module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.NumberTheory.ZetaValues | {
"line": 189,
"column": 4
} | {
"line": 189,
"column": 24
} | {
"line": 189,
"column": 25
} | [
{
"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... | [
"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 : ℝ), f (k ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ZetaValues | {
"line": 206,
"column": 58
} | {
"line": 206,
"column": 68
} | {
"line": 206,
"column": 69
} | [
{
"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)",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
... | [
"case neg\nk : ℕ\nk1 : ¬k = 1\nm : bernoulliFun k 1 = 2 ^ k / 2 * (0 + bernoulliFun k (1 / 2 + 0 / 2) + bernoulliFun k (1 / 2 + 1 / 2))\n⊢ bernoulliFun k 2⁻¹ = (2 / 2 ^ k - 1) * ↑(bernoulli k)"
] | ← one_div, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.ZetaValues | {
"line": 241,
"column": 2
} | {
"line": 241,
"column": 13
} | {
"line": 241,
"column": 14
} | [
{
"pp": "n : ℤ\nhn : n ≠ 0\n⊢ bernoulliFourierCoeff 0 n = 0",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℤ\nhn : n ≠ 0\n⊢ bernoulliFourierCoeff 0 n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaValues | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 13
} | {
"line": 183,
"column": 14
} | [
{
"pp": "x : ℝ\nhx : x ∈ Icc 0 1\nk : ℕ\nhk : k.succ ≠ 0\n⊢ hurwitzZetaEven (↑x) (-(2 * ↑k.succ)) = 0",
"ppTerm": "?m.111",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
... | [
"x : ℝ\nhx : x ∈ Icc 0 1\nk : ℕ\nhk : k.succ ≠ 0\n⊢ hurwitzZetaEven (↑x) (-(2 * (↑k + 1))) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 42,
"column": 2
} | {
"line": 42,
"column": 49
} | {
"line": 42,
"column": 50
} | [
{
"pp": "f g : ℕ → ℂ\ns : ℂ\nhf : LSeriesSummable f s\nhg : LSeriesSummable g s\n⊢ LSeriesSummable (f + g) s",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"id",
"instHAdd",
"Pi.instAdd",
"HAdd.hAdd",
"Nat",
"Complex.instAdd",
"Complex",
"LSe... | [
"f g : ℕ → ℂ\ns : ℂ\nhf : LSeriesSummable f s\nhg : LSeriesSummable g s\n⊢ Summable (term (f + g) s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 33
} | {
"line": 47,
"column": 34
} | [
{
"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",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"congrArg",
"SummationFilter",
"Co... | [
"f g : ℕ → ℂ\ns : ℂ\nhf : LSeriesSummable f s\nhg : LSeriesSummable g s\n⊢ ∑' (n : ℕ), (term f s n + term g s n) = ∑' (n : ℕ), term f s n + ∑' (n : ℕ), term g s n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 49
} | {
"line": 96,
"column": 50
} | [
{
"pp": "f g : ℕ → ℂ\ns : ℂ\nhf : LSeriesSummable f s\nhg : LSeriesSummable g s\n⊢ LSeriesSummable (f - g) s",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"HSub.hSub",
"id",
"instHSub",
"Nat",
"Pi.instSub",
"Complex.instSub",
"Complex",
"LSe... | [
"f g : ℕ → ℂ\ns : ℂ\nhf : LSeriesSummable f s\nhg : LSeriesSummable g s\n⊢ Summable (term (f - g) s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 33
} | {
"line": 101,
"column": 34
} | [
{
"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",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"congrArg",
"SummationFilter",
"Co... | [
"f g : ℕ → ℂ\ns : ℂ\nhf : LSeriesSummable f s\nhg : LSeriesSummable g s\n⊢ ∑' (n : ℕ), (term f s n - term g s n) = ∑' (n : ℕ), term f s n - ∑' (n : ℕ), term g s n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 18
} | {
"line": 125,
"column": 19
} | [
{
"pp": "f : ℕ → ℂ\nc s : ℂ\nhc : c ≠ 0\nhf : LSeriesSummable (c • f) s\n⊢ LSeriesSummable f s",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f : ℕ → ℂ\nc s : ℂ\nhc : c ≠ 0\nhf : LSeriesSummable (c • f) s\n⊢ LSeriesSummable f s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 77
} | {
"line": 156,
"column": 78
} | [
{
"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)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"LSeries.term_sum",
"NormedCommRing.toSeminormedCommR... | [
"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⊢ HasSum (fun a ↦ ∑ c ∈ S, term (f c) s a) (∑ i ∈ S, a i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 49
} | {
"line": 160,
"column": 50
} | [
{
"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",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Pi.addCommMonoid",
"id",
"Nat",
"Complex",
"Complex.instAddCommMonoid",
... | [
"ι : Type u_1\nf : ι → ℕ → ℂ\nS : Finset ι\ns : ℂ\nhf : ∀ i ∈ S, LSeriesSummable (f i) s\n⊢ Summable (term (∑ i ∈ S, f i) s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 165,
"column": 2
} | {
"line": 165,
"column": 33
} | {
"line": 165,
"column": 34
} | [
{
"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",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Pi.addCommMonoid",
... | [
"ι : Type u_1\nf : ι → ℕ → ℂ\nS : Finset ι\ns : ℂ\nhf : ∀ i ∈ S, LSeriesSummable (f i) s\n⊢ ∑' (n : ℕ), ∑ i ∈ S, term (f i) s n = ∑ i ∈ S, ∑' (n : ℕ), term (f i) s n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Nonvanishing | {
"line": 86,
"column": 2
} | {
"line": 87,
"column": 7
} | {
"line": 87,
"column": 8
} | [
{
"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",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Nat.instMulZeroClass",
"Real.instLE",
"Real",
"toAr... | [
"N : ℕ\nχ : DirichletCharacter ℂ N\ns : ℂ\nhs : 1 < s.re\nn : ℕ\nhn : n ≠ 0\n⊢ ‖χ ↑n‖ ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.ZMod | {
"line": 439,
"column": 72
} | {
"line": 442,
"column": 70
} | {
"line": 444,
"column": 2
} | [
{
"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",
"ppTerm": "?m.103",
"assigned": true,
"usedConstants"... | [] | 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.Injectivity | {
"line": 73,
"column": 4
} | {
"line": 73,
"column": 32
} | {
"line": 73,
"column": 33
} | [
{
"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 = (... | [
