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.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": 637,
"column": 22
} | {
"line": 637,
"column": 33
} | {
"line": 637,
"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.Finite | {
"line": 222,
"column": 65
} | {
"line": 222,
"column": 88
} | {
"line": 222,
"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.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.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.Padics.RingHoms | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 13
} | {
"line": 133,
"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": 143,
"column": 2
} | {
"line": 143,
"column": 37
} | {
"line": 143,
"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": 166,
"column": 2
} | {
"line": 166,
"column": 13
} | {
"line": 166,
"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.ClassNumber.Finite | {
"line": 250,
"column": 2
} | {
"line": 253,
"column": 23
} | {
"line": 254,
"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.Padics.RingHoms | {
"line": 325,
"column": 2
} | {
"line": 325,
"column": 29
} | {
"line": 325,
"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": 332,
"column": 4
} | {
"line": 332,
"column": 50
} | {
"line": 332,
"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": 343,
"column": 45
} | {
"line": 343,
"column": 56
} | {
"line": 343,
"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.Chebyshev | {
"line": 695,
"column": 46
} | {
"line": 695,
"column": 89
} | {
"line": 696,
"column": 2
} | [
{
"pp": "x : ℝ\nhx : 2 ≤ x\na : ℕ → ℝ := (Set.ofPred 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",
... | [] | by rw [deriv_log, mul_comm, div_eq_mul_inv] | [anonymous] | Lean.Parser.Term.byTactic |
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": 456,
"column": 8
} | {
"line": 456,
"column": 19
} | {
"line": 456,
"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": 457,
"column": 8
} | {
"line": 457,
"column": 19
} | {
"line": 457,
"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.Padics.RingHoms | {
"line": 486,
"column": 6
} | {
"line": 486,
"column": 42
} | {
"line": 487,
"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": 479,
"column": 98
} | {
"line": 493,
"column": 71
} | {
"line": 495,
"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.NumberTheory.Chebyshev | {
"line": 809,
"column": 7
} | {
"line": 809,
"column": 41
} | {
"line": 809,
"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.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.RingTheory.Discriminant | {
"line": 234,
"column": 4
} | {
"line": 234,
"column": 51
} | {
"line": 234,
"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": 259,
"column": 4
} | {
"line": 259,
"column": 29
} | {
"line": 259,
"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": 767,
"column": 6
} | {
"line": 767,
"column": 17
} | {
"line": 767,
"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.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.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.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.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.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.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.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": 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.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.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": 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.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.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.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.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.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.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 |
