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.ModularForms.JacobiTheta.Bounds | {
"line": 188,
"column": 4
} | {
"line": 188,
"column": 61
} | {
"line": 189,
"column": 2
} | [
{
"pp": "a : ℝ\nha : 0 ≤ a\n⊢ (fun t ↦ ((1 - rexp (-π * t)) ^ 2)⁻¹) =O[atTop] fun x ↦ 1",
"ppTerm": "?m.340",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"one_pow",
"NormedCommRing.toSeminormedCommRing",
"MulOne.toOne",
"SeminormedRing.toNorm",
... | [] | simpa only [inv_pow, one_pow] using! isBigO_one_aux.pow 2 | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.NumberTheory.LSeries.MellinEqDirichlet | {
"line": 88,
"column": 4
} | {
"line": 88,
"column": 42
} | {
"line": 88,
"column": 43
} | [
{
"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\nthis : ∀ (i : ι), ‖a i‖... | [
"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\nthis : ∀ (i : ι), ‖a i‖ / (π * q i)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 336,
"column": 37
} | {
"line": 336,
"column": 48
} | {
"line": 336,
"column": 49
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nP : WeakFEPair E\ninst✝ : CompleteSpace E\ns : ℂ\nhs : P.k < s.re\n⊢ -1 < (s - 1).re",
"ppTerm": "?m.84",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"AddGroup.... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nP : WeakFEPair E\ninst✝ : CompleteSpace E\ns : ℂ\nhs : P.k < s.re\n⊢ 0 < s.re"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 337,
"column": 43
} | {
"line": 337,
"column": 54
} | {
"line": 337,
"column": 55
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nP : WeakFEPair E\ninst✝ : CompleteSpace E\ns : ℂ\nhs : P.k < s.re\nh_re1 : -1 < (s - 1).re\n⊢ -1 < (s - ↑P.k - 1).re",
"ppTerm": "?m.120",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_o... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nP : WeakFEPair E\ninst✝ : CompleteSpace E\ns : ℂ\nhs : P.k < s.re\nh_re1 : -1 < (s - 1).re\n⊢ P.k < s.re"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 363,
"column": 29
} | {
"line": 364,
"column": 11
} | {
"line": 364,
"column": 12
} | [
{
"pp": "z τ : ℂ\nhτ : 0 < τ.im\nT : ℝ\nhT : 0 < T\nhτ' : T < τ.im\nS : ℝ\nhz : |z.im| < S\nV : Set (ℂ × ℂ) := {u | |u.im| < S} ×ˢ {v | T < v.im}\nhVo : IsOpen V\nu : ℤ → ℝ := fun n ↦ 2 * π * ↑|n| * rexp (-π * (T * ↑n ^ 2 - 2 * S * ↑|n|))\n⊢ Summable u",
"ppTerm": "?m.208",
"assigned": true,
"usedCo... | [
"z τ : ℂ\nhτ : 0 < τ.im\nT : ℝ\nhT : 0 < T\nhτ' : T < τ.im\nS : ℝ\nhz : |z.im| < S\nV : Set (ℂ × ℂ) := {u | |u.im| < S} ×ˢ {v | T < v.im}\nhVo : IsOpen V\nu : ℤ → ℝ := fun n ↦ 2 * π * ↑|n| * rexp (-π * (T * ↑n ^ 2 - 2 * S * ↑|n|))\n⊢ Summable fun n ↦ 2 * (π * (↑|n| * rexp (-π * (T * ↑n ^ 2 - 2 * (S * ↑|n|)))))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 457,
"column": 10
} | {
"line": 457,
"column": 21
} | {
"line": 457,
"column": 22
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\n⊢ ↑P.k - 0 ≠ 0",
