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.LegendreSymbol.Basic | {
"line": 63,
"column": 25
} | {
"line": 63,
"column": 45
} | {
"line": 63,
"column": 45
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\na : ZMod p\nha : a ≠ 0\n⊢ Units.mk0 a ha ^ (p / 2) = 1 ↔ a ^ (p / 2) = 1",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"instHDiv",
"Monoid.toMulOneClass",
"congrArg",
"AddGroupWithOne.toAddMonoidWit... | [] | simp [Units.ext_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.LegendreSymbol.Basic | {
"line": 63,
"column": 25
} | {
"line": 63,
"column": 45
} | {
"line": 63,
"column": 45
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\na : ZMod p\nha : a ≠ 0\n⊢ Units.mk0 a ha ^ (p / 2) = 1 ↔ a ^ (p / 2) = 1",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"instHDiv",
"Monoid.toMulOneClass",
"congrArg",
"AddGroupWithOne.toAddMonoidWit... | [] | simp [Units.ext_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LegendreSymbol.Basic | {
"line": 63,
"column": 25
} | {
"line": 63,
"column": 45
} | {
"line": 63,
"column": 45
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\na : ZMod p\nha : a ≠ 0\n⊢ Units.mk0 a ha ^ (p / 2) = 1 ↔ a ^ (p / 2) = 1",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"instHDiv",
"Monoid.toMulOneClass",
"congrArg",
"AddGroupWithOne.toAddMonoidWit... | [] | simp [Units.ext_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LegendreSymbol.Basic | {
"line": 119,
"column": 6
} | {
"line": 120,
"column": 75
} | {
"line": 121,
"column": 6
} | [
{
"pp": "case pos\np : ℕ\ninst✝ : Fact (Nat.Prime p)\na : ℤ\nhc : ringChar (ZMod p) = 2\nha : ↑a = 0\n⊢ ↑(legendreSym p a) = ↑a ^ (p / 2)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Nat.Prime",
"instHDiv",
"legendreSym._proof_1",
... | [
"case pos\np : ℕ\ninst✝ : Fact (Nat.Prime p)\na : ℤ\nhc : ringChar (ZMod p) = 2\nha : ↑a = 0\n⊢ ↑0 = 0"
] | rw [legendreSym, ha, quadraticChar_zero,
zero_pow (Nat.div_pos (@Fact.out p.Prime).two_le (succ_pos 1)).ne'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.FLT.Three | {
"line": 118,
"column": 4
} | {
"line": 118,
"column": 20
} | {
"line": 118,
"column": 21
} | [
{
"pp": "case inr.inr\na b c : ℤ\nha : a ≠ 0\nh3a : 3 ∣ a\nHgcd : {a, b, c}.gcd id = 1\nH : ∀ (a b c : ℤ), c ≠ 0 → ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → a ^ 3 + b ^ 3 ≠ c ^ 3\nHF : a ^ 3 + b ^ 3 + c ^ 3 = 0\nx : ℤ\nh3b : 3 ∣ b\nhx : x = c\n⊢ 3 ∣ id x",
"ppTerm": "?inr.inr",
"assigned": true,
"us... | [
"case inr.inr\na b c : ℤ\nha : a ≠ 0\nh3a : 3 ∣ a\nHgcd : {a, b, c}.gcd id = 1\nH : ∀ (a b c : ℤ), c ≠ 0 → ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → a ^ 3 + b ^ 3 ≠ c ^ 3\nHF : a ^ 3 + b ^ 3 + c ^ 3 = 0\nx : ℤ\nh3b : 3 ∣ b\nhx : x = c\n⊢ 3 ∣ c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 803,
"column": 6
} | {
"line": 803,
"column": 95
} | {
"line": 804,
"column": 8
} | [
{
"pp": "case neg.refine_1.smul\nA : Type u_1\nB : Type u_3\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\ninst✝⁶ : Algebra A B\ninst✝⁵ : IsDomain A\ninst✝⁴ : IsDedekindDomain A\ninst✝³ : IsDedekindDomain B\ninst✝² : IsTorsionFree A B\ninst✝¹ : Module.Finite A B\ninst✝ : Algebra.IsSeparable (FractionRing A) (Fracti... | [
"case neg.refine_1.smul\nA : Type u_1\nB : Type u_3\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\ninst✝⁶ : Algebra A B\ninst✝⁵ : IsDomain A\ninst✝⁴ : IsDedekindDomain A\ninst✝³ : IsDedekindDomain B\ninst✝² : IsTorsionFree A B\ninst✝¹ : Module.Finite A B\ninst✝ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\np... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum | {
"line": 85,
"column": 4
} | {
"line": 85,
"column": 87
} | {
"line": 86,
"column": 6
} | [
{
"pp": "F : Type u_1\ninst✝⁵ : Field F\ninst✝⁴ : Fintype F\ninst✝³ : DecidableEq F\nhF : ringChar F ≠ 2\nF' : Type u_2\ninst✝² : Field F'\ninst✝¹ : Fintype F'\ninst✝ : DecidableEq F'\nhF' : ringChar F' ≠ 2\nh : ringChar F' ≠ ringChar F\nχ : MulChar F F' := (quadraticChar F).ringHomComp (algebraMap ℤ F')\na : F... | [
"F : Type u_1\ninst✝⁵ : Field F\ninst✝⁴ : Fintype F\ninst✝³ : DecidableEq F\nhF : ringChar F ≠ 2\nF' : Type u_2\ninst✝² : Field F'\ninst✝¹ : Fintype F'\ninst✝ : DecidableEq F'\nhF' : ringChar F' ≠ 2\nh : ringChar F' ≠ ringChar F\nχ : MulChar F F' := (quadraticChar F).ringHomComp (algebraMap ℤ F')\na : Fˣ\nha : (qua... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic | {
"line": 236,
"column": 6
} | {
"line": 236,
"column": 54
} | {
"line": 236,
"column": 55
} | [
{
"pp": "case neg\nF : Type u_1\ninst✝² : Field F\ninst✝¹ : Fintype F\ninst✝ : DecidableEq F\nhF : ringChar F ≠ 2\na : F\nh₀ : ¬a = 0\ns : Finset F := {x | x ^ 2 = a}.toFinset\nh : ¬IsSquare a\n⊢ ∀ (x : F), x ∉ s",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset... | [
"case neg\nF : Type u_1\ninst✝² : Field F\ninst✝¹ : Fintype F\ninst✝ : DecidableEq F\nhF : ringChar F ≠ 2\na : F\nh₀ : ¬a = 0\ns : Finset F := {x | x ^ 2 = a}.toFinset\nh : ¬IsSquare a\n⊢ ∀ (x : F), ¬a = x ^ 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.FLT.Three | {
"line": 178,
"column": 4
} | {
"line": 178,
"column": 15
} | {
"line": 178,
"column": 16
} | [
{
"pp": "case refine_5\nK : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nH : FermatLastTheoremForThreeGen hζ\na b c : ℤ\nhc : c ≠ 0\nha : ¬3 ∣ a\nhb : ¬3 ∣ b\nx✝ : 3 ∣ c\nhcoprime : IsCoprime a b\nh : a ^ 3 + b ^ 3 = c ^ 3\nx : ℤ\nhx... | [
"case refine_5\nK : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nH : FermatLastTheoremForThreeGen hζ\na b c : ℤ\nhc : c ≠ 0\nha : ¬3 ∣ a\nhb : ¬3 ∣ b\nx✝ : 3 ∣ c\nhcoprime : IsCoprime a b\nh : a ^ 3 + b ^ 3 = c ^ 3\nx : ℤ\nhx : c = 3 * x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 826,
"column": 4
} | {
"line": 826,
"column": 35
} | {
"line": 826,
"column": 36
} | [
{
"pp": "A : Type u_1\nB : Type u_3\ninst✝¹² : CommRing A\ninst✝¹¹ : CommRing B\ninst✝¹⁰ : Algebra A B\ninst✝⁹ : IsDomain A\ninst✝⁸ : IsDedekindDomain A\ninst✝⁷ : IsDedekindDomain B\ninst✝⁶ : IsTorsionFree A B\ninst✝⁵ : Module.Finite A B\ninst✝⁴ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\np : Ideal... | [
"A : Type u_1\nB : Type u_3\ninst✝¹² : CommRing A\ninst✝¹¹ : CommRing B\ninst✝¹⁰ : Algebra A B\ninst✝⁹ : IsDomain A\ninst✝⁸ : IsDedekindDomain A\ninst✝⁷ : IsDedekindDomain B\ninst✝⁶ : IsTorsionFree A B\ninst✝⁵ : Module.Finite A B\ninst✝⁴ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\np : Ideal A\ninst✝³ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 831,
"column": 4
} | {
"line": 831,
"column": 51
} | {
"line": 831,
"column": 52
} | [
{
"pp": "case refine_1\nA : Type u_1\nB : Type u_3\ninst✝¹² : CommRing A\ninst✝¹¹ : CommRing B\ninst✝¹⁰ : Algebra A B\ninst✝⁹ : IsDomain A\ninst✝⁸ : IsDedekindDomain A\ninst✝⁷ : IsDedekindDomain B\ninst✝⁶ : IsTorsionFree A B\ninst✝⁵ : Module.Finite A B\ninst✝⁴ : Algebra.IsSeparable (FractionRing A) (FractionRin... | [
"case refine_1\nA : Type u_1\nB : Type u_3\ninst✝¹² : CommRing A\ninst✝¹¹ : CommRing B\ninst✝¹⁰ : Algebra A B\ninst✝⁹ : IsDomain A\ninst✝⁸ : IsDedekindDomain A\ninst✝⁷ : IsDedekindDomain B\ninst✝⁶ : IsTorsionFree A B\ninst✝⁵ : Module.Finite A B\ninst✝⁴ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\np : Id... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 844,
"column": 6
} | {
"line": 844,
"column": 17
} | {
"line": 844,
"column": 18
} | [
{
"pp": "case refine_1\nA : Type u_1\nB : Type u_3\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : Algebra A B\ninst✝⁸ : IsDomain A\ninst✝⁷ : IsDedekindDomain A\ninst✝⁶ : IsDedekindDomain B\ninst✝⁵ : IsTorsionFree A B\ninst✝⁴ : Module.Finite A B\ninst✝³ : Algebra.IsSeparable (FractionRing A) (FractionRing... | [
"case refine_1\nA : Type u_1\nB : Type u_3\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : Algebra A B\ninst✝⁸ : IsDomain A\ninst✝⁷ : IsDedekindDomain A\ninst✝⁶ : IsDedekindDomain B\ninst✝⁵ : IsTorsionFree A B\ninst✝⁴ : Module.Finite A B\ninst✝³ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\np : Ide... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 857,
