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.RingTheory.IsAdjoinRoot | {
"line": 407,
"column": 2
} | {
"line": 407,
"column": 42
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring S\nf : R[X]\ninst✝ : Algebra R S\nh : IsAdjoinRootMonic S f\nn : ℕ\na✝ : Nontrivial R\nhdeg : f.degree ≤ ↑n\n⊢ f.natDegree ≤ n",
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"WithBot",
"CommSemiring.toSemirin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IsAdjoinRoot | {
"line": 411,
"column": 2
} | {
"line": 411,
"column": 13
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring S\nf : R[X]\ninst✝ : Algebra R S\nh : IsAdjoinRootMonic S f\nhdeg : 1 < f.natDegree\n⊢ h.modByMonicHom h.root = X",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IsAdjoinRoot | {
"line": 604,
"column": 4
} | {
"line": 604,
"column": 42
} | [
{
"pp": "case h\nR : Type u\nS : Type v\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nf : R[X]\nh : IsAdjoinRoot S f\ninst✝³ : IsDomain R\ninst✝² : IsDomain S\ninst✝¹ : IsTorsionFree R S\ninst✝ : IsIntegrallyClosed R\nα : S\nhα : IsIntegral R α\nhα₂ : R[α] = ⊤\nx✝ : R[X]\n⊢ x✝ ∈ RingHom.ker (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.DirichletContinuation | {
"line": 144,
"column": 6
} | {
"line": 144,
"column": 45
} | [
{
"pp": "case refine_3.refine_2\nM N : ℕ\ninst✝¹ : NeZero M\ninst✝ : NeZero N\nhMN : M ∣ N\nχ : DirichletCharacter ℂ M\ns : ℂ\nhs : s ≠ 1\nhpc : IsPreconnected {1}ᶜ\nhne : 2 ∈ {1}ᶜ\nt : ℂ\nht : 1 < t.re\n⊢ LFunction ((changeLevel hMN) χ) t = (fun s ↦ LFunction χ s * ∏ p ∈ N.primeFactors, (1 - χ ↑p * ↑p ^ (-s)))... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.ZMod | {
"line": 439,
"column": 72
} | {
"line": 442,
"column": 70
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nΦ : ZMod N → ℂ\nhΦ : Function.Odd Φ\ns : ℂ\nhs : 1 < s.re\nthis : ∑ x, Φ x * sinZeta (toAddCircle x) s = I * LFunction (𝓕 Φ) s\n⊢ ∑ x, Φ x * completedSinZeta (toAddCircle x) s = I * completedLFunction (𝓕 Φ) s",
"usedConstants": [
"ZMod.completedLFunction",
"If... | by
have hs' : 0 < re (s + 1) := by simp only [add_re, one_re]; linarith
simpa only [sinZeta, ← mul_div_assoc, ← sum_div, div_left_inj' (Gammaℝ_ne_zero_of_re_pos hs'),
LFunction_eq_completed_div_gammaFactor_odd (dft_odd_iff.mpr hΦ)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LSeries.ZMod | {
"line": 502,
"column": 4
} | {
"line": 502,
"column": 46
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nΦ : ZMod N → ℂ\nhΦ : Function.Even Φ\ns : ℂ\nhs₀ : s ≠ 0 ∨ ∑ j, Φ j = 0\nhs₁ : s ≠ 1 ∨ Φ 0 = 0\nF : ℂ → ℂ := fun t ↦ completedLFunction Φ (1 - t)\nG : ℂ → ℂ := fun t ↦ ↑N ^ (t - 1) * completedLFunction (𝓕 Φ) t\nU : Set ℂ := {t | (t ≠ 0 ∨ ∑ j, Φ j = 0) ∧ (t ≠ 1 ∨ Φ 0 = 0)}\nhsU... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 621,
"column": 40
} | {
"line": 621,
"column": 62
} | [
{
"pp": "K : Type u_4\ninst✝¹ : Field K\ninst✝ : AdmissibleAbsValues K\nC₁ : ℝ\nhC₁ : ∀ (a b c d : K), logHeight ![a * c, a * d + b * c, b * d] ≤ C₁ + logHeight ![a, b] + logHeight ![c, d]\nC₂ : ℝ\nhC₂ :\n ∀ {a b c d : K},\n ![a, b] ≠ 0 → ![c, d] ≠ 0 → C₂ + logHeight ![a, b] + logHeight ![c, d] ≤ logHeight ... | specialize hC₂ hab hcd | Lean.Elab.Tactic.evalSpecialize | Lean.Parser.Tactic.specialize |
Mathlib.NumberTheory.LSeries.DirichletContinuation | {
"line": 345,
"column": 2
} | {
"line": 345,
"column": 27
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\np : ℕ\nhp : p ∈ n.primeFactors\n⊢ 1 - (↑p)⁻¹ ≠ 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"AddGroupWithOne.toAddGroup",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.DirichletContinuation | {
"line": 355,
"column": 2
} | {
"line": 355,
"column": 13
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\n⊢ ContinuousWithinAt (LFunctionTrivChar₁ n) {1}ᶜ 1",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"ContinuousWithinAt",
"HMul.hMul",
"Complex.instNormedAddCommGroup",
"Complex.commRing",
"ContinuousAt",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.DirichletContinuation | {
"line": 392,
"column": 2
} | {
"line": 392,
"column": 23
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\nχ : DirichletCharacter ℂ n\nhχ : χ ≠ 1\nh : Differentiable ℂ (LFunction χ)\n⊢ ContinuousOn (fun s ↦ -deriv (LFunction χ) s / LFunction χ s) {s | LFunction χ s ≠ 0}",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"neg_div",
"inst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 37,
"column": 2
} | {
"line": 37,
"column": 39
} | [
{
"pp": "f g : ℕ → ℂ\ns a b : ℂ\nhf : LSeriesHasSum f s a\nhg : LSeriesHasSum g s b\n⊢ LSeriesHasSum (f + g) s (a + b)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"congrArg",
"SummationFilter",
"Co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 42,
"column": 2
} | {
"line": 42,
"column": 49
} | [
{
"pp": "f g : ℕ → ℂ\ns : ℂ\nhf : LSeriesSummable f s\nhg : LSeriesSummable g s\n⊢ LSeriesSummable (f + g) s",
"usedConstants": [
"id",
"instHAdd",
"Pi.instAdd",
"HAdd.hAdd",
"Nat",
"Complex.instAdd",
"Complex",
"LSeriesSummable"
]
