 66
} | [
{
"pp": "case h.e'_3.h.e'_5\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : Nonempty ι\np : ℝ\nhp : 1 ≤ p\nr : ℝ\nh₁ : ∀ (x : ι → ℂ), 0 ≤ ∑ i, ‖x i‖ ^ p\nh₂ : ∀ (x : ι → ℂ), 0 ≤ (∑ i, ‖x i‖ ^ p) ^ (1 / p)\nhr : 0 < r\n⊢ ENNReal.ofReal (r ^ (2 * card ι)) = ENNReal.ofReal (r ^ ∑ i, finrank ℝ ℂ)",
"usedConstants": ... | Complex.finrank_real_complex, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 243,
"column": 6
} | {
"line": 255,
"column": 50
} | [
{
"pp": "E : Type u_1\ninst✝³ : MeasurableSpace E\ninst✝² : TopologicalSpace E\ninst✝¹ : T2Space E\ninst✝ : BorelSpace E\nu : ℕ → ℝ≥0\nK : ℕ → Set E\nC : ℝ≥0\nhu : Tendsto u atTop (𝓝 0)\nhK : ∀ (n : ℕ), IsCompact (K n)\nh : NormalSpace E ∨ Monotone K\nI :\n ∀ (μ : FiniteMeasure E) (n : ℕ),\n ∑ i ∈ Finset.r... | simp only [Tendsto]
rw [← Ultrafilter.coe_map]
apply IsCompact.ultrafilter_le_nhds'
(isCompact_setOf_finiteMeasure_le_of_isCompact C (A n))
simp only [null_iff_toMeasure_null, Ultrafilter.mem_map, preimage_setOf_eq]
filter_upwards [hf] with ρ hρ
simp only [restrict_mass, restrict_m... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 243,
"column": 6
} | {
"line": 255,
"column": 50
} | [
{
"pp": "E : Type u_1\ninst✝³ : MeasurableSpace E\ninst✝² : TopologicalSpace E\ninst✝¹ : T2Space E\ninst✝ : BorelSpace E\nu : ℕ → ℝ≥0\nK : ℕ → Set E\nC : ℝ≥0\nhu : Tendsto u atTop (𝓝 0)\nhK : ∀ (n : ℕ), IsCompact (K n)\nh : NormalSpace E ∨ Monotone K\nI :\n ∀ (μ : FiniteMeasure E) (n : ℕ),\n ∑ i ∈ Finset.r... | simp only [Tendsto]
rw [← Ultrafilter.coe_map]
apply IsCompact.ultrafilter_le_nhds'
(isCompact_setOf_finiteMeasure_le_of_isCompact C (A n))
simp only [null_iff_toMeasure_null, Ultrafilter.mem_map, preimage_setOf_eq]
filter_upwards [hf] with ρ hρ
simp only [restrict_mass, restrict_m... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 290,
"column": 6
} | {
"line": 290,
"column": 49
} | [
{
"pp": "case h\nE : Type u_1\ninst✝³ : MeasurableSpace E\ninst✝² : TopologicalSpace E\ninst✝¹ : T2Space E\ninst✝ : BorelSpace E\nu : ℕ → ℝ≥0\nK : ℕ → Set E\nC : ℝ≥0\nhu : Tendsto u atTop (𝓝 0)\nhK : ∀ (n : ℕ), IsCompact (K n)\nh : NormalSpace E ∨ Monotone K\nI :\n ∀ (μ : FiniteMeasure E) (n : ℕ),\n ∑ i ∈ ... | exact tendsto_finsetSum _ (fun i hi ↦ hν i) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.HasOuterApproxClosedProd | {
"line": 93,
"column": 4
} | {
"line": 94,
"column": 41
} | [
{
"pp": "case refine_1\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) → ... | · simp only [Set.mem_pi, mem_univ, mem_setOf_eq, forall_const] at hs₁ hs₂ ⊢
exact fun i ↦ (hs₁ i).inter (hs₂ i) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.ResolventTransform | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 36
} | [
{
"pp": "case neg\n𝕜 : Type u_1\nA : Type u_2\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : HereditarilyLindelofSpace 𝕜\ninst✝⁵ : CompleteSpace 𝕜\ninst✝⁴ : MeasurableSpace 𝕜\ninst✝³ : BorelSpace 𝕜\ninst✝² : RCLike A\ninst✝¹ : NormedAlgebra 𝕜 A\nμ : Measure 𝕜\ninst✝ : IsFiniteMeasure μ\na : A\nha : a ∉ ⇑... | · simp [support_eq_empty_iff.mp h] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 450,
"column": 6
} | {
"line": 450,
"column": 49
} | [
{
"pp": "case h\nE : Type u_1\ninst✝³ : MeasurableSpace E\ninst✝² : TopologicalSpace E\ninst✝¹ : T2Space E\ninst✝ : BorelSpace E\nu : ℕ → ℝ≥0\nK : ℕ → Set E\nC : ℝ≥0\nhu : Tendsto u atTop (𝓝 0)\nhK : ∀ (n : ℕ), IsCompact (K n)\nI :\n ∀ (μ : FiniteMeasure E) (n : ℕ),\n ∑ i ∈ Finset.range (n + 1), μ.restrict... | exact tendsto_finsetSum _ (fun i hi ↦ hν i) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.SeparableMeasure | {
"line": 138,
"column": 51
} | {
"line": 164,
"column": 52
} | [
{
"pp": "X : Type u_1\nE : Type u_2\nm : MeasurableSpace X\ninst✝ : NormedAddCommGroup E\nμ : Measure X\np : ℝ≥0∞\none_le_p : Fact (1 ≤ p)\np_ne_top : Fact (p ≠ ∞)\n𝒜 : Set (Set X)\nh𝒜 : μ.MeasureDense 𝒜\nc : E\n⊢ {x | ∃ s, ∃ (hs : MeasurableSet s) (hμs : μ s ≠ ∞), indicatorConstLp p hs hμs c = x} ⊆\n clo... | by
obtain rfl | hc := eq_or_ne c 0
· refine Subset.trans ?_ subset_closure
rintro - ⟨s, ms, hμs, rfl⟩
obtain ⟨t, ht, hμt⟩ := h𝒜.nonempty'
refine ⟨t, ht, hμt, ?_⟩
simp_rw [indicatorConstLp]
simp
· have p_pos : 0 < p := lt_of_lt_of_le (by simp) one_le_p.elim
rintro - ⟨s, ms, hμs, rfl⟩
r... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.SeparableMeasure | {
"line": 185,
"column": 4
} | {
"line": 263,
"column": 75
} | [
{
"pp": "X : Type u_1\nm : MeasurableSpace X\nμ : Measure X\n𝒜 : Set (Set X)\ninst✝ : IsFiniteMeasure μ\nh𝒜 : IsSetAlgebra 𝒜\nhgen : m = MeasurableSpace.generateFrom 𝒜\ns : Set X\nms : MeasurableSet s\n⊢ μ s ≠ ∞ → ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, μ (s ∆ t) < ENNReal.ofReal ε",
"usedConstants": [
"IsRi... | have : MeasurableSet s ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, μ.real (s ∆ t) < ε := by
rw [hgen] at ms
induction s, ms using generateFrom_induction with
-- If `t ∈ 𝒜`, then `μ (t ∆ t) = 0` which is less than any `ε > 0`.