"column": 4
} | {
"line": 857,
"column": 15
} | {
"line": 857,
"column": 16
} | [
{
"pp": "case pos\nA : Type u_1\nB : Type u_3\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : Algebra A B\ninst✝⁸ : IsDomain A\ninst✝⁷ : IsDedekindDomain A\ninst✝⁶ : IsDedekindDomain B\ninst✝⁵ : IsTorsionFree A B\ninst✝⁴ : Module.Finite A B\ninst✝³ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\n... | [
"case pos\nA : Type u_1\nB : Type u_3\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : Algebra A B\ninst✝⁸ : IsDomain A\ninst✝⁷ : IsDedekindDomain A\ninst✝⁶ : IsDedekindDomain B\ninst✝⁵ : IsTorsionFree A B\ninst✝⁴ : Module.Finite A B\ninst✝³ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\np : Ideal A\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 863,
"column": 42
} | {
"line": 863,
"column": 59
} | {
"line": 863,
"column": 60
} | [
{
"pp": "A : Type u_1\nB : Type u_3\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : Algebra A B\ninst✝⁸ : IsDomain A\ninst✝⁷ : IsDedekindDomain A\ninst✝⁶ : IsDedekindDomain B\ninst✝⁵ : IsTorsionFree A B\ninst✝⁴ : Module.Finite A B\ninst✝³ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\np : Ideal ... | [
"A : Type u_1\nB : Type u_3\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : Algebra A B\ninst✝⁸ : IsDomain A\ninst✝⁷ : IsDedekindDomain A\ninst✝⁶ : IsDedekindDomain B\ninst✝⁵ : IsTorsionFree A B\ninst✝⁴ : Module.Finite A B\ninst✝³ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\np : Ideal A\ninst✝² : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity | {
"line": 129,
"column": 4
} | {
"line": 129,
"column": 67
} | {
"line": 129,
"column": 68
} | [
{
"pp": "case inr\np q : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Fact (Nat.Prime q)\nhp : p ≠ 2\nhq : q ≠ 2\nh : p ≠ q\nqr : legendreSym q ↑p * legendreSym p ↑q * legendreSym p ↑q = (-1) ^ (p / 2 * (q / 2)) * legendreSym p ↑q\nthis : ↑↑q ≠ 0\n⊢ legendreSym q ↑p = (-1) ^ (p / 2 * (q / 2)) * legendreSym p ↑q",
... | [
"case inr\np q : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Fact (Nat.Prime q)\nhp : p ≠ 2\nhq : q ≠ 2\nh : p ≠ q\nqr : legendreSym q ↑p * legendreSym p ↑q * legendreSym p ↑q = (-1) ^ (p / 2 * (q / 2)) * legendreSym p ↑q\nthis : ↑↑q ≠ 0\n⊢ legendreSym q ↑p = (-1) ^ (p / 2 * (q / 2)) * legendreSym p ↑q"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Fermat | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 35
} | {
"line": 105,
"column": 36
} | [
{
"pp": "case inr\nm n : ℕ\nhmn : m ≠ n\nthis : ∀ {m n : ℕ}, m ≠ n → m < n → m.fermatNumber.Coprime n.fermatNumber\nhmn' : ¬m < n\n⊢ m.fermatNumber.Coprime n.fermatNumber",
"ppTerm": "?inr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case inr\nm n : ℕ\nhmn : m ≠ n\nthis : ∀ {m n : ℕ}, m ≠ n → m < n → m.fermatNumber.Coprime n.fermatNumber\nhmn' : ¬m < n\n⊢ m.fermatNumber.Coprime n.fermatNumber"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 876,
"column": 49
} | {
"line": 876,
"column": 60
} | {
"line": 876,
"column": 61
} | [
{
"pp": "A : Type u_1\nB : Type u_3\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : Algebra A B\ninst✝⁸ : IsDomain A\ninst✝⁷ : IsDedekindDomain A\ninst✝⁶ : IsDedekindDomain B\ninst✝⁵ : IsTorsionFree A B\ninst✝⁴ : Module.Finite A B\ninst✝³ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\np : Ideal ... | [
"A : Type u_1\nB : Type u_3\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : Algebra A B\ninst✝⁸ : IsDomain A\ninst✝⁷ : IsDedekindDomain A\ninst✝⁶ : IsDedekindDomain B\ninst✝⁵ : IsTorsionFree A B\ninst✝⁴ : Module.Finite A B\ninst✝³ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\np : Ideal A\ninst✝² : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 878,
"column": 59
} | {
"line": 878,
"column": 70
} | {
"line": 878,
"column": 71
} | [
{
"pp": "A : Type u_1\nB : Type u_3\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : Algebra A B\ninst✝⁸ : IsDomain A\ninst✝⁷ : IsDedekindDomain A\ninst✝⁶ : IsDedekindDomain B\ninst✝⁵ : IsTorsionFree A B\ninst✝⁴ : Module.Finite A B\ninst✝³ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\np : Ideal ... | [
"A : Type u_1\nB : Type u_3\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : Algebra A B\ninst✝⁸ : IsDomain A\ninst✝⁷ : IsDedekindDomain A\ninst✝⁶ : IsDedekindDomain B\ninst✝⁵ : IsTorsionFree A B\ninst✝⁴ : Module.Finite A B\ninst✝³ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\np : Ideal A\ninst✝² : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 879,
"column": 74
} | {
"line": 879,
"column": 85
} | {
"line": 879,
"column": 86
} | [
{
"pp": "A : Type u_1\nB : Type u_3\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : Algebra A B\ninst✝⁸ : IsDomain A\ninst✝⁷ : IsDedekindDomain A\ninst✝⁶ : IsDedekindDomain B\ninst✝⁵ : IsTorsionFree A B\ninst✝⁴ : Module.Finite A B\ninst✝³ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\np : Ideal ... | [
"A : Type u_1\nB : Type u_3\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : Algebra A B\ninst✝⁸ : IsDomain A\ninst✝⁷ : IsDedekindDomain A\ninst✝⁶ : IsDedekindDomain B\ninst✝⁵ : IsTorsionFree A B\ninst✝⁴ : Module.Finite A B\ninst✝³ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\np : Ideal A\ninst✝² : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Fermat | {
"line": 170,
"column": 2
} | {
"line": 170,
"column": 70
} | {
"line": 170,
"column": 71
} | [
{
"pp": "a n p : ℕ\nhp : Prime p\nhp2 : p ≠ 2\nhpdvd : p ∣ a ^ 2 ^ n + 1\nthis✝ : Fact (2 < p)\nthis : Fact (Prime p)\nha1 : ↑a ^ 2 ^ n = -1\nha0 : ↑a ≠ 0\nha : orderOf ↑a = 2 ^ (n + 1)\n⊢ ∃ k, p = k * 2 ^ (n + 1) + 1",
"ppTerm": "?m.187",
"assigned": false,
"usedConstants": [],
"usedFVars": [],... | [
"a n p : ℕ\nhp : Prime p\nhp2 : p ≠ 2\nhpdvd : p ∣ a ^ 2 ^ n + 1\nthis✝ : Fact (2 < p)\nthis : Fact (Prime p)\nha1 : ↑a ^ 2 ^ n = -1\nha0 : ↑a ≠ 0\nha : orderOf ↑a = 2 ^ (n + 1)\n⊢ ∃ k, p = k * 2 ^ (n + 1) + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.FermatPsp | {
"line": 145,
"column": 20
} | {
"line": 145,
"column": 41
} | {
"line": 146,
"column": 4
} | [
{
"pp": "a b : ℕ\nha : 2 ≤ a\nhb : 2 < b\n⊢ a ^ 2 * a = a ^ 3",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"instPowNat",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"Nat.instMonoid",
"Nat.pow_succ",
"id",
"instMulNat",
"instOfNatNat",
"... | [] | rw [Nat.pow_succ a 2] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.FermatPsp | {
"line": 145,
"column": 20
} | {
"line": 145,
"column": 41
} | {
"line": 146,
"column": 4
} | [
{
"pp": "a b : ℕ\nha : 2 ≤ a\nhb : 2 < b\n⊢ a ^ 2 * a = a ^ 3",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"instPowNat",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"Nat.instMonoid",
"Nat.pow_succ",
"id",
"instMulNat",
"instOfNatNat",
"... | [] | rw [Nat.pow_succ a 2] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.FermatPsp | {
"line": 145,
"column": 20
} | {
"line": 145,
"column": 41
} | {
"line": 146,
"column": 4
} | [
{
"pp": "a b : ℕ\nha : 2 ≤ a\nhb : 2 < b\n⊢ a ^ 2 * a = a ^ 3",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"instPowNat",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"Nat.instMonoid",
"Nat.pow_succ",
"id",
"instMulNat",
"instOfNatNat",
"... | [] | rw [Nat.pow_succ a 2] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.FermatPsp | {
"line": 150,
"column": 36
} | {
"line": 150,
"column": 62
} | {
"line": 150,
"column": 63
} | [
{
"pp": "b p : ℕ\nx✝ : 2 ≤ b\nhp : Odd p\n⊢ b - 1 ∣ b ^ p - 1",
"ppTerm": "?m.108",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"b p : ℕ\nx✝ : 2 ≤ b\nhp : Odd p\n⊢ b - 1 ∣ b ^ p - 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.FermatPsp | {
"line": 151,
"column": 36
} | {
"line": 151,
"column": 62
} | {
"line": 151,
"column": 63
} | [
{
"pp": "b p : ℕ\nx✝ : 2 ≤ b\nhp : Odd p\nq₁ : b - 1 ∣ b ^ p - 1\n⊢ b + 1 ∣ b ^ p + 1",
"ppTerm": "?m.140",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"b p : ℕ\nx✝ : 2 ≤ b\nhp : Odd p\nq₁ : b - 1 ∣ b ^ p - 1\n⊢ b + 1 ∣ b ^ p + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.FLT.Three | {