"ppTerm": "?m.242",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Real",
"Real.instZero",
"congrArg",
"sub_zero",
... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\n⊢ ¬P.k = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 170,
"column": 2
} | {
"line": 170,
"column": 20
} | {
"line": 170,
"column": 21
} | [
{
"pp": "a t : ℝ\nht : 0 < t\nthis :\n ∀ (n : ℤ), cexp (-(↑π * (↑n + ↑a) ^ 2 * ↑t)) = cexp (-(↑π * ↑a ^ 2 * ↑t)) * jacobiTheta₂_term n (↑a * I * ↑t) (I * ↑t)\n⊢ HasSum (fun x ↦ ↑(rexp (-π * (↑x + a) ^ 2 * t))) (cexp (-↑π * ↑a ^ 2 * ↑t) * jacobiTheta₂ (↑a * I * ↑t) (I * ↑t))",
"ppTerm": "?m.120",
"assig... | [
"a t : ℝ\nht : 0 < t\nthis :\n ∀ (n : ℤ), cexp (-(↑π * (↑n + ↑a) ^ 2 * ↑t)) = cexp (-(↑π * ↑a ^ 2 * ↑t)) * jacobiTheta₂_term n (↑a * I * ↑t) (I * ↑t)\n⊢ HasSum (fun x ↦ cexp (-(↑π * ↑a ^ 2 * ↑t)) * jacobiTheta₂_term x (↑a * I * ↑t) (I * ↑t))\n (cexp (-(↑π * ↑a ^ 2 * ↑t)) * jacobiTheta₂ (↑a * I * ↑t) (I * ↑t))"
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 180,
"column": 2
} | {
"line": 180,
"column": 20
} | {
"line": 180,
"column": 21
} | [
{
"pp": "a t : ℝ\nht : 0 < t\nthis : ∀ (n : ℤ), cexp (2 * ↑π * I * ↑a * ↑n) * cexp (-(↑π * ↑n ^ 2 * ↑t)) = jacobiTheta₂_term n (↑a) (I * ↑t)\n⊢ HasSum (fun n ↦ cexp (2 * ↑π * I * ↑a * ↑n) * ↑(rexp (-π * ↑n ^ 2 * t))) (jacobiTheta₂ (↑a) (I * ↑t))",
"ppTerm": "?m.118",
"assigned": true,
"usedConstants... | [
"a t : ℝ\nht : 0 < t\nthis : ∀ (n : ℤ), cexp (2 * ↑π * I * ↑a * ↑n) * cexp (-(↑π * ↑n ^ 2 * ↑t)) = jacobiTheta₂_term n (↑a) (I * ↑t)\n⊢ HasSum (fun n ↦ jacobiTheta₂_term n (↑a) (I * ↑t)) (jacobiTheta₂ (↑a) (I * ↑t))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 190,
"column": 4
} | {
"line": 191,
"column": 11
} | {
"line": 191,
"column": 12
} | [
{
"pp": "case pos\nt : ℝ\nht : 0 < t\nthis : (a : Prop) → Decidable a\nk : ℤ\n⊢ HasSum (fun n ↦ if ↑n + ↑k = 0 then 0 else rexp (-π * (↑n + ↑k) ^ 2 * t)) (evenKernel (↑↑k) t - 1)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"... | [
"case pos\nt : ℝ\nht : 0 < t\nthis : (a : Prop) → Decidable a\nk : ℤ\n⊢ HasSum (fun n ↦ if n = -k then 0 else rexp (-(π * ↑(n + k) ^ 2 * t))) (evenKernel (↑↑k) t - 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 192,
"column": 37
} | {
"line": 192,
"column": 55
} | {
"line": 192,
"column": 56
} | [
{
"pp": "a t : ℝ\nht : 0 < t\nthis✝ : (a : Prop) → Decidable a\nh : ¬∃ n, ↑n = a\nthis : ∀ (n : ℤ), ↑n + a ≠ 0\n⊢ HasSum (fun n ↦ if ↑n + a = 0 then 0 else rexp (-π * (↑n + a) ^ 2 * t)) (evenKernel (↑a) t - 0)",
"ppTerm": "?m.137",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr... | [
"a t : ℝ\nht : 0 < t\nthis✝ : (a : Prop) → Decidable a\nh : ¬∃ n, ↑n = a\nthis : ∀ (n : ℤ), ↑n + a ≠ 0\n⊢ HasSum (fun n ↦ rexp (-(π * (↑n + a) ^ 2 * t))) (evenKernel (↑a) t)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 192,
"column": 2
} | {