| hC t t_mem _ =>
exact ⟨hgen ▸ measurableSet_generateFrom t_mem, fun ε ε_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.SeparableMeasure | {
"line": 185,
"column": 4
} | {
"line": 263,
"column": 75
} | [
{
"pp": "X : Type u_1\nm : MeasurableSpace X\nμ : Measure X\n𝒜 : Set (Set X)\ninst✝ : IsFiniteMeasure μ\nh𝒜 : IsSetAlgebra 𝒜\nhgen : m = MeasurableSpace.generateFrom 𝒜\ns : Set X\nms : MeasurableSet s\n⊢ μ s ≠ ∞ → ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, μ (s ∆ t) < ENNReal.ofReal ε",
"usedConstants": [
"IsRi... | have : MeasurableSet s ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, μ.real (s ∆ t) < ε := by
rw [hgen] at ms
induction s, ms using generateFrom_induction with
-- If `t ∈ 𝒜`, then `μ (t ∆ t) = 0` which is less than any `ε > 0`.
| hC t t_mem _ =>
exact ⟨hgen ▸ measurableSet_generateFrom t_mem, fun ε ε_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.SeparableMeasure | {
"line": 495,
"column": 74
} | {
"line": 495,
"column": 92
} | [
{
"pp": "X : Type u_1\nE : Type u_2\nm : MeasurableSpace X\ninst✝² : NormedAddCommGroup E\nμ : Measure X\np : ℝ≥0∞\none_le_p : Fact (1 ≤ p)\np_ne_top : Fact (p ≠ ∞)\n𝒜✝ : Set (Set X)\ninst✝¹ : IsSeparable μ\ninst✝ : SeparableSpace E\n𝒜 : Set (Set X)\ncount_𝒜 : 𝒜.Countable\nh𝒜 : μ.MeasureDense 𝒜\n𝒜₀ : Set... | one_div_mul_eq_div | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.AddContent | {
"line": 79,
"column": 4
} | {
"line": 80,
"column": 80
} | [
{
"pp": "α : Type u_1\nhα : MeasurableSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\nμ : Measure α\nm : Set α → E\nhm : ∀ (s : Set α), ‖m s‖ₑ ≤ μ s\ninst✝ : IsFiniteMeasure μ\nh'm : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → Disjoint s t → m (s ∪ t) = m s + m t\nh''m :... | apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds
(h := fun n ↦ μ (⋃ i ∈ Set.Ici n, f i)) ?_ (fun i ↦ bot_le) (fun i ↦ hm _) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Measure.SeparableMeasure | {
"line": 505,
"column": 6
} | {
"line": 505,
"column": 84
} | [
{
"pp": "case refine_2.refine_2\nX : Type u_1\nE : Type u_2\nm : MeasurableSpace X\ninst✝² : NormedAddCommGroup E\nμ : Measure X\np : ℝ≥0∞\none_le_p : Fact (1 ≤ p)\np_ne_top : Fact (p ≠ ∞)\n𝒜✝ : Set (Set X)\ninst✝¹ : IsSeparable μ\ninst✝ : SeparableSpace E\n𝒜 : Set (Set X)\ncount_𝒜 : 𝒜.Countable\nh𝒜 : μ.Me... | rcases f_mem (ε / 2) (by linarith [ε_pos]) with ⟨bf, ⟨nf, df, sf, bf_eq⟩, hbf⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | {
"line": 436,
"column": 4
} | {
"line": 436,
"column": 91
} | [
{
"pp": "case mpr\ns : Set ℕ\nk p : ℕ\nhp : p > 0\nhs : ∀ x ≥ k, x ∈ s ↔ x + p ∈ s\n⊢ IsSemilinearSet s",
"usedConstants": [
"setOf",
"Set.Finite",
"Set.sep_subset_setOf",
"Membership.mem",
"Set.Finite.subset",
"And",
"Nat",
"LT.lt",
"instLTNat",
"... | have h₁ : {x ∈ s | x < k}.Finite := (Set.finite_lt_nat k).subset (sep_subset_setOf _ _) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 373,
"column": 4
} | {
"line": 373,
"column": 55
} | [
{
"pp": "ι : Type u_3\ns : Set (ι → ℕ)\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\nhs : ∀ (f g : (ι → ℕ) → ℕ), ∑ i ∈ t, f i • id i = ∑ i ∈ t, g i • id i → ∀ i ∈ t, f i = g i\n⊢ ∑ x ∈ t, toRatVec ((Int.toNat ∘ f) x • id x) ... | rw [← sub_eq_zero, ← Finset.sum_sub_distrib, ← heq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.ModelTheory.PartialEquiv | {
"line": 134,
"column": 2
} | {
"line": 139,
"column": 84
} | [
{
"pp": "L : Language\nM : Type w\nN : Type w'\ninst✝¹ : L.Structure M\ninst✝ : L.Structure N\nf g : M ≃ₚ[L] N\n⊢ f ≤ g ↔\n ∃ (dom_le_dom : f.dom ≤ g.dom) (cod_le_cod : f.cod ≤ g.cod),\n ∀ (x : ↥f.dom), (inclusion cod_le_cod) (f.toEquiv x) = g.toEquiv ((inclusion dom_le_dom) x)",
"usedConstants": [
... | constructor
· exact fun h ↦ ⟨dom_le_dom h, cod_le_cod h,
by intro x; apply (subtype _).inj'; rwa [toEquiv_inclusion_apply]⟩
· rintro ⟨dom_le_dom, le_cod, h_eq⟩
rw [le_def]
exact ⟨dom_le_dom, by ext; change subtype _ (g.toEquiv _) = _; rw [← h_eq]; rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.PartialEquiv | {
"line": 134,
"column": 2
} | {
"line": 139,
"column": 84
} | [
{
"pp": "L : Language\nM : Type w\nN : Type w'\ninst✝¹ : L.Structure M\ninst✝ : L.Structure N\nf g : M ≃ₚ[L] N\n⊢ f ≤ g ↔\n ∃ (dom_le_dom : f.dom ≤ g.dom) (cod_le_cod : f.cod ≤ g.cod),\n ∀ (x : ↥f.dom), (inclusion cod_le_cod) (f.toEquiv x) = g.toEquiv ((inclusion dom_le_dom) x)",
"usedConstants": [
... | constructor
· exact fun h ↦ ⟨dom_le_dom h, cod_le_cod h,
by intro x; apply (subtype _).inj'; rwa [toEquiv_inclusion_apply]⟩
· rintro ⟨dom_le_dom, le_cod, h_eq⟩
rw [le_def]
exact ⟨dom_le_dom, by ext; change subtype _ (g.toEquiv _) = _; rw [← h_eq]; rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.DirectLimit | {
"line": 372,
"column": 42
} | {
"line": 372,
"column": 59
} | [
{
"pp": "L : Language\nι : Type v\ninst✝⁵ : Preorder ι\nG : ι → Type w\ninst✝⁴ : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝³ : IsDirectedOrder ι\ninst✝² : DirectedSystem G fun i j h ↦ ⇑(f i j h)\ninst✝¹ : Nonempty ι\nP : Type u₁\ninst✝ : L.Structure P\nF : DirectLimit G f ↪[L] P\nx... | rw [lift_of]; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.DirectLimit | {
"line": 372,
"column": 42
} | {
"line": 372,
"column": 59
} | [
{
"pp": "L : Language\nι : Type v\ninst✝⁵ : Preorder ι\nG : ι → Type w\ninst✝⁴ : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝³ : IsDirectedOrder ι\ninst✝² : DirectedSystem G fun i j h ↦ ⇑(f i j h)\ninst✝¹ : Nonempty ι\nP : Type u₁\ninst✝ : L.Structure P\nF : DirectLimit G f ↪[L] P\nx... | rw [lift_of]; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.PartialEquiv | {
"line": 442,
"column": 6
} | {
"line": 443,
"column": 75
} | [
{