"line": 289,
"column": 2
} | {
"line": 289,
"column": 38
} | {
"line": 290,
"column": 4
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nS' : Solution' hζ\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\n⊢ 2 ≤ S'.multiplicity",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"id",
"instOfNatNat",
"LE.le",
"instLENa... | [
"K : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nS' : Solution' hζ\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\n⊢ 2 ≤ _root_.multiplicity λ S'.c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.NatInt | {
"line": 63,
"column": 4
} | {
"line": 63,
"column": 46
} | {
"line": 64,
"column": 4
} | [
{
"pp": "case refine_1.inr.inl\n⊢ (maximalIdeal ℕ).IsPrime",
"ppTerm": "?refine_1.inr.inl",
"assigned": true,
"usedConstants": [
"IsLocalRing.maximalIdeal",
"Ideal.IsMaximal.isPrime",
"instIsLocalRingNat",
"IsLocalRing.maximalIdeal.isMaximal",
"Nat",
"Nat.instComm... | [
"case refine_1.inr.inr\np : ℕ\nhp : Nat.Prime p\n⊢ (span {p}).IsPrime"
] | · exact (maximalIdeal.isMaximal ℕ).isPrime | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.FLT.Three | {
"line": 307,
"column": 32
} | {
"line": 307,
"column": 49
} | {
"line": 309,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nS' : Solution' hζ\n⊢ S'.a ^ 3 + S'.a ^ 2 * S'.b * (↑η ^ 2 + ↑η + 1) + S'.a * S'.b ^ 2 * (↑η ^ 2 + ↑η + 1) + S'.b ^ 3 = S'.a ^ 3 + S'.b ^ 3",
"ppTerm": "?m.606",
"assigned": true,
"usedConstants": [
"IsPrimitiveRoot.toInte... | [] | rw [eta_sq]; ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.FLT.Three | {
"line": 307,
"column": 32
} | {
"line": 307,
"column": 49
} | {
"line": 309,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nS' : Solution' hζ\n⊢ S'.a ^ 3 + S'.a ^ 2 * S'.b * (↑η ^ 2 + ↑η + 1) + S'.a * S'.b ^ 2 * (↑η ^ 2 + ↑η + 1) + S'.b ^ 3 = S'.a ^ 3 + S'.b ^ 3",
"ppTerm": "?m.606",
"assigned": true,
"usedConstants": [
"IsPrimitiveRoot.toInte... | [] | rw [eta_sq]; ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.FermatPsp | {
"line": 215,
"column": 4
} | {
"line": 215,
"column": 39
} | {
"line": 215,
"column": 40
} | [
{
"pp": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : ¬p ∣ b * (b ^ 2 - 1)\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nhA : p < A\nhi_A : 1 < A\nhi_B : 1 < B\nhi_b : 0 < b\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 ≤ b ^ (2 * p)\nhi_bpowpsubone : 1 ≤ b ^... | [
"b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : ¬p ∣ b * (b ^ 2 - 1)\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nhA : p < A\nhi_A : 1 < A\nhi_B : 1 < B\nhi_b : 0 < b\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 ≤ b ^ (2 * p)\nhi_bpowpsubone : 1 ≤ b ^ (p - 1)\np_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.FrobeniusNumber | {
"line": 142,
"column": 28
} | {
"line": 142,
"column": 63
} | {
"line": 142,
"column": 64
} | [
{
"pp": "s : Set ℕ\nh0 : ¬setGcd s = 0\nt : Finset ℕ\nhts : ↑t ⊆ s\na : ↥t → ℤ\neq : ∑ i, a i • ↑↑i = ↑(setGcd s)\nx : ℕ\nhxs : x ∈ s\nhx : x ≠ 0\nn : ℕ := x / setGcd s * ∑ i, (-a i).toNat * ↑i\n⊢ ↑(insert x t) ⊆ s",
"ppTerm": "?m.213",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"con... | [
"s : Set ℕ\nh0 : ¬setGcd s = 0\nt : Finset ℕ\nhts : ↑t ⊆ s\na : ↥t → ℤ\neq : ∑ i, a i • ↑↑i = ↑(setGcd s)\nx : ℕ\nhxs : x ∈ s\nhx : x ≠ 0\nn : ℕ := x / setGcd s * ∑ i, (-a i).toNat * ↑i\n⊢ x ∈ s ∧ ↑t ⊆ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.FermatPsp | {
"line": 248,
"column": 6
} | {
"line": 248,
"column": 32
} | {
"line": 248,
"column": 33
} | [
{
"pp": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : ¬p ∣ b * (b ^ 2 - 1)\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nhA : p < A\nhi_A : 1 < A\nhi_B : 1 < B\nhi_b : 0 < b\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 ≤ b ^ (2 * p)\nhi_bpowpsubone : 1 ≤ b ^... | [
"b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : ¬p ∣ b * (b ^ 2 - 1)\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nhA : p < A\nhi_A : 1 < A\nhi_B : 1 < B\nhi_b : 0 < b\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 ≤ b ^ (2 * p)\nhi_bpowpsubone : 1 ≤ b ^ (p - 1)\nAB... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.FrobeniusNumber | {
"line": 191,
"column": 6
} | {
"line": 191,
"column": 17
} | {
"line": 191,
"column": 18
} | [
{
"pp": "case refine_1\nm n✝¹ : ℕ\ns✝ t✝ : Set ℕ\nn✝ : ℕ\ns : Submodule ℕ ℕ\nt : Finset ℕ\nn : ℕ\nhts : ↑t ⊆ ↑s\nhn : ∀ m ≥ n, setGcd ↑s ∣ m → m ∈ Ideal.span ↑t\n⊢ ↑(t ∪ {m ∈ Finset.range n | m ∈ s}) ⊆ ↑s",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule"... | [
"case refine_1\nm n✝¹ : ℕ\ns✝ t✝ : Set ℕ\nn✝ : ℕ\ns : Submodule ℕ ℕ\nt : Finset ℕ\nn : ℕ\nhts : ↑t ⊆ ↑s\nhn : ∀ m ≥ n, setGcd ↑s ∣ m → m ∈ Ideal.span ↑t\n⊢ ↑t ⊆ ↑s ∧ {x | x < n ∧ x ∈ s} ⊆ ↑s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.FrobeniusNumber | {
"line": 194,
"column": 38
} | {
"line": 194,
"column": 49
} | {
"line": 194,
"column": 50
} | [
{
"pp": "m✝ n✝¹ : ℕ\ns✝ t✝ : Set ℕ\nn✝ : ℕ\ns : Submodule ℕ ℕ\nt : Finset ℕ\nn : ℕ\nhts : ↑t ⊆ ↑s\nhn : ∀ m ≥ n, setGcd ↑s ∣ m → m ∈ Ideal.span ↑t\nm : ℕ\nhm : m ∈ s\ngt : m < n\n⊢ m ∈ ↑(t ∪ {m ∈ Finset.range n | m ∈ s})",
"ppTerm": "?m.124",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"m✝ n✝¹ : ℕ\ns✝ t✝ : Set ℕ\nn✝ : ℕ\ns : Submodule ℕ ℕ\nt : Finset ℕ\nn : ℕ\nhts : ↑t ⊆ ↑s\nhn : ∀ m ≥ n, setGcd ↑s ∣ m → m ∈ Ideal.span ↑t\nm : ℕ\nhm : m ∈ s\ngt : m < n\n⊢ m ∈ t ∨ m < n ∧ m ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.FermatPsp | {
"line": 276,
"column": 4
} | {
"line": 276,
"column": 86
} | {
"line": 276,
"column": 87
} | [
{
"pp": "case h\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : ¬p ∣ b * (b ^ 2 - 1)\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nhA : p < A\nhi_A : 1 < A\nhi_B : 1 < B\nhi_b : 0 < b\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 ≤ b ^ (2 * p)\nhi_bpowpsubone :... | [
"case h\nb : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : ¬p ∣ b * (b ^ 2 - 1)\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nhA : p < A\nhi_A : 1 < A\nhi_B : 1 < B\nhi_b : 0 < b\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 ≤ b ^ (2 * p)\nhi_bpowpsubone : 1 ≤ b ^ (p ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.FermatPsp | {
"line": 282,
"column": 4
} | {
"line": 282,
"column": 43
} | {
"line": 282,
"column": 44
} | [
{
"pp": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : ¬p ∣ b * (b ^ 2 - 1)\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nhA : p < A\nhi_A : 1 < A\nhi_B : 1 < B\nhi_b : 0 < b\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 ≤ b ^ (2 * p)\nhi_bpowpsubone : 1 ≤ b ^... | [
"b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : ¬p ∣ b * (b ^ 2 - 1)\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nhA : p < A\nhi_A : 1 < A\nhi_B : 1 < B\nhi_b : 0 < b\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 ≤ b ^ (2 * p)\nhi_bpowpsubone : 1 ≤ b ^ (p - 1)\np_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.FLT.Three | {
"line": 386,
"column": 65
} | {
"line": 389,
"column": 22
} | {
"line": 391,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nS : Solution hζ\n⊢ λ ∣ S.a + ↑η ^ 2 * S.b",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"IsPrimitiveRoot.toInteger_isPrimitiveRoot",
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"_private.Mathlib.Numb... | [] | by
rw [show S.a + η ^ 2 * S.b = (S.a + S.b) + λ ^ 2 * S.b + 2 * λ * S.b by rw [coe_eta]; ring]
exact dvd_add (dvd_add (dvd_trans (dvd_pow_self _ (by decide)) S.hab) ⟨λ * S.b, by ring⟩)
⟨2 * S.b, by ring⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 946,
"column": 10
} | {
"line": 946,
"column": 40
} | {
"line": 946,
"column": 41
} | [
{
"pp": "case hgt\nA : Type u_1\nB : Type u_3\ninst✝⁹ : CommRing A\ninst✝⁸ : CommRing B\ninst✝⁷ : Algebra A B\ninst✝⁶ : IsDomain A\ninst✝⁵ : IsDedekindDomain A\ninst✝⁴ : IsDedekindDomain B\ninst✝³ : IsTorsionFree A B\ninst✝² : Module.Finite A B\ninst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\nP ... | [