"line": 195,
"column": 49
} | {
"line": 197,
"column": 0
} | [
{
"pp": "case neg\na t : ℝ\nht : 0 < t\nthis : (a : Prop) → Decidable a\nh : ¬∃ n, ↑n = a\n⊢ HasSum (fun n ↦ if ↑n + a = 0 then 0 else rexp (-π * (↑n + a) ^ 2 * t)) (evenKernel (↑a) t - 0)",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_forall_eq",
"Add... | [] | · suffices ∀ (n : ℤ), n + a ≠ 0 by simpa [this] using hasSum_int_evenKernel a ht
contrapose! h
let ⟨n, hn⟩ := h
exact ⟨-n, by simpa [neg_eq_iff_add_eq_zero]⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 200,
"column": 2
} | {
"line": 200,
"column": 13
} | {
"line": 200,
"column": 14
} | [
{
"pp": "a t : ℝ\nht : 0 < t\n⊢ HasSum (fun n ↦ if n = 0 then 0 else cexp (2 * ↑π * I * ↑a * ↑n) * ↑(rexp (-π * ↑n ^ 2 * t)))\n (↑(cosKernel (↑a) t) - 1)",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
... | [
"a t : ℝ\nht : 0 < t\n⊢ HasSum (fun n ↦ if n = 0 then 0 else cexp (2 * ↑π * I * ↑a * ↑n) * cexp (-(↑π * ↑n ^ 2 * ↑t)))\n (↑(cosKernel (↑a) t) - 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 13
} | {
"line": 147,
"column": 14
} | [
{
"pp": "x : ℝ\n⊢ oddKernel 0 x = 0",
"ppTerm": "?m.6",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℝ\n⊢ oddKernel 0 x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 13
} | {
"line": 156,
"column": 14
} | [
{
"pp": "x : ℝ\n⊢ sinKernel 0 x = 0",
"ppTerm": "?m.6",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℝ\n⊢ sinKernel 0 x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 474,
"column": 4
} | {
"line": 474,
"column": 96
} | {
"line": 474,
"column": 97
} | [
{
"pp": "z τ : ℂ\nhτ : 0 < τ.im\n⊢ 0 < (-I * τ).re",
"ppTerm": "?m.159",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Complex.mul_re",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"Real.instZero",
... | [
"z τ : ℂ\nhτ : 0 < τ.im\n⊢ 0 < τ.im"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 275,
"column": 4
} | {
"line": 275,
"column": 15
} | {
"line": 275,
"column": 16
} | [
{
"pp": "a : UnitAddCircle\nr p : ℝ\nhp : 0 < p\nhp' : (fun x ↦ cosKernel a x - 1) =O[atTop] fun x ↦ rexp (-p * x)\n⊢ (fun x ↦ (ofReal ∘ cosKernel a) x - 1) =O[atTop] fun x ↦ x ^ r",
"ppTerm": "?m.223",
"assigned": true,
"usedConstants": [
"Real.instPow",
"Real",
"Complex.instNorme... | [
"a : UnitAddCircle\nr p : ℝ\nhp : 0 < p\nhp' : (fun x ↦ cosKernel a x - 1) =O[atTop] fun x ↦ rexp (-p * x)\n⊢ (fun x ↦ ↑(cosKernel a x) - 1) =O[atTop] fun x ↦ x ^ r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.RiemannZeta | {
"line": 102,
"column": 6
} | {
"line": 102,
"column": 39
} | {
"line": 102,
"column": 40
} | [
{
"pp": "s : ℂ\n⊢ completedRiemannZeta₀ (1 - s) = completedRiemannZeta₀ s",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"HurwitzZeta.completedHurwitzZetaEven₀",
"congrArg",
"HSub.hSub",
"AddCommGroup.toAddGroup",
"id",
"Sub... | [
"s : ℂ\n⊢ completedHurwitzZetaEven₀ 0 (1 - s) = completedRiemannZeta₀ s"