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.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.SmoothNumbers | {
"line": 335,
"column": 2
} | {
"line": 336,
"column": 9
} | {
"line": 336,
"column": 10
} | [
{
"pp": "N : ℕ\n⊢ N.smoothNumbersᶜ \\ {0} ⊆ {n | N ≤ n}",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.ofPred",
"Compl.compl",
"Set.instSingletonSet",
"id",
"instOfNatNat",
"LE.le",
"instLENat",
"Set.in... | [
"N : ℕ\n⊢ (factoredNumbers (Finset.range N))ᶜ \\ {0} ⊆ {n | N ≤ n}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.EulerProduct.ExpLog | {
"line": 28,
"column": 38
} | {
"line": 34,
"column": 31
} | {
"line": 36,
"column": 0
} | [
{
"pp": "α : Type u_1\nf : α → ℂ\nhsum : Summable f\n⊢ Summable fun n ↦ log (1 - f n)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Complex.log_one",
"NormedCommRing.toSeminormedCommRing",
"Complex.log",
"Semiring.toModule",
"Complex.instNormedAddCommGroup"... | [] | by
have hg : DifferentiableAt ℂ (fun z ↦ log (1 - z)) 0 := by
have : 1 - 0 ∈ slitPlane := (sub_zero (1 : ℂ)).symm ▸ one_mem_slitPlane
fun_prop
have : (fun z ↦ log (1 - z)) =O[𝓝 0] id := by
simpa only [sub_zero, log_one] using! hg.isBigO_sub
exact this.comp_summable hsum | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LSeries.Convergence | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 48
} | {
"line": 49,
"column": 49
} | [
{
"pp": "f : ℕ → ℂ\ns : ℂ\nhs : abscissaOfAbsConv f < ↑s.re\n⊢ ∃ a, LSeriesSummable f ↑a ∧ a < s.re",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f : ℕ → ℂ\ns : ℂ\nhs : abscissaOfAbsConv f < ↑s.re\n⊢ ∃ a, LSeriesSummable f ↑a ∧ a < s.re"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Convergence | {
"line": 56,
"column": 15
} | {
"line": 56,
"column": 26
} | {
"line": 56,
"column": 27
} | [
{
"pp": "f : ℕ → ℂ\ns : ℂ\nhs : abscissaOfAbsConv f < ↑s.re\nx : ℝ\nhx₁ : abscissaOfAbsConv f < ↑x\nhx₂ : ↑x < ↑s.re\n⊢ x < s.re",
"ppTerm": "?m.30",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f : ℕ → ℂ\ns : ℂ\nhs : abscissaOfAbsConv f < ↑s.re\nx : ℝ\nhx₁ : abscissaOfAbsConv f < ↑x\nhx₂ : ↑x < ↑s.re\n⊢ x < s.re"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Convergence | {
"line": 60,
"column": 16
} | {
"line": 60,
"column": 27
} | {
"line": 60,
"column": 28
} | [
{
"pp": "f : ℕ → ℂ\ns : ℂ\nh : LSeriesSummable f s\n⊢ ↑s.re ∈ Real.toEReal '' {x | LSeriesSummable f ↑x}",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Set.ofPred",
"Set.mem_image._simp_1",
"EReal",
"Membership.mem",
... | [
"f : ℕ → ℂ\ns : ℂ\nh : LSeriesSummable f s\n⊢ LSeriesSummable f ↑s.re"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Convergence | {
"line": 66,
"column": 60
} | {
"line": 66,
"column": 89
} | {
"line": 66,
"column": 90
} | [
{
"pp": "f : ℕ → ℂ\nx : ℝ\nh : ∀ (y : ℝ), x < y → LSeriesSummable f ↑y\ny : EReal\nhy : y ∈ lowerBounds (Real.toEReal '' {x | LSeriesSummable f ↑x})\na : EReal\n⊢ ∀ (a : ℝ), LSeriesSummable f ↑a → y ≤ ↑a",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals... | [
"f : ℕ → ℂ\nx : ℝ\nh : ∀ (y : ℝ), x < y → LSeriesSummable f ↑y\ny : EReal\nhy : y ∈ lowerBounds (Real.toEReal '' {x | LSeriesSummable f ↑x})\na : EReal\n⊢ ∀ (a : ℝ), LSeriesSummable f ↑a → y ≤ ↑a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Convergence | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 13
} | {
"line": 110,
"column": 14
