module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.VectorMeasure.WithDensityVec | {
"line": 385,
"column": 4
} | {
"line": 385,
"column": 15
} | {
"line": 385,
"column": 16
} | [
{
"pp": "case e_f.inr\nX : Type u_1\nE : Type u_2\nmX : MeasurableSpace X\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure X\nf : X → E\nhf : Integrable f μ\nhE : Nontrivial E\nthis : IsFiniteMeasure (μ.withDensity fun x ↦ ‖f x‖ₑ)\nI : VectorMeasure.Integrable (μ.wi... | [
"case e_f.inr\nX : Type u_1\nE : Type u_2\nmX : MeasurableSpace X\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure X\nf : X → E\nhf : Integrable f μ\nhE : Nontrivial E\nthis : IsFiniteMeasure (μ.withDensity fun x ↦ ‖f x‖ₑ)\nI : VectorMeasure.Integrable (μ.withDensity fu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.WithDensityVec | {
"line": 395,
"column": 33
} | {
"line": 395,
"column": 44
} | {
"line": 395,
"column": 45
} | [
{
"pp": "X : Type u_1\nE : Type u_2\nmX : MeasurableSpace X\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure X\nf : X → E\nhf : Integrable f μ\nhE : Nontrivial E\nthis✝ : IsFiniteMeasure (μ.withDensity fun x ↦ ‖f x‖ₑ)\nI : VectorMeasure.Integrable (μ.withDensity fun... | [
"X : Type u_1\nE : Type u_2\nmX : MeasurableSpace X\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure X\nf : X → E\nhf : Integrable f μ\nhE : Nontrivial E\nthis✝ : IsFiniteMeasure (μ.withDensity fun x ↦ ‖f x‖ₑ)\nI : VectorMeasure.Integrable (μ.withDensity fun x ↦ ‖f x‖ₑ)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.AlmostPrime | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 13
} | {
"line": 66,
"column": 14
} | [
{
"pp": "p : ℕ\nhp : Prime p\n⊢ IsAlmostPrime 1 p",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.Prime",
"Nat.isAlmostPrime_one_iff._simp_1",
"id",
"instOfNatNat",
"Nat.IsAlmostPrime",
"Nat",
"OfNat.ofNat",
"Eq"
],
... | [
"p : ℕ\nhp : Prime p\n⊢ Prime p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.AlmostPrime | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 13
} | {
"line": 85,
"column": 14
} | [
{
"pp": "p q : ℕ\nhp : Prime p\nhq : Prime q\n⊢ IsAlmostPrime 2 (p * q)",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p q : ℕ\nhp : Prime p\nhq : Prime q\n⊢ IsAlmostPrime 2 (p * q)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.AlmostPrime | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 23
} | {
"line": 88,
"column": 24
} | [
{
"pp": "p : ℕ\nhp : Prime p\n⊢ IsAlmostPrime 2 (p ^ 2)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"pow_two",
"Monoid.toMulOneClass",
"congrArg",
"Nat.instMonoid",
"id",
"MulOne.toMul",
"instOfNatNat",
... | [
"p : ℕ\nhp : Prime p\n⊢ IsAlmostPrime 2 (p * p)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ADEInequality | {
"line": 209,
"column": 59
} | {
"line": 209,
"column": 70
} | {
"line": 209,
"column": 71
} | [
{
"pp": "p q r : ℕ+\nhs : [p, q, r].SortedLE\nx✝ : [p, q, r].length = 3\nH : 1 < sumInv ↑[p, q, r]\n⊢ (p ≤ q ∧ p ≤ r) ∧ q ≤ r",
"ppTerm": "?m.225",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p q r : ℕ+\nhs : [p, q, r].SortedLE\nx✝ : [p, q, r].length = 3\nH : 1 < sumInv ↑[p, q, r]\n⊢ (p ≤ q ∧ p ≤ r) ∧ q ≤ r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ArithmeticFunction.VonMangoldt | {
"line": 120,
"column": 9
} | {
"line": 120,
"column": 49
} | {
"line": 120,
"column": 49
} | [
{
"pp": "n : ℕ\n⊢ (Λ * ↑ζ) n = log n",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"ArithmeticFunction.vonMangoldt",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"HMul.hMul",
"ArithmeticFunction.instFunLikeNat",
"ArithmeticFunctio... | [
"n : ℕ\n⊢ Real.log ↑n = log n"
] | rw [coe_mul_zeta_apply, vonMangoldt_sum] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.ArithmeticFunction.VonMangoldt | {
"line": 145,
"column": 19
} | {
"line": 145,
"column": 30
} | {
"line": 145,
"column": 31
} | [
{
"pp": "n : ℕ\nhn : ¬n = 0\nmn : 0 ∣ n\n⊢ n = 0",
"ppTerm": "?m.108",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\nhn : ¬n = 0\nmn : 0 ∣ n\n⊢ n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ArithmeticFunction.Carmichael | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 87
} | {
"line": 130,
"column": 0
} | [
{
"pp": "n : ℕ\nhn : n ≤ 2\n⊢ exponent (ZMod (2 ^ n))ˣ = Nat.card (ZMod (2 ^ n))ˣ",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"ZMod.commRing",
"CommSemiring.toSemiring",
"Nat.instMonoid",
"DivInvMonoid.toZPow",
"Units",
"Nat.card",
... | [] | exact IsCyclic.iff_exponent_eq_card.mp <| ZMod.isCyclic_units_two_pow_iff n |>.mpr hn | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.ArithmeticFunction.Carmichael | {
"line": 133,
"column": 40
} | {
"line": 135,
"column": 29
} | {
"line": 137,
"column": 0
} | [
{
"pp": "n : ℕ\nhn : n ≤ 2\n⊢ λ (2 ^ n) = 2 ^ (n - 1)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.zero_le",
"Nat.instMulZeroClass",
"ArithmeticFunction.instFunLikeNat",
"of_decide_eq_true",
"Nat.rawCast",
"congrArg",
"Nat.i... | [] | by
rw [carmichael_two_pow_of_le_two_eq_totient hn]
interval_cases n <;> decide | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.PowerSeries.Derivative | {
"line": 96,
"column": 36
} | {
"line": 96,
"column": 54
} | {
"line": 96,
"column": 55
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nr : R\nf : R⟦X⟧\n⊢ (C r * f).derivativeFun = C r * f.derivativeFun",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"instSMulOfMul",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
... | [
"R : Type u_1\ninst✝ : CommSemiring R\nr : R\nf : R⟦X⟧\n⊢ C r • f.derivativeFun + f • (C r).derivativeFun = C r * f.derivativeFun"
] | derivativeFun_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 194,
"column": 4
} | {
"line": 194,
"column": 15
} | {
"line": 194,
"column": 16
} | [
{
"pp": "p : ℕ\nhp : Nat.Prime p\nhp2 : p ≠ 2\nn : ℕ\nH : ↑p ∣ 1\n⊢ p = 1",
"ppTerm": "?m.199",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℕ\nhp : Nat.Prime p\nhp2 : p ≠ 2\nn : ℕ\nH : ↑p ∣ 1\n⊢ p = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.Exp | {
"line": 159,
"column": 4
} | {
"line": 160,
"column": 34
} | {
"line": 160,
"column": 35
} | [
{
"pp": "case succ\nA : Type u_4\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nk : ℕ\nh : exp A ^ k = (rescale ↑k) (exp A)\n⊢ exp A ^ (k + 1) = (rescale ↑(k + 1)) (exp A)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
... | [
"case succ\nA : Type u_4\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nk : ℕ\nh : exp A ^ k = (rescale ↑k) (exp A)\n⊢ exp A ^ (k + 1) = exp A ^ k * exp A"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 267,
"column": 27
} | {
"line": 267,
"column": 38
} | {
"line": 267,
"column": 39
} | [
{
"pp": "n : ℕ\nhn0 : 2 * n ≠ 0\nhn1 : 2 * n ≠ 1\n⊢ n ≠ 0",
"ppTerm": "?m.112",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"id",
"Ne",
"Nat",
"Zero.toOfNat0",
"OfNat.ofNat",
"MulZeroClass.toZero"
],
"usedFVars": [
"n"
],
... | [
"n : ℕ\nhn0 : 2 * n ≠ 0\nhn1 : 2 * n ≠ 1\n⊢ ¬n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 275,
"column": 25
} | {
"line": 275,
"column": 60
} | {
"line": 275,
"column": 61
} | [
{
"pp": "n : ℕ\nhn0 : n ≠ 0\nhn1 : n ≠ 1\nh2n : ¬2 ∣ n\nthis✝ : Nat.Coprime 4 n\nh : (Nat.card (ZMod 4)ˣ).Coprime (Nat.card (ZMod n)ˣ)\nthis : NeZero n\n⊢ Odd (φ n)",
"ppTerm": "?m.229",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\nhn0 : n ≠ 0\nhn1 : n ≠ 1\nh2n : ¬2 ∣ n\nthis✝ : Nat.Coprime 4 n\nh : (Nat.card (ZMod 4)ˣ).Coprime (Nat.card (ZMod n)ˣ)\nthis : NeZero n\n⊢ Odd (φ n)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.BernoulliPolynomials | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 20
} | {
"line": 92,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ (if n = 1 then 1 else 0) + _root_.bernoulli n = bernoulli' n",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"AddMonoid.toAddZeroClass",