] | ← completedHurwitzZetaEven₀_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.RiemannZeta | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 49
} | {
"line": 131,
"column": 50
} | [
{
"pp": "s : ℂ\n⊢ hurwitzZeta 0 s = riemannZeta s",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"riemannZeta",
"AddLeftCancelSemigroup.toIsLeftCancelAdd",
"AddMonoid.toAddZeroClass",
"AddGroupWithOne.toAddMonoidWithOne",
"Hurwit... | [
"s : ℂ\n⊢ hurwitzZetaOdd 0 s = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.RiemannZeta | {
"line": 209,
"column": 2
} | {
"line": 210,
"column": 21
} | {
"line": 210,
"column": 22
} | [
{
"pp": "s : ℂ\nhs : 1 < s.re\n⊢ riemannZeta s = ∑' (n : ℕ), 1 / ↑n ^ s",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"s : ℂ\nhs : 1 < s.re\n⊢ riemannZeta s = ∑' (n : ℕ), 1 / ↑n ^ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 430,
"column": 2
} | {
"line": 430,
"column": 18
} | {
"line": 431,
"column": 2
} | [
{
"pp": "a b : UnitAddCircle\nthis :\n ∀ (s : ℂ),\n completedHurwitzZetaEven a s - completedHurwitzZetaEven b s =\n completedHurwitzZetaEven₀ a s - completedHurwitzZetaEven₀ b s -\n ((if a = 0 then 1 else 0) - if b = 0 then 1 else 0) / s\n⊢ DifferentiableAt ℂ (fun s ↦ completedHurwitzZetaEven a ... | [
"a b : UnitAddCircle\nthis :\n ∀ (s : ℂ),\n completedHurwitzZetaEven a s - completedHurwitzZetaEven b s =\n completedHurwitzZetaEven₀ a s - completedHurwitzZetaEven₀ b s -\n ((if a = 0 then 1 else 0) - if b = 0 then 1 else 0) / s\n⊢ DifferentiableAt ℂ\n (fun x ↦\n completedHurwitzZetaEven₀ a... | rw [funext this] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 442,
"column": 6
} | {
"line": 442,
"column": 29
} | {
"line": 442,
"column": 30
} | [
{
"pp": "case refine_2.refine_1\na : UnitAddCircle\ns : ℂ\nhs : s ≠ 0\nhs' : s ≠ 1 ∨ a ≠ 0\nh : s ≠ 1\n⊢ s / 2 ≠ ↑(1 / 2)",
"ppTerm": "?refine_2.refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"congrArg",
"Real.i... | [
"case refine_2.refine_1\na : UnitAddCircle\ns : ℂ\nhs : s ≠ 0\nhs' : s ≠ 1 ∨ a ≠ 0\nh : s ≠ 1\n⊢ ¬s / 2 = 2⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 80,
"column": 2
} | {
"line": 81,
"column": 9
} | {
"line": 81,
"column": 10
} | [
{
"pp": "⊢ (abscissaOfAbsConv fun n ↦ ↑(μ n)) = 1",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"instInfSetEReal",
"Real",
"Set.Ioi",
"ArithmeticFunction.instFunLikeNat",
"congrArg",
"Set.ofPred",
"_private.Mathli... | [
"⊢ sInf (Set.Ioo 1 ⊤) = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 153,
"column": 2
} | {
"line": 153,
"column": 57
} | {
"line": 153,
"column": 58
} | [
{
"pp": "n : ℕ\nχ : DirichletCharacter ℂ n\nthis : (1 ⍟ fun x ↦ ↑(μ x)) = δ\n⊢ (fun n_1 ↦ χ ↑n_1) * 1 ⍟ ((fun n_1 ↦ χ ↑n_1) * fun n ↦ ↑(μ n)) = δ",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"ArithmeticFunc... | [