"case hgt\nA : Type u_1\nB : Type u_3\ninst✝⁹ : CommRing A\ninst✝⁸ : CommRing B\ninst✝⁷ : Algebra A B\ninst✝⁶ : IsDomain A\ninst✝⁵ : IsDedekindDomain A\ninst✝⁴ : IsDedekindDomain B\ninst✝³ : IsTorsionFree A B\ninst✝² : Module.Finite A B\ninst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\nP : Ideal B\ni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.FLT.Three | {
"line": 489,
"column": 4
} | {
"line": 489,
"column": 86
} | {
"line": 489,
"column": 86
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nS : Solution hζ\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nh : λ ^ (3 * S.multiplicity - 2) * λ * λ ∣ (S.a + S.b) * (λ * S.y) * (λ * S.z)\nthis : 2 ≤ S.multiplicity\n⊢ λ ^ (3 * S.multiplicity - 2) ∣ S.y * (S.z * (S.a ... | [
"K : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nS : Solution hζ\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nh : λ ^ (3 * S.multiplicity - 2) * λ * λ ∣ (S.a + S.b) * S.y * S.z * λ * λ\nthis : 2 ≤ S.multiplicity\n⊢ λ ^ (3 * S.multiplicity - 2) ∣ S.y * (S.z * (S.a + S.b))"
] | show (S.a + S.b) * (λ * y S) * (λ * z S) = (S.a + S.b) * y S * z S * λ * λ by ring | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.FLT.Three | {
"line": 551,
"column": 2
} | {
"line": 565,
"column": 6
} | {
"line": 567,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nS : Solution hζ\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\n⊢ S.x * S.y * S.z = ↑S.u * S.w ^ 3",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"IsPrimitiveRoot.toInteger_isPrimitiveRoot",
... | [] | suffices hh : λ ^ (3 * S.multiplicity - 2) * S.x * λ * S.y * λ * S.z =
S.u * λ ^ (3 * S.multiplicity) * S.w ^ 3 by
rw [show λ ^ (3 * multiplicity S - 2) * x S * λ * y S * λ * z S =
λ ^ (3 * multiplicity S - 2) * λ * λ * x S * y S * z S by ring] at hh
have := S.two_le_multiplicity
rw [mul_comm _ ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.FLT.Three | {
"line": 551,
"column": 2
} | {
"line": 565,
"column": 6
} | {
"line": 567,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nS : Solution hζ\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\n⊢ S.x * S.y * S.z = ↑S.u * S.w ^ 3",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"IsPrimitiveRoot.toInteger_isPrimitiveRoot",
... | [] | suffices hh : λ ^ (3 * S.multiplicity - 2) * S.x * λ * S.y * λ * S.z =
S.u * λ ^ (3 * S.multiplicity) * S.w ^ 3 by
rw [show λ ^ (3 * multiplicity S - 2) * x S * λ * y S * λ * z S =
λ ^ (3 * multiplicity S - 2) * λ * λ * x S * y S * z S by ring] at hh
have := S.two_le_multiplicity
rw [mul_comm _ ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.FLT.Three | {
"line": 623,
"column": 14
} | {
"line": 623,
"column": 31
} | {
"line": 625,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nS : Solution hζ\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\n⊢ S.a + S.b + ↑η ^ 2 * S.b - S.a + ↑η ^ 2 * S.b + 2 * ↑η * S.b + S.b = 0",
"ppTerm": "?m.428",
"assigned": true,
"usedConstants": [
"IsPrimi... | [] | rw [eta_sq]; ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.FLT.Three | {
"line": 623,
"column": 14
} | {
"line": 623,
"column": 31
} | {
"line": 625,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nS : Solution hζ\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\n⊢ S.a + S.b + ↑η ^ 2 * S.b - S.a + ↑η ^ 2 * S.b + 2 * ↑η * S.b + S.b = 0",
"ppTerm": "?m.428",
"assigned": true,
"usedConstants": [
"IsPrimi... | [] | rw [eta_sq]; ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Height.Basic | {
"line": 356,
"column": 4
} | {
"line": 356,
"column": 15
} | {
"line": 356,
"column": 16
} | [
{
"pp": "case inr.refine_1\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : AdmissibleAbsValues K\nι : Type u_2\ninst✝ : Finite ι\nx : ι → K\nhx : x ≠ 0\ni : ι\nhi : x i ≠ 0\nhx' : (x i)⁻¹ • x ≠ 0\nv : AbsoluteValue K ℝ\nx✝ : v ∈ archAbsVal\n⊢ 1 = v (((x i)⁻¹ • x) i)",
"ppTerm": "?inr.refine_1",
"assigned": tr... | [
"case inr.refine_1\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : AdmissibleAbsValues K\nι : Type u_2\ninst✝ : Finite ι\nx : ι → K\nhx : x ≠ 0\ni : ι\nhi : x i ≠ 0\nhx' : (x i)⁻¹ • x ≠ 0\nv : AbsoluteValue K ℝ\nx✝ : v ∈ archAbsVal\n⊢ 1 = (v (x i))⁻¹ * v (x i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.Basic | {
"line": 373,
"column": 4
} | {
"line": 373,
"column": 20
} | {
"line": 373,
"column": 21
} | [
{
"pp": "case inl\nK : Type u_1\ninst✝³ : Field K\ninst✝² : AdmissibleAbsValues K\nι : Type u_2\nι' : Type u_3\ninst✝¹ : Finite ι\ninst✝ : Finite ι'\nf : ι → ι'\nx : ι' → K\nh₀ : x ∘ f = 0\n⊢ mulHeight (x ∘ f) ≤ mulHeight x",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"case inl\nK : Type u_1\ninst✝³ : Field K\ninst✝² : AdmissibleAbsValues K\nι : Type u_2\nι' : Type u_3\ninst✝¹ : Finite ι\ninst✝ : Finite ι'\nf : ι → ι'\nx : ι' → K\nh₀ : x ∘ f = 0\n⊢ 1 ≤ mulHeight x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 53,
"column": 8
} | {
"line": 53,
"column": 19
} | {
"line": 53,
"column": 20
} | [
{
"pp": "case insert.refine_2.inl\nα : Type u_2\nβ : Type u_3\nF : Type u_4\ninst✝³ : AddCommMonoid β\ninst✝² : FunLike F β ℝ\ninst✝¹ : NonnegHomClass F β ℝ\ninst✝ : ZeroHomClass F β ℝ\nv : F\nl : α → β\nhv : IsNonarchimedean ⇑v\na : α\ns : Finset α\nha : a ∉ s\nih : v (∑ i ∈ s, l i) ≤ ⨆ i, v (l ↑i)\nhs : IsEmp... | [
"case insert.refine_2.inl\nα : Type u_2\nβ : Type u_3\nF : Type u_4\ninst✝³ : AddCommMonoid β\ninst✝² : FunLike F β ℝ\ninst✝¹ : NonnegHomClass F β ℝ\ninst✝ : ZeroHomClass F β ℝ\nv : F\nl : α → β\nhv : IsNonarchimedean ⇑v\na : α\ns : Finset α\nha : a ∉ s\nih : v (∑ i ∈ s, l i) ≤ ⨆ i, v (l ↑i)\nhs : IsEmpty ↥s\n⊢ 0 ≤... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.Basic | {
"line": 392,
"column": 2
} | {
"line": 392,
"column": 42
} | {
"line": 392,
"column": 43
} | [
{
"pp": "K : Type u_1\ninst✝³ : Field K\ninst✝² : AdmissibleAbsValues K\nι : Type u_2\nι' : Type u_3\ninst✝¹ : Finite ι\ninst✝ : Finite ι'\nf : ι → ι'\nx : ι' → K\n⊢ logHeight (x ∘ f) ≤ logHeight x",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Real.instLE",
"Real",
"Fu... | [
"K : Type u_1\ninst✝³ : Field K\ninst✝² : AdmissibleAbsValues K\nι : Type u_2\nι' : Type u_3\ninst✝¹ : Finite ι\ninst✝ : Finite ι'\nf : ι → ι'\nx : ι' → K\n⊢ log (mulHeight (x ∘ f)) ≤ log (mulHeight x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.Basic | {
"line": 401,
"column": 67
} | {
"line": 401,
"column": 78
} | {
"line": 401,
"column": 79
} | [
{
"pp": "K : Type u_1\ninst✝³ : Field K\ninst✝² : AdmissibleAbsValues K\nι : Type u_2\nι' : Type u_3\ninst✝¹ : Finite ι\ninst✝ : Finite ι'\nx : ι → K\nhx : x ≠ 0\ni : ι\nhi : x i ≠ 0 i\nthis : Nonempty ι\n⊢ Sum.elim x 0 (Sum.inl i) ≠ 0 (Sum.inl i)",
"ppTerm": "?m.77",
"assigned": true,
"usedConstant... | [
"K : Type u_1\ninst✝³ : Field K\ninst✝² : AdmissibleAbsValues K\nι : Type u_2\nι' : Type u_3\ninst✝¹ : Finite ι\ninst✝ : Finite ι'\nx : ι → K\nhx : x ≠ 0\ni : ι\nhi : x i ≠ 0 i\nthis : Nonempty ι\n⊢ ¬x i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.Basic | {
"line": 408,
"column": 15
} | {
"line": 408,
"column": 26
} | {
"line": 408,
"column": 27
} | [
{
"pp": "case inr\nK : Type u_1\ninst✝³ : Field K\ninst✝² : AdmissibleAbsValues K\nι : Type u_2\nι' : Type u_3\ninst✝¹ : Finite ι\ninst✝ : Finite ι'\nx : ι → K\nhx : x ≠ 0\ni : ι\nhi : x i ≠ 0 i\nthis : Nonempty ι\nhx' : Sum.elim x 0 ≠ 0\nv : AbsoluteValue K ℝ\nval✝ : ι'\n⊢ v (Sum.elim x 0 (Sum.inr val✝)) ≤ ⨆ i... | [
"case inr\nK : Type u_1\ninst✝³ : Field K\ninst✝² : AdmissibleAbsValues K\nι : Type u_2\nι' : Type u_3\ninst✝¹ : Finite ι\ninst✝ : Finite ι'\nx : ι → K\nhx : x ≠ 0\ni : ι\nhi : x i ≠ 0 i\nthis : Nonempty ι\nhx' : Sum.elim x 0 ≠ 0\nv : AbsoluteValue K ℝ\nval✝ : ι'\n⊢ 0 ≤ ⨆ i, v (x i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.Basic | {
"line": 444,
"column": 2
} | {
"line": 444,
"column": 37
} | {
"line": 444,
"column": 38
} | [
{
"pp": "case e'_5\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : AdmissibleAbsValues K\nι : Type u_5\ninst✝ : Subsingleton ι\nx : ι → K\nhx : x ≠ 0\ni : ι\nhi : x i ≠ 0 i\nthis : Nonempty ι\nj : ι\n⊢ ((x i)⁻¹ • x) j = 1 j",