"bernoulli",
"Rat",
"AddZeroClass.toAddZero",
"instOfNatNat",
"dite",
"... | [
"case pos\nn : ℕ\nh : n = 1\n⊢ (if n = 1 then 1 else 0) + _root_.bernoulli n = bernoulli' n",
"case neg\nn : ℕ\nh : ¬n = 1\n⊢ (if n = 1 then 1 else 0) + _root_.bernoulli n = bernoulli' n"
] | by_cases h : n = 1 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 315,
"column": 6
} | {
"line": 315,
"column": 61
} | {
"line": 315,
"column": 62
} | [
{
"pp": "case neg.refine_1.refine_1\nn : ℕ\nhn : Odd n\nhn0 : n ≠ 0\nh1 : n ≠ 1\np : ℕ\nhp : Nat.Prime p\ndvd : p ∣ n\nodd : Odd p\nhnp : ¬n = p ^ n.factorization p\nthis : p ^ n.factorization p ∣ n\n⊢ p ^ n.factorization p ≠ 1",
"ppTerm": "?neg.refine_1.refine_1✝",
"assigned": true,
"usedConstants"... | [
"case neg.refine_1.refine_1\nn : ℕ\nhn : Odd n\nhn0 : n ≠ 0\nh1 : n ≠ 1\np : ℕ\nhp : Nat.Prime p\ndvd : p ∣ n\nodd : Odd p\nhnp : ¬n = p ^ n.factorization p\nthis : p ^ n.factorization p ∣ n\n⊢ ¬p = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.AbelSummation | {
"line": 153,
"column": 2
} | {
"line": 157,
"column": 71
} | {
"line": 158,
"column": 2
} | [
{
"pp": "case inr\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\nf : ℝ → 𝕜\na b : ℝ\nha : 0 ≤ a\nhab : a ≤ b\nhf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t\nhf_int : IntegrableOn (deriv f) (Set.Icc a b) volume\naux1 : ↑⌊a⌋₊ ≤ a\naux2 : b ≤ ↑⌊b⌋₊ + 1\nhb : ⌊a⌋₊ < ⌊b⌋₊\naux3 : a ≤ ↑⌊a⌋₊ + 1\naux4 : ↑⌊a⌋₊ +... | [
"case inr\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\nf : ℝ → 𝕜\na b : ℝ\nha : 0 ≤ a\nhab : a ≤ b\nhf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t\nhf_int : IntegrableOn (deriv f) (Set.Icc a b) volume\naux1 : ↑⌊a⌋₊ ≤ a\naux2 : b ≤ ↑⌊b⌋₊ + 1\nhb : ⌊a⌋₊ < ⌊b⌋₊\naux3 : a ≤ ↑⌊a⌋₊ + 1\naux4 : ↑⌊a⌋₊ + 1 ≤ b\naux5... | rw [this, sum_integral_adjacent_intervals_Ico hb, Nat.cast_add, Nat.cast_one,
← integral_interval_sub_left (a := a) (c := ⌊a⌋₊ + 1),
← integral_add_adjacent_intervals (b := ⌊b⌋₊) (c := b),
integralmulsum c hf_diff hf_int _ _ _ aux3 aux1 le_rfl le_rfl aux4,
integralmulsum c hf_diff hf_int _ _ _ aux5 le_r... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 363,
"column": 2
} | {
"line": 364,
"column": 7
} | {
"line": 365,
"column": 2
} | [
{
"pp": "case neg.inr.inr.ha\nn : ℕ\nh0 : ¬2 * (2 * n) = 0\nh1 : ¬2 * (2 * n) = 1\nh2 : ¬2 * (2 * n) = 2\nh4 : ¬2 * (2 * n) = 4\nhn✝ : Even (2 * (2 * n))\nhn : Even (2 * n)\n⊢ ¬IsCyclic (ZMod (2 * (2 * n)))ˣ",
"ppTerm": "?neg.inr.inr.ha✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"case neg.inr.inr.hb\nn : ℕ\nh0 : ¬2 * (2 * n) = 0\nh1 : ¬2 * (2 * n) = 1\nh2 : ¬2 * (2 * n) = 2\nh4 : ¬2 * (2 * n) = 4\nhn✝ : Even (2 * (2 * n))\nhn : Even (2 * n)\n⊢ ¬((∃ x x_1, Nat.Prime x ∧ Odd x ∧ 1 ≤ x_1 ∧ 2 * (2 * n) = x ^ x_1) ∨\n ∃ x x_1, Nat.Prime x ∧ Odd x ∧ 1 ≤ x_1 ∧ 2 * (2 * n) = 2 * x ^ x_1)"
] | · rw [← mul_assoc, show 2 * 2 = 4 from rfl, isCyclic_units_four_mul_iff]
lia | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.BernoulliPolynomials | {
"line": 215,
"column": 2
} | {
"line": 215,
"column": 13
} | {
"line": 215,
"column": 14
} | [
{
"pp": "n : ℕ\nx : ℚ\nthis : (bernoulli n).comp (1 + X) = bernoulli n + n • X ^ (n - 1)\n⊢ eval (1 + x) (bernoulli n) = eval x (bernoulli n) + ↑n * x ^ (n - 1)",
"ppTerm": "?m.32",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\nx : ℚ\nthis : (bernoulli n).comp (1 + X) = bernoulli n + n • X ^ (n - 1)\n⊢ eval (1 + x) (bernoulli n) = eval x (bernoulli n) + ↑n * x ^ (n - 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Bernoulli | {
"line": 169,
"column": 4
} | {
"line": 169,
"column": 40
} | {
"line": 169,
"column": 41
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nthis : ∑ p ∈ antidiagonal n, bernoulli' p.1 / ↑p.1! * ((↑p.2 + 1) * ↑p.2!)⁻¹ = (↑n !)⁻¹\n⊢ (coeff (n + 1, 0).1) (PowerSeries.mk fun n ↦ (algebraMap ℚ A) (bernoulli' n / ↑n !)) *\n (coeff (n + 1, 0).2) (exp A - 1) +\n ∑ p ∈ antid... | [
"A : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nthis : ∑ p ∈ antidiagonal n, bernoulli' p.1 / ↑p.1! * ((↑p.2 + 1) * ↑p.2!)⁻¹ = (↑n !)⁻¹\n⊢ ∑ x ∈ antidiagonal n,\n (algebraMap ℚ A) (bernoulli' x.1 / ↑x.1!) * ((algebraMap ℚ A) (↑x.2!)⁻¹ * (algebraMap ℚ A) (↑x.2 + 1)⁻¹) =\n (algebraMap ℚ A) (↑n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.BernoulliPolynomials | {
"line": 238,
"column": 2
} | {
"line": 238,
"column": 23
} | {
"line": 238,
"column": 24
} | [
{
"pp": "n : ℕ\nx : ℚ\n⊢ eval (-x) (bernoulli n) = (-1) ^ n * (eval x (bernoulli n) + ↑n * x ^ (n - 1))",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"Distrib.leftDistribClass",
"Eq.mpr",
"Polynomial.eval",
"Rat.instMul",
"HMul.hMul",
... | [
"n : ℕ\nx : ℚ\n⊢ eval (-x) (bernoulli n) = (-1) ^ n * eval x (bernoulli n) + (-1) ^ n * (↑n * x ^ (n - 1))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.BernoulliPolynomials | {
"line": 241,
"column": 59
} | {
"line": 248,
"column": 8
} | {
"line": 250,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ (bernoulli n).comp (1 - X) = (-1) ^ n * bernoulli n",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Rat.addCommMonoid",
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"AddGroup.toSubtractionMonoid",
"Int.cast_neg",
"Int.cast",
"Mathlib.Tact... | [] | by
cases n with
| zero => simp
| succ n =>
trans ((bernoulli (n + 1)).comp (1 + X)).comp (-X)
· simp [comp_assoc, sub_eq_add_neg]
simp [bernoulli_comp_one_add_X, bernoulli_comp_neg_X, neg_pow (X : Polynomial ℚ)]
ring | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.BernoulliPolynomials | {
"line": 252,
"column": 2
} | {
"line": 252,
"column": 13
} | {
"line": 252,
"column": 14
} | [
{
"pp": "n : ℕ\nx : ℚ\n⊢ eval (1 - x) (bernoulli n) = (-1) ^ n * eval x (bernoulli n)",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\nx : ℚ\n⊢ eval (1 - x) (bernoulli n) = (-1) ^ n * eval x (bernoulli n)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Bernoulli | {
"line": 187,
"column": 6
} | {
"line": 187,
"column": 20
} | {
"line": 187,
"column": 21
} | [
{
"pp": "case inl\nn : ℕ\nh_odd : Odd n\nhlt : 1 < n\nB : ℚ⟦X⟧ := PowerSeries.mk fun n ↦ bernoulli' n / ↑n !\nthis : (B - evalNegHom B) * (exp ℚ - 1) = X * (exp ℚ - 1)\nh : (coeff n) (B - (rescale (-1)) B) = if n = 1 then 1 else 0\n⊢ -bernoulli' n = bernoulli' n",
"ppTerm": "?inl",
"assigned": true,
... | [
"case pos\nn : ℕ\nh_odd : Odd n\nhlt : 1 < n\nB : ℚ⟦X⟧ := PowerSeries.mk fun n ↦ bernoulli' n / ↑n !\nthis : (B - evalNegHom B) * (exp ℚ - 1) = X * (exp ℚ - 1)\nh✝ : n = 1\nh : (coeff n) (B - (rescale (-1)) B) = 1\n⊢ -bernoulli' n = bernoulli' n",
"case neg\nn : ℕ\nh_odd : Odd n\nhlt : 1 < n\nB : ℚ⟦X⟧ := PowerSer... | split_ifs at h | Mathlib.Tactic._aux_Mathlib_Tactic_SplitIfs___elabRules_Mathlib_Tactic_splitIfs_1 | Mathlib.Tactic.splitIfs |
Mathlib.NumberTheory.Bernoulli | {
"line": 188,
"column": 6
} | {
"line": 188,
"column": 41
} | {
"line": 188,
"column": 42
} | [
{
"pp": "case inr\nn : ℕ\nh_odd : Odd n\nhlt : 1 < n\nB : ℚ⟦X⟧ := PowerSeries.mk fun n ↦ bernoulli' n / ↑n !\nthis : (B - evalNegHom B) * (exp ℚ - 1) = X * (exp ℚ - 1)\nh : ∀ (n : ℕ), (coeff n) (exp ℚ - 1) = (coeff n) 0\n⊢ bernoulli' n = 0",
"ppTerm": "?inr",
"assigned": false,
"usedConstants": [],
... | [
"case inr\nn : ℕ\nh_odd : Odd n\nhlt : 1 < n\nB : ℚ⟦X⟧ := PowerSeries.mk fun n ↦ bernoulli' n / ↑n !\nthis : (B - evalNegHom B) * (exp ℚ - 1) = X * (exp ℚ - 1)\nh : ∀ (n : ℕ), (coeff n) (exp ℚ - 1) = (coeff n) 0\n⊢ bernoulli' n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Bernoulli | {
"line": 190,
"column": 4
} | {
"line": 190,
"column": 39
} | {
"line": 190,
"column": 40
} | [
{
"pp": "n : ℕ\nh_odd : Odd n\nhlt : 1 < n\nB : ℚ⟦X⟧ := PowerSeries.mk fun n ↦ bernoulli' n / ↑n !\n⊢ B * (exp ℚ - 1) = X * exp ℚ",