"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 = (↑n + 1) ^ ↑x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Nonvanishing | {
"line": 185,
"column": 4
} | {
"line": 185,
"column": 50
} | {
"line": 185,
"column": 51
} | [
{
"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",
"ppTerm": "?inr.refine_2.hf",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"HMul.hMul",
"ri... | [
"case inr.refine_2.hf\nN : ℕ\ninst✝ : NeZero N\nB : BadChar N\nG : ℂ → ℂ := Function.update (fun s ↦ (s - 1) * riemannZeta s) 1 1\nH : ℂ → ℂ := Function.update (fun s ↦ (LFunction B.χ s - LFunction B.χ 1) / (s - 1)) 1 (deriv (LFunction B.χ) 1)\nthis : B.F = G * H\n⊢ Tendsto (fun s ↦ (s - 1) * riemannZeta s) (𝓝[≠] ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.ZMod | {
"line": 502,
"column": 4
} | {
"line": 502,
"column": 46
} | {
"line": 502,
"column": 47
} | [
{
"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... | [
"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 : s ∈ U\nh2... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Nonvanishing | {
"line": 271,
"column": 4
} | {
"line": 272,
"column": 11
} | {
"line": 272,
"column": 12
} | [
{
"pp": "N : ℕ\nχ : DirichletCharacter ℂ N\ns : ℂ\nhs : 1 < s.re\np : Nat.Primes\n⊢ ‖χ ↑↑p * ↑↑p ^ (-s)‖ ≤ ↑↑p ^ (-s).re",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instPow",
"Real.instLE",
"Real",
"Nat.Prime",
"HMu... | [
"N : ℕ\nχ : DirichletCharacter ℂ N\ns : ℂ\nhs : 1 < s.re\np : Nat.Primes\n⊢ ‖χ ↑↑p‖ * ↑↑p ^ (-s).re ≤ ↑↑p ^ (-s).re"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Nonvanishing | {
"line": 301,
"column": 2
} | {
"line": 302,
"column": 44
} | {
"line": 303,
"column": 6
} | [
{
"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... | [
"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 ↦ -log (1 - (χ ^ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Nonvanishing | {
"line": 323,
"column": 8
} | {
"line": 323,
"column": 99
} | {
"line": 324,
"column": 10
} | [
{
"pp": "case convert_2\nN : ℕ\ninst✝ : NeZero N\n⊢ Tendsto (fun w ↦ 1 + w) (𝓝[≠] 0) (𝓝[≠] 1)",
"ppTerm": "?convert_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Compl.compl",
"nhdsWithin",
"PartialOrder.toPreorder",
... | [
"case convert_2\nN : ℕ\ninst✝ : NeZero N\n⊢ 𝓝[≠] 0 ≤ comap (fun w ↦ 1 + w) (𝓝[≠] 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Nonvanishing | {
"line": 331,
"column": 2
} | {
"line": 332,
"column": 9
} | {
"line": 332,
"column": 10
} | [
{
"pp": "N : ℕ\nχ : DirichletCharacter ℂ N\ny : ℝ\nhy : y ≠ 0 ∨ χ ≠ 1\n⊢ 1 + I * ↑y ≠ 1 ∨ χ ≠ 1",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"False",
"Real",
"HMul.hMul",
"ZMod.commRing",
"MulZeroClass.to... | [
"N : ℕ\nχ : DirichletCharacter ℂ N\ny : ℝ\nhy : y ≠ 0 ∨ χ ≠ 1\n⊢ ¬y = 0 ∨ ¬χ = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Injectivity | {
"line": 104,
"column": 28
} | {
"line": 104,
"column": 49
} | {
"line": 105,
"column": 6
} | [
{
"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 = (... | [
"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 = (↑n + 1) ^ ↑x... | 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": 232,
"column": 2
} | {
"line": 233,
"column": 9
} | {
"line": 233,
"column": 10
} | [
{
"pp": "q : 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",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"ArithmeticFunction.vonMangoldt",
"DirichletCharacter.fintype",
"Eq.mpr",
... | [
"q : ℕ\na : ZMod q\ninst✝ : NeZero q\nha : IsUnit a\nn : ℕ\n⊢ ↑(residueClass a n) = (↑q.totient)⁻¹ * ∑ x, x a⁻¹ * x ↑n * ↑(Λ n)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Nonvanishing | {
"line": 348,
"column": 2
} | {
"line": 349,
"column": 9
} | {
"line": 349,
"column": 10
} | [
{
"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",
"ppTerm": "?m.86",
... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Nonvanishing | {
"line": 365,
"column": 8
} | {
"line": 365,
"column": 64
} | {
"line": 365,
"column": 65
} | [
{
"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",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
... | [
"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⊢ 1 * ↑x * 1 = ↑x"
] | inv_mul_cancel₀ <| pow_ne_zero 3 (ofReal_ne_zero.mpr h), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.PrimesInAP | {
"line": 277,
"column": 29
} | {
"line": 277,
"column": 93
} | {
"line": 277,
"column": 94
} | [
{
"pp": "q : ℕ\na : ZMod q\ninst✝ : NeZero q\nχ : DirichletCharacter ℂ q\nhχ : χ ∈ {1}ᶜ\n⊢ χ ≠ 1",
"ppTerm": "?m.168",
"assigned": true,
"usedConstants": [
"ZMod.commRing",
"MulChar.hasOne",
"id",
"Ne",
"Field.toSemifield",
"ZMod",
"Semifield.toCommGroupWith... | [
"q : ℕ\na : ZMod q\ninst✝ : NeZero q\nχ : DirichletCharacter ℂ q\nhχ : χ ∈ {1}ᶜ\n⊢ ¬χ = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.PrimesInAP | {
"line": 367,
"column": 8
} | {
"line": 367,
"column": 55
} | {
"line": 367,
"column": 56
} | [
{
"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}",
"ppTerm": "?m.180",
"assigned": true,
"usedConstants... | [
"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⊢ 1 ≤ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Injectivity | {
"line": 226,
"column": 2
} | {
"line": 227,
"column": 9
} | {
"line": 227,
"column": 10
} | [
{
"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",
"ppTerm": "?m.56",
"assigned": false,
"us... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.ZetaZeros | {
"line": 43,
"column": 2
} | {
"line": 45,
"column": 64
} | {
"line": 47,
"column": 0
} | [
{
"pp": "⊢ riemannZetaZerosᶜ ∈ Filter.codiscreteWithin {1}ᶜ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | 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": 43,
"column": 2
} | {
"line": 45,
"column": 64
} | {
"line": 47,
"column": 0
} | [
{
"pp": "⊢ riemannZetaZerosᶜ ∈ Filter.codiscreteWithin {1}ᶜ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | 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": 60,