"ppTerm": "?e'_5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"D... | [
"case e'_5\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : AdmissibleAbsValues K\nι : Type u_5\ninst✝ : Subsingleton ι\nx : ι → K\nhx : x ≠ 0\ni : ι\nhi : x i ≠ 0 i\nthis : Nonempty ι\nj : ι\n⊢ (x i)⁻¹ * x i = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 64
} | {
"line": 131,
"column": 65
} | [
{
"pp": "case inl\nK : Type u_1\ninst✝⁴ : Field K\nι : Type u_2\nι' : Type u_3\ninst✝³ : Fintype ι\ninst✝² : Finite ι'\ninst✝¹ : AdmissibleAbsValues K\ninst✝ : Nonempty ι\nA : ι' × ι → K\nx : ι → K\nH₀ : 1 ≤ ↑(Nat.card ι) ^ totalWeight K * mulHeight A * mulHeight x\nhι' : IsEmpty ι'\n⊢ (mulHeight fun j ↦ ∑ i, A... | [
"case inl\nK : Type u_1\ninst✝⁴ : Field K\nι : Type u_2\nι' : Type u_3\ninst✝³ : Fintype ι\ninst✝² : Finite ι'\ninst✝¹ : AdmissibleAbsValues K\ninst✝ : Nonempty ι\nA : ι' × ι → K\nx : ι → K\nH₀ : 1 ≤ ↑(Nat.card ι) ^ totalWeight K * mulHeight A * mulHeight x\nhι' : IsEmpty ι'\n⊢ 1 ≤ ↑(Nat.card ι) ^ totalWeight K * m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 133,
"column": 4
} | {
"line": 133,
"column": 40
} | {
"line": 133,
"column": 41
} | [
{
"pp": "case inr.inl\nK : Type u_1\ninst✝⁴ : Field K\nι : Type u_2\nι' : Type u_3\ninst✝³ : Fintype ι\ninst✝² : Finite ι'\ninst✝¹ : AdmissibleAbsValues K\ninst✝ : Nonempty ι\nA : ι' × ι → K\nx : ι → K\nH₀ : 1 ≤ ↑(Nat.card ι) ^ totalWeight K * mulHeight A * mulHeight x\nhι' : Nonempty ι'\nh : (fun j ↦ ∑ i, A (j... | [
"case inr.inl\nK : Type u_1\ninst✝⁴ : Field K\nι : Type u_2\nι' : Type u_3\ninst✝³ : Fintype ι\ninst✝² : Finite ι'\ninst✝¹ : AdmissibleAbsValues K\ninst✝ : Nonempty ι\nA : ι' × ι → K\nx : ι → K\nH₀ : 1 ≤ ↑(Nat.card ι) ^ totalWeight K * mulHeight A * mulHeight x\nhι' : Nonempty ι'\nh : (fun j ↦ ∑ i, A (j, i) * x i) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 135,
"column": 4
} | {
"line": 135,
"column": 15
} | {
"line": 135,
"column": 16
} | [
{
"pp": "case inr.inr.inl\nK : Type u_1\ninst✝⁴ : Field K\nι : Type u_2\nι' : Type u_3\ninst✝³ : Fintype ι\ninst✝² : Finite ι'\ninst✝¹ : AdmissibleAbsValues K\ninst✝ : Nonempty ι\nx : ι → K\nhι' : Nonempty ι'\nH₀ : 1 ≤ ↑(Nat.card ι) ^ totalWeight K * mulHeight 0 * mulHeight x\nh : (fun j ↦ ∑ i, 0 (j, i) * x i) ... | [
"case inr.inr.inl\nK : Type u_1\ninst✝⁴ : Field K\nι : Type u_2\nι' : Type u_3\ninst✝³ : Fintype ι\ninst✝² : Finite ι'\ninst✝¹ : AdmissibleAbsValues K\ninst✝ : Nonempty ι\nx : ι → K\nhι' : Nonempty ι'\nH₀ : 1 ≤ ↑(Nat.card ι) ^ totalWeight K * mulHeight 0 * mulHeight x\nh : (fun j ↦ ∑ i, 0 (j, i) * x i) ≠ 0\n⊢ 1 ≤ ↑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 15
} | {
"line": 137,
"column": 16
} | [
{
"pp": "case inr.inr.inr.inl\nK : Type u_1\ninst✝⁴ : Field K\nι : Type u_2\nι' : Type u_3\ninst✝³ : Fintype ι\ninst✝² : Finite ι'\ninst✝¹ : AdmissibleAbsValues K\ninst✝ : Nonempty ι\nA : ι' × ι → K\nhι' : Nonempty ι'\nhA : A ≠ 0\nH₀ : 1 ≤ ↑(Nat.card ι) ^ totalWeight K * mulHeight A * mulHeight 0\nh : (fun j ↦ ... | [
"case inr.inr.inr.inl\nK : Type u_1\ninst✝⁴ : Field K\nι : Type u_2\nι' : Type u_3\ninst✝³ : Fintype ι\ninst✝² : Finite ι'\ninst✝¹ : AdmissibleAbsValues K\ninst✝ : Nonempty ι\nA : ι' × ι → K\nhι' : Nonempty ι'\nhA : A ≠ 0\nH₀ : 1 ≤ ↑(Nat.card ι) ^ totalWeight K * mulHeight A * mulHeight 0\nh : (fun j ↦ ∑ i, A (j, i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.Basic | {
"line": 559,
"column": 4
} | {
"line": 559,
"column": 22
} | {
"line": 559,
"column": 23
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : AdmissibleAbsValues K\nx✝ : K\nv : AbsoluteValue K ℝ\nx : K\nthis : ∀ (i : Fin 2), v (![x, 1] i) = ![v x, 1] i\n⊢ max (v x) 1 = ⨆ i, v (![x, 1] i)",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : AdmissibleAbsValues K\nx✝ : K\nv : AbsoluteValue K ℝ\nx : K\nthis : ∀ (i : Fin 2), v (![x, 1] i) = ![v x, 1] i\n⊢ max (v x) 1 = ⨆ i, ![v x, 1] i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.Basic | {
"line": 561,
"column": 2
} | {
"line": 561,
"column": 47
} | {
"line": 563,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : AdmissibleAbsValues K\nx : K\nH : ∀ (v : AbsoluteValue K ℝ) (x : K), max (v x) 1 = ⨆ i, v (![x, 1] i)\nhx : ![x, 1] ≠ 0\n⊢ mulHeight₁ x = mulHeight ![x, 1]",
"ppTerm": "?m.336",
"assigned": true,
"usedConstants": [
"Real.partialOrder",
"Re... | [] | simp only [mulHeight₁_eq, mulHeight_eq hx, H] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.FLT.Three | {
"line": 709,
"column": 34
} | {
"line": 709,
"column": 78
} | {
"line": 709,
"column": 79
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nS : Solution hζ\nS' : Solution' hζ\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nh : λ ^ (S.multiplicity - 1) * S.X = 0\n⊢ S.X = 0",
"ppTerm": "?m.117",
"assigned": false,
"usedConstants": [],
"usedFVars"... | [
"K : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nS : Solution hζ\nS' : Solution' hζ\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nh : λ ^ (S.multiplicity - 1) * S.X = 0\n⊢ S.X = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.FLT.Three | {
"line": 714,
"column": 4
} | {
"line": 714,
"column": 19
} | {
"line": 714,
"column": 20
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nS : Solution hζ\nS' : Solution' hζ\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nh : S.multiplicity = 1\n⊢ False",
"ppTerm": "?m.193",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoa... | [
"K : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nS : Solution hζ\nS' : Solution' hζ\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nh : S.multiplicity = 1\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.Basic | {
"line": 758,
"column": 4
} | {
"line": 758,
"column": 15
} | {
"line": 758,
"column": 16
} | [
{
"pp": "case inr.inl\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : AdmissibleAbsValues K\nι : Type u_2\ninst✝ : Finite ι\ny : ι → K\nhι : Nonempty ι\n⊢ mulHeight (0 * y) ≤ mulHeight 0 * mulHeight y",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"... | [
"case inr.inl\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : AdmissibleAbsValues K\nι : Type u_2\ninst✝ : Finite ι\ny : ι → K\nhι : Nonempty ι\n⊢ 1 ≤ mulHeight y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.Basic | {
"line": 760,
"column": 4
} | {
"line": 760,
"column": 15
} | {
"line": 760,
"column": 16
} | [
{
"pp": "case inr.inr.inl\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : AdmissibleAbsValues K\nι : Type u_2\ninst✝ : Finite ι\nx : ι → K\nhι : Nonempty ι\nhx : x ≠ 0\n⊢ mulHeight (x * 0) ≤ mulHeight x * mulHeight 0",
"ppTerm": "?inr.inr.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"R... | [
"case inr.inr.inl\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : AdmissibleAbsValues K\nι : Type u_2\ninst✝ : Finite ι\nx : ι → K\nhι : Nonempty ι\nhx : x ≠ 0\n⊢ 1 ≤ mulHeight x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 249,
"column": 6
} | {
"line": 249,
"column": 17
} | {
"line": 249,
"column": 18
} | [
{
"pp": "case pos\nK : Type u_4\ninst✝¹ : Field K\nι : Type u_5\nι' : Type u_6\ninst✝ : AdmissibleAbsValues K\nh✝¹ : IsEmpty ι'\nh✝ : archAbsVal.card = 0\n⊢ 1 * ∏ᶠ (v : ↑nonarchAbsVal), ⨆ j, 1 ≤ 1",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"... | [
"case pos\nK : Type u_4\ninst✝¹ : Field K\nι : Type u_5\nι' : Type u_6\ninst✝ : AdmissibleAbsValues K\nh✝¹ : IsEmpty ι'\nh✝ : archAbsVal.card = 0\n⊢ ∏ᶠ (v : ↑nonarchAbsVal), 0 ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.Basic | {
"line": 793,
"column": 61
} | {
"line": 800,
"column": 30
} | {
"line": 802,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : AdmissibleAbsValues K\nι : Type u_2\ns : Finset ι\nx : ι → K\n⊢ mulHeight₁ (∏ i ∈ s, x i) ≤ ∏ i ∈ s, mulHeight₁ (x i)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"MulOne.toOne",
... | [] | by
classical
induction s using Finset.induction with
| empty => simp
| insert b s hb ih =>
simp only [Finset.prod_insert hb]
grw [← ih]