"ppTerm": "?m.154",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\nh_odd : Odd n\nhlt : 1 < n\nB : ℚ⟦X⟧ := PowerSeries.mk fun n ↦ bernoulli' n / ↑n !\n⊢ B * (exp ℚ - 1) = X * exp ℚ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Bernoulli | {
"line": 315,
"column": 29
} | {
"line": 315,
"column": 40
} | {
"line": 315,
"column": 41
} | [
{
"pp": "n p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\nf : ℕ → ℕ → ℚ := fun a b ↦ bernoulli a / ↑a ! * (coeff (b + 1)) (exp ℚ ^ n)\nm : ℕ\nh : m ∈ range q.succ\n⊢ m < q + 1",
"ppTerm": "?m.213",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instOne",
"PartialOrder.toPreorder",
... | [
"n p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\nf : ℕ → ℕ → ℚ := fun a b ↦ bernoulli a / ↑a ! * (coeff (b + 1)) (exp ℚ ^ n)\nm : ℕ\nh : m ∈ range q.succ\n⊢ m ≤ q"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.AbelSummation | {
"line": 278,
"column": 4
} | {
"line": 279,
"column": 28
} | {
"line": 281,
"column": 0
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\na : ℝ\nm : ℕ\nha : 0 ≤ a\ng : ℝ → 𝕜\nhg : LocallyIntegrableOn g (Set.Ici a) volume\nK : Set ℝ\nhK₁ : K ⊆ Set.Ici a\nhK₂ : IsCompact K\nhK₃ : ¬K.Nonempty\n⊢ IntegrableOn (fun t ↦ g t * ∑ k ∈ Icc m ⌊t⌋₊, c k) K volume",
"ppTerm": "?neg✝",
"... | [] | rw [Set.not_nonempty_iff_eq_empty.mp hK₃]
exact integrableOn_empty | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.AbelSummation | {
"line": 278,
"column": 4
} | {
"line": 279,
"column": 28
} | {
"line": 281,
"column": 0
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\na : ℝ\nm : ℕ\nha : 0 ≤ a\ng : ℝ → 𝕜\nhg : LocallyIntegrableOn g (Set.Ici a) volume\nK : Set ℝ\nhK₁ : K ⊆ Set.Ici a\nhK₂ : IsCompact K\nhK₃ : ¬K.Nonempty\n⊢ IntegrableOn (fun t ↦ g t * ∑ k ∈ Icc m ⌊t⌋₊, c k) K volume",
"ppTerm": "?neg✝",
"... | [] | rw [Set.not_nonempty_iff_eq_empty.mp hK₃]
exact integrableOn_empty | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Bertrand | {
"line": 151,
"column": 41
} | {
"line": 151,
"column": 80
} | {
"line": 151,
"column": 81
} | [
{
"pp": "n : ℕ\nn_large : 2 < n\nno_prime : ∀ (p : ℕ), Nat.Prime p → n < p → 2 * n < p\nx : ℕ\nhx : x ≤ 2 * n\nh2x : 2 * n < 3 * x\n⊢ ¬Nat.Prime x ∨ x ≤ n",
"ppTerm": "?m.174",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\nn_large : 2 < n\nno_prime : ∀ (p : ℕ), Nat.Prime p → n < p → 2 * n < p\nx : ℕ\nhx : x ≤ 2 * n\nh2x : 2 * n < 3 * x\n⊢ ¬Nat.Prime x ∨ x ≤ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.AbelSummation | {
"line": 291,
"column": 37
} | {
"line": 291,
"column": 78
} | {
"line": 291,
"column": 78
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\nf : ℝ → 𝕜\nhf_diff : ∀ t ∈ Set.Ici 0, DifferentiableAt ℝ f t\nhf_int : LocallyIntegrableOn (deriv f) (Set.Ici 0) volume\nl : 𝕜\nh_lim : Tendsto (fun n ↦ f ↑n * ∑ k ∈ Icc 0 n, c k) atTop (𝓝 l)\ng : ℝ → 𝕜\nhg_dom : (fun t ↦ deriv f t * ∑ k ∈ Icc 0 ⌊t⌋₊, c... | [] | rw [← integral_of_le (Nat.cast_nonneg _)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.AbelSummation | {
"line": 291,
"column": 37
} | {
"line": 291,
"column": 78
} | {
"line": 291,
"column": 78
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\nf : ℝ → 𝕜\nhf_diff : ∀ t ∈ Set.Ici 0, DifferentiableAt ℝ f t\nhf_int : LocallyIntegrableOn (deriv f) (Set.Ici 0) volume\nl : 𝕜\nh_lim : Tendsto (fun n ↦ f ↑n * ∑ k ∈ Icc 0 n, c k) atTop (𝓝 l)\ng : ℝ → 𝕜\nhg_dom : (fun t ↦ deriv f t * ∑ k ∈ Icc 0 ⌊t⌋₊, c... | [] | rw [← integral_of_le (Nat.cast_nonneg _)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.AbelSummation | {
"line": 291,
"column": 37
} | {
"line": 291,
"column": 78
} | {
"line": 291,
"column": 78
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\nf : ℝ → 𝕜\nhf_diff : ∀ t ∈ Set.Ici 0, DifferentiableAt ℝ f t\nhf_int : LocallyIntegrableOn (deriv f) (Set.Ici 0) volume\nl : 𝕜\nh_lim : Tendsto (fun n ↦ f ↑n * ∑ k ∈ Icc 0 n, c k) atTop (𝓝 l)\ng : ℝ → 𝕜\nhg_dom : (fun t ↦ deriv f t * ∑ k ∈ Icc 0 ⌊t⌋₊, c... | [] | rw [← integral_of_le (Nat.cast_nonneg _)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Bernoulli | {
"line": 320,
"column": 83
} | {
"line": 320,
"column": 93
} | {
"line": 321,
"column": 6
} | [
{
"pp": "n p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\nf : ℕ → ℕ → ℚ := fun a b ↦ bernoulli a / ↑a ! * (coeff (b + 1)) (exp ℚ ^ n)\nm : ℕ\nh✝ : m ∈ range q.succ\nh : m < q + 1\n⊢ bernoulli m * ↑((q + 1)! / (m ! * (q + 1 - m)!)) * ↑n ^ (q + 1 - m) =\n bernoulli m * ↑q.succ ! / ↑m ! * (↑(q - m + 1)!)⁻¹ * ↑n ^ (q ... | [
"n p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\nf : ℕ → ℕ → ℚ := fun a b ↦ bernoulli a / ↑a ! * (coeff (b + 1)) (exp ℚ ^ n)\nm : ℕ\nh✝ : m ∈ range q.succ\nh : m < q + 1\n⊢ bernoulli m * ↑((q + 1)! / (m ! * (q + 1 - m)!)) * ↑n ^ (q + 1 - m) =\n bernoulli m * ↑q.succ ! / ↑m ! * (1 / ↑(q - m + 1)!) * ↑n ^ (q - m + 1)"
... | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Bernoulli | {
"line": 336,
"column": 6
} | {
"line": 336,
"column": 17
} | {
"line": 336,
"column": 18
} | [
{
"pp": "n p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p ↦ bernoulli p / ↑p !) * PowerSeries.mk fun q ↦ (coeff (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p ↦ ∑ i ∈ range (p + 1), bernoulli i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nthis :\n ∀ (n_1 : ℕ),\n (coeff n_... | [
"n p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p ↦ bernoulli p / ↑p !) * PowerSeries.mk fun q ↦ (coeff (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p ↦ ∑ i ∈ range (p + 1), bernoulli i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nthis :\n ∀ (n_1 : ℕ),\n (coeff n_1) (PowerSer... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue | {
"line": 100,
"column": 12
} | {
"line": 100,
"column": 58
} | {
"line": 100,
"column": 59
} | [
{
"pp": "R : Type u_1\ninst✝ : EuclideanDomain R\nabv : AbsoluteValue R ℤ\nh : abv.IsAdmissible\nthis : DecidableEq R\nn : ℕ\nih :\n ∀ {ε : ℝ},\n 0 < ε →\n ∀ {b : R},\n b ≠ 0 →\n ∀ (A : Fin (h.card ε ^ n).succ → Fin n → R),\n ∃ i₀ i₁, i₀ ≠ i₁ ∧ ∀ (k : Fin n), ↑(abv (A i₁ k % ... | [
"R : Type u_1\ninst✝ : EuclideanDomain R\nabv : AbsoluteValue R ℤ\nh : abv.IsAdmissible\nthis : DecidableEq R\nn : ℕ\nih :\n ∀ {ε : ℝ},\n 0 < ε →\n ∀ {b : R},\n b ≠ 0 →\n ∀ (A : Fin (h.card ε ^ n).succ → Fin n → R),\n ∃ i₀ i₁, i₀ ≠ i₁ ∧ ∀ (k : Fin n), ↑(abv (A i₁ k % b - A i₀ k %... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue | {
"line": 105,
"column": 6
} | {
"line": 105,
"column": 72
} | {
"line": 105,
"column": 73
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝ : EuclideanDomain R\nabv : AbsoluteValue R ℤ\nh✝ : abv.IsAdmissible\nthis✝ : DecidableEq R\nn : ℕ\nih :\n ∀ {ε : ℝ},\n 0 < ε →\n ∀ {b : R},\n b ≠ 0 →\n ∀ (A : Fin (h✝.card ε ^ n).succ → Fin n → R),\n ∃ i₀ i₁, i₀ ≠ i₁ ∧ ∀ (k : Fin n)... | [
"case refine_1\nR : Type u_1\ninst✝ : EuclideanDomain R\nabv : AbsoluteValue R ℤ\nh✝ : abv.IsAdmissible\nthis✝ : DecidableEq R\nn : ℕ\nih :\n ∀ {ε : ℝ},\n 0 < ε →\n ∀ {b : R},\n b ≠ 0 →\n ∀ (A : Fin (h✝.card ε ^ n).succ → Fin n → R),\n ∃ i₀ i₁, i₀ ≠ i₁ ∧ ∀ (k : Fin n), ↑(abv (A i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.AbelSummation | {
"line": 358,
"column": 8
} | {
"line": 364,
"column": 17
} | {
"line": 366,
"column": 0
} | [
{
"pp": "case succ.calc_4\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\nf : ℝ → 𝕜\nm : ℕ\nh_bdd : (fun n ↦ ‖f ↑n‖ * ∑ k ∈ Icc 0 n, ‖c k‖) =O[atTop] fun x ↦ 1\nhf_int : LocallyIntegrableOn (deriv fun t ↦ ‖f t‖) (Set.Ici ↑m) volume\nhf :\n ∀ (n : ℕ),\n ∑ k ∈ Icc 0 n, ‖f ↑k‖ * ‖c k‖ =\n ‖f ↑n‖ * ∑ k ∈ I... | [] | unfold C₂
grw [setIntegral_mono_set ?_ (.of_forall fun _ ↦ norm_nonneg _)
Set.Ioc_subset_Ioi_self.eventuallyLE]
rw [← integrableOn_Ici_iff_integrableOn_Ioi, IntegrableOn,
integrable_norm_iff (by fun_prop)]