"column": 2
} | {
"line": 60,
"column": 13
} | {
"line": 60,
"column": 14
} | [
{
"pp": "⊢ IsClosed riemannZetaZeros",
"ppTerm": "?m.3",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ IsClosed riemannZetaZeros"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.ZetaZeros | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 13
} | {
"line": 63,
"column": 14
} | [
{
"pp": "⊢ IsDiscrete riemannZetaZeros",
"ppTerm": "?m.3",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ IsDiscrete riemannZetaZeros"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LegendreSymbol.Complex | {
"line": 26,
"column": 2
} | {
"line": 26,
"column": 37
} | {
"line": 26,
"column": 38
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : Finite F\n⊢ ringChar ℂ ≠ ringChar F",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Ne",
"instOfNatNat",
"Field.toSemifield",
"Semifield.toDivisionSemiring",
"ringCh... | [
"F : Type u_1\ninst✝¹ : Field F\ninst✝ : Finite F\n⊢ 0 ≠ ringChar F"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.PrimesInAP | {
"line": 406,
"column": 8
} | {
"line": 407,
"column": 43
} | {
"line": 407,
"column": 44
} | [
{
"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' : ... | [
"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' : ℝ\nhC' : ∀ {... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.PrimesInAP | {
"line": 458,
"column": 2
} | {
"line": 458,
"column": 45
} | {
"line": 458,
"column": 46
} | [
{
"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]",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Int.cast_natCast",
"ZMod.commRing... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.PrimesInAP | {
"line": 465,
"column": 2
} | {
"line": 465,
"column": 13
} | {
"line": 465,
"column": 14
} | [
{
"pp": "n q a : ℕ\nhq : q ≠ 0\nh : a.Coprime q\n⊢ ∃ p > n, Prime p ∧ p ≡ a [MOD q]",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.Prime",
"congrArg",
"Exists",
"id",
"funext",
"GT.gt",
"And",
"Nat.ModEq",
"Nat",
... | [
"n q a : ℕ\nhq : q ≠ 0\nh : a.Coprime q\n⊢ ∃ p, n < p ∧ Prime p ∧ p ≡ a [MOD q]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LegendreSymbol.GaussEisensteinLemmas | {
"line": 91,
"column": 6
} | {
"line": 91,
"column": 17
} | {
"line": 91,
"column": 18
} | [
{
"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)!",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast_neg",
"Int.cast",... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\na : ℤ\nhap : ↑a ≠ 0\n⊢ ↑a ^ (p / 2) = (-1) ^ #({x ∈ Ico 1 (p / 2 + 1) | p / 2 < (↑a * ↑x).val}) ∨ ↑(p / 2)! = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LegendreSymbol.GaussEisensteinLemmas | {
"line": 139,
"column": 4
} | {
"line": 140,
"column": 50
} | {
"line": 141,
"column": 6
} | [
{
"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",
"ppTerm": "?m.104",
"assigned": true,
"usedConstants": [
"AddGrou... | [
"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 + 1) | p / 2 < (↑a * ↑x).val})) + ∑ x ∈ Ico 1 (p / 2 + 1), ↑(a * x / p) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 204,
"column": 4
} | {
"line": 205,
"column": 28
} | {
"line": 205,
"column": 29
} | [
{
"pp": "s 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)‖",
"ppTerm": "?m.222",
... | [
"s 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) / t) ≤ ε * t ^ (-s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LegendreSymbol.GaussEisensteinLemmas | {
"line": 188,
"column": 6
} | {
"line": 188,
"column": 23
} | {
"line": 188,
"column": 24
} | [
{
"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... | [
"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 * q ≤ x.2 *... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 271,
"column": 4
} | {
"line": 271,
"column": 15
} | {
"line": 271,
"column": 16
} | [
{
"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... | [
"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 Set.Ioc 1 T, ‖... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 272,
"column": 2
} | {
"line": 273,
"column": 40
} | {
"line": 274,
"column": 2
} | [
{
"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... | [
"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 Set.Ioc 1 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.Topology.Algebra.Valued.ValuativeRel | {
"line": 41,
"column": 4
} | {
"line": 41,
"column": 90
} | {
"line": 42,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝³ : Ring 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",
"ppTerm": "?m.35",
"assigned": true,
... | [
"R : Type u_1\ninst✝³ : Ring 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⊢ (fun x_1 ↦ x + x_1) ⁻¹' s ∈ 𝓝 0 ↔ ∃ γ, {a | v (-x + a) < ↑γ} ⊆ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.ValuativeRel | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 58
} | {
"line": 62,
"column": 6
} | [
{
"pp": "R✝ : Type u_1\ninst✝⁸ : Ring R✝\ninst✝⁷ : ValuativeRel R✝\ninst✝⁶ : TopologicalSpace R✝\ninst✝⁵ : IsValuativeTopology R✝\nR : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : ValuativeRel R\ninst✝² : UniformSpace R\ninst✝¹ : IsUniformAddGroup R\ninst✝ : IsValuativeTopology R\na✝ : Set R\n⊢ failed to pretty print ex... | [
"R✝ : Type u_1\ninst✝⁸ : Ring R✝\ninst✝⁷ : ValuativeRel R✝\ninst✝⁶ : TopologicalSpace R✝\ninst✝⁵ : IsValuativeTopology R✝\nR : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : ValuativeRel R\ninst✝² : UniformSpace R\ninst✝¹ : IsUniformAddGroup R\ninst✝ : IsValuativeTopology R\na✝ : Set R\n⊢ failed to pretty print expression (us... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 453,
"column": 4
} | {
"line": 453,
"column": 21
} | {
"line": 453,
"column": 22
} | [
{
"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)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