} | [
{
"pp": "f : ℕ → ℂ\nh : ∃ C, ∀ (n : ℕ), n ≠ 0 → ‖f n‖ ≤ C\n⊢ abscissaOfAbsConv f ≤ 1",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f : ℕ → ℂ\nh : ∃ C, ∀ (n : ℕ), n ≠ 0 → ‖f n‖ ≤ C\n⊢ abscissaOfAbsConv f ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Convergence | {
"line": 110,
"column": 69
} | {
"line": 110,
"column": 80
} | {
"line": 110,
"column": 81
} | [
{
"pp": "f : ℕ → ℂ\nh : ∃ C, ∀ (n : ℕ), n ≠ 0 → ‖f n‖ ≤ C\n⊢ ∃ C, ∀ (n : ℕ), n ≠ 0 → ‖f n‖ ≤ C * ↑n ^ 0",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instPow",
"Real.instLE",
"Real",
"HMul.hMul",
"Real.instZero",
... | [
"f : ℕ → ℂ\nh : ∃ C, ∀ (n : ℕ), n ≠ 0 → ‖f n‖ ≤ C\n⊢ ∃ C, ∀ (n : ℕ), ¬n = 0 → ‖f n‖ ≤ C"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Convergence | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 13
} | {
"line": 116,
"column": 14
} | [
{
"pp": "f : ℕ → ℂ\nh : f =O[atTop] fun x ↦ 1\n⊢ abscissaOfAbsConv f ≤ 1",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f : ℕ → ℂ\nh : f =O[atTop] fun x ↦ 1\n⊢ abscissaOfAbsConv f ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Convergence | {
"line": 116,
"column": 63
} | {
"line": 116,
"column": 74
} | {
"line": 116,
"column": 75
} | [
{
"pp": "f : ℕ → ℂ\nh : f =O[atTop] fun x ↦ 1\n⊢ f =O[atTop] fun n ↦ ↑n ^ 0",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instPow",
"Real.instLE",
"Real",
"Real.instZero",
"congrArg",
"Asymptotics.IsBigO",
... | [
"f : ℕ → ℂ\nh : f =O[atTop] fun x ↦ 1\n⊢ IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) atTop fun x ↦ ‖f x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Convergence | {
"line": 130,
"column": 21
} | {
"line": 130,
"column": 32
} | {
"line": 130,
"column": 33
} | [
{
"pp": "f : ℕ → ℝ\nx : ℝ\nh : (abscissaOfAbsConv fun x ↦ ↑(f x)) < ↑x\naux : term (fun x ↦ ↑(f x)) ↑x = fun n ↦ ↑(if n = 0 then 0 else f n / ↑n ^ x)\nthis : Summable fun x_1 ↦ if x_1 = 0 then 0 else f x_1 / ↑x_1 ^ x\nn : ℕ\nhn : n ∈ {0}ᶜ\n⊢ ¬n = 0",
"ppTerm": "?m.91",
"assigned": false,
"usedConsta... | [
"f : ℕ → ℝ\nx : ℝ\nh : (abscissaOfAbsConv fun x ↦ ↑(f x)) < ↑x\naux : term (fun x ↦ ↑(f x)) ↑x = fun n ↦ ↑(if n = 0 then 0 else f n / ↑n ^ x)\nthis : Summable fun x_1 ↦ if x_1 = 0 then 0 else f x_1 / ↑x_1 ^ x\nn : ℕ\nhn : n ∈ {0}ᶜ\n⊢ ¬n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Convolution | {
"line": 59,
"column": 2
} | {
"line": 59,
"column": 41
} | {
"line": 61,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Zero R\nf : ArithmeticFunction R\nn : ℕ\n⊢ (toArithmeticFunction ⇑f) n = f n",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"ite_eq_right_iff._simp_1",
"ArithmeticFunction.instFunLikeNat",
"congrArg",
"instOfNatNat",
"Arithmetic... | [] | simp +contextual [toArithmeticFunction] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.LSeries.Convolution | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 49
} | {
"line": 88,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nf g : ℕ → R\nn : ℕ\n⊢ (f ⍟ g) n = ∑ p ∈ n.divisorsAntidiagonal, f p.1 * g p.2",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"HMul.hMul",
"Nat.divisorsAntidiagonal",
"ArithmeticFunct... | [
"R : Type u_1\ninst✝ : Semiring R\nf g : ℕ → R\nn : ℕ\n⊢ (∑ x ∈ n.divisorsAntidiagonal, if x.2 = 0 then 0 else if x.1 = 0 then 0 else f x.1 * g x.2) =\n ∑ p ∈ n.divisorsAntidiagonal, f p.1 * g p.2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Convolution | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 53
} | {
"line": 147,
"column": 54
} | [
{
"pp": "f g : ℕ → ℂ\ns a b : ℂ\nhf : LSeriesHasSum f s a\nhg : LSeriesHasSum g s b\nhsum : Summable fun x ↦ term f s x.1 * term g s x.2\n⊢ LSeriesHasSum (f ⍟ g) s (a * b)",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"LS... | [