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 33
} | [
{
"pp": "f g : ℕ → ℂ\ns : ℂ\nhf : LSeriesSummable f s\nhg : LSeriesSummable g s\n⊢ LSeries (f + g) s = LSeries f s + LSeries g s",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"congrArg",
"SummationFilter",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 39
} | [
{
"pp": "f : ℕ → ℂ\ns a : ℂ\nhf : LSeriesHasSum f s a\n⊢ LSeriesHasSum (-f) s (-a)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Pi.instNeg",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"congrArg",
"SummationFilter",
"Complex.instNormed... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 41
} | [
{
"pp": "f : ℕ → ℂ\ns : ℂ\nhf : LSeriesSummable f s\n⊢ LSeriesSummable (-f) s",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Pi.instNeg",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"congrArg",
"SummationFilter",
"Complex.instNormedField... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 39
} | [
{
"pp": "f g : ℕ → ℂ\ns a b : ℂ\nhf : LSeriesHasSum f s a\nhg : LSeriesHasSum g s b\n⊢ LSeriesHasSum (f - g) s (a - b)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"congrArg",
"SummationFilter",
"Co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 49
} | [
{
"pp": "f g : ℕ → ℂ\ns : ℂ\nhf : LSeriesSummable f s\nhg : LSeriesSummable g s\n⊢ LSeriesSummable (f - g) s",
"usedConstants": [
"HSub.hSub",
"id",
"instHSub",
"Nat",
"Pi.instSub",
"Complex.instSub",
"Complex",
"LSeriesSummable"
]
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 33
} | [
{
"pp": "f g : ℕ → ℂ\ns : ℂ\nhf : LSeriesSummable f s\nhg : LSeriesSummable g s\n⊢ LSeries (f - g) s = LSeries f s - LSeries g s",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"congrArg",
"SummationFilter",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ZetaValues | {
"line": 137,
"column": 2
} | {
"line": 138,
"column": 9
} | [
{
"pp": "k : ℕ\nx : ℝ\n⊢ bernoulliFun k (1 - x) = (-1) ^ k * bernoulliFun k x",
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"Real.instRCLike",
"congrArg",
"CommSemiring.toSemiring",
"Polynomial.bernoulli",
"AlgHom",
"Real.instSub",
"AlgHom.fu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 40
} | [
{
"pp": "f : ℕ → ℂ\nc s a : ℂ\nhf : LSeriesHasSum f s a\n⊢ LSeriesHasSum (c • f) s (c * a)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"instHSMul",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"instSMulOfMul",
"HMul.hMul",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 42
} | [
{
"pp": "f : ℕ → ℂ\nc s : ℂ\nhf : LSeriesSummable f s\n⊢ LSeriesSummable (c • f) s",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"instHSMul",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"instSMulOfMul",
"congrArg",
"SummationFilter",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 18
} | [
{
"pp": "f : ℕ → ℂ\nc s : ℂ\nhc : c ≠ 0\nhf : LSeriesSummable (c • f) s\n⊢ LSeriesSummable f s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 77
} | [
{
"pp": "ι : Type u_1\nf : ι → ℕ → ℂ\nS : Finset ι\ns : ℂ\na : ι → ℂ\nhf : ∀ i ∈ S, LSeriesHasSum (f i) s (a i)\n⊢ LSeriesHasSum (∑ i ∈ S, f i) s (∑ i ∈ S, a i)",
"usedConstants": [
"Eq.mpr",
"LSeries.term_sum",
"NormedCommRing.toSeminormedCommRing",
"NonUnitalCommRing.toNonUnitalNon... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 49
} | [
{
"pp": "ι : Type u_1\nf : ι → ℕ → ℂ\nS : Finset ι\ns : ℂ\nhf : ∀ i ∈ S, LSeriesSummable (f i) s\n⊢ LSeriesSummable (∑ i ∈ S, f i) s",
"usedConstants": [
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Pi.addCommMonoid",
"Complex.instNormedField",
"id",
"NonUnitalNonAssocRing... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 165,
"column": 2
} | {
"line": 165,
"column": 33
} | [
{
"pp": "ι : Type u_1\nf : ι → ℕ → ℂ\nS : Finset ι\ns : ℂ\nhf : ∀ i ∈ S, LSeriesSummable (f i) s\n⊢ LSeries (∑ i ∈ S, f i) s = ∑ i ∈ S, LSeries (f i) s",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Pi.addCommMo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ZetaValues | {
"line": 189,
"column": 4
} | {
"line": 189,
"column": 24
} | [
{
"pp": "case succ\nm : ℕ\nm0 : m ≠ 0\nx : ℝ\nm0' : ↑m ≠ 0\nf : ℕ → ℝ → ℝ := fun k x ↦ bernoulliFun k (↑m * x) - ↑m ^ k / ↑m * ∑ i ∈ Finset.range m, bernoulliFun k (x + ↑i / ↑m)\nk : ℕ\nh : ∀ (x : ℝ), f k x = 0\nd : ∀ (x : ℝ), HasDerivAt (f (k + 1)) 0 x\nc : ℝ\nfc : ∀ (x : ℝ), f (k + 1) x = c\ni : c = 0\n⊢ ∀ (x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ZetaValues | {
"line": 206,
"column": 58
} | {
"line": 206,
"column": 68
} | [
{
"pp": "case neg\nk : ℕ\nk1 : ¬k = 1\nm : bernoulliFun k 1 = 2 ^ k / 2 * (0 + bernoulliFun k (2⁻¹ + 0 / 2) + bernoulliFun k (2⁻¹ + 1 / 2))\n⊢ bernoulliFun k 2⁻¹ = (2 / 2 ^ k - 1) * ↑(bernoulli k)",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"MulOne.toOne",
"Real",
"instHDi... | ← one_div, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaValues | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 13
} | [
{
"pp": "x : ℝ\nhx : x ∈ Icc 0 1\nk : ℕ\nhk : k.succ ≠ 0\n⊢ hurwitzZetaEven (↑x) (-(2 * ↑k.succ)) = 0",