... | [
"case inl\na b : ℕ\nha2 : a % 2 = 1\nhb2 : b % 2 = 1\nha1 : a % 4 = 1\n⊢ J(↑b | a) = J(↑a | b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 455,
"column": 4
} | {
"line": 455,
"column": 21
} | {
"line": 455,
"column": 22
} | [
{
"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)",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"congrArg",
"Nat.instAtL... | [
"case inr.inl\na b : ℕ\nha2 : a % 2 = 1\nhb2 : b % 2 = 1\nha3 : a % 4 = 3\nhb1 : b % 4 = 1\n⊢ J(↑b | a) = J(↑a | b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 456,
"column": 2
} | {
"line": 456,
"column": 24
} | {
"line": 456,
"column": 25
} | [
{
"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)",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"and_self",
"id",
... | [
"case inr.inr\na b : ℕ\nha2 : a % 2 = 1\nhb2 : b % 2 = 1\nha3 : a % 4 = 3\nhb3 : b % 4 = 3\n⊢ -J(↑b | a) = J(↑a | b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.LocallyCompact | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 36
} | {
"line": 89,
"column": 37
} | [
{
"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",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"Real",... | [
"K : Type u_1\ninst✝¹ : NontriviallyNormedField K\ninst✝ : IsUltrametricDist K\nx : K\nhx : 0 < ‖x‖\nhx' : ‖x‖ < 1\n⊢ ¬x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.ValuativeRel | {
"line": 73,
"column": 4
} | {
"line": 74,
"column": 11
} | {
"line": 74,
"column": 12
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝³ : Ring 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))",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"WithZeroTopology.topologicalSpace",
"Uni... | [
"case pos\nR : Type u_1\ninst✝³ : Ring R\ninst✝² : ValuativeRel R\ninst✝¹ : TopologicalSpace R\ninst✝ : IsValuativeTopology R\nx : R\nhx : v x = 0\n⊢ ∀ (ib : ValueGroupWithZero R), ¬ib = 0 → ∃ ia, ∀ (x_1 : R), v.restrict (x_1 - x) < ↑ia → v x_1 < ib"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.ValuativeRel | {
"line": 76,
"column": 4
} | {
"line": 77,
"column": 11
} | {
"line": 77,
"column": 12
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝³ : Ring 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))",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"WithZeroTopology.topologicalSpace",
"Un... | [
"case neg\nR : Type u_1\ninst✝³ : Ring R\ninst✝² : ValuativeRel R\ninst✝¹ : TopologicalSpace R\ninst✝ : IsValuativeTopology R\nx : R\nhx : ¬v x = 0\n⊢ ∃ ia, ∀ (x_1 : R), v.restrict (x_1 - x) < ↑ia → v x_1 = v x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.ValuativeRel | {
"line": 78,
"column": 19
} | {
"line": 78,
"column": 61
} | {
"line": 78,
"column": 62
} | [
{
"pp": "R : Type u_1\ninst✝³ : Ring 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",
"ppTerm": "?m.95",
"assigned":... | [
"R : Type u_1\ninst✝³ : Ring R\ninst✝² : ValuativeRel R\ninst✝¹ : TopologicalSpace R\ninst✝ : IsValuativeTopology R\nx : R\nhx : ¬v x = 0\nx✝ : R\n⊢ v (x✝ - x) < v x → v x✝ = v x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.LocallyCompact | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 70
} | {
"line": 139,
"column": 4
} | [
{
"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... | [
"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, t.Finite ∧ ⋃ y ∈ t, Metric.ball y ‖p‖ = Set.univ\n⊢ Finite �... | simp only [Set.subset_univ, Set.univ_subset_iff, true_and] at this | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 296,
"column": 71
} | {
"line": 296,
"column": 82
} | {
"line": 296,
"column": 83
} | [
{
"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... | [
"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 Set.Ioc 1 T, ‖... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.LocallyCompact | {
"line": 231,
"column": 26
} | {
"line": 231,
"column": 44
} | {
"line": 231,
"column": 45
} | [
{
"pp": "K : 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 (MonoidWithZeroHom.ofClass v)))ˣ\na : K\nha : (MonoidWithZeroHom.ofClass v) a = ↑↑z\nhz1 : z ≤ 1\nz0' : 0 < ↑z\nz0 : 0 < ↑↑z\n⊢ 0 < v a",
"p... | [
"K : 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 (MonoidWithZeroHom.ofClass v)))ˣ\na : K\nha : (MonoidWithZeroHom.ofClass v) a = ↑↑z\nhz1 : z ≤ 1\nz0' : 0 < ↑z\nz0 : 0 < ↑↑z\n⊢ 0 < v a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LocalField.Basic | {
"line": 83,
"column": 24
} | {
"line": 83,
"column": 35
} | {
"line": 83,
"column": 36
} | [
{
"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 ... | [
"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 0\nhs' : IsC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LocalField.Basic | {
"line": 86,
"column": 6
} | {
"line": 89,
"column": 41
} | {
"line": 90,
"column": 2
} | [
{
"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
} | {
"line": 90,
"column": 2
} | [
{
"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.NumberTheory.LocalField.Basic | {
"line": 181,
"column": 6
} | {
"line": 181,
"column": 21
} | {
"line": 181,
"column": 22
} | [
{
"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",
"ppTerm... | [
"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)\nn : ℕ\n⊢ f (n + 1) ≡ f n [SMOD 𝓂[K] ^ n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LocalField.Basic | {
"line": 187,
"column": 4
} | {
"line": 187,
"column": 32
} | {
"line": 187,
"column": 33
} | [
{
"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) ⊆ ... | [
"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) ⊆ S n\nh : ∀ (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.SimpleRing.Field | {
"line": 33,
"column": 30
} | {
"line": 33,
"column": 59
} | {
"line": 33,
"column": 60
} | [
{
"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",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"id",
"Ne",
"Zero.toOfNat0",
"OfNat.ofNat",