"f g : ℕ → ℂ\ns a b : ℂ\nhf : LSeriesHasSum f s a\nhg : LSeriesHasSum g s b\nhsum : Summable fun x ↦ term f s x.1 * term g s x.2\n⊢ HasSum (fun n ↦ ∑' (b : ↑((fun p ↦ p.1 * p.2) ⁻¹' {n})), term f s (↑b).1 * term g s (↑b).2) (a * b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.EulerProduct.Basic | {
"line": 213,
"column": 2
} | {
"line": 213,
"column": 48
} | {
"line": 214,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NormedCommRing R\nf : ℕ → R\ninst✝ : CompleteSpace R\nhf₁ : f 1 = 1\nhmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n\nhsum : Summable fun x ↦ ‖f x‖\nhf₀ : f 0 = 0\nthis :\n Tendsto (fun n ↦ ∏ i ∈ range n, {p | Nat.Prime p}.mulIndicator (fun p ↦ ∑' (e : ℕ), f (p ^ e)) i) ... | [
"R : Type u_1\ninst✝¹ : NormedCommRing R\nf : ℕ → R\ninst✝ : CompleteSpace R\nhf₁ : f 1 = 1\nhmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n\nhsum : Summable fun x ↦ ‖f x‖\nhf₀ : f 0 = 0\nthis :\n Tendsto (fun n ↦ ∏ i ∈ range n, {p | Nat.Prime p}.mulIndicator (fun p ↦ ∑' (e : ℕ), f (p ^ e)) i) atTop\n (... | let F : ℕ → R := fun p ↦ ∑' (e : ℕ), f (p ^ e) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.NumberTheory.EulerProduct.Basic | {
"line": 291,
"column": 2
} | {
"line": 292,
"column": 9
} | {
"line": 292,
"column": 10
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedField F\ninst✝ : CompleteSpace F\nf : ℕ →*₀ F\np : ℕ\nhp : Nat.Prime p\nhsum : Summable fun x ↦ ‖f x‖\n⊢ Summable fun a ↦ ‖f p ^ a‖",
"ppTerm": "?m.51",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"F : Type u_1\ninst✝¹ : NormedField F\ninst✝ : CompleteSpace F\nf : ℕ →*₀ F\np : ℕ\nhp : Nat.Prime p\nhsum : Summable fun x ↦ ‖f x‖\n⊢ Summable fun a ↦ ‖f p ^ a‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.EulerProduct.Basic | {
"line": 313,
"column": 2
} | {
"line": 313,
"column": 91
} | {
"line": 315,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedField F\ninst✝ : CompleteSpace F\nf : ℕ →* F\nh : ∀ {p : ℕ}, Nat.Prime p → ‖f p‖ < 1\ns : Finset ℕ\nhmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n\nH₁ : ∏ p ∈ s with Nat.Prime p, ∑' (n : ℕ), f (p ^ n) = ∏ p ∈ s with Nat.Prime p, (1 - f p)⁻¹\nH₂ : ∀ {p : ℕ}, Nat.Pri... | [] | exact H₁ ▸ summable_and_hasSum_factoredNumbers_prod_filter_prime_tsum f.map_one hmul H₂ s | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.EulerProduct.Basic | {
"line": 350,
"column": 2
} | {
"line": 351,
"column": 9
} | {
"line": 351,
"column": 10
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedField F\ninst✝ : CompleteSpace F\nf : ℕ →*₀ F\nhsum : Summable fun x ↦ ‖f x‖\nH : (fun p ↦ (1 - f ↑p)⁻¹) = fun p ↦ ∑' (e : ℕ), f (↑p ^ e)\n⊢ HasProd (fun p ↦ (1 - f ↑p)⁻¹) (∑' (n : ℕ), f n)",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Eq.mpr"... | [
"F : Type u_1\ninst✝¹ : NormedField F\ninst✝ : CompleteSpace F\nf : ℕ →*₀ F\nhsum : Summable fun x ↦ ‖f x‖\nH : (fun p ↦ (1 - f ↑p)⁻¹) = fun p ↦ ∑' (e : ℕ), f (↑p ^ e)\n⊢ HasProd (fun p ↦ ∑' (e : ℕ), f ↑p ^ e) (∑' (n : ℕ), f n)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Deriv | {
"line": 51,
"column": 39
} | {
"line": 52,
"column": 9
} | {
"line": 52,
"column": 10
} | [
{
"pp": "f : ℕ → ℂ\nn : ℕ\ns : ℂ\nhn : n ≠ 0\n⊢ HasDerivAt (fun z ↦ ↑n ^ (-z)) (-log ↑n * ↑n ^ (-s)) s",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"NormedCommRi... | [
"f : ℕ → ℂ\nn : ℕ\ns : ℂ\nhn : n ≠ 0\n⊢ HasDerivAt (fun z ↦ ↑n ^ (-z)) (log ↑n * ↑n ^ (-s) * -1) s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Deriv | {
"line": 70,
"column": 32
} | {
"line": 70,