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"AddGroupWithOne.toAddMonoidWithOne",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ZetaValues | {
"line": 241,
"column": 2
} | {
"line": 241,
"column": 13
} | [
{
"pp": "n : ℤ\nhn : n ≠ 0\n⊢ bernoulliFourierCoeff 0 n = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ZetaValues | {
"line": 286,
"column": 2
} | {
"line": 286,
"column": 29
} | [
{
"pp": "k : ℕ\nhk : k ≠ 0\nn : ℤ\nthis : ofReal ∘ periodizedBernoulli k = AddCircle.liftIco 1 0 (ofReal ∘ bernoulliFun k)\n⊢ fourierCoeffOn ⋯ (ofReal ∘ bernoulliFun k) n = -↑k ! / (2 * ↑π * I * ↑n) ^ k",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"lt... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Injectivity | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 32
} | [
{
"pp": "f : ℕ → ℂ\nn : ℕ\nh : ∀ m ≤ n, f m = 0\nha : abscissaOfAbsConv f < ⊤\ny : ℝ\nhay : abscissaOfAbsConv f < ↑y\nhyt : ↑y < ⊤\nF : ℝ → ℕ → ℂ := fun x ↦ {m | n + 1 < m}.indicator fun m ↦ f m / (↑m / (↑n + 1)) ^ ↑x\nhF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0\nhF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Injectivity | {
"line": 103,
"column": 28
} | {
"line": 103,
"column": 49
} | [
{
"pp": "f : ℕ → ℂ\nn : ℕ\nh : ∀ m ≤ n, f m = 0\nha : abscissaOfAbsConv f < ⊤\ny : ℝ\nhay : abscissaOfAbsConv f < ↑y\nhyt : ↑y < ⊤\nF : ℝ → ℕ → ℂ := fun x ↦ {m | n + 1 < m}.indicator fun m ↦ f m / (↑m / (↑n + 1)) ^ ↑x\nhF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0\nhF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (... | rw [← mul_zero (f k)] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.NumberTheory.LSeries.PrimesInAP | {
"line": 92,
"column": 6
} | {
"line": 93,
"column": 13
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : CommGroup α\ninst✝³ : UniformSpace α\ninst✝² : IsUniformGroup α\ninst✝¹ : CompleteSpace α\ninst✝ : T0Space α\nf : ℕ → α\nhfm : Multipliable f\nhf : Function.mulSupport f ⊆ {n | IsPrimePow n}\n⊢ Multipliable ((fun pk ↦ f (↑pk.1 ^ (pk.2 + 1))) ∘ ⇑prodNatEquiv.symm)",
"usedConst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Nonvanishing | {
"line": 86,
"column": 2
} | {
"line": 87,
"column": 7
} | [
{
"pp": "N : ℕ\nχ : DirichletCharacter ℂ N\ns : ℂ\nhs : 1 < s.re\nn : ℕ\nhn : n ≠ 0\n⊢ ‖(toArithmeticFunction fun x ↦ χ ↑x) n‖ ≤ 1",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Nat.instMulZeroClass",
"Real.instLE",
"Real",
"toArithmeticFunction",
"ArithmeticFunction.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Nonvanishing | {
"line": 109,
"column": 4
} | {
"line": 109,
"column": 83
} | [
{
"pp": "case inr\nN : ℕ\nχ : DirichletCharacter ℂ N\nhχ : χ ^ 2 = 1\nn : ℕ\nhn : n ≠ 0\n⊢ 0 ≤ χ.zetaMul n",
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Eq.mpr",
"Nat.instMulZeroClass",
"ArithmeticFunction.instFunLikeNat",
"congrArg",
"Nat.instMonoid",
"P... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Nonvanishing | {
"line": 184,
"column": 4
} | {
"line": 184,
"column": 50
} | [
{
"pp": "case inr.refine_2.hf\nN : ℕ\ninst✝ : NeZero N\nB : BadChar N\nG : ℂ → ℂ := ⋯\nH : ℂ → ℂ := ⋯\nthis : B.F = G * H\n⊢ ContinuousAt G 1",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"HMul.hMul",
"riemannZeta",
"Complex.instNormedAddCommGroup",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Injectivity | {
"line": 145,
"column": 4
} | {
"line": 145,
"column": 19
} | [
{
"pp": "case pos\nf : ℕ → ℂ\nh : abscissaOfAbsConv f = ⊤\n⊢ (fun x ↦ LSeries f ↑x) =ᶠ[atTop] 0 ↔ (∀ (n : ℕ), n ≠ 0 → f n = 0) ∨ abscissaOfAbsConv f = ⊤",
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"iff_true",
"EReal",
"Complex.instZero",
"Filter.EventuallyEq"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Injectivity | {
"line": 170,
"column": 10
} | {
"line": 170,
"column": 21
} | [
{
"pp": "case neg.refine_1.ind.succ\nf : ℕ → ℂ\nh : ¬abscissaOfAbsConv f = ⊤\nH : (fun x ↦ LSeries f ↑x) =ᶠ[atTop] 0\nF : ℕ → ℂ := fun n ↦ if n = 0 then 0 else f n\nhF₀ : F 0 = 0\nhF : ∀ {n : ℕ}, n ≠ 0 → F n = f n\nha : ¬abscissaOfAbsConv F = ⊤\nh' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x\nH' : ∀ (n : ℕ), (fun ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Injectivity | {
"line": 180,
"column": 4
} | {
"line": 180,
"column": 19
} | [
{
"pp": "case pos\nf : ℕ → ℂ\nhf : f 0 = 0\nh : abscissaOfAbsConv f = ⊤\n⊢ LSeries f = 0 ↔ f = 0 ∨ abscissaOfAbsConv f = ⊤",
"usedConstants": [
"Eq.mpr",
"congrArg",
"iff_true",
"EReal",
"Complex.instZero",
"instTopEReal",
"id",
"Pi.instZero",
"LSeries",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.PrimesInAP | {
"line": 276,
"column": 2
} | {
"line": 277,
"column": 9
} | [
{
"pp": "case h\nq : ℕ\na : ZMod q\ninst✝ : NeZero q\nha : IsUnit a\nn : ℕ\n⊢ ↑(residueClass a n) = ((↑q.totient)⁻¹ • ∑ χ, χ a⁻¹ • fun n ↦ χ ↑n * ↑(Λ n)) n",
"usedConstants": [
"ArithmeticFunction.vonMangoldt",
"DirichletCharacter.fintype",
"Eq.mpr",
"Semigroup.toMul",
"Real",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Injectivity | {
"line": 188,
"column": 6
} | {
"line": 188,
"column": 26
} | [
{