"Ring.t... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.SimpleRing.Field | {
"line": 40,
"column": 57
} | {
"line": 40,
"column": 81
} | {
"line": 40,
"column": 82
} | [
{
"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",
"ppTerm": "?m.239",
"assigned": true,
"usedConstants": [
"Distrib.leftD... | [
"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⊢ ∃ y, x * y = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LucasLehmer | {
"line": 208,
"column": 10
} | {
"line": 208,
"column": 57
} | {
"line": 208,
"column": 58
} | [
{
"pp": "p : ℕ\nw : 1 < p\nh : ↑(sMod p (p - 2)) = 0\n⊢ ?m.28 ∣ sMod p (p - 2)",
"ppTerm": "?m.31",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℕ\nw : 1 < p\nh : ↑(sMod p (p - 2)) = 0\n⊢ ?m.28 ∣ sMod p (p - 2)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.MahlerMeasure | {
"line": 61,
"column": 6
} | {
"line": 61,
"column": 17
} | {
"line": 61,
"column": 18
} | [
{
"pp": "case refine_1\nn : ℕ\nB₁ B₂ : Fin (n + 1) → ℝ\np : ↑(boxPoly n B₁ B₂)\nprop : ∀ (i : Fin (n + 1)), B₁ i ≤ ↑((↑p).coeff ↑i) ∧ ↑((↑p).coeff ↑i) ≤ B₂ i\n⊢ (toFn (n + 1)) ↑p ∈ ↑(Finset.Icc (fun x ↦ ⌈B₁ x⌉) fun x ↦ ⌊B₂ x⌋)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.m... | [
"case refine_1\nn : ℕ\nB₁ B₂ : Fin (n + 1) → ℝ\np : ↑(boxPoly n B₁ B₂)\nprop : ∀ (i : Fin (n + 1)), B₁ i ≤ ↑((↑p).coeff ↑i) ∧ ↑((↑p).coeff ↑i) ≤ B₂ i\n⊢ (fun x ↦ ⌈B₁ x⌉) ≤ (toFn (n + 1)) ↑p ∧ (toFn (n + 1)) ↑p ≤ fun x ↦ ⌊B₂ x⌋"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LucasLehmer | {
"line": 411,
"column": 4
} | {
"line": 411,
"column": 44
} | {
"line": 411,
"column": 45
} | [
{
"pp": "k : ℕ\ninst✝ : Fact (Nat.Prime (2 * k + 1))\nleg3 : legendreSym (2 * k + 1) 3 = -1\nq : ℕ := 2 * k + 1\n⊢ 3 ^ k = -1",
"ppTerm": "?m.78",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Nat.instAtLeastTwoHAddOfNat",
"AddGroupWithOne.toAddMonoidW... | [
"k : ℕ\ninst✝ : Fact (Nat.Prime (2 * k + 1))\nleg3 : legendreSym (2 * k + 1) 3 = -1\nq : ℕ := 2 * k + 1\n⊢ -1 = 3 ^ k"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LucasLehmer | {
"line": 418,
"column": 36
} | {
"line": 420,
"column": 10
} | {
"line": 422,
"column": 0
} | [
{
"pp": "q : ℕ\ninst✝ : Fact (Nat.Prime q)\nodd : Odd q\nleg3 : legendreSym q 3 = -1\n⊢ (1 + α) ^ (q + 1) = -2",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonUnitalComm... | [] | by
rw [pow_succ, one_add_α_pow_q odd leg3, mul_comm, ← _root_.sq_sub_sq, α_sq]
norm_num | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.MahlerMeasure | {
"line": 214,
"column": 2
} | {
"line": 214,
"column": 70
} | {
"line": 215,
"column": 2
} | [
{
"pp": "p : ℤ[X]\nh : (map (castRingHom ℂ) p).mahlerMeasure = 1\nz : ℂ\nhz₀ : z ≠ 0\nhz : z ∈ p.aroots ℂ\n⊢ ∃ n, 0 < n ∧ IsPrimitiveRoot z n",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Complex.commRing",
"Exists",
"instOfNatNat",
"Polynomial.pow_eq_one_of_mahl... | [
"p : ℤ[X]\nh : (map (castRingHom ℂ) p).mahlerMeasure = 1\nz : ℂ\nhz₀ : z ≠ 0\nhz : z ∈ p.aroots ℂ\nw✝ : ℕ\nleft✝ : 0 < w✝\nhz_pow : z ^ w✝ = 1\n⊢ ∃ n, 0 < n ∧ IsPrimitiveRoot z n"
] | obtain ⟨_, _, hz_pow⟩ := pow_eq_one_of_mahlerMeasure_eq_one h hz₀ hz | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.Algebra.IsUniformGroup.DiscreteSubgroup | {
"line": 37,
"column": 4
} | {
"line": 37,
"column": 29
} | {
"line": 37,
"column": 30
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : TopologicalSpace G\nH K : Subgroup G\nhHK : H ≤ K\n⊢ ∀ (s : Set ↥H),\n (∃ a,\n IsOpen[inst✝] a ∧\n Subtype.val ⁻¹' Subtype.val ⁻¹' a =\n ⇑{ toFun := fun g ↦ ⟨↑↑g, ⋯⟩, invFun := fun g ↦ ⟨⟨↑g, ⋯⟩, ⋯⟩, left_inv := ⋯, right_inv := ⋯,\n ... | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : TopologicalSpace G\nH K : Subgroup G\nhHK : H ≤ K\n⊢ ∀ (s : Set ↥H),\n (∃ a, IsOpen[inst✝] a ∧ ∀ (a_1 : G) (b : a_1 ∈ K) (b_1 : ⟨a_1, b⟩ ∈ H.subgroupOf K), a_1 ∈ a ↔ ⟨a_1, ⋯⟩ ∈ s) ↔\n ∃ t, IsOpen[inst✝] t ∧ ∀ (a : G) (b : a ∈ H), a ∈ t ↔ ⟨a, b⟩ ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LucasLehmer | {
"line": 502,
"column": 6
} | {
"line": 502,
"column": 19
} | {
"line": 502,
"column": 19
} | [
{
"pp": "p' : ℕ\nh : sZMod (p' + 2) (p' + 2 - 2) = 0\n⊢ ∃ k, ω ^ 2 ^ (p' + 1) = ↑k * ↑(mersenne (p' + 2)) * ω ^ 2 ^ p' - 1",
"ppTerm": "?m.120",
"assigned": true,
"usedConstants": [
"Int.cast",
"ZMod.commRing",
"congrArg",
"CommSemiring.toSemiring",
"Nat.instMonoid",
... | [
"p' : ℕ\nh : ↑(s (p' + 2 - 2)) = 0\n⊢ ∃ k, ω ^ 2 ^ (p' + 1) = ↑k * ↑(mersenne (p' + 2)) * ω ^ 2 ^ p' - 1"
] | sZMod_eq_s p' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LucasLehmer | {
"line": 503,
"column": 44
} | {
"line": 503,
"column": 91
} | {
"line": 503,
"column": 92
} | [
{
"pp": "p' : ℕ\nh : ↑(s (p' + 2 - 2)) = 0\n⊢ 2 ^ (p' + 2) - 1 ∣ s p'",
"ppTerm": "?m.151",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p' : ℕ\nh : ↑(s (p' + 2 - 2)) = 0\n⊢ 2 ^ (p' + 2) - 1 ∣ s p'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LucasLehmer | {
"line": 526,
"column": 2
} | {
"line": 526,
"column": 13
} | {
"line": 526,
"column": 14
} | [
{
"pp": "p' : ℕ\nh : lucasLehmerResidue (p' + 2) = 0\nk : ℤ\nw : ω ^ 2 ^ (p' + 1) = ↑k * 0 * ω ^ 2 ^ p' - 1\n⊢ ω ^ 2 ^ (p' + 1) = -1",
"ppTerm": "?m.61",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p' : ℕ\nh : lucasLehmerResidue (p' + 2) = 0\nk : ℤ\nw : ω ^ 2 ^ (p' + 1) = ↑k * 0 * ω ^ 2 ^ p' - 1\n⊢ ω ^ 2 ^ (p' + 1) = -1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LucasLehmer | {
"line": 668,
"column": 2
} | {
"line": 668,
"column": 39
} | {
"line": 669,
"column": 0
} | [
{
"pp": "q i : ℕ\n⊢ sModNatTR q i = sModNat q i",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Nat.instMod",
"instHMod",
"instOfNatNat",
"LucasLehmer.norm_num_ext.sModNatAux",
"_private.Mathlib.NumberTheory.LucasL... | [] | rw [sModNatTR, helper, sModNatAux_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.LucasLehmer | {
"line": 668,