exact (locallyIntegrableOn_mul_sum_Icc _ m.cast_nonneg hf_int).integra... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.AbelSummation | {
"line": 358,
"column": 8
} | {
"line": 364,
"column": 17
} | {
"line": 366,
"column": 0
} | [
{
"pp": "case succ.calc_4\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\nf : ℝ → 𝕜\nm : ℕ\nh_bdd : (fun n ↦ ‖f ↑n‖ * ∑ k ∈ Icc 0 n, ‖c k‖) =O[atTop] fun x ↦ 1\nhf_int : LocallyIntegrableOn (deriv fun t ↦ ‖f t‖) (Set.Ici ↑m) volume\nhf :\n ∀ (n : ℕ),\n ∑ k ∈ Icc 0 n, ‖f ↑k‖ * ‖c k‖ =\n ‖f ↑n‖ * ∑ k ∈ I... | [] | unfold C₂
grw [setIntegral_mono_set ?_ (.of_forall fun _ ↦ norm_nonneg _)
Set.Ioc_subset_Ioi_self.eventuallyLE]
rw [← integrableOn_Ici_iff_integrableOn_Ioi, IntegrableOn,
integrable_norm_iff (by fun_prop)]
exact (locallyIntegrableOn_mul_sum_Icc _ m.cast_nonneg hf_int).integra... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Chebyshev | {
"line": 95,
"column": 4
} | {
"line": 95,
"column": 15
} | {
"line": 95,
"column": 16
} | [
{
"pp": "case refine_2\nx : ℝ\nhy : 2 ≤ x\nthis : 0 ≤ x\n⊢ 2 ∈ {p ∈ Ioc 0 ⌊x⌋₊ | Nat.Prime p}",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"Real.partialOrder",
"Real",
"Finset.mem_filter._simp_1",
"Nat.Prime",
... | [
"case refine_2\nx : ℝ\nhy : 2 ≤ x\nthis : 0 ≤ x\n⊢ 2 ≤ ⌊x⌋₊ ∧ Nat.Prime 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ClassNumber.AdmissibleCardPowDegree | {
"line": 45,
"column": 4
} | {
"line": 45,
"column": 15
} | {
"line": 45,
"column": 16
} | [
{
"pp": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Semiring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nhb : b.natDegree ≤ d\nA : Fin m.succ → Fq[X]\nhA : ∀ (i : Fin m.succ), (A i).degree < b.degree\nf : Fin m.succ → Fin d → Fq := fun i j ↦ (A i).coeff ↑j\n⊢ Fintype.card (Fin d → Fq) < Fintype.card ... | [
"Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Semiring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nhb : b.natDegree ≤ d\nA : Fin m.succ → Fq[X]\nhA : ∀ (i : Fin m.succ), (A i).degree < b.degree\nf : Fin m.succ → Fin d → Fq := fun i j ↦ (A i).coeff ↑j\n⊢ Fintype.card Fq ^ d ≤ m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ClassNumber.AdmissibleCardPowDegree | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 15
} | {
"line": 75,
"column": 16
} | [
{
"pp": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin m.succ → Fq[X]\nhA : ∀ (i : Fin m.succ), (A i).degree < b.degree\nhb : b ≠ 0\nf : Fin m.succ → Fin d → Fq := fun i j ↦ (A i).coeff (b.natDegree - ↑j.succ)\n⊢ Fintype.card (Fin d → Fq) < Fintyp... | [
"Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin m.succ → Fq[X]\nhA : ∀ (i : Fin m.succ), (A i).degree < b.degree\nhb : b ≠ 0\nf : Fin m.succ → Fin d → Fq := fun i j ↦ (A i).coeff (b.natDegree - ↑j.succ)\n⊢ Fintype.card Fq ^ d ≤ m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Chebyshev | {
"line": 326,
"column": 4
} | {
"line": 326,
"column": 68
} | {
"line": 327,
"column": 2
} | [
{
"pp": "case h\nn k : ℕ\nhk : k ∈ range (n + 1)\n⊢ n.choose k ≤ n.lcmUpto",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Nat.choose",
"Nat.lcmUpto_pos",
"Nat.lcmUpto",
"_private.Mathlib.NumberTheory.Chebyshev.0.Chebyshev.two_pow_le_mul_lcmUpto._proof_1_1",
"Ch... | [] | exact le_of_dvd (lcmUpto_pos n) (choose_dvd_lcmUpto <| by grind) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.ClassNumber.Finite | {
"line": 173,
"column": 4
} | {
"line": 176,
"column": 14
} | {
"line": 177,
"column": 2
} | [
{
"pp": "case mp\nR : Type u_1\nS : Type u_2\ninst✝⁷ : EuclideanDomain R\ninst✝⁶ : CommRing S\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R S\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nbS : Basis ι R S\nadm : abv.IsAdmissible\ninst✝¹ : Infinite R\ninst✝ : DecidableEq R\nx : R... | [] | rintro ⟨hx, ⟨i, j⟩, _, rfl⟩
refine ⟨i, j, ?_, rfl⟩
rintro rfl
simp at hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ClassNumber.Finite | {
"line": 173,
"column": 4
} | {
"line": 176,
"column": 14
} | {
"line": 177,
"column": 2
} | [
{
"pp": "case mp\nR : Type u_1\nS : Type u_2\ninst✝⁷ : EuclideanDomain R\ninst✝⁶ : CommRing S\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R S\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nbS : Basis ι R S\nadm : abv.IsAdmissible\ninst✝¹ : Infinite R\ninst✝ : DecidableEq R\nx : R... | [] | rintro ⟨hx, ⟨i, j⟩, _, rfl⟩
refine ⟨i, j, ?_, rfl⟩
rintro rfl
simp at hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Chebyshev | {
"line": 406,
"column": 2
} | {
"line": 406,
"column": 21
} | {
"line": 407,
"column": 2
} | [
{
"pp": "x : ℝ\n⊢ θ x ≤ ψ x",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Real.instLE",
"Real",
"PartialOrder.toPreorder",
"Nat.instAtLeastTwoHAddOfNat",
"Real.decidableLT",
"Real.instLT",
"Preorder.toLE",
"Eq.mp",
"instOfNatNat",
... | [
"case pos\nx : ℝ\nh : x < 2\n⊢ θ x ≤ ψ x",
"case neg\nx : ℝ\nh : 2 ≤ x\n⊢ θ x ≤ ψ x"
] | by_cases! h : x < 2 | Mathlib.Tactic.ByCases._aux_Mathlib_Tactic_ByCases___macroRules_Mathlib_Tactic_ByCases_byCases!_1 | Mathlib.Tactic.ByCases.byCases! |
Mathlib.NumberTheory.Bernoulli | {
"line": 640,
"column": 22
} | {
"line": 640,
"column": 33
} | {
"line": 640,
"column": 34
} | [
{
"pp": "case h.e_a\nk p : ℕ\nhk : k > 0\ninst✝ : Fact (Nat.Prime p)\nhcast : ↑(∑ v ∈ Ico 1 p, ↑v ^ (2 * k) + if p - 1 ∣ 2 * k then 1 else 0) = 0\nT : ℤ\nhT_int : (∑ v ∈ Ico 1 p, ↑v ^ (2 * k) + if p - 1 ∣ 2 * k then 1 else 0) = ↑p * T\nhT : ∑ v ∈ Ico 1 p, ↑v ^ (2 * k) + vonStaudtIndicator (2 * k) p = ↑p * ↑T\nh... | [
"case h.e_a\nk p : ℕ\nhk : k > 0\ninst✝ : Fact (Nat.Prime p)\nhcast : ↑(∑ v ∈ Ico 1 p, ↑v ^ (2 * k) + if p - 1 ∣ 2 * k then 1 else 0) = 0\nT : ℤ\nhT_int : (∑ v ∈ Ico 1 p, ↑v ^ (2 * k) + if p - 1 ∣ 2 * k then 1 else 0) = ↑p * T\nhT : ∑ v ∈ Ico 1 p, ↑v ^ (2 * k) + vonStaudtIndicator (2 * k) p = ↑p * ↑T\nhp_ne : ↑p ≠ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ClassNumber.AdmissibleCardPowDegree | {
"line": 192,
"column": 6
} | {
"line": 192,
"column": 17
} | {
"line": 192,
"column": 18
} | [
{
"pp": "case succ.refine_1.refine_1\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Field Fq\nε : ℝ\nhε : 0 < ε\nb : Fq[X]\nhb : b ≠ 0\nhbε : 0 < cardPowDegree b • ε\nn : ℕ\nih :\n ∀ (A : Fin n → Fq[X]),\n ∃ t, ∀ (i₀ i₁ : Fin n), t i₀ = t i₁ ↔ ↑(cardPowDegree (A i₁ % b - A i₀ % b)) < cardPowDegree b • ε\nA : ... | [
"case succ.refine_1.refine_1\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Field Fq\nε : ℝ\nhε : 0 < ε\nb : Fq[X]\nhb : b ≠ 0\nhbε : 0 < cardPowDegree b • ε\nn : ℕ\nih :\n ∀ (A : Fin n → Fq[X]),\n ∃ t, ∀ (i₀ i₁ : Fin n), t i₀ = t i₁ ↔ ↑(cardPowDegree (A i₁ % b - A i₀ % b)) < cardPowDegree b • ε\nA : Fin (n + 1) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ClassNumber.Finite | {
"line": 221,
"column": 65
} | {
"line": 221,
"column": 88
} | {
"line": 221,
"column": 89
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : EuclideanDomain R\ninst✝⁶ : CommRing S\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R S\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nbS : Basis ι R S\nadm : abv.IsAdmissible\ninst✝¹ : Infinite R\ninst✝ : DecidableEq R\na : S\nb : R\n... | [
"R : Type u_1\nS : Type u_2\ninst✝⁷ : EuclideanDomain R\ninst✝⁶ : CommRing S\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R S\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nbS : Basis ι R S\nadm : abv.IsAdmissible\ninst✝¹ : Infinite R\ninst✝ : DecidableEq R\na : S\nb : R\nhb : b ≠ 0\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ClassNumber.Finite | {
"line": 249,
"column": 2
} | {
"line": 252,
"column": 23
} | {
"line": 253,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁸ : EuclideanDomain R\ninst✝⁷ : CommRing S\ninst✝⁶ : IsDomain S\ninst✝⁵ : Algebra R S\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝⁴ : DecidableEq ι\ninst✝³ : Fintype ι\nbS : Basis ι R S\nadm : abv.IsAdmissible\ninst✝² : Infinite R\ninst✝¹ : DecidableEq R\ninst✝ : Algeb... | [