"pp": "case neg\nf : ℕ → ℂ\nhf : f 0 = 0\nh : ¬abscissaOfAbsConv f = ⊤\nH : LSeries f = 0\n⊢ (fun x ↦ LSeries f ↑x) =ᶠ[Filter.atTop] 0",
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Complex.instZero",
"Filter.EventuallyEq",
"id",
"Pi.instZero",
"Complex... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.PrimesInAP | {
"line": 321,
"column": 29
} | {
"line": 321,
"column": 93
} | [
{
"pp": "q : ℕ\na : ZMod q\ninst✝ : NeZero q\nχ : DirichletCharacter ℂ q\nhχ : χ ∈ {1}ᶜ\n⊢ χ ≠ 1",
"usedConstants": [
"ZMod.commRing",
"MulChar.hasOne",
"id",
"Ne",
"Field.toSemifield",
"ZMod",
"Semifield.toCommGroupWithZero",
"DirichletCharacter",
"One.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Injectivity | {
"line": 225,
"column": 2
} | {
"line": 226,
"column": 9
} | [
{
"pp": "f g : ℕ → ℂ\nhf : abscissaOfAbsConv f < ⊤\nhg : abscissaOfAbsConv g < ⊤\nh : (fun x ↦ LSeries f ↑x) =ᶠ[atTop] fun x ↦ LSeries g ↑x\nn : ℕ\nhn : n ≠ 0\nhsub : (fun x ↦ LSeries (f - g) ↑x) =ᶠ[atTop] 0\nha : abscissaOfAbsConv (f - g) ≠ ⊤\n⊢ f n = g n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Nonvanishing | {
"line": 270,
"column": 4
} | {
"line": 271,
"column": 11
} | [
{
"pp": "N : ℕ\nχ : DirichletCharacter ℂ N\ns : ℂ\nhs : 1 < s.re\np : Nat.Primes\n⊢ ‖χ ↑↑p * ↑↑p ^ (-s)‖ ≤ ↑↑p ^ (-s).re",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instPow",
"Real.instLE",
"Real",
"Nat.Prime",
"HMul.hMul",
"ZMod.commRing",
"congrA... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.ZetaZeros | {
"line": 41,
"column": 2
} | {
"line": 43,
"column": 64
} | [
{
"pp": "⊢ riemannZetaZerosᶜ ∈ Filter.codiscreteWithin {1}ᶜ",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
"False",
"Real.partialOrder",
"i... | refine analyticOn_riemannZeta.preimage_zero_mem_codiscreteWithin (x := 2) ?_ (by simp) ?_
· exact riemannZeta_ne_zero_of_one_le_re Nat.one_le_ofNat
· exact isConnected_compl_singleton_of_one_lt_rank (by simp) 1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LSeries.ZetaZeros | {
"line": 41,
"column": 2
} | {
"line": 43,
"column": 64
} | [
{
"pp": "⊢ riemannZetaZerosᶜ ∈ Filter.codiscreteWithin {1}ᶜ",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
"False",
"Real.partialOrder",
"i... | refine analyticOn_riemannZeta.preimage_zero_mem_codiscreteWithin (x := 2) ?_ (by simp) ?_
· exact riemannZeta_ne_zero_of_one_le_re Nat.one_le_ofNat
· exact isConnected_compl_singleton_of_one_lt_rank (by simp) 1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LSeries.ZetaZeros | {
"line": 58,
"column": 5
} | {
"line": 58,
"column": 16
} | [
{
"pp": "⊢ IsClosed riemannZetaZeros",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.ZetaZeros | {
"line": 61,
"column": 5
} | {
"line": 61,
"column": 16
} | [
{
"pp": "⊢ IsDiscrete riemannZetaZeros",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LegendreSymbol.Complex | {
"line": 26,
"column": 2
} | {
"line": 26,
"column": 37
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : Finite F\n⊢ ringChar ℂ ≠ ringChar F",
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Ne",
"instOfNatNat",
"Field.toSemifield",
"Semifield.toDivisionSemiring",
"ringChar",
"Nat",
"DivisionSemiring.to... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.PrimesInAP | {
"line": 411,
"column": 8
} | {
"line": 411,
"column": 54
} | [
{
"pp": "q : ℕ\ninst✝ : NeZero q\na : ZMod q\nha : IsUnit a\nH :\n ∀ {x : ℝ},\n 1 < x → ∑' (n : ℕ), residueClass a n / ↑n ^ x = (LFunctionResidueClassAux a ↑x).re + (↑q.totient)⁻¹ / (x - 1)\nx : ℝ\nhx : x ∈ Set.Icc 1 2\n⊢ ↑x ∈ {s | 1 ≤ s.re}",
"usedConstants": [
"Real.instLE",
"Real",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Nonvanishing | {
"line": 300,
"column": 2
} | {
"line": 301,
"column": 44
} | [
{
"pp": "N : ℕ\nχ : DirichletCharacter ℂ N\nx : ℝ\nhx : 0 < x\ny : ℝ\nh₀ : 1 < (1 + ↑x).re\nh₁ : 1 < (1 + ↑x + I * ↑y).re\nh₂ : 1 < (1 + ↑x + 2 * I * ↑y).re\nH₀ : Summable fun p ↦ -log (1 - 1 ↑↑p * ↑↑p ^ (-(1 + ↑x)))\nH₁ : Summable fun p ↦ -log (1 - χ ↑↑p * ↑↑p ^ (-(1 + ↑x + I * ↑y)))\nH₂ : Summable fun p ↦ -lo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Nonvanishing | {
"line": 322,
"column": 8
} | {
"line": 322,
"column": 99
} | [
{
"pp": "case convert_2\nN : ℕ\ninst✝ : NeZero N\n⊢ Tendsto (fun w ↦ 1 + w) (𝓝[≠] 0) (𝓝[≠] 1)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Compl.compl",
"nhdsWithin",
"PartialOrder.toPreorder",
"Complex.instNormedField",
"PseudoMetricSpac... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Nonvanishing | {
"line": 330,
"column": 2
} | {
"line": 331,
"column": 9
} | [
{
"pp": "N : ℕ\nχ : DirichletCharacter ℂ N\ny : ℝ\nhy : y ≠ 0 ∨ χ ≠ 1\n⊢ 1 + I * ↑y ≠ 1 ∨ χ ≠ 1",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"False",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"ZMod.commRing",
"MulZe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LegendreSymbol.GaussEisensteinLemmas | {
"line": 91,
"column": 6
} | {
"line": 91,
"column": 17
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\na : ℤ\nhap : ↑a ≠ 0\n⊢ ↑a ^ (p / 2) * ↑(p / 2)! = ↑((-1) ^ #({x ∈ Ico 1 (p / 2).succ | p / 2 < (↑a * ↑x).val})) * ↑(p / 2)!",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast_neg",