"pp": "case refine_1.refine_2.h\nL : Language\nM : Type w\nN : Type w'\ninst✝¹ : L.Structure M\ninst✝ : L.Structure N\nh : L.IsExtensionPair M N\nS : L.Substructure M\nS_FG : S.FG\nf : ↥S ↪[L] N\nm : M\nf' : M ≃ₚ[L] N\nhf' : f'.dom.FG\nmf' : m ∈ (↑⟨f', hf'⟩).dom\nff'1 : (↑⟨{ dom := S, cod := f.toHom.range, to... | simp only [← ff'2, Embedding.comp_apply, Substructure.coe_inclusion, inclusion_mk,
Equiv.coe_toEmbedding, coe_subtype, PartialEquiv.toEmbedding_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Graph | {
"line": 100,
"column": 51
} | {
"line": 102,
"column": 5
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\n⊢ Language.simpleGraphOfStructure V = G",
"usedConstants": [
"SimpleGraph.Adj",
"Iff.rfl",
"FirstOrder.Language.simpleGraph_model",
"funext",
"SimpleGraph.structure",
"propext",
"SimpleGraph.ext",
"FirstOrder.Language.si... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.CountableDenseLinearOrder | {
"line": 159,
"column": 10
} | {
"line": 160,
"column": 32
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : Finset (α × β)\nhf : ∀ p ∈ f, ∀ q ∈ f, cmp p.1 q.1 = cmp p.2 q.2\np : β × α\nhp : p ∈ Finset.image (⇑(Equiv.prodComm α β)) f\nq : β × α\nhq : q ∈ Finset.image (⇑(Equiv.prodComm α β)) f\n⊢ (Equiv.prodComm α β).symm p ∈ f",
... | rw [← Finset.mem_coe, Finset.coe_image, Equiv.image_eq_preimage_symm] at hp
rwa [← Finset.mem_coe] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.CountableDenseLinearOrder | {
"line": 159,
"column": 10
} | {
"line": 160,
"column": 32
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : Finset (α × β)\nhf : ∀ p ∈ f, ∀ q ∈ f, cmp p.1 q.1 = cmp p.2 q.2\np : β × α\nhp : p ∈ Finset.image (⇑(Equiv.prodComm α β)) f\nq : β × α\nhq : q ∈ Finset.image (⇑(Equiv.prodComm α β)) f\n⊢ (Equiv.prodComm α β).symm p ∈ f",
... | rw [← Finset.mem_coe, Finset.coe_image, Equiv.image_eq_preimage_symm] at hp
rwa [← Finset.mem_coe] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Order | {
"line": 294,
"column": 2
} | {
"line": 296,
"column": 24
} | [
{
"pp": "L : Language\nα : Type w\nM : Type w'\nn : ℕ\ninst✝³ : L.IsOrdered\ninst✝² : L.Structure M\ninst✝¹ : Preorder M\ninst✝ : L.OrderedStructure M\n⊢ M ⊨ L.preorderTheory",
"usedConstants": [
"FirstOrder.Language.Sentence.Realize",
"Eq.mpr",
"FirstOrder.Language.preorderTheory",
... | simp only [preorderTheory, Theory.model_insert_iff, Relations.realize_reflexive, relMap_leSymb,
Theory.model_singleton_iff, Relations.realize_transitive, Matrix.cons_val_zero,
Matrix.cons_val_one] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.VectorMeasure.Integral | {
"line": 365,
"column": 4
} | {
"line": 365,
"column": 29
} | [
{
"pp": "case pos.refine_2\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 : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf : X → E\nμ : VectorMeasure X F\n... | · exact hB.add_measure hC | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.VectorMeasure.Integral | {
"line": 398,
"column": 4
} | {
"line": 398,
"column": 29
} | [
{
"pp": "case pos.refine_2\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 : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf : X → E\nμ : VectorMeasure X F\n... | · exact hB.add_measure hC | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.ModelTheory.Topology.Types | {
"line": 88,
"column": 6
} | {
"line": 88,
"column": 37
} | [
{
"pp": "case isCompact_univ.refine_2.refine_1\nL : Language\nT✝ : L.Theory\nα : Type u_1\nF : Ultrafilter (T✝.CompleteType α)\na✝ : ↑F ≤ Filter.principal univ\nx : Finset L[[α]].Sentence\nhx : ↑x ⊆ {φ | T✝.typesWith φ ∈ F}\nthis : ⋂ i ∈ x, T✝.typesWith i ∈ ↑F\nT : T✝.CompleteType α\nT_inter : T ∈ ⋂ i ∈ x, T✝.t... | exact T.isMaximal.1.mono subset | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.ArithmeticFunction.VonMangoldt | {
"line": 141,
"column": 12
} | {
"line": 141,
"column": 14
} | [
{
"pp": "n m : ℕ\n⊢ m ∣ n → ¬n = 0 → ↑(μ m) * -Real.log ↑(n / m) = ↑(μ m) * Real.log ↑m - ↑(μ m) * Real.log ↑n",
"usedConstants": [
"Dvd.dvd",
"Nat.instDvd",
"Nat"
]
}
] | mn | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.NumberTheory.ArithmeticFunction.VonMangoldt | {
"line": 148,
"column": 4
} | {
"line": 148,
"column": 24
} | [
{
"pp": "n : ℕ\nthis : ∑ i ∈ n.divisors, ↑(μ i) * -Real.log ↑(n / i) = ∑ i ∈ n.divisors, (↑(μ i) * Real.log ↑i - ↑(μ i) * Real.log ↑n)\n⊢ ↑((μ * ↑ζ) n) * Real.log ↑n = 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOn... | moebius_mul_coe_zeta | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PowerSeries.Derivative | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 34
} | [
{
"pp": "R : Type u_1\ninst✝ : Field R\nf : R⟦X⟧\n⊢ (d⁄dX R) f⁻¹ = -f⁻¹ ^ 2 * (d⁄dX R) f",
"usedConstants": [
"Derivation",
"MvPowerSeries.instAddCommGroup",
"NegZeroClass.toNeg",
"Semiring.toModule",
"HMul.hMul",
"MvPowerSeries.instCommSemiring",
"CommSemiring.toSe... | by_cases h : constantCoeff f = 0 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 227,
"column": 2
} | {
"line": 227,
"column": 40
} | [
{
"pp": "case h.H1\np : ℕ\nhp : Nat.Prime p\nhp2 : p ≠ 2\nthis✝¹ : Fact (Nat.Prime p)\nn : ℕ\nx✝ : NeZero (p ^ (n + 1))\na : ZMod (p ^ (n + 1)) := ⋯\nha_def : a = 1 + ↑p\nha : IsUnit a\nha' : orderOf ha.unit = p ^ n\nb : (ZMod (p ^ (n + 1)))ˣ\nhc : orderOf ((unitsMap ⋯) b) = p - 1\nthis✝ : p - 1 ∣ orderOf b\nk ... | rw [Nat.coprime_self_sub_right hp.pos] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.PowerSeries.Derivative | {
"line": 194,
"column": 50
} | {
"line": 194,
"column": 78
} | [
{