"column": 65
} | {
"line": 70,
"column": 65
} | [
{
"pp": "f : ℕ → ℂ\ns : ℂ\nh : abscissaOfAbsConv f < ↑s.re\nx : ℝ\nhxs : x < s.re\nhf : LSeriesSummable f ↑x\ny : ℝ\nhxy : x < y\nhys : y < s.re\nS : Set ℂ := {z | y < z.re}\nh₀ : Summable fun n ↦ ‖term f (↑x) n‖\nh₁ : ∀ (n : ℕ), DifferentiableOn ℂ (fun x ↦ term f x n) S\nh₂ : IsOpen S\nn : ℕ\nz : ℂ\nhz : z ∈ S... | [] | by simpa using! (hxy.trans hz).le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LSeries.Positivity | {
"line": 63,
"column": 57
} | {
"line": 63,
"column": 85
} | {
"line": 63,
"column": 86
} | [
{
"pp": "a : ℕ → ℂ\nha₀ : 0 ≤ a\nha₁ : 0 < a 1\nx : ℝ\nhx : abscissaOfAbsConv a < ↑x\n⊢ abscissaOfAbsConv a < ↑(↑x).re",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"PartialOrder.toPreorder",
"EReal",
"id",
"Complex.ofReal",
"Complex.r... | [
"a : ℕ → ℂ\nha₀ : 0 ≤ a\nha₁ : 0 < a 1\nx : ℝ\nhx : abscissaOfAbsConv a < ↑x\n⊢ abscissaOfAbsConv a < ↑x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.EulerProduct.Basic | {
"line": 378,
"column": 2
} | {
"line": 378,
"column": 28
} | {
"line": 378,
"column": 29
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedField F\ninst✝ : CompleteSpace F\nf : ℕ →*₀ F\nhsum : Summable fun x ↦ ‖f x‖\nhmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n\nthis :\n Tendsto (fun n ↦ ∏ i ∈ range n, {p | Nat.Prime p}.mulIndicator (fun p ↦ ∑' (e : ℕ), f (p ^ e)) i) atTop\n (𝓝 (∑' (n : ℕ), f n... | [
"F : Type u_1\ninst✝¹ : NormedField F\ninst✝ : CompleteSpace F\nf : ℕ →*₀ F\nhsum : Summable fun x ↦ ‖f x‖\nhmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n\nthis :\n Tendsto (fun n ↦ ∏ i ∈ range n, {p | Nat.Prime p}.mulIndicator (fun p ↦ ∑' (e : ℕ), f (p ^ e)) i) atTop\n (𝓝 (∑' (n : ℕ), f n))\nH : ∀ (n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.EulerProduct.Basic | {
"line": 397,
"column": 4
} | {
"line": 397,
"column": 15
} | {
"line": 397,
"column": 16
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : CommGroup α\ninst✝³ : UniformSpace α\ninst✝² : IsUniformGroup α\ninst✝¹ : CompleteSpace α\ninst✝ : T0Space α\nf : ℕ → α\nhf : Multipliable fun n ↦ f ↑n\n⊢ ∏' (p : Primes) (n : ℕ), f (↑p ^ (n + 1)) = ∏' (p : Primes × ℕ), f ↑(prodNatEquiv p)",
"ppTerm": "?m.56",
"assigned":... | [
"α : Type u_1\ninst✝⁴ : CommGroup α\ninst✝³ : UniformSpace α\ninst✝² : IsUniformGroup α\ninst✝¹ : CompleteSpace α\ninst✝ : T0Space α\nf : ℕ → α\nhf : Multipliable fun n ↦ f ↑n\n⊢ ∏' (p : Primes) (n : ℕ), f (↑p ^ (n + 1)) = ∏' (p : Primes × ℕ), f ↑(prodNatEquiv p)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Positivity | {
"line": 77,
"column": 4
} | {
"line": 77,
"column": 54
} | {
"line": 77,
"column": 55
} | [
{
"pp": "a : ℕ → ℂ\nha₀ : 0 ≤ a\nha₁ : 0 < a 1\nf : ℂ → ℂ\nhf : Differentiable ℂ f\nx : ℝ\nhx : abscissaOfAbsConv a ≤ ↑x\nhf' : Set.EqOn f (LSeries a) {s | x < s.re}\ny : ℝ\nhxy : x < max x y + 1\nhxy' : abscissaOfAbsConv a < ↑(max x y) + 1\nhys : ↑(max x y) + 1 ∈ {s | x < s.re}\n⊢ 0 < f (↑(max x y) + 1)",
... | [
"a : ℕ → ℂ\nha₀ : 0 ≤ a\nha₁ : 0 < a 1\nf : ℂ → ℂ\nhf : Differentiable ℂ f\nx : ℝ\nhx : abscissaOfAbsConv a ≤ ↑x\nhf' : Set.EqOn f (LSeries a) {s | x < s.re}\ny : ℝ\nhxy : x < max x y + 1\nhxy' : abscissaOfAbsConv a < ↑(max x y) + 1\nhys : ↑(max x y) + 1 ∈ {s | x < s.re}\n⊢ 0 < LSeries a (↑(max x y) + 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Deriv | {
"line": 115,
"column": 4
} | {
"line": 115,
"column": 15
} | {
"line": 115,
"column": 16
} | [
{
"pp": "f : ℕ → ℂ\ns : ℝ\nhs : abscissaOfAbsConv (logMul f) < ↑s\nn : ℕ\nhn : max 1 ⌈Real.exp 1⌉₊ ≤ n\n⊢ 1 ≤ Real.log ↑n",