"Int.cast",
"Eq.mpr",
"NegZeroClass.toNeg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Nonvanishing | {
"line": 347,
"column": 2
} | {
"line": 348,
"column": 9
} | [
{
"pp": "N : ℕ\nχ : DirichletCharacter ℂ N\ninst✝ : NeZero N\ny : ℝ\nhy : y ≠ 0 ∨ χ ≠ 1\nh : LFunction χ (1 + I * ↑y) = 0\nthis : HasDerivAt (LFunction χ) (deriv (LFunction χ) (0 + (1 + I * ↑y))) (0 + (1 + I * ↑y))\n⊢ (fun x ↦ LFunction χ (↑x + (1 + I * ↑y))) =O[𝓝[>] 0] fun x ↦ ↑x",
"usedConstants": []
}... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Nonvanishing | {
"line": 364,
"column": 8
} | {
"line": 364,
"column": 64
} | [
{
"pp": "case inr\nN : ℕ\nχ : DirichletCharacter ℂ N\ninst✝ : NeZero N\nt : ℝ\nh✝ : χ ^ 2 ≠ 1 ∨ t ≠ 0\nHz : LFunction χ (1 + I * ↑t) = 0\nhz₁ : t ≠ 0 ∨ χ ≠ 1\nhz₂ : 2 * t ≠ 0 ∨ χ ^ 2 ≠ 1\nx : ℝ\nh : x ≠ 0\n⊢ (↑x ^ 3)⁻¹ * ↑x ^ 3 * ↑x * 1 = ↑x",
"usedConstants": [
"Iff.mpr",
"NormedCommRing.toNorm... | inv_mul_cancel₀ <| pow_ne_zero 3 (ofReal_ne_zero.mpr h), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 123,
"column": 12
} | {
"line": 123,
"column": 23
} | [
{
"pp": "f : ℕ → ℂ\nhf : f 0 = 0\nr : ℝ\nhr : 0 ≤ r\ns : ℂ\nhs : r < s.re\nhS : LSeriesSummable f s\nh₁ : (-s - 1).re + r < -1\nh₂ : s ≠ 0\nh₃ : ∀ t ∈ Set.Ici 1, DifferentiableAt ℝ (fun x ↦ ↑x ^ (-s)) t\nh₄ : ∀ (n : ℕ), ∑ k ∈ Icc 0 n, f k = ∑ k ∈ Icc 1 n, f k\nhO : (fun n ↦ ∑ k ∈ Icc 0 n, f k) =O[atTop] fun n ↦... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.PrimesInAP | {
"line": 450,
"column": 8
} | {
"line": 451,
"column": 43
} | [
{
"pp": "case refine_1\nq : ℕ\ninst✝ : NeZero q\na : ZMod q\nha : IsUnit a\nH : Summable fun n ↦ (if Nat.Prime n then residueClass a n else 0) / ↑n\nkey : Summable fun n ↦ residueClass a n / ↑n\nC : ℝ := ∑' (n : ℕ), residueClass a n / ↑n\nH₁✝ : ∀ {x : ℝ}, 1 < x → ∑' (n : ℕ), residueClass a n / ↑n ^ x ≤ C\nC' : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LegendreSymbol.GaussEisensteinLemmas | {
"line": 139,
"column": 4
} | {
"line": 140,
"column": 50
} | [
{
"pp": "p : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Fact (p % 2 = 1)\na : ℕ\nha2✝ : a % 2 = 1\nhap : ↑a ≠ 0\nha2 : ↑a = ↑1\n⊢ ↑(#({x ∈ Ico 1 (p / 2).succ | p / 2 < (↑a * ↑x).val})) - ↑(∑ x ∈ Ico 1 (p / 2).succ, x * a / p) = 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LegendreSymbol.GaussEisensteinLemmas | {
"line": 188,
"column": 6
} | {
"line": 188,
"column": 23
} | [
{
"pp": "p q : ℕ\nhp : Fact (Nat.Prime p)\nhq0 : ↑q ≠ 0\nhswap :\n #({x ∈ Ico 1 (q / 2).succ ×ˢ Ico 1 (p / 2).succ | x.2 * q ≤ x.1 * p}) =\n #({x ∈ Ico 1 (p / 2).succ ×ˢ Ico 1 (q / 2).succ | x.1 * q ≤ x.2 * p})\nx : ℕ × ℕ\nhx : x ∈ Ico 1 (p / 2).succ ×ˢ Ico 1 (q / 2).succ\nhpq : x.2 * p ≤ x.1 * q\nhqp : x.1... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 130,
"column": 6
} | {
"line": 130,
"column": 42
} | [
{
"pp": "case convert_3.refine_1\nf : ℕ → ℂ\nhf : f 0 = 0\nr : ℝ\nhr : 0 ≤ r\ns : ℂ\nhs : r < s.re\nhS : LSeriesSummable f s\nh₁ : (-s - 1).re + r < -1\nh₂ : s ≠ 0\nh₃ : ∀ t ∈ Set.Ici 1, DifferentiableAt ℝ (fun x ↦ ↑x ^ (-s)) t\nh₄ : ∀ (n : ℕ), ∑ k ∈ Icc 0 n, f k = ∑ k ∈ Icc 1 n, f k\nhO : (fun n ↦ ∑ k ∈ Icc 0 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.PrimesInAP | {
"line": 499,
"column": 2
} | {
"line": 499,
"column": 45
} | [
{
"pp": "n q : ℕ\na : ℤ\nhq : q ≠ 0\nh : IsCoprime a ↑q\nthis✝ : NeZero q\nthis : IsUnit ↑a\np : ℕ\nhpn : p > n\nhpp : Prime p\nheq : ↑p = ↑a\n⊢ ↑p ≡ a [ZMOD ↑q]",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Int.cast_natCast",
"ZMod.commRing",
"congrArg",
"AddGroupWithOne.t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.PrimesInAP | {
"line": 506,
"column": 2
} | {
"line": 506,
"column": 13
} | [
{
"pp": "n q a : ℕ\nhq : q ≠ 0\nh : a.Coprime q\n⊢ ∃ p > n, Prime p ∧ p ≡ a [MOD q]",
"usedConstants": [
"Eq.mpr",
"Nat.Prime",
"congrArg",
"Exists",
"id",
"funext",
"GT.gt",
"And",
"Nat.ModEq",
"Nat",
"LT.lt",
"congrFun'",
"instL... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 187,
"column": 6
} | {
"line": 187,
"column": 17
} | [
{
"pp": "f : ℕ → ℂ\nl : ℂ\nhlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)\ns : ℝ\nhs : 1 < s\nh₁ : LocallyIntegrableOn (fun t ↦ (∑ k ∈ Icc 1 ⌊t⌋₊, f k) * ↑t ^ (-↑s - 1)) (Set.Ici 1) volume\n⊢ Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑(↑n ^ 1)) atTop (𝓝 l)",
"usedConstants": [
"Eq.mpr"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 188,
"column": 4
} | {
"line": 188,
"column": 15
} | [
{
"pp": "f : ℕ → ℂ\nl : ℂ\nhlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)\ns : ℝ\nhs : 1 < s\nh₁ : LocallyIntegrableOn (fun t ↦ (∑ k ∈ Icc 1 ⌊t⌋₊, f k) * ↑t ^ (-↑s - 1)) (Set.Ici 1) volume\nthis : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑(↑n ^ 1)) atTop (𝓝 l)\n⊢ (fun t ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 204,
"column": 4
} | {
"line": 205,
"column": 28
} | [
{
"pp": "case h\ns T ε : ℝ\nS : ℝ → ℂ\nhs : 1 < s\nhS₁ : LocallyIntegrableOn (fun t ↦ S t) (Set.Ici 1) volume\nhS₂ : ∀ t ≥ T, ‖S t‖ ≤ ε * t\nh : LocallyIntegrableOn (fun t ↦ ‖S t‖ * t ^ (-s - 1)) (Set.Ici 1) volume\nt : ℝ\nht : T ≤ t\nht' : 0 < t\n⊢ ‖‖S t‖ * t ^ (-s - 1)‖ ≤ ε * ‖t ^ (-s)‖",