exact mulHeight₁_mul_le .. | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 228,
"column": 2
} | {
"line": 228,
"column": 19
} | {
"line": 229,
"column": 2
} | [
{
"pp": "s : ℝ\nhs : 1 < s\n⊢ Tendsto\n (fun N ↦\n 1 / (s - 1) * (1 - 1 / (↑N + 1) ^ (s - 1)) - 1 / s * (∑ n ∈ Finset.range N, 1 / (↑n + 1) ^ s - ↑N / (↑N + 1) ^ s))\n atTop (𝓝 (1 / (s - 1) - 1 / s * ∑' (n : ℕ), 1 / (↑n + 1) ^ s))",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [... | [
"case hf\ns : ℝ\nhs : 1 < s\n⊢ Tendsto (fun x ↦ 1 / (s - 1) * (1 - 1 / (↑x + 1) ^ (s - 1))) atTop (𝓝 (1 / (s - 1)))",
"case hg\ns : ℝ\nhs : 1 < s\n⊢ Tendsto (fun x ↦ 1 / s * (∑ n ∈ Finset.range x, 1 / (↑n + 1) ^ s - ↑x / (↑x + 1) ^ s)) atTop\n (𝓝 (1 / s * ∑' (n : ℕ), 1 / (↑n + 1) ^ s))"
] | apply Tendsto.sub | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 304,
"column": 6
} | {
"line": 304,
"column": 17
} | {
"line": 304,
"column": 18
} | [
{
"pp": "case inl.inr\nK : Type u_4\ninst✝³ : Field K\nι : Type u_5\nι' : Type u_6\ninst✝² : AdmissibleAbsValues K\ninst✝¹ : Finite ι'\ninst✝ : Finite ι\nN : ℕ\np : ι' → MvPolynomial ι K\nhp : ∀ (i : ι'), (p i).IsHomogeneous N\nh : (fun j ↦ constantCoeff (p j)) ≠ 0\n⊢ (mulHeight fun j ↦ (eval 0) (p j)) ≤ max (m... | [
"case inl.inr\nK : Type u_4\ninst✝³ : Field K\nι : Type u_5\nι' : Type u_6\ninst✝² : AdmissibleAbsValues K\ninst✝¹ : Finite ι'\ninst✝ : Finite ι\nN : ℕ\np : ι' → MvPolynomial ι K\nhp : ∀ (i : ι'), (p i).IsHomogeneous N\nh : (fun j ↦ constantCoeff (p j)) ≠ 0\n⊢ (mulHeight fun j ↦ constantCoeff (p j)) ≤ mulHeightBoun... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 307,
"column": 4
} | {
"line": 307,
"column": 36
} | {
"line": 307,
"column": 37
} | [
{
"pp": "case inr.inl\nK : Type u_4\ninst✝³ : Field K\nι : Type u_5\nι' : Type u_6\ninst✝² : AdmissibleAbsValues K\ninst✝¹ : Finite ι'\ninst✝ : Finite ι\nN : ℕ\np : ι' → MvPolynomial ι K\nhp : ∀ (i : ι'), (p i).IsHomogeneous N\nx : ι → K\nhx : x ≠ 0\nh₀ : (fun j ↦ (eval x) (p j)) = 0\n⊢ (mulHeight fun j ↦ (eval... | [
"case inr.inl\nK : Type u_4\ninst✝³ : Field K\nι : Type u_5\nι' : Type u_6\ninst✝² : AdmissibleAbsValues K\ninst✝¹ : Finite ι'\ninst✝ : Finite ι\nN : ℕ\np : ι' → MvPolynomial ι K\nhp : ∀ (i : ι'), (p i).IsHomogeneous N\nx : ι → K\nhx : x ≠ 0\nh₀ : (fun j ↦ (eval x) (p j)) = 0\n⊢ 1 ≤ mulHeight x ^ N"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.Projectivization | {
"line": 34,
"column": 4
} | {
"line": 34,
"column": 19
} | {
"line": 34,
"column": 20
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\ninst✝¹ : AdmissibleAbsValues K\nι : Type u_2\ninst✝ : Finite ι\na b : { v // v ≠ 0 }\nt : K\nh : t = 0\n⊢ ↑a ≠ t • ↑b",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"instSMulOfMul",
"congrArg",
"... | [
"K : Type u_1\ninst✝² : Field K\ninst✝¹ : AdmissibleAbsValues K\nι : Type u_2\ninst✝ : Finite ι\na b : { v // v ≠ 0 }\nt : K\nh : t = 0\n⊢ ¬↑a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.Lemmas | {
"line": 39,
"column": 2
} | {
"line": 40,
"column": 83
} | {
"line": 40,
"column": 84
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nμ : R\nhμ : IsPrimitiveRoot μ (n + 1)\nthis : eval 1 (∏ k ∈ range n, (X - C (μ ^ (k + 1) * 1))) = eval 1 (∑ i ∈ range (n + 1), X ^ i)\n⊢ ∏ k ∈ range n, (1 - μ ^ (k + 1)) = ↑n + 1",
"ppTerm": "?m.186",
"assigned": false,
"usedCons... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nμ : R\nhμ : IsPrimitiveRoot μ (n + 1)\nthis : eval 1 (∏ k ∈ range n, (X - C (μ ^ (k + 1) * 1))) = eval 1 (∑ i ∈ range (n + 1), X ^ i)\n⊢ ∏ k ∈ range n, (1 - μ ^ (k + 1)) = ↑n + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 324,
"column": 50
} | {
"line": 324,
"column": 61
} | {
"line": 324,
"column": 62
} | [
{
"pp": "this : Tendsto (fun s ↦ ∑' (n : ℕ), 1 / (↑n + 1) ^ s - 1 / (s - 1)) (𝓝[>] 1) (𝓝 γ)\ns : ℝ\nhs : s ∈ Ioi 1\n⊢ 1 < (↑s).re",
"ppTerm": "?m.130",
"assigned": true,
"usedConstants": [
"Real",
"Real.instLT",
"id",
"Complex.ofReal",
"Complex.re",
"Real.instOn... | [
"this : Tendsto (fun s ↦ ∑' (n : ℕ), 1 / (↑n + 1) ^ s - 1 / (s - 1)) (𝓝[>] 1) (𝓝 γ)\ns : ℝ\nhs : s ∈ Ioi 1\n⊢ 1 < s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 332,
"column": 4
} | {
"line": 332,
"column": 27
} | {
"line": 332,
"column": 28
} | [
{
"pp": "aux2 : Tendsto (fun s ↦ ∑' (n : ℕ), 1 / (↑n + 1) ^ s - 1 / (s - 1)) (𝓝[>] 1) (𝓝 (1 - termTSum 1))\nthis : γ = 1 - termTSum 1\n⊢ Tendsto (fun s ↦ ∑' (n : ℕ), 1 / (↑n + 1) ^ s - 1 / (s - 1)) (𝓝[>] 1) (𝓝 γ)",
"ppTerm": "?m.266",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"R... | [
"aux2 : Tendsto (fun s ↦ ∑' (n : ℕ), 1 / (↑n + 1) ^ s - 1 / (s - 1)) (𝓝[>] 1) (𝓝 (1 - termTSum 1))\nthis : γ = 1 - termTSum 1\n⊢ Tendsto (fun s ↦ ∑' (n : ℕ), 1 / (↑n + 1) ^ s - 1 / (s - 1)) (𝓝[>] 1) (𝓝 (1 - termTSum 1))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.Lemmas | {
"line": 64,
"column": 2
} | {
"line": 71,
"column": 34
} | {
"line": 72,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nk n : ℕ\nμ : R\nm : ℕ\nhn : k < m + k + 1\nhμ : IsPrimitiveRoot μ (m + k + 1)\nhdvd : ∀ (k : ℕ), ∃ z ∈ ℤ[μ], μ ^ k - 1 = z * (μ - 1)\nZ : ℕ → R := fun k ↦ Classical.choose ⋯\nZdef : ∀ (k : ℕ), Z k ∈ ℤ[μ] ∧ μ ^ k - 1 = Z k * (μ - 1)\n... | [
"case refine_2\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nk n : ℕ\nμ : R\nm : ℕ\nhn : k < m + k + 1\nhμ : IsPrimitiveRoot μ (m + k + 1)\nhdvd : ∀ (k : ℕ), ∃ z ∈ ℤ[μ], μ ^ k - 1 = z * (μ - 1)\nZ : ℕ → R := fun k ↦ Classical.choose ⋯\nZdef : ∀ (k : ℕ), Z k ∈ ℤ[μ] ∧ μ ^ k - 1 = Z k * (μ - 1)\n⊢ ↑(m + k + ... | · apply Subalgebra.mul_mem
· apply Subalgebra.mul_mem
· exact Subalgebra.pow_mem _ (Subalgebra.neg_mem _ <| Subalgebra.one_mem _) _
· exact Subalgebra.prod_mem _ fun _ _ ↦ (Zdef _).1
· refine Subalgebra.prod_mem _ fun _ _ ↦ ?_
apply Subalgebra.sub_mem
· exact Subalgebra.pow_mem _ (self_m... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 367,
"column": 6
} | {
"line": 367,
"column": 23
} | {
"line": 368,
"column": 6
} | [
{
"pp": "case refine_2.refine_2\nf : ℂ → ℂ := fun s ↦ riemannZeta s - 1 / (s - 1)\n⊢ Tendsto (fun x ↦ f x * (x - 1) - f 1 * (x - 1)) (𝓝[≠] 1) (𝓝 (1 - 1 - f 1 * (1 - 1)))",
"ppTerm": "?refine_2.refine_2",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"NonUn... | [
"case refine_2.refine_2.hf\nf : ℂ → ℂ := ⋯\n⊢ Tendsto (fun x ↦ f x * (x - 1)) (𝓝[≠] 1) (𝓝 (1 - 1))",
"case refine_2.refine_2.hg\nf : ℂ → ℂ := ⋯\n⊢ Tendsto (fun x ↦ f 1 * (x - 1)) (𝓝[≠] 1) (𝓝 (f 1 * (1 - 1)))"
] | apply Tendsto.sub | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 377,
"column": 2
} | {
"line": 377,
"column": 55
} | {
"line": 378,
"column": 5
} | [
{
"pp": "F : Type u_1\ninst✝² : Norm F\ninst✝¹ : One F\ninst✝ : NormOneClass F\n⊢ (fun s ↦ riemannZeta s - 1 / (s - 1)) =O[𝓝 1] fun x ↦ 1",
"ppTerm": "?m.33",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"F : Type u_1\ninst✝² : Norm F\ninst✝¹ : One F\ninst✝ : NormOneClass F\n⊢ (fun s ↦ riemannZeta s - 1 / (s - 1)) =O[𝓝 1] fun x ↦ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.NumberField | {
"line": 63,
"column": 2
} | {
"line": 64,
"column": 54
} | {
"line": 64,
"column": 55
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\ninst✝¹ : NumberField K\ninst✝ : DecidableEq (AbsoluteValue K ℝ)\nv : InfinitePlace K\nthis : DecidableEq (InfinitePlace K)\n⊢ Multiset.count (↑v) (multisetInfinitePlace K) = v.mult",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Multiset.sum"... | [
"K : Type u_1\ninst✝² : Field K\ninst✝¹ : NumberField K\ninst✝ : DecidableEq (AbsoluteValue K ℝ)\nv : InfinitePlace K\nthis : DecidableEq (InfinitePlace K)\n⊢ (∑ x, if x = v then x.mult else 0) = v.mult"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.NumberField | {
"line": 164,
"column": 2
} | {
"line": 165,
"column": 9
} | {
"line": 165,
"column": 10
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nthis : Inhabited (InfinitePlace K)\n⊢ 0 < totalWeight K",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Multiset.sum",
"Eq.mpr",
"Real.partialOrder",
"Real",
"Finset.univ",
"Multiset.map",
... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nthis : Inhabited (InfinitePlace K)\n⊢ 0 < ∑ x, x.mult"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.JacobiSum.Basic | {
"line": 111,
"column": 4
} | {
"line": 111,
"column": 94
} | {
"line": 111,
"column": 95
} | [
{
"pp": "case hc\nF : Type u_1\nR : Type u_2\ninst✝² : Field F\ninst✝¹ : Fintype F\ninst✝ : CommRing R\nx : F\nhx : x ≠ 0 ∧ x ≠ 1\n⊢ IsUnit (x * (1 - x))",
"ppTerm": "?hc",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"GroupWithZero.toMonoidWithZer... | [
"case hc\nF : Type u_1\nR : Type u_2\ninst✝² : Field F\ninst✝¹ : Fintype F\ninst✝ : CommRing R\nx : F\nhx : x ≠ 0 ∧ x ≠ 1\n⊢ ¬x = 0 ∧ ¬x = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 482,
"column": 4
} | {
"line": 482,
"column": 15
} | {
"line": 482,
"column": 16
} | [
{
"pp": "case inr\ns : ℂ\nhs : s ≠ 1\nhg_an : AnalyticOnNhd ℂ (fun z ↦ (starRingEnd ℂ) (riemannZeta ((starRingEnd ℂ) z))) {1}ᶜ\nhgz : ∀ (z : ℂ), 1 < z.re → (starRingEnd ℂ) (riemannZeta ((starRingEnd ℂ) z)) = riemannZeta z\nheq : EqOn (fun z ↦ (starRingEnd ℂ) (riemannZeta ((starRingEnd ℂ) z))) riemannZeta {1}ᶜ\n... | [
"case inr\ns : ℂ\nhs : s ≠ 1\nhg_an : AnalyticOnNhd ℂ (fun z ↦ (starRingEnd ℂ) (riemannZeta ((starRingEnd ℂ) z))) {1}ᶜ\nhgz : ∀ (z : ℂ), 1 < z.re → (starRingEnd ℂ) (riemannZeta ((starRingEnd ℂ) z)) = riemannZeta z\nheq : EqOn (fun z ↦ (starRingEnd ℂ) (riemannZeta ((starRingEnd ℂ) z))) riemannZeta {1}ᶜ\n⊢ riemannZet... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 453,
"column": 4
} | {
"line": 453,
"column": 22
} | {
"line": 453,
"column": 23
} | [
{
"pp": "K : Type u_6\ninst✝³ : Field K\nι : Type u_7\nι' : Type u_8\ninst✝² : Fintype ι'\ninst✝¹ : AdmissibleAbsValues K\ninst✝ : Finite ι\nM N : ℕ\nq : ι × ι' → MvPolynomial ι K\nhq : ∀ (a : ι × ι'), (q a).IsHomogeneous M\np : ι' → MvPolynomial ι K\nx : ι → K\nh : ∀ (k : ι), ∑ j, (eval x) (q (k, j)) * (eval x... | [
"K : Type u_6\ninst✝³ : Field K\nι : Type u_7\nι' : Type u_8\ninst✝² : Fintype ι'\ninst✝¹ : AdmissibleAbsValues K\ninst✝ : Finite ι\nM N : ℕ\nq : ι × ι' → MvPolynomial ι K\nhq : ∀ (a : ι × ι'), (q a).IsHomogeneous M\np : ι' → MvPolynomial ι K\nx : ι → K\nh : ∀ (k : ι), ∑ j, (eval x) (q (k, j)) * (eval x) (p j) = x ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.NumberField | {
"line": 272,
"column": 4
} | {
"line": 272,
"column": 47
} | {
"line": 272,
"column": 48
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : K\nhx : IsAlgebraic ℤ x\nm : ℕ\nr : 𝓞 K\nhm : m ≠ 0\nhmr : m • x = ↑r\nn : ℕ := (Submodule.toAddSubgroup (span {↑m})).relIndex (Submodule.toAddSubgroup (span {↑m, r}))\nhndef : n = (Submodule.toAddSubgroup (span {↑m})).relIndex (Submodule.toAd... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : K\nhx : IsAlgebraic ℤ x\nm : ℕ\nr : 𝓞 K\nhm : m ≠ 0\nhmr : m • x = ↑r\nn : ℕ := (Submodule.toAddSubgroup (span {↑m})).relIndex (Submodule.toAddSubgroup (span {↑m, r}))\nhndef : n = (Submodule.toAddSubgroup (span {↑m})).relIndex (Submodule.toAddSubgroup (s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.NumberField | {
"line": 288,
"column": 6
} | {
"line": 288,
"column": 78
} | {
"line": 288,
"column": 78
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nn : ℕ\nhn : ¬n = 0\na : 𝓞 K\nHw : 0 < absNorm (span {↑n})\n⊢ 1 ≤ ↑(absNorm (span {↑n})) * ∏ᶠ (v : FinitePlace K), ⨆ i, v (![↑a, ↑n] i)",
"ppTerm": "?m.104",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.t... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nn : ℕ\nhn : ¬n = 0\na : 𝓞 K\nHw : 0 < absNorm (span {↑n})\n⊢ ↑(absNorm (span (Set.range ![a, ↑n]))) * ∏ᶠ (v : FinitePlace K), ⨆ i, v ↑(![a, ↑n] i) ≤\n ↑(absNorm (span {↑n})) * ∏ᶠ (v : FinitePlace K), ⨆ i, v (![↑a, ↑n] i)"
] | ← absNorm_mul_finprod_finitePlace_eq_one (show ![a, n] ≠ 0 by simp [hn]) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Height.NumberField | {
"line": 316,
"column": 4
} | {
"line": 316,
"column": 25
} | {
"line": 316,
"column": 26
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : K\nn : ℕ\nhn : n ≠ 0\na : 𝓞 K\nha₁ : ↑n * x = ↑a\nha₂ : Ideal.absNorm (Ideal.span {↑n, a}) = n ^ (totalWeight K - 1)\nhv : ∀ (i : Fin 2), ↑(![a, ↑n] i) = ![↑a, ↑n] i\n⊢ ↑n ^ (totalWeight K - 1) * ∏ᶠ (v : FinitePlace K), ⨆ i, v (![↑a, ↑n] i) = ... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : K\nn : ℕ\nhn : n ≠ 0\na : 𝓞 K\nha₁ : ↑n * x = ↑a\nha₂ : Ideal.absNorm (Ideal.span {↑n, a}) = n ^ (totalWeight K - 1)\nhv : ∀ (i : Fin 2), ↑(![a, ↑n] i) = ![↑a, ↑n] i\n⊢ ↑n ^ (totalWeight K - 1) * ∏ᶠ (v : FinitePlace K), ⨆ i, v (![↑a, ↑n] i) = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 566,
"column": 2
} | {
"line": 566,
"column": 13
} | {
"line": 566,
"column": 14
} | [
{
"pp": "⊢ Tendsto riemannZeta₁ (𝓝 1) (𝓝 1)",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ Tendsto riemannZeta₁ (𝓝 1) (𝓝 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 611,
"column": 4
} | {
"line": 611,
"column": 15
} | {
"line": 611,
"column": 16
} | [
{
"pp": "this : DifferentiableAt ℝ (fun s ↦ Real.log (riemannZeta₁ ↑s).re) 1\n⊢ (fun s ↦ Real.log (riemannZeta₁ ↑s).re) =O[𝓝 1] fun x ↦ x - 1",
"ppTerm": "?m.101",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"this : DifferentiableAt ℝ (fun s ↦ Real.log (riemannZeta₁ ↑s).re) 1\n⊢ (fun s ↦ Real.log (riemannZeta₁ ↑s).re) =O[𝓝 1] fun x ↦ x - 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 635,
"column": 4
} | {
"line": 635,
"column": 15
} | {
"line": 635,
"column": 16
} | [
{
"pp": "this : DifferentiableAt ℂ (fun s ↦ deriv riemannZeta₁ s / riemannZeta₁ s) 1\n⊢ (fun s ↦ deriv riemannZeta₁ s / riemannZeta₁ s - ↑γ) =O[𝓝 1] fun x ↦ x - 1",
"ppTerm": "?m.151",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"this : DifferentiableAt ℂ (fun s ↦ deriv riemannZeta₁ s / riemannZeta₁ s) 1\n⊢ (fun s ↦ deriv riemannZeta₁ s / riemannZeta₁ s - ↑γ) =O[𝓝 1] fun x ↦ x - 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 662,
"column": 4
} | {
"line": 662,
"column": 25
} | {
"line": 662,
"column": 26
} | [
{
"pp": "this : (fun s ↦ (riemannZeta₁ s)⁻¹ - 1) =O[𝓝 1] fun x ↦ x - 1\n⊢ (fun s ↦ (s - 1) * ((riemannZeta₁ s)⁻¹ - 1)) =O[𝓝 1] fun s ↦ (s - 1) ^ 2",
"ppTerm": "?m.187",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"riemannZeta₁",
"HM... | [
"this : (fun s ↦ (riemannZeta₁ s)⁻¹ - 1) =O[𝓝 1] fun x ↦ x - 1\n⊢ (fun s ↦ (s - 1) * ((riemannZeta₁ s)⁻¹ - 1)) =O[𝓝 1] fun s ↦ (s - 1) * (s - 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 663,
"column": 2
} | {
"line": 663,
"column": 13
} | {
"line": 663,
"column": 14
} | [
{
"pp": "⊢ (fun s ↦ (riemannZeta₁ s)⁻¹ - 1) =O[𝓝 1] fun x ↦ x - 1",
"ppTerm": "?m.186",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ (fun s ↦ (riemannZeta₁ s)⁻¹ - 1) =O[𝓝 1] fun x ↦ x - 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 669,
"column": 4
} | {
"line": 669,
"column": 25
} | {
"line": 669,
"column": 26
} | [
{
"pp": "this : (fun x ↦ x - 1) =o[𝓝 1] fun x ↦ 1\n⊢ (fun s ↦ (s - 1) ^ 2) =o[𝓝[≠] 1] fun x ↦ x - 1",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"HMul.hMul",
"pow_two",
"Monoid.toMulOneClass",
"congrA... | [
"this : (fun x ↦ x - 1) =o[𝓝 1] fun x ↦ 1\n⊢ (fun s ↦ (s - 1) * (s - 1)) =o[𝓝[≠] 1] fun x ↦ x - 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IsAdjoinRoot | {
"line": 117,
"column": 56
} | {
"line": 117,
"column": 67
} | {
"line": 117,
"column": 68
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring S\nf : R[X]\ninst✝ : Algebra R S\nh : IsAdjoinRoot S f\np : R[X]\n⊢ h.map p = 0 ↔ f ∣ p",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring S\nf : R[X]\ninst✝ : Algebra R S\nh : IsAdjoinRoot S f\np : R[X]\n⊢ h.map p = 0 ↔ f ∣ p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IsAdjoinRoot | {
"line": 176,
"column": 74
} | {
"line": 177,
"column": 92
} | {
"line": 179,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring S\nf : R[X]\ninst✝ : Algebra R S\nh : IsAdjoinRoot S f\na : AdjoinRoot f\n⊢ h.adjoinRootAlgEquiv a = h.map ⋯.choose",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Exists.choose_spec",
"AdjoinR... | [] | by