"pp": "case h\nA : Type u_1\ninst✝ : CommRing A\nf g : A⟦X⟧\nhg : HasSubst g\nn m : ℕ\nhm : ∀ b ≥ m, ∀ n' ≤ n + 1, (coeff n') (g ^ b) = 0\nthis : (coeff (n + 1)) (subst g f) = (coeff (n + 1)) (subst g ↑((trunc (m + 1)) f))\n⊢ (coeff n) ((d⁄dX A) (subst g ↑((trunc (m + 1)) f))) = (coeff n) (subst g ((d⁄dX A) f... | derivative_subst_coe A _ hg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.BernoulliPolynomials | {
"line": 100,
"column": 2
} | {
"line": 102,
"column": 76
} | [
{
"pp": "⊢ _root_.bernoulli 0 * ↑(Nat.choose 3 0) * (1 / 4) ^ (3 - 0) + -1 / 2 * ↑(Nat.choose 3 1) * (1 / 4) ^ (3 - 1) +\n _root_.bernoulli 2 * ↑(Nat.choose 3 2) * (1 / 4) ^ (3 - 2) +\n _root_.bernoulli 3 * ↑(Nat.choose 3 3) * (1 / 4) ^ (3 - 3) =\n 3 / 64",
"usedConstants": [
"Rat.instO... | rw [bernoulli_eq_bernoulli'_of_ne_one zero_ne_one, bernoulli'_zero,
bernoulli_eq_bernoulli'_of_ne_one (by decide : 2 ≠ 1), bernoulli'_two,
bernoulli_eq_bernoulli'_of_ne_one (by decide : 3 ≠ 1), bernoulli'_three] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.BernoulliPolynomials | {
"line": 169,
"column": 4
} | {
"line": 169,
"column": 37
} | [
{
"pp": "n p : ℕ\n⊢ ∑ x ∈ range (p + 1), _root_.bernoulli (p.succ - (p + 1 - x)) * ↑(p.succ.choose (p + 1 - x)) * ↑n ^ (p + 1 - x) =\n ∑ i ∈ range (p + 1), _root_.bernoulli i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i)",
"usedConstants": [
"Rat.instMul",
"Nat.choose",
"HMul.hMul",
"F... | apply sum_congr rfl fun x hx => _ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.NumberTheory.Bernoulli | {
"line": 99,
"column": 75
} | {
"line": 99,
"column": 95
} | [
{
"pp": "n : ℕ\n⊢ 1 -\n ∑ x ∈ range n,\n (↑(n.choose x) / (↑n - ↑x + 1) * bernoulli' x - ↑(n.choose (n - x)) / (↑n - ↑x + 1) * bernoulli' x) -\n 1 =\n 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Rat.instOfNat",
"Rat.instSub",
"Eq.mpr",
"NegZe... | sub_sub_cancel_left, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Bernoulli | {
"line": 101,
"column": 47
} | {
"line": 101,
"column": 87
} | [
{
"pp": "n x : ℕ\nhx : x ∈ range n\n⊢ ↑(n.choose x) / (↑n - ↑x + 1) * bernoulli' x - ↑(n.choose (n - x)) / (↑n - ↑x + 1) * bernoulli' x = 0",
"usedConstants": [
"Rat.instOfNat",
"Rat.instSub",
"Eq.mpr",
"Rat.instMul",
"Nat.choose",
"instHDiv",
"HMul.hMul",
"co... | choose_symm (le_of_lt (mem_range.1 hx)), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ClassNumber.AdmissibleCardPowDegree | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 29
} | [
{
"pp": "case pos\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Field Fq\nb : Fq[X]\nhb : b ≠ 0\nε : ℝ\nhε : 0 < ε\nA : Fin (Fintype.card Fq ^ ⌈-log ε / log ↑(Fintype.card Fq)⌉₊).succ → Fq[X]\nhbε : 0 < cardPowDegree b • ε\none_lt_q : 1 < Fintype.card Fq\none_lt_q' : 1 < ↑(Fintype.card Fq)\nq_pos : 0 < Fintype.c... | refine ⟨i₀, i₁, i_ne, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.ClassNumber.Finite | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 54
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : EuclideanDomain R\ninst✝⁴ : CommRing S\ninst✝³ : IsDomain S\ninst✝² : Algebra R S\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nbS : Basis ι R S\na : S\ny : ℤ\nhy : ∀ (k : ι), abv ((bS.repr a) k) ≤ y\n⊢ abv ((Algebra.norm R) (∑ i... | rw [Algebra.norm_apply, ← LinearMap.det_toMatrix bS] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Chebyshev | {
"line": 496,
"column": 63
} | {
"line": 496,
"column": 74
} | [
{
"pp": "x : ℝ\nhx : 2 ≤ x\na : ℕ → ℝ := (setOf Nat.Prime).indicator fun n ↦ log ↑n\n⊢ a 0 = 0",
"usedConstants": [
"CharP.cast_eq_zero",
"Real",
"Nat.Prime",
"Real.instZero",
"Real.instRCLike",
"congrArg",
"Set.indicator",
"AddMonoid.toAddZeroClass",
"s... | by simp [a] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Chebyshev | {
"line": 496,
"column": 77
} | {
"line": 496,
"column": 88
} | [
{
"pp": "x : ℝ\nhx : 2 ≤ x\na : ℕ → ℝ := (setOf Nat.Prime).indicator fun n ↦ log ↑n\n⊢ a 1 = 0",
"usedConstants": [
"Real",
"Nat.Prime",
"Real.instZero",
"congrArg",
"Set.indicator",
"setOf",
"AddGroupWithOne.toAddMonoidWithOne",
"Membership.mem",
"Real.... | by simp [a] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ClassNumber.Finite | {
"line": 275,
"column": 2
} | {
"line": 275,
"column": 68
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁹ : EuclideanDomain R\ninst✝⁸ : CommRing S\ninst✝⁷ : IsDomain S\ninst✝⁶ : Algebra R S\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Fintype ι\nbS : Basis ι R S\nadm : abv.IsAdmissible\ninst✝³ : Infinite R\ninst✝² : DecidableEq R\ninst✝¹ : IsDe... | have hM : algebraMap R S M ≠ 0 := prod_finsetApprox_ne_zero bS adm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 107,
"column": 4
} | {
"line": 108,
"column": 19
} | [
{
"pp": "case h.e'_4\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nh : ‖↑r‖ ≤ 1\nn : ℤ := modPart p r\nthis : ↑p ∣ r.num - n * ↑r.den\n⊢ (⟨↑r, h⟩ - ↑(modPart p r)) * ↑r.den = (Int.castRingHom ℤ_[p]) (r.num - n * ↑r.den)",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Int.... | simp only [n, sub_mul, Int.cast_natCast, eq_intCast, Int.cast_mul, sub_left_inj,
Int.cast_sub] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.ClassNumber.Finite | {
"line": 284,
"column": 6
} | {
"line": 285,
"column": 52
} | [
{
"pp": "case refine_2\nR : Type u_1\nS : Type u_2\ninst✝⁹ : EuclideanDomain R\ninst✝⁸ : CommRing S\ninst✝⁷ : IsDomain S\ninst✝⁶ : Algebra R S\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Fintype ι\nbS : Basis ι R S\nadm : abv.IsAdmissible\ninst✝³ : Infinite R\ninst✝² : DecidableEq R... | rw [ClassGroup.mk0_eq_mk0_iff]
exact ⟨algebraMap _ _ M, b, hM, b_ne_zero, hJ⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ClassNumber.Finite | {
"line": 284,
"column": 6
} | {
"line": 285,
"column": 52
} | [
{