"ppTerm": "?m.85",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f : ℕ → ℂ\ns : ℝ\nhs : abscissaOfAbsConv (logMul f) < ↑s\nn : ℕ\nhn : max 1 ⌈Real.exp 1⌉₊ ≤ n\n⊢ 1 ≤ Real.log ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Positivity | {
"line": 80,
"column": 4
} | {
"line": 80,
"column": 83
} | {
"line": 81,
"column": 6
} | [
{
"pp": "case refine_1\na : ℕ → ℂ\nha₀ : 0 ≤ a\nha₁ : 0 < a 1\nf : ℂ → ℂ\nhf : Differentiable ℂ f\nx : ℝ\nhx : abscissaOfAbsConv a ≤ ↑x\nhf' : Set.EqOn f (LSeries a) {s | x < s.re}\ny : ℝ\nhxy : x < max x y + 1\nhxy' : abscissaOfAbsConv a < ↑(max x y) + 1\nhys : ↑(max x y) + 1 ∈ {s | x < s.re}\nhfx : 0 < f (↑(m... | [
"case refine_1\na : ℕ → ℂ\nha₀ : 0 ≤ a\nha₁ : 0 < a 1\nf : ℂ → ℂ\nhf : Differentiable ℂ f\nx : ℝ\nhx : abscissaOfAbsConv a ≤ ↑x\nhf' : Set.EqOn f (LSeries a) {s | x < s.re}\ny : ℝ\nhxy : x < max x y + 1\nhxy' : abscissaOfAbsConv a < ↑(max x y) + 1\nhys : ↑(max x y) + 1 ∈ {s | x < s.re}\nhfx : 0 < f (↑(max x y) + 1)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 506,
"column": 6
} | {
"line": 522,
"column": 55
} | {
"line": 523,
"column": 2
} | [
{
"pp": "case h.inr.inl.inr\nξ : ℝ\nu : ℤ\nih : ∀ m < 1, ∀ {ξ : ℝ} {u : ℤ}, ContfracLegendre.Ass ξ u ↑m → ∃ n, ↑u / ↑m = ξ.convergent n\nleft✝ : IsCoprime u ↑1\nh₁ : ↑1 = 1 → -(1 / 2) < ξ - ↑u\nh₂ : |ξ - ↑u / ↑↑1| < (↑↑1 * (2 * ↑↑1 - 1))⁻¹\nht : ξ < ↑u\n⊢ ∃ n, ↑u = ξ.convergent n",
"ppTerm": "?h.inr.inl.inr... | [] | replace h₁ := lt_sub_iff_add_lt'.mp (h₁ rfl)
have hξ₁ : ⌊ξ⌋ = u - 1 := by
rw [floor_eq_iff, cast_sub, cast_one, sub_add_cancel]
exact ⟨(((sub_lt_sub_iff_left _).mpr one_half_lt_one).trans h₁).le, ht⟩
rcases eq_or_ne ξ ⌊ξ⌋ with Hξ | Hξ
· rw [Hξ, hξ₁, cast_sub, cast_one, ← sub_eq_add_neg... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 506,
"column": 6
} | {
"line": 522,
"column": 55
} | {
"line": 523,
"column": 2
} | [
{
"pp": "case h.inr.inl.inr\nξ : ℝ\nu : ℤ\nih : ∀ m < 1, ∀ {ξ : ℝ} {u : ℤ}, ContfracLegendre.Ass ξ u ↑m → ∃ n, ↑u / ↑m = ξ.convergent n\nleft✝ : IsCoprime u ↑1\nh₁ : ↑1 = 1 → -(1 / 2) < ξ - ↑u\nh₂ : |ξ - ↑u / ↑↑1| < (↑↑1 * (2 * ↑↑1 - 1))⁻¹\nht : ξ < ↑u\n⊢ ∃ n, ↑u = ξ.convergent n",
"ppTerm": "?h.inr.inl.inr... | [] | replace h₁ := lt_sub_iff_add_lt'.mp (h₁ rfl)
have hξ₁ : ⌊ξ⌋ = u - 1 := by
rw [floor_eq_iff, cast_sub, cast_one, sub_add_cancel]
exact ⟨(((sub_lt_sub_iff_left _).mpr one_half_lt_one).trans h₁).le, ht⟩
rcases eq_or_ne ξ ⌊ξ⌋ with Hξ | Hξ
· rw [Hξ, hξ₁, cast_sub, cast_one, ← sub_eq_add_neg... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 541,
"column": 4
} | {
"line": 541,
"column": 74
} | {
"line": 542,
"column": 2
} | [
{
"pp": "case refine_1\nξ : ℝ\nq : ℚ\nh : |ξ - ↑q| < 1 / (2 * ↑q.den ^ 2)\n⊢ IsCoprime q.num ↑q.den",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Nat.Coprime",
"Rat.reduced",
"Rat.num",
"Int.isCoprime_iff_nat_coprime",
"Rat.den",
... | [] | exact isCoprime_iff_nat_coprime.mpr (natAbs_natCast q.den ▸ q.reduced) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 541,
"column": 4
} | {
"line": 541,
"column": 74
} | {
"line": 542,
"column": 2
} | [
{
"pp": "case refine_1\nξ : ℝ\nq : ℚ\nh : |ξ - ↑q| < 1 / (2 * ↑q.den ^ 2)\n⊢ IsCoprime q.num ↑q.den",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Nat.Coprime",
"Rat.reduced",
"Rat.num",
"Int.isCoprime_iff_nat_coprime",
"Rat.den",
... | [] | exact isCoprime_iff_nat_coprime.mpr (natAbs_natCast q.den ▸ q.reduced) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 541,
"column": 4
} | {
"line": 541,
"column": 74
} | {
"line": 542,
"column": 2
} | [
{
"pp": "case refine_1\nξ : ℝ\nq : ℚ\nh : |ξ - ↑q| < 1 / (2 * ↑q.den ^ 2)\n⊢ IsCoprime q.num ↑q.den",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Nat.Coprime",