"R : Type u_1\nS : Type u_2\ninst✝⁸ : EuclideanDomain R\ninst✝⁷ : CommRing S\ninst✝⁶ : IsDomain S\ninst✝⁵ : Algebra R S\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝⁴ : DecidableEq ι\ninst✝³ : Fintype ι\nbS : Basis ι R S\nadm : abv.IsAdmissible\ninst✝² : Infinite R\ninst✝¹ : DecidableEq R\ninst✝ : Algebra.IsAlgebra... | refine
lt_of_le_of_lt (le_of_eq ?_)
(mul_lt_mul hqr le_rfl (abv.pos ((Algebra.norm_ne_zero_iff_of_basis bS).mpr hb))
(abv.nonneg _)) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.FunctionField | {
"line": 121,
"column": 2
} | {
"line": 122,
"column": 39
} | {
"line": 123,
"column": 4
} | [
{
"pp": "F : Type u_1\nK : Type u_2\ninst✝⁴ : Field F\ninst✝³ : Field K\ninst✝² : Algebra F[X] K\ninst✝¹ : Algebra F⟮X⟯ K\ninst✝ : IsScalarTower F[X] F⟮X⟯ K\n⊢ ¬IsField ↥(ringOfIntegers F K)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
... | [
"F : Type u_1\nK : Type u_2\ninst✝⁴ : Field F\ninst✝³ : Field K\ninst✝² : Algebra F[X] K\ninst✝¹ : Algebra F⟮X⟯ K\ninst✝ : IsScalarTower F[X] F⟮X⟯ K\n⊢ ¬IsField F[X]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Chebyshev | {
"line": 693,
"column": 46
} | {
"line": 693,
"column": 89
} | {
"line": 694,
"column": 2
} | [
{
"pp": "x : ℝ\nhx : 2 ≤ x\na : ℕ → ℝ := (setOf Nat.Prime).indicator fun n ↦ 1\nf : ℝ → ℝ\nu : ℝ\nx✝ : u ∈ Set.uIcc 2 x\n⊢ deriv (fun x ↦ log x) u * f u = f u / u",
"ppTerm": "?m.230",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.m... | [] | by rw [deriv_log, mul_comm, div_eq_mul_inv] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 13
} | {
"line": 132,
"column": 14
} | [
{
"pp": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nn : ℕ\nx : ℤ_[p]\na b : ℕ\nha : x - ↑a ∈ Ideal.span {↑p ^ n}\nhb : x - ↑b ∈ Ideal.span {↑p ^ n}\n⊢ ↑a = ↑b",
"ppTerm": "?m.41",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℕ\nhp_prime : Fact (Nat.Prime p)\nn : ℕ\nx : ℤ_[p]\na b : ℕ\nha : x - ↑a ∈ Ideal.span {↑p ^ n}\nhb : x - ↑b ∈ Ideal.span {↑p ^ n}\n⊢ ↑a = ↑b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 37
} | {
"line": 142,
"column": 38
} | [
{
"pp": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nx : ℤ_[p]\nm n : ℕ\nhm : x - ↑m ∈ Ideal.span {↑p}\nhn : x - ↑n ∈ Ideal.span {↑p}\nthis : ↑↑m = ↑↑n\n⊢ ↑m = ↑n",
"ppTerm": "?m.146",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℕ\nhp_prime : Fact (Nat.Prime p)\nx : ℤ_[p]\nm n : ℕ\nhm : x - ↑m ∈ Ideal.span {↑p}\nhn : x - ↑n ∈ Ideal.span {↑p}\nthis : ↑↑m = ↑↑n\n⊢ ↑m = ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 13
} | {
"line": 164,
"column": 14
} | [
{
"pp": "case h.right\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nx : ℤ_[p]\nr : ℚ\nhr : ‖↑x - ↑r‖ < 1\nH : ‖↑r‖ ≤ 1\nn : ℕ\nhnp : ↑n < ↑p\nhn : ‖↑r - ↑↑↑n‖ < 1\n⊢ ‖↑r - ↑n‖ < 1",
"ppTerm": "?h.right",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case h.right\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nx : ℤ_[p]\nr : ℚ\nhr : ‖↑x - ↑r‖ < 1\nH : ‖↑r‖ ≤ 1\nn : ℕ\nhnp : ↑n < ↑p\nhn : ‖↑r - ↑↑↑n‖ < 1\n⊢ ‖↑r - ↑n‖ < 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Chebyshev | {
"line": 807,
"column": 7
} | {
"line": 807,
"column": 41
} | {
"line": 807,
"column": 42
} | [
{
"pp": "ε : ℝ\nεpos : 0 < ε\nthis✝ : ∀ᶠ (x : ℝ) in atTop, ‖∫ (t : ℝ) in 2..x, θ t / (t * log t ^ 2)‖ ≤ ε * ‖x / log x‖\nx : ℝ\nhx : 2 ≤ x\nhx2 : ‖∫ (t : ℝ) in 2..x, θ t / (t * log t ^ 2)‖ ≤ ε * x / log x\nthis : 0 ≤ log x\n⊢ θ x / log x + ∫ (t : ℝ) in 2..x, θ t / (t * log t ^ 2) ≤ log 4 * x / log x + ε * x / l... | [
"ε : ℝ\nεpos : 0 < ε\nthis✝ : ∀ᶠ (x : ℝ) in atTop, ‖∫ (t : ℝ) in 2..x, θ t / (t * log t ^ 2)‖ ≤ ε * ‖x / log x‖\nx : ℝ\nhx : 2 ≤ x\nhx2 : ‖∫ (t : ℝ) in 2..x, θ t / (t * log t ^ 2)‖ ≤ ε * x / log x\nthis : 0 ≤ log x\n⊢ log 4 * x / log x + ∫ (t : ℝ) in 2..x, θ t / (t * log t ^ 2) ≤ log 4 * x / log x + ε * x / log x"
... | theta_le_log4_mul_x (by linarith), | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 323,
"column": 2
} | {
"line": 323,
"column": 29
} | {
"line": 323,
"column": 30
} | [
{
"pp": "p' : ℕ\nhp_prime : Fact (Nat.Prime (0 + p' + 1))\nx : ℤ_[0 + p' + 1]\n⊢ x.zmodRepr < (0 + p').succ",
"ppTerm": "?m.199",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PadicInt.zmodRepr",
"congrArg",
"AddMonoid.toAddZeroClass",
"Nat.instAddMonoid",
"id",... | [
"p' : ℕ\nhp_prime : Fact (Nat.Prime (0 + p' + 1))\nx : ℤ_[0 + p' + 1]\n⊢ x.zmodRepr < p'.succ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 330,
"column": 4
} | {
"line": 330,
"column": 50
} | {
"line": 330,
"column": 51
} | [
{
"pp": "case mp\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nx : ℤ_[p]\nh : toZMod x = 0\n⊢ x ∈ maximalIdeal ℤ_[p]",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case mp\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nx : ℤ_[p]\nh : toZMod x = 0\n⊢ x ∈ maximalIdeal ℤ_[p]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 341,
"column": 45
} | {
"line": 341,
"column": 56
} | {
"line": 341,
"column": 57
} | [
{
"pp": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nx : ℤ_[p]\n⊢ x - ↑(toZMod x).val ∈ maximalIdeal ℤ_[p]",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"mem_nonunits_iff._simp_1",
"Semiring.toModule",
"LinearOrderedCommMonoidWith... | [
"p : ℕ\nhp_prime : Fact (Nat.Prime p)\nx : ℤ_[p]\n⊢ ¬IsUnit (x - (toZMod x).cast)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.CyclotomicCharacter | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 13
} | {
"line": 88,
"column": 14
} | [
{
"pp": "L : Type u\ninst✝² : CommRing L\ninst✝¹ : IsDomain L\nn : ℕ\ninst✝ : NeZero n\ng : L ≃+* L\n⊢ ∃ m, ∀ t ∈ rootsOfUnity n L, g ↑t = ↑(t ^ m)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"congrArg",
"CommSemiring.toSemiring",
"R... | [
"L : Type u\ninst✝² : CommRing L\ninst✝¹ : IsDomain L\nn : ℕ\ninst✝ : NeZero n\ng : L ≃+* L\n⊢ ∃ m, ∀ (t : Lˣ), t ^ n = 1 → g ↑t = ↑(t ^ m)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 454,
"column": 8
} | {
"line": 454,
"column": 19
} | {
"line": 454,
"column": 20
} | [
{
"pp": "case ha\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝ : ℤ_[p]\nn : ℕ\nx : ℤ_[p]\na b : ℕ\nha : x - ↑a ∈ Ideal.span {↑(p ^ n)}\nhb : x - ↑b ∈ Ideal.span {↑(p ^ n)}\n⊢ x - ↑a ∈ Ideal.span {↑p ^ n}",
"ppTerm": "?ha",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoal... | [
"case ha\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝ : ℤ_[p]\nn : ℕ\nx : ℤ_[p]\na b : ℕ\nha : x - ↑a ∈ Ideal.span {↑(p ^ n)}\nhb : x - ↑b ∈ Ideal.span {↑(p ^ n)}\n⊢ x - ↑a ∈ Ideal.span {↑p ^ n}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 455,
"column": 8
} | {
"line": 455,
"column": 19
} | {
"line": 455,
"column": 20
} | [
{
"pp": "case hb\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝ : ℤ_[p]\nn : ℕ\nx : ℤ_[p]\na b : ℕ\nha : x - ↑a ∈ Ideal.span {↑(p ^ n)}\nhb : x - ↑b ∈ Ideal.span {↑(p ^ n)}\n⊢ x - ↑b ∈ Ideal.span {↑p ^ n}",
"ppTerm": "?hb",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoal... | [
"case hb\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝ : ℤ_[p]\nn : ℕ\nx : ℤ_[p]\na b : ℕ\nha : x - ↑a ∈ Ideal.span {↑(p ^ n)}\nhb : x - ↑b ∈ Ideal.span {↑(p ^ n)}\n⊢ x - ↑b ∈ Ideal.span {↑p ^ n}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.CyclotomicCharacter | {
"line": 248,
"column": 2
} | {
"line": 252,
"column": 9
} | {
"line": 252,
"column": 10
} | [
{
"pp": "L : Type u\ninst✝⁴ : CommRing L\ninst✝³ : IsDomain L\nn : ℕ\ninst✝² : NeZero n\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Algebra R L\nμ : L\nhμ : IsPrimitiveRoot μ n\ng : L ≃ₐ[R] L\n⊢ μ ^ (↑((autToPow R hμ) g)).val = μ ^ (↑((modularCyclotomicCharacter L ⋯) g.toRingEquiv)).val",
"ppTerm": "?m.59",... | [