rw (occs := [1]) [← (AdjoinRoot.mk_surjective a).choose_spec, adjoinRootAlgEquiv_apply_mk] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.JacobiSum.Basic | {
"line": 297,
"column": 18
} | {
"line": 297,
"column": 35
} | {
"line": 297,
"column": 35
} | [
{
"pp": "case pos\nF : Type u_1\nR : Type u_2\ninst✝³ : Field F\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Fintype F\nn : ℕ\nhn : 2 < n\nχ ψ : MulChar F R\nμ : R\nhχ : χ ^ n = 1\nhψ : ψ ^ n = 1\nhμ : IsPrimitiveRoot μ n\nq : ℕ\nhq : Fintype.card F = n * q + 1\nz₁ : R\nhz₁ : z₁ ∈ ℤ[μ]\nHz₁ : ↑n = z₁ * (μ... | [
"case pos\nF : Type u_1\nR : Type u_2\ninst✝³ : Field F\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Fintype F\nn : ℕ\nhn : 2 < n\nχ ψ : MulChar F R\nμ : R\nhχ : χ ^ n = 1\nhψ : ψ ^ n = 1\nhμ : IsPrimitiveRoot μ n\nq : ℕ\nhq : Fintype.card F = n * q + 1\nz₁ : R\nhz₁ : z₁ ∈ ℤ[μ]\nHz₁ : ↑n = z₁ * (μ - 1) ^ 2\nh... | jacobiSum_one_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Height.NumberField | {
"line": 503,
"column": 4
} | {
"line": 503,
"column": 23
} | {
"line": 503,
"column": 23
} | [
{
"pp": "ι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : Nonempty ι\nx : ι → ℤ\nhx : Finset.univ.gcd x = 1\nhx₀ : Int.cast ∘ x ≠ 0\n⊢ (∏ v, (⨆ i, v ((Int.cast ∘ x) i)) ^ v.mult) * ∏ᶠ (v : FinitePlace ℚ), ⨆ i, v ((Int.cast ∘ x) i) = ⨆ i, ↑|x i|",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
... | [
"ι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : Nonempty ι\nx : ι → ℤ\nhx : Finset.univ.gcd x = 1\nhx₀ : Int.cast ∘ x ≠ 0\n⊢ (∏ x_1, (⨆ i, ↑|(Int.cast ∘ x) i|) ^ x_1.mult) * ∏ᶠ (v : FinitePlace ℚ), ⨆ i, v ((Int.cast ∘ x) i) = ⨆ i, ↑|x i|"
] | infinitePlace_apply | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.Height.NumberField | {
"line": 528,
"column": 4
} | {
"line": 528,
"column": 84
} | {
"line": 529,
"column": 6
} | [
{
"pp": "q : ℚ\n⊢ Finset.univ.gcd ![q.num, ↑q.den] = 1",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Int.gcd",
"Eq.mpr",
"Rat.num",
"Finset.univ",
"congrArg",
"Int.instStrongNormalizedGCDMonoid",
"Finset",
"instNormalizedGCDMonoidOfStrongN... | [
"q : ℚ\n⊢ q.num.gcd ↑q.den = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IsAdjoinRoot | {
"line": 409,
"column": 2
} | {
"line": 409,
"column": 42
} | {
"line": 409,
"column": 43
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring S\nf : R[X]\ninst✝ : Algebra R S\nh : IsAdjoinRootMonic S f\nn : ℕ\na✝ : Nontrivial R\nhdeg : f.degree ≤ ↑n\n⊢ f.natDegree ≤ n",
"ppTerm": "?m.90",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
... | [
"R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring S\nf : R[X]\ninst✝ : Algebra R S\nh : IsAdjoinRootMonic S f\nn : ℕ\na✝ : Nontrivial R\nhdeg : f.degree ≤ ↑n\n⊢ f.degree ≤ ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IsAdjoinRoot | {
"line": 413,
"column": 2
} | {
"line": 413,
"column": 13
} | {
"line": 413,
"column": 14
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring S\nf : R[X]\ninst✝ : Algebra R S\nh : IsAdjoinRootMonic S f\nhdeg : 1 < f.natDegree\n⊢ h.modByMonicHom h.root = X",
"ppTerm": "?m.30",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring S\nf : R[X]\ninst✝ : Algebra R S\nh : IsAdjoinRootMonic S f\nhdeg : 1 < f.natDegree\n⊢ h.modByMonicHom h.root = X"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IsAdjoinRoot | {
"line": 608,
"column": 4
} | {
"line": 608,
"column": 42
} | {
"line": 608,
"column": 43
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nf : R[X]\nh : IsAdjoinRoot S f\ninst✝³ : IsDomain R\ninst✝² : IsDomain S\ninst✝¹ : IsTorsionFree R S\ninst✝ : IsIntegrallyClosed R\nα : S\nhα : IsIntegral R α\nhα₂ : R[α] = ⊤\nx✝ : R[X]\n⊢ x✝ ∈ RingHom.ker (aeval α)... | [
"R : Type u\nS : Type v\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nf : R[X]\nh : IsAdjoinRoot S f\ninst✝³ : IsDomain R\ninst✝² : IsDomain S\ninst✝¹ : IsTorsionFree R S\ninst✝ : IsIntegrallyClosed R\nα : S\nhα : IsIntegral R α\nhα₂ : R[α] = ⊤\nx✝ : R[X]\n⊢ (aeval α) x✝ = 0 ↔ minpoly R α ∣ x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 621,
"column": 40
} | {
"line": 621,
"column": 62
} | {
"line": 621,
"column": 62
} | [
{
"pp": "K : Type u_4\ninst✝¹ : Field K\ninst✝ : AdmissibleAbsValues K\nC₁ : ℝ\nhC₁ : ∀ (a b c d : K), logHeight ![a * c, a * d + b * c, b * d] ≤ C₁ + logHeight ![a, b] + logHeight ![c, d]\nC₂ : ℝ\nhC₂ :\n ∀ {a b c d : K},\n ![a, b] ≠ 0 → ![c, d] ≠ 0 → C₂ + logHeight ![a, b] + logHeight ![c, d] ≤ logHeight ... | [
"K : Type u_4\ninst✝¹ : Field K\ninst✝ : AdmissibleAbsValues K\nC₁ : ℝ\nhC₁ : ∀ (a b c d : K), logHeight ![a * c, a * d + b * c, b * d] ≤ C₁ + logHeight ![a, b] + logHeight ![c, d]\nC₂ : ℝ\na✝ b✝ c✝ d✝ : K\nhC₂ : C₂ + logHeight ![a✝, b✝] + logHeight ![c✝, d✝] ≤ logHeight ![a✝ * c✝, a✝ * d✝ + b✝ * c✝, b✝ * d✝]\nhab ... | specialize hC₂ hab hcd | Lean.Elab.Tactic.evalSpecialize | Lean.Parser.Tactic.specialize |
Mathlib.NumberTheory.LSeries.DirichletContinuation | {
"line": 144,
"column": 6
} | {
"line": 144,
"column": 45
} | {
"line": 144,
"column": 46
} | [
{
"pp": "case refine_3.refine_2\nM N : ℕ\ninst✝¹ : NeZero M\ninst✝ : NeZero N\nhMN : M ∣ N\nχ : DirichletCharacter ℂ M\ns : ℂ\nhs : s ≠ 1\nhpc : IsPreconnected {1}ᶜ\nhne : 2 ∈ {1}ᶜ\nt : ℂ\nht : 1 < t.re\n⊢ LFunction ((changeLevel hMN) χ) t = (fun s ↦ LFunction χ s * ∏ p ∈ N.primeFactors, (1 - χ ↑p * ↑p ^ (-s)))... | [
"case refine_3.refine_2\nM N : ℕ\ninst✝¹ : NeZero M\ninst✝ : NeZero N\nhMN : M ∣ N\nχ : DirichletCharacter ℂ M\ns : ℂ\nhs : s ≠ 1\nhpc : IsPreconnected {1}ᶜ\nhne : 2 ∈ {1}ᶜ\nt : ℂ\nht : 1 < t.re\n⊢ LSeries (fun x ↦ ((changeLevel hMN) χ) ↑x) t = LSeries (fun x ↦ χ ↑x) t * ∏ p ∈ N.primeFactors, (1 - χ ↑p * ↑p ^ (-t))... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.ZMod | {
"line": 196,
"column": 6
} | {
"line": 196,
"column": 41
} | {
"line": 196,
"column": 42
} | [
{
"pp": "case neg\nN : ℕ\ninst✝ : NeZero N\nj : ZMod N\ns : ℂ\nhjs : j ≠ 0 ∨ s ≠ 1\nU : Set ℂ := if j = 0 then {z | z ≠ 1} else Set.univ\nV : Set ℂ := {z | 1 < z.re}\nhUo : IsOpen U\nf : ℂ → ℂ := LFunction fun k ↦ 𝕖 (j * k)\ng : ℂ → ℂ := expZeta (toAddCircle j)\nhU : ∀ {u : ℂ}, u ∈ U ↔ u ≠ 1 ∨ j ≠ 0\nhf : Anal... | [
"case neg\nN : ℕ\ninst✝ : NeZero N\nj : ZMod N\ns : ℂ\nhjs : j ≠ 0 ∨ s ≠ 1\nU : Set ℂ := if j = 0 then {z | z ≠ 1} else Set.univ\nV : Set ℂ := {z | 1 < z.re}\nhUo : IsOpen U\nf : ℂ → ℂ := LFunction fun k ↦ 𝕖 (j * k)\ng : ℂ → ℂ := expZeta (toAddCircle j)\nhU : ∀ {u : ℂ}, u ∈ U ↔ u ≠ 1 ∨ j ≠ 0\nhf : AnalyticOnNhd ℂ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.DirichletContinuation | {
"line": 345,
"column": 2
} | {
"line": 345,
"column": 27
} | {
"line": 345,
"column": 28
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\np : ℕ\nhp : p ∈ n.primeFactors\n⊢ 1 - (↑p)⁻¹ ≠ 0",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneCl... | [
"n : ℕ\ninst✝ : NeZero n\np : ℕ\nhp : p ∈ n.primeFactors\n⊢ ¬p = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.DirichletContinuation | {
"line": 355,
"column": 2
} | {
"line": 355,
"column": 13
} | {
"line": 355,
"column": 14
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\n⊢ ContinuousWithinAt (LFunctionTrivChar₁ n) {1}ᶜ 1",
"ppTerm": "?m.106",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"ContinuousWithinAt",
"HMul.hMul",
"Complex.instNormedAddCommGroup",
"C... | [
"n : ℕ\ninst✝ : NeZero n\n⊢ Tendsto (fun s ↦ (s - 1) * LFunctionTrivChar n s) (𝓝[≠] 1) (𝓝 (∏ p ∈ n.primeFactors, (1 - (↑p)⁻¹)))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ZetaValues | {
"line": 137,
"column": 2
} | {
"line": 138,
"column": 9
} | {
"line": 138,
"column": 10
} | [
{
"pp": "k : ℕ\nx : ℝ\n⊢ bernoulliFun k (1 - x) = (-1) ^ k * bernoulliFun k x",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"bernoulliFun._proof_1",
"congrArg",
"CommSemiring.toSemiring",
"Polynomial.bernoulli",
... | [
"k : ℕ\nx : ℝ\n⊢ (Polynomial.aeval (1 - x)) (Polynomial.bernoulli k) = (-1) ^ k * (Polynomial.aeval x) (Polynomial.bernoulli k)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.