"pp": "case refine_2\nR : Type u_1\nS : Type u_2\ninst✝⁹ : EuclideanDomain R\ninst✝⁸ : CommRing S\ninst✝⁷ : IsDomain S\ninst✝⁶ : Algebra R S\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Fintype ι\nbS : Basis ι R S\nadm : abv.IsAdmissible\ninst✝³ : Infinite R\ninst✝² : DecidableEq R... | rw [ClassGroup.mk0_eq_mk0_iff]
exact ⟨algebraMap _ _ M, b, hM, b_ne_zero, hJ⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 7
} | [
{
"pp": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nx : ℤ_[p]\nr : ℚ\nhr : ‖↑x - ↑r‖ < 1\nH : ‖↑r‖ ≤ 1\nn : ℕ\nhnp : ↑n < ↑p\nhn : ‖⟨↑r, H⟩ - ↑↑n‖ < 1\n⊢ ∃ n < p, ‖x - ↑n‖ < 1",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Real",
"NormedRing.toRing",
"PadicInt",
... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 591,
"column": 2
} | {
"line": 591,
"column": 7
} | [
{
"pp": "R : Type u_1\ninst✝ : NonAssocSemiring R\np : ℕ\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nhp_prime : Fact (Nat.Prime p)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\n⊢ ∃ i, ∀ j ≥ i, padicNorm p (↑(nthHomSe... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 606,
"column": 2
} | {
"line": 606,
"column": 7
} | [
{
"pp": "R : Type u_1\ninst✝ : NonAssocSemiring R\np : ℕ\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nhp_prime : Fact (Nat.Prime p)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\n⊢ ∃ i, ∀ j ≥ i, padicNorm p (↑(nthHomSe... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.NumberTheory.Cyclotomic.CyclotomicCharacter | {
"line": 312,
"column": 6
} | {
"line": 314,
"column": 50
} | [
{
"pp": "case pos\nL : Type u\ninst✝³ : CommRing L\ninst✝² : IsDomain L\nn : ℕ\ninst✝¹ : NeZero n\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nH : ∀ (i : ℕ), ∃ ζ, IsPrimitiveRoot ζ (p ^ i)\n⊢ cyclotomicCharacter.toFun p 1 = 1",
"usedConstants": [
"Units.val",
"RingHom.instRingHomClass",
"MonoidHom.... | · haveI _ (i) : HasEnoughRootsOfUnity L (p ^ i) := ⟨H i, rootsOfUnity.isCyclic _ _⟩
refine PadicInt.ext_of_toZModPow.mp fun n ↦ ?_
simp [cyclotomicCharacter.toZModPow_toFun] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.Cyclotomic.CyclotomicCharacter | {
"line": 318,
"column": 6
} | {
"line": 320,
"column": 50
} | [
{
"pp": "case pos\nL : Type u\ninst✝³ : CommRing L\ninst✝² : IsDomain L\nn : ℕ\ninst✝¹ : NeZero n\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf g : L ≃+* L\nH : ∀ (i : ℕ), ∃ ζ, IsPrimitiveRoot ζ (p ^ i)\n⊢ cyclotomicCharacter.toFun p (f * g) = cyclotomicCharacter.toFun p f * cyclotomicCharacter.toFun p g",
"usedCon... | · haveI _ (i) : HasEnoughRootsOfUnity L (p ^ i) := ⟨H i, rootsOfUnity.isCyclic _ _⟩
refine PadicInt.ext_of_toZModPow.mp fun n ↦ ?_
simp [cyclotomicCharacter.toZModPow_toFun] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 766,
"column": 43
} | {
"line": 766,
"column": 59
} | [
{
"pp": "f : ℕ → ℤ\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nhi : ∀ (i : ℕ), ↑p ^ i ∣ f (i + 1) - f i\nn : ℕ\nx : ℤ_[p] := ofIntSeq f ⋯\ns : PadicSeq p := ⟨fun x ↦ ↑(f x), ⋯⟩\nhs : ↑x = mk s\ne : ℤ_[p]\nhe : x = ↑p ^ n * e + ↑(x.appr n)\nN : ℕ\nhN : ‖↑p ^ n * ↑e + ↑(↑(x.appr n) - f (N + n))‖ < ↑p ^ (-↑n)\nthis : ↑(f ... | Int.cast_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 772,
"column": 8
} | {
"line": 772,
"column": 13
} | [
{
"pp": "case succ\nf : ℕ → ℤ\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nhi : ∀ (i : ℕ), ↑p ^ i ∣ f (i + 1) - f i\nn N : ℕ\nIH : ↑(f (N + n)) = ↑(f n)\n⊢ ↑(f (N + 1 + n)) = ↑(f n)",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"ZMod.commRing",
"congrArg",
"Nat.instMonoid",
"id",
... | ← IH, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 356,
"column": 9
} | {
"line": 356,
"column": 33
} | [
{
"pp": "a : ℕ\na1 : 1 < a\nm n : ℕ\nh : n ≤ m\nt : pellZd a1 m = pellZd a1 n * pellZd a1 (m - n)\n⊢ pellZd a1 (m - n) = pellZd a1 n * pellZd a1 (m - n) * star (pellZd a1 n)",
"usedConstants": [
"Zsqrtd.instMul",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
... | mul_comm (pellZd _ n) _, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Dioph | {
"line": 176,
"column": 21
} | {
"line": 176,
"column": 55
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nx✝¹ x✝ : Poly α\n⊢ x✝¹ + x✝ = x✝ + x✝¹",
"usedConstants": [
"Poly.instAdd",
"Poly.ext",
"Poly",
"congrArg",
"Poly.instFunLike",
"id",
"Int",
"add_comm",
"instHAdd",
"HAdd.hAdd",
"Nat",
"True",
... | ext; simp_rw [add_apply, add_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Dioph | {
"line": 176,
"column": 21
} | {
"line": 176,
"column": 55
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nx✝¹ x✝ : Poly α\n⊢ x✝¹ + x✝ = x✝ + x✝¹",
"usedConstants": [
"Poly.instAdd",
"Poly.ext",
"Poly",
"congrArg",
"Poly.instFunLike",
"id",
"Int",
"add_comm",
"instHAdd",
"HAdd.hAdd",
"Nat",
"True",
... | ext; simp_rw [add_apply, add_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 444,
"column": 8
} | {
"line": 445,
"column": 65
} | [
{
"pp": "case h₂\na : ℕ\na1 : 1 < a\nn k : ℕ\nhx : xn a1 (n * k) ≡ xn a1 n ^ k [MOD yn a1 n ^ 2]\nhy : yn a1 (n * k) ≡ k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3]\nL : xn a1 (n * k) * xn a1 n + d a1 * yn a1 (n * k) * yn a1 n ≡ xn a1 n ^ k * xn a1 n + 0 [MOD yn a1 n ^ 2]\n⊢ yn a1 (n * k) * xn a1 n ≡ k * xn... | have : k * xn a1 n ^ (k - 1) * yn a1 n * xn a1 n = k * xn a1 n ^ k * yn a1 n := by
rcases k with - | k <;> simp [_root_.pow_succ]; ring_nf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 471,
"column": 2
} | {
"line": 471,
"column": 39
} | [
{
"pp": "a : ℕ\na1 : 1 < a\nn : ℕ\n⊢ pellZd a1 (n + 2) + pellZd a1 n = ↑(2 * a) * pellZd a1 (n + 1)",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Zsqrtd.instMul",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.to... | ext <;> simp [dz_val, az] <;> ring_nf | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 471,
"column": 2
} | {
"line": 471,