"L : Type u\ninst✝⁴ : CommRing L\ninst✝³ : IsDomain L\nn : ℕ\ninst✝² : NeZero n\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Algebra R L\nμ : L\nhμ : IsPrimitiveRoot μ n\ng : L ≃ₐ[R] L\n⊢ g μ = μ ^ (modularCyclotomicCharacter.toFun n g.toRingEquiv).val"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 484,
"column": 6
} | {
"line": 484,
"column": 42
} | {
"line": 485,
"column": 4
} | [
{
"pp": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nm n : ℕ\nh : m ≤ n\nx : ℤ_[p]\n⊢ (↑(x.appr n)).cast = 0 ↔ ↑(x.appr m) = 0",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ZMod.cast",
"ZMod.commRing",
"congrArg",
"CommSemiring.toSemiring",
"Nat.i... | [
"p : ℕ\nhp_prime : Fact (Nat.Prime p)\nm n : ℕ\nh : m ≤ n\nx : ℤ_[p]\n⊢ ↑(x.appr n) = 0 ↔ ↑(x.appr m) = 0"
] | ZMod.cast_natCast (pow_dvd_pow p h), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 477,
"column": 98
} | {
"line": 491,
"column": 71
} | {
"line": 493,
"column": 0
} | [
{
"pp": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nm n : ℕ\nh : m ≤ n\n⊢ (ZMod.castHom ⋯ (ZMod (p ^ m))).comp (toZModPow n) = toZModPow m",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"PadicInt.toZModHom._proof_2",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | by
apply ZMod.ringHom_eq_of_ker_eq
ext x
rw [RingHom.mem_ker, RingHom.mem_ker]
simp only [Function.comp_apply, ZMod.castHom_apply, RingHom.coe_comp]
simp only [toZModPow, toZModHom, RingHom.coe_mk]
dsimp
rw [ZMod.cast_natCast (pow_dvd_pow p h),
zmod_congr_of_sub_mem_span m (x.appr n) (x.appr n) (x.app... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Discriminant | {
"line": 235,
"column": 4
} | {
"line": 235,
"column": 51
} | {
"line": 235,
"column": 52
} | [
{
"pp": "case e_a.refine_1\nK : Type u\nL : Type v\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : Module.Finite K L\npb : PowerBasis K L\ninst✝ : Algebra.IsSeparable K L\nE : Type v := AlgebraicClosure L\nthis : (a b : E) → Decidable (a = b) := fun a b ↦ Classical.propDecidable (a = b)\ne :... | [
"case e_a.refine_1\nK : Type u\nL : Type v\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : Module.Finite K L\npb : PowerBasis K L\ninst✝ : Algebra.IsSeparable K L\nE : Type v := AlgebraicClosure L\nthis : (a b : E) → Decidable (a = b) := fun a b ↦ Classical.propDecidable (a = b)\ne : Fin pb.dim ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Discriminant | {
"line": 261,
"column": 4
} | {
"line": 261,
"column": 29
} | {
"line": 261,
"column": 30
} | [
{
"pp": "K : Type u\nL : Type v\ninst✝¹⁰ : Field K\ninst✝⁹ : Field L\ninst✝⁸ : Algebra K L\ninst✝⁷ : Module.Finite K L\nR : Type z\ninst✝⁶ : CommRing R\ninst✝⁵ : Algebra R K\ninst✝⁴ : Algebra R L\ninst✝³ : IsScalarTower R K L\ninst✝² : Algebra.IsSeparable K L\ninst✝¹ : IsIntegrallyClosed R\ninst✝ : IsFractionRi... | [
"K : Type u\nL : Type v\ninst✝¹⁰ : Field K\ninst✝⁹ : Field L\ninst✝⁸ : Algebra K L\ninst✝⁷ : Module.Finite K L\nR : Type z\ninst✝⁶ : CommRing R\ninst✝⁵ : Algebra R K\ninst✝⁴ : Algebra R L\ninst✝³ : IsScalarTower R K L\ninst✝² : Algebra.IsSeparable K L\ninst✝¹ : IsIntegrallyClosed R\ninst✝ : IsFractionRing R K\nB : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 762,
"column": 6
} | {
"line": 762,
"column": 17
} | {
"line": 762,
"column": 18
} | [
{
"pp": "f : ℕ → ℤ\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nhi : ∀ (i : ℕ), ↑p ^ i ∣ f (i + 1) - f i\nn : ℕ\nx : ℤ_[p] := ofIntSeq f ⋯\ns : PadicSeq p := ⟨fun x ↦ ↑(f x), ⋯⟩\nhs : ↑x = mk s\ne : ℤ_[p]\nhe : x = ↑p ^ n * e + ↑(x.appr n)\nN : ℕ\nhN : ‖↑p ^ n * ↑e + ↑(↑(x.appr n) - f (N + n))‖ < ↑p ^ (-↑n)\nH : ↑p ^ (-... | [
"f : ℕ → ℤ\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nhi : ∀ (i : ℕ), ↑p ^ i ∣ f (i + 1) - f i\nn : ℕ\nx : ℤ_[p] := ofIntSeq f ⋯\ns : PadicSeq p := ⟨fun x ↦ ↑(f x), ⋯⟩\nhs : ↑x = mk s\ne : ℤ_[p]\nhe : x = ↑p ^ n * e + ↑(x.appr n)\nN : ℕ\nhN : ‖↑p ^ n * ↑e + ↑(↑(x.appr n) - f (N + n))‖ < ↑p ^ (-↑n)\nH : ↑p ^ (-↑n) < ‖↑(↑(x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 217,
"column": 4
} | {
"line": 217,
"column": 15
} | {
"line": 217,
"column": 16
} | [
{
"pp": "a : ℕ\na1 : 1 < a\nn : ℕ\no : IsPell { re := ↑a, im := 1 } := isPell_one a1\n⊢ IsPell (pellZd a1 (n + 1))",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Zsqrtd.instMul",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"Pell.pellZd_succ",
"id",
"_priv... | [
"a : ℕ\na1 : 1 < a\nn : ℕ\no : IsPell { re := ↑a, im := 1 } := isPell_one a1\n⊢ IsPell (pellZd a1 n * { re := ↑a, im := 1 })"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 302,
"column": 26
} | {
"line": 302,
"column": 61
} | {
"line": 302,
"column": 62
} | [
{
"pp": "d a : ℤ\nb c : ℤ√d\nha : a ≠ 0\nh : (↑a * b).re = (↑a * c).re ∧ (↑a * b).im = (↑a * c).im\n⊢ (↑a * b).re = (↑a * c).re → b.re = c.re",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Zsqrtd.instMul",
"Int.cast",
"Eq.mpr",
"Zsqrtd.re",
"HMul.hMul",
... | [
"d a : ℤ\nb c : ℤ√d\nha : a ≠ 0\nh : (↑a * b).re = (↑a * c).re ∧ (↑a * b).im = (↑a * c).im\n⊢ a * b.re = a * c.re → b.re = c.re"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 302,
"column": 26
} | {
"line": 302,
"column": 61
} | {
"line": 302,
"column": 62
} | [
{
"pp": "d a : ℤ\nb c : ℤ√d\nha : a ≠ 0\nh : (↑a * b).re = (↑a * c).re ∧ (↑a * b).im = (↑a * c).im\n⊢ (↑a * b).im = (↑a * c).im → b.im = c.im",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Zsqrtd.instMul",
"Int.cast",
"Eq.mpr",
"HMul.hMul",
"congrArg",
... | [
"d a : ℤ\nb c : ℤ√d\nha : a ≠ 0\nh : (↑a * b).re = (↑a * c).re ∧ (↑a * b).im = (↑a * c).im\n⊢ a * b.im = a * c.im → b.im = c.im"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 351,
"column": 4
} | {
"line": 351,
"column": 41
} | {
"line": 351,
"column": 42
} | [
{
"pp": "c d x y z w : ℕ\nxy : SqLe x c y d\nzw : SqLe z c w d\n⊢ c * (x * z) * (c * (x * z)) ≤ d * (y * w) * (d * (y * w))",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"CommSemiring.toNonUnitalCommSemiring",
"congrArg",
"id",
"... | [
"c d x y z w : ℕ\nxy : SqLe x c y d\nzw : SqLe z c w d\n⊢ c * (c * (x * (x * (z * z)))) ≤ d * (d * (y * (y * (w * w))))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 371,
"column": 2
} | {
"line": 371,
"column": 46
} | {
"line": 371,
"column": 47
} | [
{
"pp": "c d x y n : ℕ\nxy : SqLe x c y d\n⊢ SqLe (n * x) c (n * y) d",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"HMul.hMul",
"congrArg",
"mul_assoc",
"id",
"CommMagma.toMul",
"instMulNat",
"Nat.instSemi... | [
"c d x y n : ℕ\nxy : SqLe x c y d\n⊢ c * (x * (n * (n * x))) ≤ d * (y * (n * (n * y)))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.Discriminant | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 53
} | {
"line": 126,
"column": 54
} | [
{
"pp": "p k : ℕ\nK : Type u\nL : Type v\nζ : L\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime p)\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhirr : Irreducible (cyclotomic (p ^ (k + 1)) K)\nhk : p ^ (k + 1) ≠ 2\n⊢ discr K ⇑(IsPrimitiv... | [
"p k : ℕ\nK : Type u\nL : Type v\nζ : L\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime p)\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhirr : Irreducible (cyclotomic (p ^ (k + 1)) K)\nhk : p ^ (k + 1) ≠ 2\n⊢ (discr K fun i ↦ ζ ^ ↑i) = (-1) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Dioph | {
"line": 229,
"column": 17
} | {
"line": 229,
"column": 28
} | {
"line": 229,
"column": 29
} | [
{
"pp": "α✝ : Type u_1\nβ✝ : Type u_2\nα : Type ?u.9\nβ : Type ?u.11\nf : α → β\ng : Poly α\ni : α\n⊢ IsPoly fun v ↦ (proj i) (v ∘ f)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Poly",
"IsPoly",
"Function.comp",
"Poly.instFunLike",
"id",
"Poly.proj"... | [
"α✝ : Type u_1\nβ✝ : Type u_2\nα : Type ?u.9\nβ : Type ?u.11\nf : α → β\ng : Poly α\ni : α\n⊢ IsPoly fun v ↦ ↑(v (f i))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Dioph | {
"line": 229,