"usedConstants":... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 271,
"column": 4
} | {
"line": 271,
"column": 15
} | [
{
"pp": "f : ℕ → ℂ\nl : ℂ\nhlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)\nhfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s\nε : ℝ\nhε : ε > 0\nT : ℝ\nhT₁ : 1 ≤ T\nhT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * ↑y‖ < ε * y\nS : ℝ → ℂ := fun t ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k\nC : ℝ := ∫ (t : ℝ) in Se... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 272,
"column": 2
} | {
"line": 273,
"column": 40
} | [
{
"pp": "case h\nf : ℕ → ℂ\nl : ℂ\nhlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)\nhfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s\nε : ℝ\nhε : ε > 0\nT : ℝ\nhT₁ : 1 ≤ T\nhT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * ↑y‖ < ε * y\nS : ℝ → ℂ := fun t ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k\nC : ℝ := ∫ (t : ... | have h₁ : IntegrableOn (fun t ↦ ‖S t - l * t‖ * t ^ (-s - 1)) (Set.Ici 1) :=
lemma₂ hs h₀ fun t ht ↦ (hT t ht).le | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 296,
"column": 71
} | {
"line": 296,
"column": 82
} | [
{
"pp": "f : ℕ → ℂ\nl : ℂ\nhlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)\nhfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s\nε : ℝ\nhε : ε > 0\nT : ℝ\nhT₁ : 1 ≤ T\nhT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * ↑y‖ < ε * y\nS : ℝ → ℂ := fun t ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k\nC : ℝ := ∫ (t : ℝ) in Se... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 453,
"column": 4
} | {
"line": 453,
"column": 21
} | [
{
"pp": "case inl\na b : ℕ\nha2 : a % 2 = 1\nhb2 : b % 2 = 1\nha1 : a % 4 = 1\n⊢ (if a % 4 = 3 ∧ b % 4 = 3 then -J(↑b | a) else J(↑b | a)) = J(↑a | b)",
"usedConstants": [
"Eq.mpr",
"False",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"false_and",
"Nat.instCharZero",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 455,
"column": 4
} | {
"line": 455,
"column": 21
} | [
{
"pp": "case inr.inl\na b : ℕ\nha2 : a % 2 = 1\nhb2 : b % 2 = 1\nha3 : a % 4 = 3\nhb1 : b % 4 = 1\n⊢ (if a % 4 = 3 ∧ b % 4 = 3 then -J(↑b | a) else J(↑b | a)) = J(↑a | b)",
"usedConstants": [
"Eq.mpr",
"False",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"Nat.instCharZero",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 456,
"column": 2
} | {
"line": 456,
"column": 24
} | [
{
"pp": "case inr.inr\na b : ℕ\nha2 : a % 2 = 1\nhb2 : b % 2 = 1\nha3 : a % 4 = 3\nhb3 : b % 4 = 3\n⊢ (if a % 4 = 3 ∧ b % 4 = 3 then -J(↑b | a) else J(↑b | a)) = J(↑a | b)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"and_self",
"id",
"Nat.instMod",
"instHMod",
"Int.i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.LocallyCompact | {
"line": 54,
"column": 2
} | {
"line": 54,
"column": 32
} | [
{
"pp": "K : Type u_1\ninst✝¹ : NontriviallyNormedField K\ninst✝ : IsUltrametricDist K\nu : (↥𝒪[K])ˣ\n⊢ ‖↑↑u‖ = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.LocallyCompact | {
"line": 61,
"column": 2
} | {
"line": 61,
"column": 32
} | [
{
"pp": "K : Type u_1\ninst✝¹ : NontriviallyNormedField K\ninst✝ : IsUltrametricDist K\nu : ↥𝒪[K]\n⊢ IsUnit u ↔ ‖u‖ = 1",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Real",
"Subring.instSetLike",
"Valued.integer",
"CommSemiring.toSemiring",
"Nor... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.LocallyCompact | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 36
} | [
{
"pp": "K : Type u_1\ninst✝¹ : NontriviallyNormedField K\ninst✝ : IsUltrametricDist K\nx : K\nhx : 0 < ‖x‖\nhx' : ‖x‖ < 1\n⊢ 0 < ‖↑⟨x, ⋯⟩‖ ∧ ‖↑⟨x, ⋯⟩‖ < 1",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"Real",
"Subring.instSetLike",
"and_true... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.LocallyCompact | {
"line": 136,
"column": 4
} | {
"line": 136,
"column": 70
} | [
{
"pp": "case mp\nK : Type u_1\nΓ₀ : Type u_2\ninst✝⁴ : Field K\ninst✝³ : LinearOrderedCommGroupWithZero Γ₀\ninst✝² : Valued K Γ₀\ninst✝¹ : v.RankOne\ninst✝ : IsDiscreteValuationRing ↥𝒪[K]\nH : TotallyBounded Set.univ\np : ↥𝒪[K]\nhp : Irreducible p\nthis : ∃ t ⊆ Set.univ, t.Finite ∧ Set.univ ⊆ ⋃ y ∈ t, Metric... | simp only [Set.subset_univ, Set.univ_subset_iff, true_and] at this | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Algebra.Valued.LocallyCompact | {
"line": 156,
"column": 4
} | {
"line": 156,
"column": 30
} | [
{
"pp": "case mp.mk\nK : Type u_1\nΓ₀ : Type u_2\ninst✝⁴ : Field K\ninst✝³ : LinearOrderedCommGroupWithZero Γ₀\ninst✝² : Valued K Γ₀\ninst✝¹ : v.RankOne\ninst✝ : IsDiscreteValuationRing ↥𝒪[K]\nH : TotallyBounded Set.univ\np : ↥𝒪[K]\nhp : Irreducible p\nt : Set ↥𝒪[K]\nht : t.Finite\na✝ : IsLocalRing.ResidueFi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.ValuativeRel | {
"line": 41,
"column": 4
} | {
"line": 41,
"column": 90