"column": 2
} | {
"line": 668,
"column": 39
} | {
"line": 669,
"column": 0
} | [
{
"pp": "q i : ℕ\n⊢ sModNatTR q i = sModNat q i",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Nat.instMod",
"instHMod",
"instOfNatNat",
"LucasLehmer.norm_num_ext.sModNatAux",
"_private.Mathlib.NumberTheory.LucasL... | [] | rw [sModNatTR, helper, sModNatAux_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LucasLehmer | {
"line": 668,
"column": 2
} | {
"line": 668,
"column": 39
} | {
"line": 669,
"column": 0
} | [
{
"pp": "q i : ℕ\n⊢ sModNatTR q i = sModNat q i",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Nat.instMod",
"instHMod",
"instOfNatNat",
"LucasLehmer.norm_num_ext.sModNatAux",
"_private.Mathlib.NumberTheory.LucasL... | [] | rw [sModNatTR, helper, sModNatAux_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LucasLehmer | {
"line": 694,
"column": 2
} | {
"line": 694,
"column": 13
} | {
"line": 694,
"column": 14
} | [
{
"pp": "p : ℕ\nhp : 1 < p\nh : sModNatTR (2 ^ p - 1) (p - 2) ≠ 0\n⊢ ¬↑(sModNatTR (2 ^ p - 1) (p - 2)) = 0",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Nat.instMonoid",
"HSub.hSub",
"id",
"instSubNat",
"instOfNatNat",
... | [
"p : ℕ\nhp : 1 < p\nh : sModNatTR (2 ^ p - 1) (p - 2) ≠ 0\n⊢ ¬sModNatTR (2 ^ p - 1) (p - 2) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.ArithmeticSubgroups | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 69
} | {
"line": 48,
"column": 70
} | [
{
"pp": "n : Type u_1\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\nR : Type u_2\ninst✝² : CommRing R\nΓ : Subgroup (GL n R)\ninst✝¹ : LinearOrder R\ninst✝ : IsOrderedRing R\nh : ∀ {g : GL n R}, g ∈ Γ → |↑(GeneralLinearGroup.det g)| = 1\n⊢ ∀ {g : GL n R}, g ∈ Γ → GeneralLinearGroup.det g = 1 ∨ GeneralLinearGroup... | [
"n : Type u_1\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\nR : Type u_2\ninst✝² : CommRing R\nΓ : Subgroup (GL n R)\ninst✝¹ : LinearOrder R\ninst✝ : IsOrderedRing R\nh : ∀ {g : GL n R}, g ∈ Γ → |↑(GeneralLinearGroup.det g)| = 1\n⊢ ∀ {g : GL n R}, g ∈ Γ → GeneralLinearGroup.det g = 1 ∨ GeneralLinearGroup.det g = -1"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.ArithmeticSubgroups | {
"line": 79,
"column": 4
} | {
"line": 79,
"column": 34
} | {
"line": 79,
"column": 35
} | [
{
"pp": "n : Type u_1\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\nR : Type u_2\ninst✝¹ : CommRing R\nΓ✝ Γ : Subgroup (GL n R)\ninst✝ : Γ.HasDetOne\ng : ConjAct (GL n R)\nh : GL n R\nhh : g⁻¹ • h ∈ Γ\n⊢ GeneralLinearGroup.det h = 1",
"ppTerm": "?m.47",
"assigned": false,
"usedConstants": [],
"us... | [
"n : Type u_1\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\nR : Type u_2\ninst✝¹ : CommRing R\nΓ✝ Γ : Subgroup (GL n R)\ninst✝ : Γ.HasDetOne\ng : ConjAct (GL n R)\nh : GL n R\nhh : g⁻¹ • h ∈ Γ\n⊢ GeneralLinearGroup.det h = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Order.ArchimedeanDiscrete | {
"line": 72,
"column": 26
} | {
"line": 73,
"column": 90
} | {
"line": 74,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : CommGroup G\ninst✝⁴ : LinearOrder G\ninst✝³ : IsOrderedMonoid G\ninst✝² : TopologicalSpace G\ninst✝¹ : OrderTopology G\ninst✝ : MulArchimedean G\nH : Subgroup G\na✝ : Nontrivial G\nthis : Dense ↑H ∨ ∃ a, zpowers a = H\nhA : DiscreteTopology ↥H\nh : Dense ↑H\n⊢ H = ⊤",
"ppTerm... | [] | by
rw [← coe_eq_univ, ← (dense_iff_closure_eq.mp h), H.isClosed_of_discrete.closure_eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ModularForms.ArithmeticSubgroups | {
"line": 86,
"column": 4
} | {
"line": 86,
"column": 34
} | {
"line": 86,
"column": 35
} | [
{
"pp": "n : Type u_1\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\nR : Type u_2\ninst✝¹ : CommRing R\nΓ✝ Γ : Subgroup (GL n R)\ninst✝ : Γ.HasDetPlusMinusOne\ng : ConjAct (GL n R)\nh : GL n R\nhh : g⁻¹ • h ∈ Γ\n⊢ GeneralLinearGroup.det h = 1 ∨ GeneralLinearGroup.det h = -1",
"ppTerm": "?m.47",
"assigned"... | [
"n : Type u_1\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\nR : Type u_2\ninst✝¹ : CommRing R\nΓ✝ Γ : Subgroup (GL n R)\ninst✝ : Γ.HasDetPlusMinusOne\ng : ConjAct (GL n R)\nh : GL n R\nhh : g⁻¹ • h ∈ Γ\n⊢ GeneralLinearGroup.det h = 1 ∨ GeneralLinearGroup.det h = -1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.ArithmeticSubgroups | {
"line": 110,
"column": 4
} | {
"line": 111,
"column": 59
} | {
"line": 111,
"column": 60
} | [
{
"pp": "case mp\nΓ : Subgroup SL(2, ℤ)\nx✝ : (map (mapGL ℝ) Γ).IsArithmetic\nh : (map (mapGL ℝ) Γ).Commensurable (mapGL ℝ).range\n⊢ Γ.index ≠ 0",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Matrix.SpecialLinearGroup",
"instDecidableEqFin",
"Matrix.SpecialLinearGroup.ins... | [
"case mp\nΓ : Subgroup SL(2, ℤ)\nx✝ : (map (mapGL ℝ) Γ).IsArithmetic\nh : (map (mapGL ℝ) Γ).Commensurable (mapGL ℝ).range\n⊢ ¬Γ.index = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.ArithmeticSubgroups | {
"line": 110,
"column": 4
} | {
"line": 111,
"column": 59
} | {
"line": 111,
"column": 60
} | [
{
"pp": "case mpr\nΓ : Subgroup SL(2, ℤ)\nx✝ : Γ.FiniteIndex\nh : Γ.index ≠ 0\n⊢ (map (mapGL ℝ) Γ).Commensurable (mapGL ℝ).range",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonoidHom.range",
"False",
"Nat.instMulZeroClass",
"Real.partialOrder",
... | [
"case mpr\nΓ : Subgroup SL(2, ℤ)\nx✝ : Γ.FiniteIndex\nh : Γ.index ≠ 0\n⊢ ¬Γ.index = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.CongruenceSubgroups | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 32
} | {
"line": 87,
"column": 33
} | [
{
"pp": "N : ℕ\na : SL(2, ℤ)\nha : a ∈ {g | ↑(↑g 1 0) = 0}\n⊢ a⁻¹ ∈ {g | ↑(↑g 1 0) = 0}",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast_neg",
"Int.cast",
"Eq.mpr",
"NegZeroClass.toNeg",
"Matrix.SpecialLinearGroup... | [
"N : ℕ\na : SL(2, ℤ)\nha : a ∈ {g | ↑(↑g 1 0) = 0}\n⊢ ↑(↑a 1 0) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.CongruenceSubgroups | {