"column": 39
} | [
{
"pp": "a : ℕ\na1 : 1 < a\nn : ℕ\n⊢ pellZd a1 (n + 2) + pellZd a1 n = ↑(2 * a) * pellZd a1 (n + 1)",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Zsqrtd.instMul",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.to... | ext <;> simp [dz_val, az] <;> ring_nf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 471,
"column": 2
} | {
"line": 471,
"column": 39
} | [
{
"pp": "a : ℕ\na1 : 1 < a\nn : ℕ\n⊢ pellZd a1 (n + 2) + pellZd a1 n = ↑(2 * a) * pellZd a1 (n + 1)",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Zsqrtd.instMul",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.to... | ext <;> simp [dz_val, az] <;> ring_nf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Dioph | {
"line": 290,
"column": 2
} | {
"line": 290,
"column": 58
} | [
{
"pp": "case cons\nα : Type u\nS : Set (α → ℕ)\nl : List (Set (α → ℕ))\nIH :\n List.Forall Dioph l →\n ∃ β pl, ∀ (v : α → ℕ), List.Forall (fun S ↦ v ∈ S) l ↔ ∃ t, List.Forall (fun p ↦ p (v ⊗ t) = 0) pl\nd : List.Forall Dioph (S :: l)\n⊢ ∃ β pl, ∀ (v : α → ℕ), List.Forall (fun S ↦ v ∈ S) (S :: l) ↔ ∃ t, Lis... | obtain ⟨⟨β, p, pe⟩, dl⟩ := (List.forall_cons _ _ _).mp d | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 686,
"column": 63
} | {
"line": 686,
"column": 79
} | [
{
"pp": "d x y : ℕ\na : ℤ√↑d\nha : a.Nonneg\n⊢ ↑x * a + sqrtd * (↑↑y * a) = ↑x * a + sqrtd * (↑y * a)",
"usedConstants": [
"Zsqrtd.instMul",
"Int.cast",
"Eq.mpr",
"Int.cast_natCast",
"Semigroup.toMul",
"HMul.hMul",
"congrArg",
"AddGroupWithOne.toAddMonoidWithO... | Int.cast_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 230,
"column": 4
} | {
"line": 230,
"column": 78
} | [
{
"pp": "ξ q : ℚ\nh : |ξ - q| * ↑q.den < 1 / ↑q.den ^ 2 * ↑q.den\nhq₀ : 0 < ↑q.den\n⊢ |ξ * ↑q.den - ↑q.num| < 1 / ↑q.den",
"usedConstants": [
"Rat.instOfNat",
"Int.cast",
"Rat.instSub",
"Rat.instMul",
"Rat.num",
"Preorder.toLT",
"instHDiv",
"NonUnitalCommRing.... | conv_lhs at h => rw [← abs_of_pos hq₀, ← abs_mul, sub_mul, mul_den_eq_num] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convLHS_1 | Mathlib.Tactic.Conv.convLHS |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 390,
"column": 2
} | {
"line": 390,
"column": 35
} | [
{
"pp": "ξ : ℝ\nu v : ℤ\nhv : 2 ≤ v\nhv₀ hv₁ : 0 < ↑v\nhv₂ : 0 < 2 * ↑v - 1\nhcop : IsCoprime u v\nleft✝ : v = 1 → -(1 / 2) < ξ - ↑u\nh : |ξ - ↑u / ↑v| < (↑v * (2 * ↑v - 1))⁻¹\n⊢ 0 < fract ξ",
"usedConstants": [
"Iff.mpr",
"Real.instIsOrderedRing",
"Int.cast",
"Real",
"Preorder... | refine fract_pos.mpr fun hf => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 802,
"column": 39
} | {
"line": 802,
"column": 94
} | [
{
"pp": "a k x✝¹ y✝ : ℕ\nx✝ : ∃ (a1 : 1 < a), xn a1 k = x✝¹ ∧ yn a1 k = y✝\na1 : 1 < a\nhx : xn a1 k = x✝¹\nhy : yn a1 k = y✝\nkpos : k > 0\nx : ℕ := xn a1 k\ny : ℕ := yn a1 k\nm : ℕ := 2 * (k * y)\nu : ℕ := xn a1 m\nv : ℕ := yn a1 m\nky : k ≤ y\nyv : y * y ∣ v\nuco : u.Coprime (4 * y)\nb : ℕ\nba : b ≡ a [MOD u... | by rw [Int.ofNat_sub (le_of_lt b1)]; exact bm1.symm.dvd | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 831,
"column": 45
} | {
"line": 831,
"column": 100
} | [
{
"pp": "a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nky : k ≤ yn a1 i\nx✝ :\n 1 < a ∧\n k ≤ y... | by rw [Int.ofNat_sub (le_of_lt b1)]; exact bm1.symm.dvd | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.EulerProduct.Basic | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 53
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NormedCommRing R\nf : ℕ → R\ninst✝ : CompleteSpace R\nhsum : Summable f\nhf₀ : f 0 = 0\nε : ℝ\nεpos : 0 < ε\nN₀ : ℕ\nhN₀ : ∀ (s : Finset ℕ), N₀.primesBelow ≤ s → ‖∑' (m : ℕ), f m - ∑' (m : ↑(factoredNumbers s)), f ↑m‖ < ε\n⊢ ∃ N₀, ∀ N ≥ N₀, ‖∑' (m : ℕ), f m - ∑' (m : ↑(factoredNu... | refine ⟨N₀, fun N hN ↦ hN₀ (range N) fun p hp ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.LSeries.Convolution | {
"line": 85,
"column": 36
} | {
"line": 85,
"column": 84
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nf g : ℕ → R\nn : ℕ\np : ℕ × ℕ\nhp : p ∈ n.divisorsAntidiagonal\n⊢ (if p.2 = 0 then 0 else if p.1 = 0 then 0 else f p.1 * g p.2) = f p.1 * g p.2",
"usedConstants": [
"HMul.hMul",
"eq_false",
"congrArg",
"Nat.ne_zero_of_mem_divisorsAntidiagona... | by simp [ne_zero_of_mem_divisorsAntidiagonal hp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 277,
"column": 4
} | {
"line": 277,
"column": 49
} | [
{
"pp": "case refine_2\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\nthis : LocallyIntegrableOn (fun x ↦ (P.ε * ↑(x ^ (-P.k))) • P.g₀) (Ioi 0) volume\nx : ℝ\nhx : x ∈ Ioi 0\n⊢ IntegrableAtFilter ((Ioo 0 1).indicator fun x ↦ P.f x - (P.ε * ↑(x ^ (-P.k))) • P.g₀) (𝓝[Ioi... | obtain ⟨s, hs, hs'⟩ := P.hf_int.sub this x hx | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 53
} | [
{
"pp": "S T : ℝ\nhT : 0 < T\nz τ : ℂ\nhz : |z.im| ≤ S\nhτ : T ≤ τ.im\nn : ℤ\n⊢ ↑|n| * -S ≤ -|↑n * z.im|",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Real.instIsOrderedRing",
"Int.cast",
"Eq.mpr",
"NegZeroClass.toNeg",
"Int.cast_abs",
"Semigroup.toMul",
... | rw [mul_neg, neg_le_neg_iff, abs_mul, Int.cast_abs] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 147,
"column": 36
} | {
"line": 147,
"column": 52
} | [
{
"pp": "case refine_2.inr.inl\nz τ : ℂ\nhτ✝ : τ.im ≤ 0\nhτ : τ.im = 0\nhz : z.im < 0\nh : Tendsto (fun x ↦ rexp (-(2 * π * ↑↑x * z.im))) atTop (𝓝 0)\n⊢ False",
"usedConstants": [
"Int.cast",
"Int.cast_natCast",
"Real",
"Real.pi",
"HMul.hMul",
"congrArg",
"Complex.... | Int.cast_natCast | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 197,
"column": 6
} | {
"line": 197,
"column": 24
} | [
{