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : ValuativeRel R\ninst✝¹ : TopologicalSpace R\ninst✝ : ContinuousConstVAdd R R\nh₀ : ∀ (s : Set R), s ∈ 𝓝 0 ↔ ∃ γ, {z | v z < ↑γ} ⊆ s\ns : Set R\nx : R\n⊢ s ∈ 𝓝 x ↔ ∃ γ, (fun x_1 ↦ x + x_1) '' {z | v z < ↑γ} ⊆ s",
"usedConstants": [
"Filter.instMemb... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.ValuativeRel | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 58
} | [
{
"pp": "case h.h.e'_2.a\nR✝ : Type u_1\ninst✝⁸ : CommRing R✝\ninst✝⁷ : ValuativeRel R✝\ninst✝⁶ : TopologicalSpace R✝\ninst✝⁵ : IsValuativeTopology R✝\nR : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : ValuativeRel R\ninst✝² : UniformSpace R\ninst✝¹ : IsUniformAddGroup R\ninst✝ : IsValuativeTopology R\na✝ : Set R\n⊢ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.ValuativeRel | {
"line": 73,
"column": 4
} | {
"line": 74,
"column": 11
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝³ : CommRing R\ninst✝² : ValuativeRel R\ninst✝¹ : TopologicalSpace R\ninst✝ : IsValuativeTopology R\nx : R\nhx : v x = 0\n⊢ Tendsto (⇑v) (𝓝 x) (𝓝 (v x))",
"usedConstants": [
"WithZeroTopology.topologicalSpace",
"Units.val",
"Eq.mpr",
"GroupWith... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.ValuativeRel | {
"line": 76,
"column": 4
} | {
"line": 77,
"column": 11
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝³ : CommRing R\ninst✝² : ValuativeRel R\ninst✝¹ : TopologicalSpace R\ninst✝ : IsValuativeTopology R\nx : R\nhx : ¬v x = 0\n⊢ Tendsto (⇑v) (𝓝 x) (𝓝 (v x))",
"usedConstants": [
"WithZeroTopology.topologicalSpace",
"Units.val",
"Eq.mpr",
"GroupWit... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.ValuativeRel | {
"line": 78,
"column": 19
} | {
"line": 78,
"column": 61
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : ValuativeRel R\ninst✝¹ : TopologicalSpace R\ninst✝ : IsValuativeTopology R\nx : R\nhx : ¬v x = 0\nx✝ : R\n⊢ v.restrict (x✝ - x) < ↑((Units.mapEquiv ↑(ValueGroupWithZero.orderMonoidIso v)) (Units.mk0 (v x) hx)) → v x✝ = v x",
"usedConstants": [
"Unit... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology | {
"line": 141,
"column": 2
} | {
"line": 141,
"column": 30
} | [
{
"pp": "case h.e'_2.a\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : ValuativeRel R\nΓ₀ : Type u_3\ninst✝³ : LinearOrderedCommGroupWithZero Γ₀\ninst✝² : TopologicalSpace R\ninst✝¹ : IsValuativeTopology R\nv : Valuation R Γ₀\ninst✝ : v.Compatible\ns : Set R\nx : R\n⊢ (∃ γ, {z | v.restrict (z - x) < ↑γ} ⊆ s) ↔ ∃ γ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LocalField.Basic | {
"line": 83,
"column": 24
} | {
"line": 83,
"column": 35
} | [
{
"pp": "K : Type u_1\ninst✝³ : Field K\ninst✝² : ValuativeRel K\ninst✝¹ : TopologicalSpace K\ninst✝ : IsNonarchimedeanLocalField K\nγ : K\nhγ : ¬γ = 0\nthis✝ : UniformSpace K := IsTopologicalAddGroup.rightUniformSpace K\nthis : IsUniformAddGroup K := isUniformAddGroup_of_addCommGroup\ns : Set K\nhs : s ∈ nhds ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology | {
"line": 180,
"column": 45
} | {
"line": 180,
"column": 56
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : ValuativeRel R\nK : Type u_2\ninst✝⁵ : Field K\ninst✝⁴ : ValuativeRel K\nΓ₀ : Type u_3\ninst✝³ : LinearOrderedCommGroupWithZero Γ₀\ninst✝² : TopologicalSpace R\ninst✝¹ : IsValuativeTopology R\nv : Valuation R Γ₀\ninst✝ : v.Compatible\nx z : R\nγ : (ValueGroup... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LocalField.Basic | {
"line": 86,
"column": 6
} | {
"line": 89,
"column": 41
} | [
{
"pp": "case inr\nK : Type u_1\ninst✝³ : Field K\ninst✝² : ValuativeRel K\ninst✝¹ : TopologicalSpace K\ninst✝ : IsNonarchimedeanLocalField K\nγ : K\nhγ : ¬γ = 0\nthis✝ : UniformSpace K := IsTopologicalAddGroup.rightUniformSpace K\nthis : IsUniformAddGroup K := isUniformAddGroup_of_addCommGroup\ns : Set K\nhs :... | refine ⟨r', hr', hr, .trans ?_ hrs⟩
intro x hx
dsimp at hx ⊢
exact hx.trans_lt (hr.trans_le hr1) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LocalField.Basic | {
"line": 86,
"column": 6
} | {
"line": 89,
"column": 41
} | [
{
"pp": "case inr\nK : Type u_1\ninst✝³ : Field K\ninst✝² : ValuativeRel K\ninst✝¹ : TopologicalSpace K\ninst✝ : IsNonarchimedeanLocalField K\nγ : K\nhγ : ¬γ = 0\nthis✝ : UniformSpace K := IsTopologicalAddGroup.rightUniformSpace K\nthis : IsUniformAddGroup K := isUniformAddGroup_of_addCommGroup\ns : Set K\nhs :... | refine ⟨r', hr', hr, .trans ?_ hrs⟩
intro x hx
dsimp at hx ⊢
exact hx.trans_lt (hr.trans_le hr1) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology | {
"line": 185,
"column": 75
} | {
"line": 185,
"column": 86
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : ValuativeRel R\nK : Type u_2\ninst✝⁵ : Field K\ninst✝⁴ : ValuativeRel K\nΓ₀ : Type u_3\ninst✝³ : LinearOrderedCommGroupWithZero Γ₀\ninst✝² : TopologicalSpace R\ninst✝¹ : IsValuativeTopology R\nv : Valuation R Γ₀\ninst✝ : v.Compatible\ncts_add : ContinuousCons... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LocalField.Basic | {
"line": 181,
"column": 6
} | {
"line": 181,
"column": 21
} | [
{