"line": 122,
"column": 4
} | {
"line": 124,
"column": 21
} | {
"line": 124,
"column": 22
} | [
{
"pp": "case mp\nN : ℕ\nA : ↥(Gamma0 N)\nha : ↑(↑↑A 1 1) = 1\nadet : ↑(↑↑A 0 0) * ↑(↑↑A 1 1) - ↑(↑↑A 0 1) * ↑(↑↑A 1 0) = 1\n⊢ ↑(↑↑A 0 0) = 1 ∧ ↑(↑↑A 1 1) = 1 ∧ ↑(↑↑A 1 0) = 0",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Matrix.SpecialLinearGroup",... | [
"case mp\nN : ℕ\nA : ↥(Gamma0 N)\nha : ↑(↑↑A 1 1) = 1\nadet : ↑(↑↑A 0 0) * ↑(↑↑A 1 1) - ↑(↑↑A 0 1) * ↑(↑↑A 1 0) = 1\n⊢ ↑(↑↑A 0 0) = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.ArithmeticSubgroups | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 25
} | {
"line": 129,
"column": 2
} | [
{
"pp": "n✝ : Type u_1\ninst✝¹ : Fintype n✝\ninst✝ : DecidableEq n✝\nΓ : Subgroup (GL (Fin 2) ℝ)\nh : Γ.IsArithmetic\ng : GL (Fin 2) ℝ\nhg : g ∈ Γ\nn : ℕ\nhn : 0 < n\nleft✝ : n ≤ (mapGL ℝ).range.relIndex Γ\nhgn : g ^ n ∈ (mapGL ℝ).range ⊓ Γ\n⊢ |↑(GeneralLinearGroup.det g ^ n)| = 1",
"ppTerm": "?m.77",
"... | [
"n✝ : Type u_1\ninst✝¹ : Fintype n✝\ninst✝ : DecidableEq n✝\nΓ : Subgroup (GL (Fin 2) ℝ)\nh : Γ.IsArithmetic\ng : GL (Fin 2) ℝ\nhg : g ∈ Γ\nn : ℕ\nhn : 0 < n\nleft✝ : n ≤ (mapGL ℝ).range.relIndex Γ\nhgn : g ^ n ∈ (mapGL ℝ).range ⊓ Γ\nt : SL(2, ℤ)\nht : (mapGL ℝ) t = g ^ n\n⊢ |↑(GeneralLinearGroup.det g ^ n)| = 1"
] | obtain ⟨t, ht⟩ := hgn.1 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.ModularForms.CongruenceSubgroups | {
"line": 245,
"column": 4
} | {
"line": 245,
"column": 55
} | {
"line": 246,
"column": 4
} | [
{
"pp": "g : GL (Fin 2) ℚ\nM : ℕ\ninst✝ : NeZero M\nA₁ : Matrix (Fin 2) (Fin 2) ℚ := ↑g\nA₂ : Matrix (Fin 2) (Fin 2) ℚ := ↑g⁻¹\nhA₁₂ : A₁ * A₂ = 1\na₁ : ℕ := A₁.den\na₂ : ℕ := A₂.den\nx✝ : SL(2, ℤ)\ny : Matrix (Fin 2) (Fin 2) ℤ\nhy : y.det = 1\nhy' : y.map Int.cast = 1\n⊢ ∃ k, y = 1 + (a₁ * a₂ * M) • k",
"p... | [
"case h\ng : GL (Fin 2) ℚ\nM : ℕ\ninst✝ : NeZero M\nA₁ : Matrix (Fin 2) (Fin 2) ℚ := ⋯\nA₂ : Matrix (Fin 2) (Fin 2) ℚ := ⋯\nhA₁₂ : A₁ * A₂ = 1\na₁ : ℕ := ⋯\na₂ : ℕ := ⋯\nx✝ : SL(2, ℤ)\ny : Matrix (Fin 2) (Fin 2) ℤ\nhy : y.det = 1\nhy' : y.map Int.cast = 1\n⊢ y = 1 + (a₁ * a₂ * M) • of fun i j ↦ (y - 1) i j / (↑a₁ *... | use Matrix.of fun i j ↦ (y - 1) i j / (a₁ * a₂ * M) | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.NumberTheory.ModularForms.ArithmeticSubgroups | {
"line": 202,
"column": 38
} | {
"line": 202,
"column": 49
} | {
"line": 202,
"column": 50
} | [
{
"pp": "n : Type u_1\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\nR : Type u_2\ninst✝ : Ring R\n𝒢 : Subgroup (GL n R)\nhG : -1 ∈ 𝒢\ng : GL n R\nhg : g ∈ 𝒢.adjoinNegOne\nh : -g ∈ 𝒢\n⊢ g ∈ 𝒢",
"ppTerm": "?m.43",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}... | [
"n : Type u_1\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\nR : Type u_2\ninst✝ : Ring R\n𝒢 : Subgroup (GL n R)\nhG : -1 ∈ 𝒢\ng : GL n R\nhg : g ∈ 𝒢.adjoinNegOne\nh : -g ∈ 𝒢\n⊢ g ∈ 𝒢"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.ArithmeticSubgroups | {
"line": 247,
"column": 4
} | {
"line": 247,
"column": 44
} | {
"line": 247,
"column": 45
} | [
{
"pp": "case inr\nn : Type u_1\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\nR : Type u_3\ninst✝ : CommRing R\n𝒢 : Subgroup (GL n R)\nhn : Even (Fintype.card n)\nx✝ : 𝒢.HasDetOne\ng : GL n R\nhg : -g ∈ 𝒢\n⊢ GeneralLinearGroup.det g = 1",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
... | [
"case inr\nn : Type u_1\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\nR : Type u_3\ninst✝ : CommRing R\n𝒢 : Subgroup (GL n R)\nhn : Even (Fintype.card n)\nx✝ : 𝒢.HasDetOne\ng : GL n R\nhg : -g ∈ 𝒢\n⊢ (↑g).det = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactification.OnePoint.ProjectiveLine | {
"line": 138,
"column": 6
} | {
"line": 139,
"column": 13
} | {
"line": 139,
"column": 14
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : DecidableEq K\ng : GL (Fin 2) K\nh : ![↑g 0 0, ↑g 1 0] = 0\n⊢ False",
"ppTerm": "?m.42",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : DecidableEq K\ng : GL (Fin 2) K\nh : ![↑g 0 0, ↑g 1 0] = 0\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactification.OnePoint.ProjectiveLine | {
"line": 207,
"column": 6
} | {
"line": 208,
"column": 13
} | {
"line": 208,
"column": 14
} | [
{
"pp": "case neg\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : DecidableEq K\ng : GL (Fin 2) K\ninst✝ : NeZero 2\nhg : ↑g ∉ Set.range ⇑(Matrix.scalar (Fin 2))\nhdisc : (↑g 0 0 + ↑g 1 1) ^ 2 - 4 * (↑g 0 0 * ↑g 1 1 - ↑g 0 1 * ↑g 1 0) = 0\nc : K\nhc : ¬↑g 1 0 = 0\nthis : discrim (↑g 1 0) (↑g 1 1 - ↑g 0 0) (-↑g 0 1) =... | [
"case neg\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : DecidableEq K\ng : GL (Fin 2) K\ninst✝ : NeZero 2\nhg : ↑g ∉ Set.range ⇑(Matrix.scalar (Fin 2))\nhdisc : (↑g 0 0 + ↑g 1 1) ^ 2 - 4 * (↑g 0 0 * ↑g 1 1 - ↑g 0 1 * ↑g 1 0) = 0\nc : K\nhc : ¬↑g 1 0 = 0\nthis : discrim (↑g 1 0) (↑g 1 1 - ↑g 0 0) (-↑g 0 1) = 0\n⊢ ↑g 1 0... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactification.OnePoint.ProjectiveLine | {
"line": 236,
"column": 8
} | {
"line": 237,
"column": 15
} | {
"line": 237,
"column": 16
} | [
{
"pp": "K : Type u_1\ninst✝³ : Field K\ninst✝² : DecidableEq K\ninst✝¹ : LinearOrder K\ninst✝ : IsStrictOrderedRing K\ng : GL (Fin 2) K\nhg : g.IsElliptic\nc : K\nh : g • ↑c = ↑c\n⊢ ↑g 1 0 * (c * c) + (↑g 1 1 + -↑g 0 0) * c + -↑g 0 1 = 0",
"ppTerm": "?m.223",
"assigned": false,
"usedConstants": [],... | [
"K : Type u_1\ninst✝³ : Field K\ninst✝² : DecidableEq K\ninst✝¹ : LinearOrder K\ninst✝ : IsStrictOrderedRing K\ng : GL (Fin 2) K\nhg : g.IsElliptic\nc : K\nh : g • ↑c = ↑c\n⊢ ↑g 1 0 * (c * c) + (↑g 1 1 + -↑g 0 0) * c + -↑g 0 1 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.SlashActions | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 39
} | {
"line": 78,
"column": 40
} | [
{