"column": 57
} | {
"line": 229,
"column": 68
} | {
"line": 229,
"column": 69
} | [
{
"pp": "α✝ : Type u_1\nβ✝ : Type u_2\nα : Type ?u.9\nβ : Type ?u.11\nf : α → β\ng : Poly α\nn : ℤ\n⊢ IsPoly fun v ↦ (const n) (v ∘ f)",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Poly",
"IsPoly",
"Function.comp",
"Poly.instFunLike",
"id",
"Int",
... | [
"α✝ : Type u_1\nβ✝ : Type u_2\nα : Type ?u.9\nβ : Type ?u.11\nf : α → β\ng : Poly α\nn : ℤ\n⊢ IsPoly fun v ↦ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Dioph | {
"line": 230,
"column": 25
} | {
"line": 230,
"column": 36
} | {
"line": 230,
"column": 37
} | [
{
"pp": "α✝ : Type u_1\nβ✝ : Type u_2\nα : Type ?u.9\nβ : Type ?u.11\nf✝ : α → β\ng✝ f g : Poly α\npf : IsPoly fun v ↦ f (v ∘ f✝)\npg : IsPoly fun v ↦ g (v ∘ f✝)\n⊢ IsPoly fun v ↦ (f - g) (v ∘ f✝)",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"Poly",
"IsPoly",
"HSub.hSu... | [
"α✝ : Type u_1\nβ✝ : Type u_2\nα : Type ?u.9\nβ : Type ?u.11\nf✝ : α → β\ng✝ f g : Poly α\npf : IsPoly fun v ↦ f (v ∘ f✝)\npg : IsPoly fun v ↦ g (v ∘ f✝)\n⊢ IsPoly fun v ↦ f (v ∘ f✝) - g (v ∘ f✝)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Dioph | {
"line": 231,
"column": 25
} | {
"line": 231,
"column": 36
} | {
"line": 231,
"column": 37
} | [
{
"pp": "α✝ : Type u_1\nβ✝ : Type u_2\nα : Type ?u.9\nβ : Type ?u.11\nf✝ : α → β\ng✝ f g : Poly α\npf : IsPoly fun v ↦ f (v ∘ f✝)\npg : IsPoly fun v ↦ g (v ∘ f✝)\n⊢ IsPoly fun v ↦ (f * g) (v ∘ f✝)",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"Poly.instMul",
"HMul.hMul",
... | [
"α✝ : Type u_1\nβ✝ : Type u_2\nα : Type ?u.9\nβ : Type ?u.11\nf✝ : α → β\ng✝ f g : Poly α\npf : IsPoly fun v ↦ f (v ∘ f✝)\npg : IsPoly fun v ↦ g (v ∘ f✝)\n⊢ IsPoly fun v ↦ f (v ∘ f✝) * g (v ∘ f✝)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.Discriminant | {
"line": 142,
"column": 29
} | {
"line": 142,
"column": 40
} | {
"line": 142,
"column": 41
} | [
{
"pp": "p : ℕ\nK : Type u\nL : Type v\nζ : L\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nhp : Fact (Nat.Prime p)\nhcycl : IsCyclotomicExtension {p ^ 0} K L\nhζ : IsPrimitiveRoot ζ (p ^ 0)\nhirr : Irreducible (cyclotomic (p ^ 0) K)\n⊢ ζ = 1",
"ppTerm": "?m.125",
"assigned": false,
"use... | [
"p : ℕ\nK : Type u\nL : Type v\nζ : L\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nhp : Fact (Nat.Prime p)\nhcycl : IsCyclotomicExtension {p ^ 0} K L\nhζ : IsPrimitiveRoot ζ (p ^ 0)\nhirr : Irreducible (cyclotomic (p ^ 0) K)\n⊢ ζ = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 450,
"column": 68
} | {
"line": 451,
"column": 53
} | {
"line": 453,
"column": 0
} | [
{
"pp": "d : ℤ\nn : ℤ√d\n⊢ ↑n.norm = n * star n",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"AddGroup.toSubtractionMonoid",
"Zsqrtd.instMul",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Int.cast_neg",
"Int.cas... | [] | by
ext <;> simp [norm, star, mul_comm, sub_eq_add_neg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 567,
"column": 47
} | {
"line": 567,
"column": 68
} | {
"line": 567,
"column": 69
} | [
{
"pp": "d x y z w : ℕ\nxy✝ : { re := ↑x, im := -↑y }.Nonneg\nzw✝ : { re := -↑z, im := ↑w }.Nonneg\nj k m n : ℕ\nxy : SqLe (n + m + 1) d k 1\nzw : SqLe (k + j + 1) 1 n d\nt : 1 * (k + j + 1) * (k + j + 1) ≤ 1 * k * k := Nat.le_trans zw (sqLe_of_le (Nat.le_add_right n (m + 1)) le_rfl xy)\n⊢ (k + j + 1) * (k + j ... | [
"d x y z w : ℕ\nxy✝ : { re := ↑x, im := -↑y }.Nonneg\nzw✝ : { re := -↑z, im := ↑w }.Nonneg\nj k m n : ℕ\nxy : SqLe (n + m + 1) d k 1\nzw : SqLe (k + j + 1) 1 n d\nt : 1 * (k + j + 1) * (k + j + 1) ≤ 1 * k * k := Nat.le_trans zw (sqLe_of_le (Nat.le_add_right n (m + 1)) le_rfl xy)\n⊢ (k + j + 1) * (k + j + 1) ≤ k * k... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Dioph | {
"line": 295,
"column": 6
} | {
"line": 295,
"column": 17
} | {
"line": 296,
"column": 8
} | [
{
"pp": "α : Type u\nS : Set (α → ℕ)\nl : List (Set (α → ℕ))\nIH :\n List.Forall Dioph l →\n ∃ β pl, ∀ (v : α → ℕ), List.Forall (fun S ↦ v ∈ S) l ↔ ∃ t, List.Forall (fun p ↦ p (v ⊗ t) = 0) pl\nd : List.Forall Dioph (S :: l)\ndl : List.Forall Dioph l\nβ : Type u\np : Poly (α ⊕ β)\npe : ∀ (v : α → ℕ), v ∈ S ↔... | [
"α : Type u\nS : Set (α → ℕ)\nl : List (Set (α → ℕ))\nIH :\n List.Forall Dioph l →\n ∃ β pl, ∀ (v : α → ℕ), List.Forall (fun S ↦ v ∈ S) l ↔ ∃ t, List.Forall (fun p ↦ p (v ⊗ t) = 0) pl\nd : List.Forall Dioph (S :: l)\ndl : List.Forall Dioph l\nβ : Type u\np : Poly (α ⊕ β)\npe : ∀ (v : α → ℕ), v ∈ S ↔ ∃ t, p (v ⊗... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 448,
"column": 53
} | {
"line": 448,
"column": 60
} | {
"line": 448,
"column": 61
} | [
{
"pp": "a : ℕ\na1 : 1 < a\nn k : ℕ\nhx : xn a1 (n * k) ≡ xn a1 n ^ k [MOD yn a1 n ^ 2]\nhy : yn a1 (n * k) ≡ k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3]\nL : xn a1 (n * k) * xn a1 n + d a1 * yn a1 (n * k) * yn a1 n ≡ xn a1 n ^ k * xn a1 n + 0 [MOD yn a1 n ^ 2]\nR :\n xn a1 (n * k) * yn a1 n + yn a1 (n *... | [
"a : ℕ\na1 : 1 < a\nn k : ℕ\nhx : xn a1 (n * k) ≡ xn a1 n ^ k [MOD yn a1 n ^ 2]\nhy : yn a1 (n * k) ≡ k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3]\nL : xn a1 (n * k) * xn a1 n + d a1 * yn a1 (n * k) * yn a1 n ≡ xn a1 n ^ k * xn a1 n + 0 [MOD yn a1 n ^ 2]\nR :\n xn a1 (n * k) * yn a1 n + yn a1 (n * k) * xn a1 ... | yn_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Cyclotomic.Discriminant | {
"line": 202,
"column": 51
} | {
"line": 202,
"column": 62
} | {
"line": 202,
"column": 63
} | [
{
"pp": "p : ℕ\nK : Type u\nL : Type v\nζ : L\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : IsCyclotomicExtension {p} K L\nhp : Fact (Nat.Prime p)\nhζ : IsPrimitiveRoot ζ p\nhirr : Irreducible (cyclotomic p K)\nhodd : p ≠ 2\nthis : IsCyclotomicExtension {p ^ (0 + 1)} K L\n⊢ IsPrimitiveRoot ... | [
"p : ℕ\nK : Type u\nL : Type v\nζ : L\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : IsCyclotomicExtension {p} K L\nhp : Fact (Nat.Prime p)\nhζ : IsPrimitiveRoot ζ p\nhirr : Irreducible (cyclotomic p K)\nhodd : p ≠ 2\nthis : IsCyclotomicExtension {p ^ (0 + 1)} K L\n⊢ IsPrimitiveRoot ζ p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 652,
"column": 4
} | {
"line": 652,
"column": 47
} | {
"line": 652,
"column": 48
} | [
{
"pp": "d : ℕ\na : ℤ√↑d\nx y✝ : ℕ\nh : a ≤ { re := ↑x, im := ↑(y✝ + 1) }\ny : ℕ\n⊢ SqLe y d (d * y) 1",
"ppTerm": "?m.235",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"CommSemiring.toNonUnitalCommSemiring",
"congrArg",
"Nat.instMulOneClass",
"id"... | [
"d : ℕ\na : ℤ√↑d\nx y✝ : ℕ\nh : a ≤ { re := ↑x, im := ↑(y✝ + 1) }\ny : ℕ\n⊢ d * (y * y) ≤ d * (d * (y * y))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 678,
"column": 4
} | {
"line": 678,
"column": 47
} | {
"line": 678,
"column": 48
} | [
{
"pp": "d : ℕ\na : ℤ√↑d\nha✝ : a.Nonneg\nx y : ℕ\nha : { re := ↑x, im := -↑y }.Nonneg\n⊢ SqLe (d * y) 1 x d",
"ppTerm": "?m.207",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"CommSemiring.toNonUnitalCommSemiring",
"congrArg",
"Nat.instMulOneClass",
... | [
"d : ℕ\na : ℤ√↑d\nha✝ : a.Nonneg\nx y : ℕ\nha : { re := ↑x, im := -↑y }.Nonneg\n⊢ d * (d * (y * y)) ≤ d * (x * x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 682,
"column": 4
} | {
"line": 682,
"column": 47
} | {
"line": 682,
"column": 48
} | [
{
"pp": "d : ℕ\na : ℤ√↑d\nha✝ : a.Nonneg\nx y : ℕ\nha : { re := -↑x, im := ↑y }.Nonneg\n⊢ SqLe x d (d * y) 1",
"ppTerm": "?m.233",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"CommSemiring.toNonUnitalCommSemiring",
"congrArg",
"Nat.instMulOneClass",
... | [
"d : ℕ\na : ℤ√↑d\nha✝ : a.Nonneg\nx y : ℕ\nha : { re := -↑x, im := ↑y }.Nonneg\n⊢ d * (x * x) ≤ d * (d * (y * y))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 545,
"column": 6
} | {
"line": 545,
"column": 13
} | {
"line": 545,
"column": 14
} | [
{