"pp": "K : Type u_1\ninst✝⁴ : Field K\ninst✝³ : ValuativeRel K\ninst✝² : UniformSpace K\ninst✝¹ : IsUniformAddGroup K\ninst✝ : IsNonarchimedeanLocalField K\nf : ℕ → ↥𝒪[K]\nhf : ∀ {m n : ℕ}, m ≤ n → f m ≡ f n [SMOD 𝓂[K] ^ m • ⊤]\nS : ℕ → Set ↥𝒪[K] := ⋯\nn : ℕ\n⊢ f (n + 1) +ᵥ ↑(𝓂[K] ^ n) ⊆ S n",
"usedCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology | {
"line": 187,
"column": 4
} | {
"line": 187,
"column": 30
} | [
{
"pp": "case refine_4\nR : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : ValuativeRel R\nK : Type u_2\ninst✝⁵ : Field K\ninst✝⁴ : ValuativeRel K\nΓ₀ : Type u_3\ninst✝³ : LinearOrderedCommGroupWithZero Γ₀\ninst✝² : TopologicalSpace R\ninst✝¹ : IsValuativeTopology R\nv : Valuation R Γ₀\ninst✝ : v.Compatible\ncts_add :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology | {
"line": 188,
"column": 4
} | {
"line": 188,
"column": 30
} | [
{
"pp": "case refine_5\nR : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : ValuativeRel R\nK : Type u_2\ninst✝⁵ : Field K\ninst✝⁴ : ValuativeRel K\nΓ₀ : Type u_3\ninst✝³ : LinearOrderedCommGroupWithZero Γ₀\ninst✝² : TopologicalSpace R\ninst✝¹ : IsValuativeTopology R\nv : Valuation R Γ₀\ninst✝ : v.Compatible\ncts_add :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LocalField.Basic | {
"line": 187,
"column": 4
} | {
"line": 187,
"column": 32
} | [
{
"pp": "K : Type u_1\ninst✝⁴ : Field K\ninst✝³ : ValuativeRel K\ninst✝² : UniformSpace K\ninst✝¹ : IsUniformAddGroup K\ninst✝ : IsNonarchimedeanLocalField K\nf : ℕ → ↥𝒪[K]\nhf : ∀ {m n : ℕ}, m ≤ n → f m ≡ f n [SMOD 𝓂[K] ^ m • ⊤]\nS : ℕ → Set ↥𝒪[K] := fun n ↦ f n +ᵥ ↑(𝓂[K] ^ n)\nhS : ∀ (n : ℕ), S (n + 1) ⊆ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.LocallyCompact | {
"line": 275,
"column": 6
} | {
"line": 277,
"column": 22
} | [
{
"pp": "case neg\nK : Type u_1\nΓ₀ : Type u_2\ninst✝² : Field K\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : Valued K Γ₀\nhc✝ : IsCompact ↑𝒪[K]\nz : (↥(MonoidHom.mrange v))ˣ\na : K\nha : v a = ↑↑z\nhz1 : z ≤ 1\nz0' : 0 < ↑z\nz0 : 0 < ↑↑z\na0 : 0 < v a\nU : K → Set K := fun y ↦ if v y ≤ ↑↑z then {w | v... | simp only [Set.mem_setOf_eq] at hj'
rw [dif_neg hcj]
simp [← hj', hc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Valued.LocallyCompact | {
"line": 275,
"column": 6
} | {
"line": 277,
"column": 22
} | [
{
"pp": "case neg\nK : Type u_1\nΓ₀ : Type u_2\ninst✝² : Field K\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : Valued K Γ₀\nhc✝ : IsCompact ↑𝒪[K]\nz : (↥(MonoidHom.mrange v))ˣ\na : K\nha : v a = ↑↑z\nhz1 : z ≤ 1\nz0' : 0 < ↑z\nz0 : 0 < ↑↑z\na0 : 0 < v a\nU : K → Set K := fun y ↦ if v y ≤ ↑↑z then {w | v... | simp only [Set.mem_setOf_eq] at hj'
rw [dif_neg hcj]
simp [← hj', hc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.SimpleRing.Field | {
"line": 32,
"column": 30
} | {
"line": 32,
"column": 59
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Ring A\ninst✝ : IsSimpleRing A\nx : A\nhx1✝ : x ∈ Subring.center A\nhx1 : ∀ (g : A), g * x = x * g\nhx2 : ⟨x, hx1✝⟩ ≠ 0\n⊢ x ≠ 0",
"usedConstants": [
"id",
"Ne",
"Zero.toOfNat0",
"OfNat.ofNat",
"Ring.toSemiring",
"MulZeroClass.toZero",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.SimpleRing.Field | {
"line": 39,
"column": 57
} | {
"line": 39,
"column": 81
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Ring A\ninst✝ : IsSimpleRing A\nx : A\nhx1✝ : x ∈ Subring.center A\nhx1 : ∀ (g : A), g * x = x * g\nhx2 : x ≠ 0\nI : TwoSidedIdeal A := mk' (Set.range fun x_1 ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯\n⊢ x ∈ I",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"NegZeroCl... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology | {
"line": 259,
"column": 46
} | {
"line": 259,
"column": 57
} | [
{
"pp": "K : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : ValuativeRel K\nΓ₀ : Type u_3\ninst✝⁴ : LinearOrderedCommGroupWithZero Γ₀\ninst✝³ : TopologicalSpace K\ninst✝² : IsValuativeTopology K\ninst✝¹ : MulArchimedean Γ₀\nv : Valuation K Γ₀\ninst✝ : v.Compatible\nr : Γ₀\nhr : r ≠ 0\nh : ∀ (x : K), v x ≠ 0 → r < v x\n⊢ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LucasLehmer | {
"line": 57,
"column": 4
} | {
"line": 57,
"column": 15
} | [
{
"pp": "p : ℕ\n⊢ Odd (mersenne (p + 1)) ↔ p + 1 ≠ 0",
"usedConstants": [
"Eq.mpr",
"False",
"Nat.instMulZeroClass",
"Nat.instOne",
"congrArg",
"Odd",
"Nat.add_eq_zero_iff._simp_1",
"iff_true",
"id",
"one_ne_zero._simp_1",
"Ne",
"instOf... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LucasLehmer | {
"line": 207,
"column": 10
} | {
"line": 207,
"column": 57
} | [
{
"pp": "p : ℕ\nw : 1 < p\nh : ↑(sMod p (p - 2)) = 0\n⊢ ?m.28 ∣ sMod p (p - 2)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.MaricaSchoenheim | {
"line": 53,
"column": 4
} | {
"line": 53,
"column": 15
} | [
{
"pp": "case calc_1\nn : ℕ\nf : ℕ → ℕ\nhf' : ∀ k < n, Squarefree (f k)\nhn : n ≠ 0\nhf : StrictMonoOn f (Set.Iio n)\nthis : ∀ i < n, ∀ j < n, f i < (f i).gcd (f j) * n\n𝒜 : Finset (Finset ℕ) := image (fun n ↦ (f n).primeFactors) (Iio n)\nhf'' : ∀ i < n, ∀ (j : ℕ), Squarefree (f i / (f i).gcd (f j))\n⊢ Set.Inj... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.