"pp": "β : Type u_1\nG : Type u_2\nα : Type u_3\ninst✝² : Group G\ninst✝¹ : AddGroup α\ninst✝ : SlashAction β G α\nk : β\ng : G\na : α\nh : (a ∣[k] g) ∣[k] g⁻¹ = 0 ∣[k] g⁻¹\n⊢ a = 0",
"ppTerm": "?m.96",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"β : Type u_1\nG : Type u_2\nα : Type u_3\ninst✝² : Group G\ninst✝¹ : AddGroup α\ninst✝ : SlashAction β G α\nk : β\ng : G\na : α\nh : (a ∣[k] g) ∣[k] g⁻¹ = 0 ∣[k] g⁻¹\n⊢ a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.Cusps | {
"line": 66,
"column": 4
} | {
"line": 66,
"column": 40
} | {
"line": 66,
"column": 41
} | [
{
"pp": "case refine_1\nc : OnePoint ℝ\n𝒢 : Subgroup (GL (Fin 2) ℝ)\ng p : GL (Fin 2) ℝ\nhp𝒢 : p ∈ 𝒢\nhpp : p.IsParabolic\nhpc : p • c = c\n⊢ (ConjAct.toConjAct g • p).IsParabolic",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Matrix.GeneralLinearGroup.isParabolic_conj_iff._... | [
"case refine_1\nc : OnePoint ℝ\n𝒢 : Subgroup (GL (Fin 2) ℝ)\ng p : GL (Fin 2) ℝ\nhp𝒢 : p ∈ 𝒢\nhpp : p.IsParabolic\nhpc : p • c = c\n⊢ p.IsParabolic"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.Cusps | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 48
} | {
"line": 69,
"column": 0
} | [
{
"pp": "case refine_2\nc : OnePoint ℝ\n𝒢 : Subgroup (GL (Fin 2) ℝ)\ng p : GL (Fin 2) ℝ\nhp𝒢 : p ∈ 𝒢\nhpp : p.IsParabolic\nhpc : p • c = c\n⊢ (ConjAct.toConjAct g • p) • g • c = g • c",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"OnePoint.instGLAction",
"Semigroup.toM... | [] | · simp [ConjAct.toConjAct_smul, mul_smul, hpc] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.ModularForms.Cusps | {
"line": 90,
"column": 4
} | {
"line": 90,
"column": 45
} | {
"line": 90,
"column": 46
} | [
{
"pp": "case h𝒢'\n𝒢 𝒢' : Subgroup (GL (Fin 2) ℝ)\nh𝒢 : 𝒢.Commensurable 𝒢'\nc : OnePoint ℝ\n⊢ (𝒢 ⊓ 𝒢').relIndex 𝒢' ≠ 0",
"ppTerm": "?h𝒢'",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Matrix",
"instDecidableEqFin",
"CompleteLattice... | [
"case h𝒢'\n𝒢 𝒢' : Subgroup (GL (Fin 2) ℝ)\nh𝒢 : 𝒢.Commensurable 𝒢'\nc : OnePoint ℝ\n⊢ ¬𝒢.relIndex 𝒢' = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.Cusps | {
"line": 91,
"column": 4
} | {
"line": 91,
"column": 44
} | {
"line": 91,
"column": 45
} | [
{
"pp": "case h𝒢'\n𝒢 𝒢' : Subgroup (GL (Fin 2) ℝ)\nh𝒢 : 𝒢.Commensurable 𝒢'\nc : OnePoint ℝ\n⊢ (𝒢 ⊓ 𝒢').relIndex 𝒢 ≠ 0",
"ppTerm": "?h𝒢'✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subgroup.inf_relIndex_left",
"Real",
"congrArg",
"Matrix",
"instDec... | [
"case h𝒢'\n𝒢 𝒢' : Subgroup (GL (Fin 2) ℝ)\nh𝒢 : 𝒢.Commensurable 𝒢'\nc : OnePoint ℝ\n⊢ ¬𝒢'.relIndex 𝒢 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.BoundedAtCusp | {
"line": 32,
"column": 2
} | {
"line": 33,
"column": 9
} | {
"line": 33,
"column": 10
} | [
{
"pp": "g : GL (Fin 2) ℝ\nf : ℍ → ℂ\nk : ℤ\nhg : ↑g 1 0 = 0\nhf : Tendsto (fun x ↦ ‖f x‖) atImInfty (nhds 0)\n⊢ Tendsto (fun x ↦ ‖(f ∣[k] g) x‖) atImInfty (nhds 0)",
"ppTerm": "?m.71",
"assigned": true,
"usedConstants": [
"UpperHalfPlane.glAction",
"Norm.norm",
"Units.val",
... | [
"g : GL (Fin 2) ℝ\nf : ℍ → ℂ\nk : ℤ\nhg : ↑g 1 0 = 0\nhf : Tendsto (fun x ↦ ‖f x‖) atImInfty (nhds 0)\n⊢ Tendsto (fun x ↦ ‖f (g • x)‖ * (|(↑g).det| ^ (k - 1) * (|↑g 1 1| ^ k)⁻¹)) atImInfty (nhds 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.Cusps | {
"line": 119,
"column": 4
} | {
"line": 119,
"column": 50
} | {
"line": 119,
"column": 51
} | [
{
"pp": "case mp\nc : OnePoint ℝ\ng : SL(2, ℤ)\nhgp : ((mapGL ℝ) g).IsParabolic\nhgc : (mapGL ℝ) g • c = c\n⊢ c ∈ Set.range (OnePoint.map Rat.cast)",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"OnePoint.instGLAction",
"Eq.mpr",
"Real.partialOrder",
"Real",
"i... | [
"case mp\nc : OnePoint ℝ\ng : SL(2, ℤ)\nhgp : ((mapGL ℝ) g).IsParabolic\nhgc : (mapGL ℝ) g • c = c\n⊢ ((mapGL ℝ) g).parabolicFixedPoint ∈ Set.range (OnePoint.map Rat.cast)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.BoundedAtCusp | {
"line": 75,
"column": 14
} | {
"line": 75,
"column": 25
} | {
"line": 75,
"column": 26
} | [
{
"pp": "f : ℍ → ℂ\nk : ℤ\nh : ∞.IsBoundedAt f k\n⊢ IsBoundedAtImInfty f",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f : ℍ → ℂ\nk : ℤ\nh : ∞.IsBoundedAt f k\n⊢ IsBoundedAtImInfty f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.BoundedAtCusp | {
"line": 78,
"column": 14
} | {
"line": 78,
"column": 25
} | {
"line": 78,
"column": 26
} | [
{
"pp": "f : ℍ → ℂ\nk : ℤ\nh : ∞.IsZeroAt f k\n⊢ IsZeroAtImInfty f",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f : ℍ → ℂ\nk : ℤ\nh : ∞.IsZeroAt f k\n⊢ IsZeroAtImInfty f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.Cusps | {
"line": 125,
"column": 47
} | {
"line": 125,
"column": 75
} | {
"line": 125,
"column": 76
} | [
{
"pp": "a✝ : SL(2, ℤ)\nx✝ : ↑((mapGL ℝ) ModularGroup.T) ∈ Set.range ⇑(Matrix.scalar (Fin 2))\na : ℝ\nha : (Matrix.scalar (Fin 2)) a = ↑((mapGL ℝ) ModularGroup.T)\n⊢ 0 = 1",
"ppTerm": "?m.184",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Real",
"NeZero.one",
... | [
"a✝ : SL(2, ℤ)\nx✝ : ↑((mapGL ℝ) ModularGroup.T) ∈ Set.range ⇑(Matrix.scalar (Fin 2))\na : ℝ\nha : (Matrix.scalar (Fin 2)) a = ↑((mapGL ℝ) ModularGroup.T)\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.SlashActions | {
"line": 258,
"column": 2
} | {
"line": 258,
"column": 36
} | {
"line": 258,
"column": 37
} | [
{
"pp": "ι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : Nonempty ι\nk : ℤ\ng : GL (Fin 2) ℝ\nf : ι → ℍ → ℂ\nthis : 0 < Fintype.card ι\n⊢ (∏ i, f i) ∣[k * ↑(Fintype.card ι)] g = |↑(Matrix.GeneralLinearGroup.det g)| ^ (Fintype.card ι - 1) • ∏ i, f i ∣[k] g",
"ppTerm": "?m.72",
"assigned": true,
"usedConsta... | [
"ι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : Nonempty ι\nk : ℤ\ng : GL (Fin 2) ℝ\nf : ι → ℍ → ℂ\nthis : 0 < Fintype.card ι\n⊢ (∏ i, f i) ∣[k * ↑(Fintype.card ι)] g = |(↑g).det| ^ (↑(Fintype.card ι) - 1) • ∏ i, f i ∣[k] g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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