"pp": "a : ℕ\na1 : 1 < a\nn j : ℕ\n⊢ d a1 * yn a1 n * yn a1 (n + j) + xn a1 j ≡ 0 [MOD xn a1 n]",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Pell.yn_add",
"Pell.xn",
"congrArg",
"id",
"_private.Mathlib.NumberTheory.Pell... | [
"a : ℕ\na1 : 1 < a\nn j : ℕ\n⊢ d a1 * yn a1 n * (xn a1 n * yn a1 j + yn a1 n * xn a1 j) + xn a1 j ≡ 0 [MOD xn a1 n]"
] | yn_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 562,
"column": 39
} | {
"line": 562,
"column": 59
} | {
"line": 562,
"column": 60
} | [
{
"pp": "a : ℕ\na1 : 1 < a\nn j : ℕ\nh : j ≤ n\nh1 : xz a1 n ∣ ↑(d a1) * yz a1 n * yz a1 (n - j) + xz a1 j\n⊢ ↑(xn a1 n) ∣ ↑(d a1 * yn a1 n * yn a1 (n - j) + xn a1 j)",
"ppTerm": "?m.147",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Dv... | [
"a : ℕ\na1 : 1 < a\nn j : ℕ\nh : j ≤ n\nh1 : xz a1 n ∣ ↑(d a1) * yz a1 n * yz a1 (n - j) + xz a1 j\n⊢ ↑(xn a1 n) ∣ ↑(d a1) * ↑(yn a1 n) * ↑(yn a1 (n - j)) + ↑(xn a1 j)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 750,
"column": 8
} | {
"line": 750,
"column": 56
} | {
"line": 750,
"column": 57
} | [
{
"pp": "d : ℕ\ndnsq : Nonsquare d\nx y : ℕ\ng : ℕ := x.gcd y\ngpos : g > 0\nm n : ℕ\nh : m * g * (m * g) = d * (n * g) * (n * g)\nco : m.Coprime n\nhx : x = m * g\nhy : y = n * g\n⊢ g * g * (m * m) = g * g * (d * (n * n))",
"ppTerm": "?m.143",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"d : ℕ\ndnsq : Nonsquare d\nx y : ℕ\ng : ℕ := x.gcd y\ngpos : g > 0\nm n : ℕ\nh : m * g * (m * g) = d * (n * g) * (n * g)\nco : m.Coprime n\nhx : x = m * g\nhy : y = n * g\n⊢ m * (m * (g * g)) = d * (n * (n * (g * g)))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 870,
"column": 6
} | {
"line": 871,
"column": 65
} | {
"line": 871,
"column": 66
} | [
{
"pp": "d : ℤ\nh_nonsquare : ∀ (n : ℤ), d ≠ n * n\na : ℤ√d\nha : 0 * 0 = d * a.im * a.im\nh : d < 0\nthis : a.re * a.re = 0\n⊢ 0 = re 0 ∧ a.im = im 0",
"ppTerm": "?m.113",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Zsqrtd.re",
"Zsqrtd.instZero",
"congrArg",
"id",
... | [
"d : ℤ\nh_nonsquare : ∀ (n : ℤ), d ≠ n * n\na : ℤ√d\nha : 0 * 0 = d * a.im * a.im\nh : d < 0\nthis : a.re * a.re = 0\n⊢ a.im = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.MulChar.Duality | {
"line": 47,
"column": 4
} | {
"line": 47,
"column": 78
} | {
"line": 47,
"column": 79
} | [
{
"pp": "case refine_2\nM : Type u_1\nR : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommRing R\na : Mˣ\nx✝ : ∃ φ, φ a ≠ 1\nφ : Mˣ →* Rˣ\nhφ : (ofUnitHom φ) ↑a = 1\n⊢ φ a = 1",
"ppTerm": "?refine_2",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case refine_2\nM : Type u_1\nR : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommRing R\na : Mˣ\nx✝ : ∃ φ, φ a ≠ 1\nφ : Mˣ →* Rˣ\nhφ : (ofUnitHom φ) ↑a = 1\n⊢ φ a = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.MulChar.Duality | {
"line": 62,
"column": 17
} | {
"line": 62,
"column": 52
} | {
"line": 62,
"column": 53
} | [
{
"pp": "M : Type u_1\nR : Type u_2\ninst✝⁴ : CommMonoid M\ninst✝³ : CommRing R\ninst✝² : Finite M\ninst✝¹ : HasEnoughRootsOfUnity R (Monoid.exponent Mˣ)\ninst✝ : Nontrivial R\na : M\nha : a ≠ 1\nhu : ¬IsUnit a\n⊢ 1 a ≠ 1",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"M : Type u_1\nR : Type u_2\ninst✝⁴ : CommMonoid M\ninst✝³ : CommRing R\ninst✝² : Finite M\ninst✝¹ : HasEnoughRootsOfUnity R (Monoid.exponent Mˣ)\ninst✝ : Nontrivial R\na : M\nha : a ≠ 1\nhu : ¬IsUnit a\n⊢ 0 ≠ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 395,
"column": 37
} | {
"line": 395,
"column": 47
} | {
"line": 395,
"column": 48
} | [
{
"pp": "ξ : ℝ\nu v : ℤ\nhv : 2 ≤ v\nhv₀ hv₁ : 0 < ↑v\nhv₂ : 0 < 2 * ↑v - 1\nhcop : IsCoprime u v\nleft✝ : v = 1 → -(1 / 2) < ξ - ↑u\nh : |↑⌊ξ⌋ - ↑u / ↑v| < (↑v * (2 * ↑v - 1))⁻¹\nhf : ξ = ↑⌊ξ⌋\nh' : ↑⌊ξ⌋ - ↑u / ↑v = (↑⌊ξ⌋ * ↑v - ↑u) / ↑v\n⊢ (↑v)⁻¹ ≤ |↑⌊ξ⌋ * ↑v - ↑u| / ↑v",
"ppTerm": "?m.191",
"assigned... | [
"ξ : ℝ\nu v : ℤ\nhv : 2 ≤ v\nhv₀ hv₁ : 0 < ↑v\nhv₂ : 0 < 2 * ↑v - 1\nhcop : IsCoprime u v\nleft✝ : v = 1 → -(1 / 2) < ξ - ↑u\nh : |↑⌊ξ⌋ - ↑u / ↑v| < (↑v * (2 * ↑v - 1))⁻¹\nhf : ξ = ↑⌊ξ⌋\nh' : ↑⌊ξ⌋ - ↑u / ↑v = (↑⌊ξ⌋ * ↑v - ↑u) / ↑v\n⊢ 1 / ↑v ≤ |↑⌊ξ⌋ * ↑v - ↑u| / ↑v"
] | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 649,
"column": 25
} | {
"line": 654,
"column": 62
} | {
"line": 655,
"column": 10
} | [
{
"pp": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nlem2 : ∀ k > n, k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 ≤ 2 * n\njnn : n + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)\no : n = n ∨ n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) ≤ xn a1 ... | [] | by
let ⟨a2, s1⟩ :=
@eq_of_xn_modEq_lem2 _ a1 (n - 1)
(by rwa [tsub_add_cancel_of_le (succ_le_of_lt npos)])
have n1 : n = 1 := le_antisymm (tsub_eq_zero_iff_le.mp s1) npos
rw [ile, a2, n1]; exact ⟨rfl, rfl, rfl, rfl⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.DirichletCharacter.Orthogonality | {
"line": 72,
"column": 4
} | {
"line": 74,
"column": 11
} | {
"line": 74,
"column": 12
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝³ : CommRing R\nn : ℕ\ninst✝² : NeZero n\ninst✝¹ : HasEnoughRootsOfUnity R (Monoid.exponent (ZMod n)ˣ)\ninst✝ : IsDomain R\na : ZMod n\nha : a = 1\n⊢ ∑ χ, χ a = ↑n.totient",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"DirichletCharacter.fintyp... | [
"case pos\nR : Type u_1\ninst✝³ : CommRing R\nn : ℕ\ninst✝² : NeZero n\ninst✝¹ : HasEnoughRootsOfUnity R (Monoid.exponent (ZMod n)ˣ)\ninst✝ : IsDomain R\na : ZMod n\nha : a = 1\n⊢ ↑(Nat.card (DirichletCharacter R n)) = ↑n.totient"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 408,
"column": 6
} | {
"line": 408,
"column": 16
} | {
"line": 408,
"column": 17
} | [
{
"pp": "ξ : ℝ\nu v : ℤ\nhv : 2 ≤ v\nhcop : IsCoprime u v\nleft✝ : v = 1 → -(1 / 2) < ξ - ↑u\nh : |ξ - ↑u / ↑v| < (↑v * (2 * ↑v - 1))⁻¹\nhv₀ : 0 < ↑v\nhv₀' : 0 < 2 * ↑v - 1\nhv₁ : 0 < 2 * v - 1\n⊢ 0 < u - ⌊ξ⌋ * v ∧ u - ⌊ξ⌋ * v < v",
"ppTerm": "?m.90",
"assigned": true,
"usedConstants": [
"Int.... | [
"ξ : ℝ\nu v : ℤ\nhv : 2 ≤ v\nhcop : IsCoprime u v\nleft✝ : v = 1 → -(1 / 2) < ξ - ↑u\nh : |ξ - ↑u / ↑v| < 1 / (↑v * (2 * ↑v - 1))\nhv₀ : 0 < ↑v\nhv₀' : 0 < 2 * ↑v - 1\nhv₁ : 0 < 2 * v - 1\n⊢ 0 < u - ⌊ξ⌋ * v ∧ u - ⌊ξ⌋ * v < v"
] | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 793,
"column": 27
} | {
"line": 793,
"column": 38
} | {
"line": 793,
"column": 39
} | [
{
"pp": "a k x✝¹ y✝ : ℕ\nx✝ : ∃ (a1 : 1 < a), xn a1 k = x✝¹ ∧ yn a1 k = y✝\na1 : 1 < a\nhx : xn a1 k = x✝¹\nhy : yn a1 k = y✝\nkpos : k > 0\nx : ℕ := xn a1 k\ny : ℕ := yn a1 k\nm : ℕ := 2 * (k * y)\nu : ℕ := xn a1 m\nv : ℕ := yn a1 m\nky : k ≤ y\nyv : y * y ∣ v\nuco : u.Coprime (4 * y)\nb : ℕ\nba : b ≡ a [MOD u... | [
"a k x✝¹ y✝ : ℕ\nx✝ : ∃ (a1 : 1 < a), xn a1 k = x✝¹ ∧ yn a1 k = y✝\na1 : 1 < a\nhx : xn a1 k = x✝¹\nhy : yn a1 k = y✝\nkpos : k > 0\nx : ℕ := xn a1 k\ny : ℕ := yn a1 k\nm : ℕ := 2 * (k * y)\nu : ℕ := xn a1 m\nv : ℕ := yn a1 m\nky : k ≤ y\nyv : y * y ∣ v\nuco : u.Coprime (4 * y)\nb : ℕ\nba : b ≡ a [MOD u]\nbm1 : b ≡... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.SmoothNumbers | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 38
} | {
"line": 134,
"column": 39
} | [
{
"pp": "s : Finset ℕ\nn : ℕ\nh₀ : (List.filter (fun x ↦ decide (x ∈ s)) n.primeFactorsList).prod ≠ 0\np : ℕ\nhp : p ∈ (List.filter (fun x ↦ decide (x ∈ s)) n.primeFactorsList).prod.primeFactorsList\nH₁ : Prime p\nH₂ : p ∣ (List.filter (fun x ↦ decide (x ∈ s)) n.primeFactorsList).prod\n⊢ p ∈ s",
"ppTerm": "... | [
"s : Finset ℕ\nn : ℕ\nh₀ : (List.filter (fun x ↦ decide (x ∈ s)) n.primeFactorsList).prod ≠ 0\np : ℕ\nhp : p ∈ (List.filter (fun x ↦ decide (x ∈ s)) n.primeFactorsList).prod.primeFactorsList\nH₁ : Prime p\nH₂ : p ∣ (List.filter (fun x ↦ decide (x ∈ s)) n.primeFactorsList).prod\n⊢ p ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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