"Rat.reduced",
"Rat.num",
"Int.isCoprime_iff_nat_coprime",
"Rat.den",
... | [] | exact isCoprime_iff_nat_coprime.mpr (natAbs_natCast q.den ▸ q.reduced) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 549,
"column": 26
} | {
"line": 549,
"column": 36
} | {
"line": 549,
"column": 37
} | [
{
"pp": "ξ : ℝ\nq : ℚ\nh : |ξ - ↑q| < 1 / (2 * ↑q.den ^ 2)\nhq₀ : 0 < ↑q.den\nhq₁ : 0 < ↑q.den * (2 * ↑q.den - 1)\nhq₂ : 0 < 2 * (↑q.den * ↑q.den)\n⊢ 1 / (2 * ↑q.den ^ 2) < (↑q.den * (2 * ↑q.den - 1))⁻¹",
"ppTerm": "?m.194",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
... | [
"ξ : ℝ\nq : ℚ\nh : |ξ - ↑q| < 1 / (2 * ↑q.den ^ 2)\nhq₀ : 0 < ↑q.den\nhq₁ : 0 < ↑q.den * (2 * ↑q.den - 1)\nhq₂ : 0 < 2 * (↑q.den * ↑q.den)\n⊢ 1 / (2 * ↑q.den ^ 2) < 1 / (↑q.den * (2 * ↑q.den - 1))"
] | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 160,
"column": 32
} | {
"line": 160,
"column": 42
} | {
"line": 160,
"column": 43
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\nr C : ℝ\nhC :\n ∀ᶠ (x : ℝ) in 𝓝 0,\n x ∈ Ioi 0 → ‖((fun x ↦ P.g x - P.g₀) ∘ fun x ↦ x⁻¹) x‖ ≤ C * ‖((fun x ↦ x ^ (-(r + P.k))) ∘ fun x ↦ x⁻¹) x‖\nx : ℝ\nhC' : ‖P.g x⁻¹ - P.g₀‖ ≤ C * ‖x⁻¹ ^ (-(r + P.k))‖\nhx : 0... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\nr C : ℝ\nhC :\n ∀ᶠ (x : ℝ) in 𝓝 0,\n x ∈ Ioi 0 → ‖((fun x ↦ P.g x - P.g₀) ∘ fun x ↦ x⁻¹) x‖ ≤ C * ‖((fun x ↦ x ^ (-(r + P.k))) ∘ fun x ↦ x⁻¹) x‖\nx : ℝ\nhx : 0 < x\nh_nv2 : ↑(x ^ P.k) ≠ 0\nh_nv : P.ε⁻¹ * ↑(x ^ P.k) ≠ 0\nhC... | ← one_div, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 57
} | {
"line": 56,
"column": 58
} | [
{
"pp": "t : ℝ\nht : 0 < t\n⊢ rexp (-π * t) < 1",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Real.partialOrder",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
"HMul.hMul",
"C... | [
"t : ℝ\nht : 0 < t\n⊢ 0 < π * t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 128,
"column": 4
} | {
"line": 128,
"column": 40
} | {
"line": 128,
"column": 41
} | [
{
"pp": "case refine_1\nz τ : ℂ\nhτ : 0 < τ.im\n⊢ ∀ (i : ℤ), ‖jacobiTheta₂_term i z τ‖ ≤ ↑|i| ^ 0 * rexp (-π * (τ.im * ↑i ^ 2 - 2 * |z.im| * ↑|i|))",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCom... | [
"case refine_1\nz τ : ℂ\nhτ : 0 < τ.im\n⊢ ∀ (i : ℤ), ‖jacobiTheta₂_term i z τ‖ ≤ rexp (-π * (τ.im * ↑i ^ 2 - 2 * |z.im| * ↑|i|))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 121,
"column": 6
} | {
"line": 121,
"column": 25
} | {
"line": 121,
"column": 26
} | [
{
"pp": "case e'_5\na : ℝ\nha : 0 ≤ a\nt : ℝ\nht : 0 < t\nn : ℕ\n⊢ (↑n + a) ^ 0 * rexp (-π * (↑n + a ^ 2) * t) = rexp (-π * a ^ 2 * t) * rexp (-π * t) ^ n",
"ppTerm": "?e'_5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | [
"case e'_5\na : ℝ\nha : 0 ≤ a\nt : ℝ\nht : 0 < t\nn : ℕ\n⊢ (↑n + a) ^ 0 * rexp (-π * (↑n + a ^ 2) * t) = rexp (-π * a ^ 2 * t) * rexp (↑n * (-π * t))"
] | ← Real.exp_nat_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 154,
"column": 8
} | {
"line": 154,
"column": 84
} | {
"line": 154,
"column": 85
} | [
{
"pp": "case refine_2.inr.inr.inl\nz τ : ℂ\nhτ✝ : τ.im ≤ 0\nhτ : τ.im = 0\nhz : z.im = 0\n⊢ (Summable fun x ↦ rexp (-(2 * π * ↑x * z.im))) → False",
"ppTerm": "?refine_2.inr.inr.inl",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"False",
"Real",
"Real.pi"... | [
"case refine_2.inr.inr.inl\nz τ : ℂ\nhτ✝ : τ.im ≤ 0\nhτ : τ.im = 0\nhz : z.im = 0\n⊢ 1 = 0 → False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 202,
"column": 39
} | {
"line": 202,
"column": 72
} | {
"line": 202,
"column": 73