"pp": "case h\na : ℝ\nha : 0 ≤ a\naux' : (fun t ↦ ((1 - rexp (-π * t)) ^ 2)⁻¹) =O[atTop] fun x ↦ 1\nha' : 0 < a\nt : ℝ\nht : 0 < t\n⊢ -π * (a ^ 2 + 1) * t ≤ -(π * a ^ 2) * t",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
... | nlinarith [pi_pos] | Mathlib.Tactic._aux_Mathlib_Tactic_Linarith_Frontend___elabRules_Mathlib_Tactic_nlinarith_1 | Mathlib.Tactic.nlinarith |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 198,
"column": 6
} | {
"line": 200,
"column": 69
} | [
{
"pp": "case inr.refine_2\na : ℝ\nha : 0 ≤ a\naux' : (fun t ↦ ((1 - rexp (-π * t)) ^ 2)⁻¹) =O[atTop] fun x ↦ 1\nha' : 0 < a\n⊢ (fun t ↦ a * rexp (-π * a ^ 2 * t) / (1 - rexp (-π * t))) =O[atTop] fun t ↦ rexp (-(π * a ^ 2) * t)",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
... | simp_rw [mul_div_assoc, ← neg_mul]
apply IsBigO.const_mul_left
simpa only [mul_one] using (isBigO_refl _ _).mul isBigO_one_aux | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 198,
"column": 6
} | {
"line": 200,
"column": 69
} | [
{
"pp": "case inr.refine_2\na : ℝ\nha : 0 ≤ a\naux' : (fun t ↦ ((1 - rexp (-π * t)) ^ 2)⁻¹) =O[atTop] fun x ↦ 1\nha' : 0 < a\n⊢ (fun t ↦ a * rexp (-π * a ^ 2 * t) / (1 - rexp (-π * t))) =O[atTop] fun t ↦ rexp (-(π * a ^ 2) * t)",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
... | simp_rw [mul_div_assoc, ← neg_mul]
apply IsBigO.const_mul_left
simpa only [mul_one] using (isBigO_refl _ _).mul isBigO_one_aux | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 209,
"column": 4
} | {
"line": 217,
"column": 54
} | [
{
"pp": "case mpr\nz τ : ℂ\n⊢ 0 < τ.im → Summable fun x ↦ jacobiTheta₂_term_fderiv x z τ",
"usedConstants": [
"IsModuleTopology.toContinuousSMul",
"Real.instIsOrderedRing",
"Norm.norm",
"Int.cast",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSemin... | intro hτ
refine ((summable_pow_mul_jacobiTheta₂_term_bound
|z.im| hτ 2).mul_left (3 * π)).of_norm_bounded (fun n ↦ ?_)
refine (norm_jacobiTheta₂_term_fderiv_le n z τ).trans
(?_ : 3 * π * |n| ^ 2 * ‖jacobiTheta₂_term n z τ‖ ≤ _)
simp_rw [mul_assoc (3 * π)]
refine mul_le_mul_of_nonneg_left ?_ ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 209,
"column": 4
} | {
"line": 217,
"column": 54
} | [
{
"pp": "case mpr\nz τ : ℂ\n⊢ 0 < τ.im → Summable fun x ↦ jacobiTheta₂_term_fderiv x z τ",
"usedConstants": [
"IsModuleTopology.toContinuousSMul",
"Real.instIsOrderedRing",
"Norm.norm",
"Int.cast",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSemin... | intro hτ
refine ((summable_pow_mul_jacobiTheta₂_term_bound
|z.im| hτ 2).mul_left (3 * π)).of_norm_bounded (fun n ↦ ?_)
refine (norm_jacobiTheta₂_term_fderiv_le n z τ).trans
(?_ : 3 * π * |n| ^ 2 * ‖jacobiTheta₂_term n z τ‖ ≤ _)
simp_rw [mul_assoc (3 * π)]
refine mul_le_mul_of_nonneg_left ?_ ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LSeries.MellinEqDirichlet | {
"line": 114,
"column": 6
} | {
"line": 114,
"column": 18
} | [
{
"pp": "case neg\nι : Type u_1\ninst✝ : Countable ι\na : ι → ℂ\np : ι → ℝ\nF : ℝ → ℂ\ns : ℂ\nhp : ∀ (i : ι), 0 ≤ p i\nhs : 0 < s.re\nh_sum : Summable fun i ↦ ‖a i‖ / p i ^ s.re\nhs' : s ≠ 0\na' : ι → ℂ := fun i ↦ if p i = 0 then 0 else a i\nhp' : ∀ (i : ι), a' i = 0 ∨ 0 < p i\nthis : ∀ (i : ι) (t : ℝ), (if p i... | have := hp i | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 383,
"column": 44
} | {
"line": 383,
"column": 91
} | [
{
"pp": "z τ : ℂ\nn : ℤ\nthis : cexp (↑π * I * ↑n ^ 2 * 2) = 1\n⊢ cexp (2 * ↑π * I * ↑n * z) * cexp (↑π * I * ↑n ^ 2 * (τ + 2)) =\n cexp (2 * ↑π * I * ↑n * z) * cexp (↑π * I * ↑n ^ 2 * τ)",
"usedConstants": [
"Distrib.leftDistribClass",
"Int.cast",
"Eq.mpr",
"MulOne.toOne",
... | by rw [mul_add, Complex.exp_add, this, mul_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 418,
"column": 44
} | {
"line": 418,
"column": 91
} | [
{
"pp": "z τ : ℂ\nn : ℤ\nthis : cexp (↑π * I * ↑n ^ 2 * 2) = 1\n⊢ 2 * ↑π * I * ↑n * (cexp (2 * ↑π * I * ↑n * z) * cexp (↑π * I * ↑n ^ 2 * (τ + 2))) =\n 2 * ↑π * I * ↑n * (cexp (2 * ↑π * I * ↑n * z) * cexp (↑π * I * ↑n ^ 2 * τ))",
"usedConstants": [
"Distrib.leftDistribClass",
"Int.cast",
... | by rw [mul_add, Complex.exp_add, this, mul_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 485,
"column": 79
} | {
"line": 491,
"column": 11
} | [
{
"pp": "z τ : ℂ\nhτ : 0 < τ.im\nh2 : 0 < (-I * τ).re\n⊢ 1 / (-I * τ) ^ (1 / 2) * ∑' (n : ℤ), cexp (-↑π / (-I * τ) * (↑n + I * (I * z)) ^ 2) =\n 1 / (-I * τ) ^ (1 / 2) * cexp (↑π * I * (-1 / τ) * z ^ 2) *\n ∑' (n : ℤ), cexp (2 * ↑π * I * ↑n * (z / τ) + ↑π * I * ↑n ^ 2 * (-1 / τ))",
"usedConstants": ... | by
simp_rw [mul_assoc _ (cexp _), ← tsum_mul_left (a := cexp _), ← Complex.exp_add]
congr 2 with n : 1; congr 1
field_simp
ring_nf
simp_rw [I_sq, I_pow_four]
ring_nf | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 495,
"column": 73
} | {
"line": 495,
"column": 89
} | [
{
"pp": "a : ℝ\ns : ℂ\nhs : 1 < s.re\nn : ℕ\n⊢ ↑(SignType.sign (↑n + a)) / ↑|↑n + a| ^ s / 2 + -(↑(SignType.sign (↑n + 1 - a)) / ↑|↑n + 1 - a| ^ s / 2) =\n ↑(SignType.sign (↑↑n + a)) / ↑|↑↑n + a| ^ s / 2 + ↑(SignType.sign (-(↑↑n + 1) + a)) / ↑|-(↑↑n + 1) + a| ^ s / 2",
"usedConstants": [
"AddGroup.... | Int.cast_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 69
} | [
{
"pp": "n : ℕ\nχ : DirichletCharacter ℂ n\nthis : (1 ⍟ fun x ↦ ↑(μ x)) = δ\n⊢ (fun n_1 ↦ χ ↑n_1) * 1 ⍟ ((fun n_1 ↦ χ ↑n_1) * fun n ↦ ↑(μ n)) = δ",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"ArithmeticFunction.instFunLikeNat",
"ZMod.commRing",
... | simpa only [mul_convolution_distrib χ 1 ↗μ, this] using mul_delta _ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 517,
"column": 6
} | {
"line": 517,
"column": 13
} | [
{
"pp": "a : ℝ\ns : ℂ\nhs : 1 < s.re\n⊢ HasSum (fun n ↦ -I * ↑n.sign * cexp (2 * ↑π * I * ↑a * ↑n) / ↑|n| ^ s / 2) (sinZeta (↑a) s)",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssoc... | sinZeta | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 418,
"column": 34
} | {
"line": 418,
"column": 57
} | [
{
"pp": "N : ℕ\nχ : DirichletCharacter ℂ N\ns : ℂ\nhs : 1 < s.re\nhN : N ≠ 0\nhχ : LSeriesSummable (fun n ↦ χ ↑n) s\nhs' : (abscissaOfAbsConv fun n ↦ χ ↑n) < ↑s.re\nhΛ : LSeriesSummable ((fun n ↦ χ ↑n) * fun n ↦ ↑(Λ n)) s\nn✝ : ℕ\nx✝ : n✝ ≠ 0\n⊢ ((fun n ↦ χ ↑n) * fun n ↦ Complex.log ↑n) n✝ = logMul (fun n ↦ χ ↑... | simp [mul_comm, logMul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 418,
"column": 34
} | {
"line": 418,
"column": 57
} | [
{
"pp": "N : ℕ\nχ : DirichletCharacter ℂ N\ns : ℂ\nhs : 1 < s.re\nhN : N ≠ 0\nhχ : LSeriesSummable (fun n ↦ χ ↑n) s\nhs' : (abscissaOfAbsConv fun n ↦ χ ↑n) < ↑s.re\nhΛ : LSeriesSummable ((fun n ↦ χ ↑n) * fun n ↦ ↑(Λ n)) s\nn✝ : ℕ\nx✝ : n✝ ≠ 0\n⊢ ((fun n ↦ χ ↑n) * fun n ↦ Complex.log ↑n) n✝ = logMul (fun n ↦ χ ↑... | simp [mul_comm, logMul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 418,
"column": 34
} | {
"line": 418,
"column": 57
} | [
{
"pp": "N : ℕ\nχ : DirichletCharacter ℂ N\ns : ℂ\nhs : 1 < s.re\nhN : N ≠ 0\nhχ : LSeriesSummable (fun n ↦ χ ↑n) s\nhs' : (abscissaOfAbsConv fun n ↦ χ ↑n) < ↑s.re\nhΛ : LSeriesSummable ((fun n ↦ χ ↑n) * fun n ↦ ↑(Λ n)) s\nn✝ : ℕ\nx✝ : n✝ ≠ 0\n⊢ ((fun n ↦ χ ↑n) * fun n ↦ Complex.log ↑n) n✝ = logMul (fun n ↦ χ ↑... | simp [mul_comm, logMul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.FLT.Basic | {
"line": 125,
"column": 4
} | {
"line": 125,
"column": 88
} | [
{
"pp": "case inr.inr.inr.inl\nn : ℕ\nh : FermatLastTheoremWith ℕ n\na b c : ℤ\nha✝ : a ≠ 0\nhb✝ : b ≠ 0\nhc✝ : c ≠ 0\nhabc : a ^ n + b ^ n = c ^ n\nhn : Odd n\nha : 0 < a\nhb : 0 < b\nhc : c < 0\n⊢ False",
"usedConstants": [
"Int.instIsStrictOrderedRing",
"pow_pos",
"Int.instLinearOrder",... | · exact (by positivity : 0 < a ^ n + b ^ n).not_gt <| habc.trans_lt <| hn.pow_neg hc | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Polynomial.Wronskian | {
"line": 139,
"column": 8
} | {
"line": 139,
"column": 17
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : NoZeroDivisors R\na b : R[X]\nhc : IsCoprime a b\nhda : derivative a = 0\nhdb : derivative b = 0\n⊢ a.wronskian b = 0",
"usedConstants": [
"Polynomial.derivative",
"Eq.mpr",
"Semiring.toModule",
"HMul.hMul",
"congrArg",
... | wronskian | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Radical | {
"line": 63,
"column": 6
} | {
"line": 63,
"column": 15
} | [
{
"pp": "k : Type u_1\ninst✝¹ : Field k\ninst✝ : DecidableEq k\na b : k[X]\n⊢ divRadical a ∣ a.wronskian b",
"usedConstants": [
"Polynomial.instNormalizationMonoid",
"Polynomial.derivative",
"Eq.mpr",
"IsDomain.to_noZeroDivisors",
"Dvd.dvd",
"Semiring.toModule",
"HM... | wronskian | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Radical.Basic | {
"line": 384,
"column": 2
} | {
"line": 384,
"column": 59
} | [
{
"pp": "case h\nE : Type u_1\ninst✝² : EuclideanDomain E\ninst✝¹ : NormalizationMonoid E\ninst✝ : UniqueFactorizationMonoid E\na b : E\nhab : IsCoprime a b\n⊢ radical (a * b) * (divRadical a * divRadical b) = a * b",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"MulZeroClass.toMul",
"c... | rw [UniqueFactorizationMonoid.radical_mul hab.isRelPrime] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.FLT.Four | {
"line": 215,
"column": 6
} | {
"line": 215,
"column": 18
} | [
{
"pp": "a b c : ℤ\nh : Minimal a b c\nha2 : a % 2 = 1\nhc : 0 < c\nht : PythagoreanTriple (a ^ 2) (b ^ 2) c\nh2 : (a ^ 2).gcd (b ^ 2) = 1\nha22 : a ^ 2 % 2 = 1\nm n : ℤ\nht1 : a ^ 2 = m ^ 2 - n ^ 2\nht2 : b ^ 2 = 2 * m * n\nht3 : c = m ^ 2 + n ^ 2\nht4 : m.gcd n = 1\nht5 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n... | Int.gcd_comm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.FLT.Four | {
"line": 232,
"column": 6
} | {
"line": 232,
"column": 18
} | [
{
"pp": "a b c : ℤ\nh : Minimal a b c\nha2 : a % 2 = 1\nhc : 0 < c\nht : PythagoreanTriple (a ^ 2) (b ^ 2) c\nh2 : (a ^ 2).gcd (b ^ 2) = 1\nha22 : a ^ 2 % 2 = 1\nm n : ℤ\nht1 : a ^ 2 = m ^ 2 - n ^ 2\nht2 : b ^ 2 = 2 * m * n\nht3 : c = m ^ 2 + n ^ 2\nht4 : m.gcd n = 1\nht5 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n... | Int.gcd_comm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 134,
"column": 6
} | {
"line": 134,
"column": 84
} | [
{
"pp": "x y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhx : x % 2 = 1\nhy : y % 2 = 1\nx0 : ℤ\nhx2 : x - 1 = x0 * 2\n⊢ ∃ x0 y0, x = x0 * 2 + 1 ∧ y = y0 * 2 + 1",
"usedConstants": [
"Int.instCommSemigroup",
"HSub.hSub",
"Int",
"instHSub",
"instOfNat",
"Int.dvd_... | obtain ⟨y0, hy2⟩ := exists_eq_mul_left_of_dvd (Int.dvd_self_sub_of_emod_eq hy) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 319,
"column": 56
} | {
"line": 319,
"column": 68
} | [
{
"pp": "m n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 1\nhn : n % 2 = 0\n⊢ n.gcd m = 1",
"usedConstants": [
"Int.gcd",
"Eq.mpr",
"congrArg",
"id",
"instOfNatNat",
"Nat",
"OfNat.ofNat",
"Int.gcd_comm",
"Eq"
]
}
] | Int.gcd_comm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 345,
"column": 6
} | {
"line": 345,
"column": 18
} | [
{
"pp": "case inr\nm n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(m.gcd n)\nhpn : p ∣ n.natAbs\n⊢ False",
"usedConstants": [
"Int.gcd",
"Dvd.dvd",
"co... | Int.gcd_comm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 357,
"column": 53
} | {
"line": 357,
"column": 65
} | [
{
"pp": "m n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 1\nhn : n % 2 = 0\n⊢ n.gcd m = 1",
"usedConstants": [
"Int.gcd",
"Eq.mpr",
"congrArg",
"id",
"instOfNatNat",
"Nat",
"OfNat.ofNat",
"Int.gcd_comm",
"Eq"
]
}
] | Int.gcd_comm | Lean.Elab.Tactic.evalRewriteSeq | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.