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\ns : ℂ\nstep1 : mellin (fun t ↦ P.g (1 / t)) (-s) = mellin P.g s\nstep2 : mellin (fun t ↦ ↑t ^ (-↑P.k) • P.g (1 / t)) (↑P.k - s) = mellin P.g s\nstep3 : mellin (fun t ↦ P.ε • ↑t ^ (-↑P.k) • P.g (1 / t)) (↑P.k - s) = ... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\ns : ℂ\nstep1 : mellin (fun t ↦ P.g (1 / t)) (-s) = mellin P.g s\nstep2 : mellin (fun t ↦ ↑t ^ (-↑P.k) • P.g (1 / t)) (↑P.k - s) = mellin P.g s\nstep3 : mellin (fun t ↦ P.ε • ↑t ^ (-↑P.k) • P.g (1 / t)) (↑P.k - s) = P.ε • mellin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 168,
"column": 6
} | {
"line": 168,
"column": 45
} | {
"line": 168,
"column": 46
} | [
{
"pp": "a : ℝ\nha : 0 ≤ a\nt : ℝ\nht : 0 < t\n⊢ ‖rexp (-π * t)‖ < 1",
"ppTerm": "?m.153",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"Real.pi",
"HMul.hMul",
"congrArg",
"Real.instLT",
"id",
"Real.exp",
"Real.instOn... | [
"a : ℝ\nha : 0 ≤ a\nt : ℝ\nht : 0 < t\n⊢ rexp (-π * t) < 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 214,
"column": 2
} | {
"line": 214,
"column": 22
} | {
"line": 214,
"column": 23
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\nhP : IsStrongFEPair P\nr : ℝ\n⊢ P.f =O[atTop] fun x ↦ x ^ r",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\nhP : IsStrongFEPair P\nr : ℝ\n⊢ P.f =O[atTop] fun x ↦ x ^ r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 13
} | {
"line": 218,
"column": 14
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\nhP : IsStrongFEPair P\nr : ℝ\n⊢ P.f =O[𝓝[>] 0] fun x ↦ x ^ r",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\nhP : IsStrongFEPair P\nr : ℝ\n⊢ P.f =O[𝓝[>] 0] fun x ↦ x ^ r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 171,
"column": 28
} | {
"line": 171,
"column": 47
} | {
"line": 171,
"column": 48
} | [
{
"pp": "case e'_5\na : ℝ\nha : 0 ≤ a\nt : ℝ\nht : 0 < t\nh0' : ‖rexp (-π * t)‖ < 1\nn : ℕ\n⊢ ↑n * rexp (-π * (↑n + a ^ 2) * t) = ↑n * rexp (-π * t) ^ n * rexp (-π * a ^ 2 * t)",
"ppTerm": "?e'_5",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring"... | [
"case e'_5\na : ℝ\nha : 0 ≤ a\nt : ℝ\nht : 0 < t\nh0' : ‖rexp (-π * t)‖ < 1\nn : ℕ\n⊢ ↑n * rexp (-π * (↑n + a ^ 2) * t) = ↑n * rexp (↑n * (-π * t)) * rexp (-π * a ^ 2 * t)"
] | ← Real.exp_nat_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.MellinEqDirichlet | {
"line": 76,
"column": 35
} | {
"line": 76,
"column": 46
} | {
"line": 76,
"column": 47
} | [
{
"pp": "ι : Type u_1\ninst✝ : Countable ι\na : ι → ℂ\nq : ι → ℝ\nF : ℝ → ℂ\ns : ℂ\nhq : ∀ (i : ι), a i = 0 ∨ 0 < q i\nhs : 0 < s.re\nhF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a i * ↑(rexp (-π * q i * t))) (F t)\nh_sum : Summable fun i ↦ ‖a i‖ / q i ^ s.re\nhp : ∀ (i : ι), a i = 0 ∨ 0 < π * q i\n⊢ ∀ t ∈ Ioi 0, HasSum (... | [
"ι : Type u_1\ninst✝ : Countable ι\na : ι → ℂ\nq : ι → ℝ\nF : ℝ → ℂ\ns : ℂ\nhq : ∀ (i : ι), a i = 0 ∨ 0 < q i\nhs : 0 < s.re\nhF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a i * ↑(rexp (-π * q i * t))) (F t)\nh_sum : Summable fun i ↦ ‖a i‖ / q i ^ s.re\nhp : ∀ (i : ι), a i = 0 ∨ 0 < π * q i\n⊢ ∀ (t : ℝ), 0 < t → HasSum (fun i ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 176,
"column": 8
} | {
"line": 176,
"column": 27
} | {
"line": 176,
"column": 28
} | [
{
"pp": "case e'_5\na : ℝ\nha : 0 ≤ a\nt : ℝ\nht : 0 < t\nn : ℕ\n⊢ a * rexp (-π * (↑n + a ^ 2) * t) = a * rexp (-π * a ^ 2 * t) * rexp (-π * t) ^ n",
"ppTerm": "?e'_5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Re... | [
"case e'_5\na : ℝ\nha : 0 ≤ a\nt : ℝ\nht : 0 < t\nn : ℕ\n⊢ a * rexp (-π * (↑n + a ^ 2) * t) = a * rexp (-π * a ^ 2 * t) * rexp (↑n * (-π * t))"
] | ← Real.exp_nat_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
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