"n : ℕ\nχ : DirichletCharacter ℂ n\nthis : (1 ⍟ fun x ↦ ↑(μ x)) = δ\n⊢ (fun x ↦ χ ↑x) * δ = δ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.SumPrimeReciprocals | {
"line": 115,
"column": 17
} | {
"line": 115,
"column": 47
} | {
"line": 115,
"column": 48
} | [
{
"pp": "h : Summable ({p | Nat.Prime p}.indicator fun n ↦ 1 / ↑n)\nk : ℕ\nhk : ∑' (x : ℕ), ({p | Nat.Prime p} ∩ {p | k ≤ p}).indicator (fun n ↦ 1 / ↑n) x < 1 / 2\nh' : Summable (({p | Nat.Prime p} ∩ {p | k ≤ p}).indicator fun n ↦ 1 / ↑n)\np : ℕ\nhp : p ∈ (4 ^ (k.primesBelow.card + 1)).succ.primesBelow \\ k.pri... | [
"h : Summable ({p | Nat.Prime p}.indicator fun n ↦ 1 / ↑n)\nk : ℕ\nhk : ∑' (x : ℕ), ({p | Nat.Prime p} ∩ {p | k ≤ p}).indicator (fun n ↦ 1 / ↑n) x < 1 / 2\nh' : Summable (({p | Nat.Prime p} ∩ {p | k ≤ p}).indicator fun n ↦ 1 / ↑n)\np : ℕ\nhp : p ∈ (4 ^ (k.primesBelow.card + 1)).succ.primesBelow \\ k.primesBelow\nhp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 204,
"column": 2
} | {
"line": 205,
"column": 9
} | {
"line": 205,
"column": 10
} | [
{
"pp": "N : ℕ\nhn : N ≠ 0\nχ : DirichletCharacter ℂ N\n⊢ (abscissaOfAbsConv fun n ↦ χ ↑n) = 1",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instInfSetEReal",
"Real",
"Set.Ioi",
"ZMod.commRing",
"congrArg",
"Set.ofPred",
"PartialO... | [
"N : ℕ\nhn : N ≠ 0\nχ : DirichletCharacter ℂ N\n⊢ sInf (Set.Ioo 1 ⊤) = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 213,
"column": 2
} | {
"line": 213,
"column": 13
} | {
"line": 213,
"column": 14
} | [
{
"pp": "N : ℕ\nχ : DirichletCharacter ℂ N\nf : ℕ → ℂ\ns : ℂ\nh : LSeriesSummable f s\nn : ℕ\n⊢ ‖((fun n ↦ χ ↑n) * f) n‖ ≤ ‖f n‖",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"ZMod.commRing",
... | [
"N : ℕ\nχ : DirichletCharacter ℂ N\nf : ℕ → ℂ\ns : ℂ\nh : LSeriesSummable f s\nn : ℕ\n⊢ ‖χ ↑n‖ * ‖f n‖ ≤ ‖f n‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 232,
"column": 13
} | {
"line": 232,
"column": 28
} | {
"line": 232,
"column": 29
} | [
{
"pp": "N : ℕ\nχ : DirichletCharacter ℂ N\ns : ℂ\nhs : 1 < s.re\nh : L (fun n ↦ χ ↑n) s = 0\n⊢ False",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"N : ℕ\nχ : DirichletCharacter ℂ N\ns : ℂ\nhs : 1 < s.re\nh : L (fun n ↦ χ ↑n) s = 0\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 301,
"column": 13
} | {
"line": 301,
"column": 41
} | {
"line": 301,
"column": 42
} | [
{
"pp": "s : ℂ\nhs : 1 < s.re\nh : L (fun n ↦ ↑(ζ n)) s = 0\n⊢ False",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"s : ℂ\nhs : 1 < s.re\nh : L (fun n ↦ ↑(ζ n)) s = 0\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 339,
"column": 34
} | {
"line": 339,
"column": 45
} | {
"line": 339,
"column": 46
} | [
{
"pp": "x : ℝ\nhx : 1 < x\n⊢ 1 < (↑x).re",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Real",
"Real.instLT",
"id",
"Complex.ofReal",
"Complex.re",
"Real.instOne",
"LT.lt",
"One.toOfNat1",
"OfNat.ofNat"
],
"usedFVars": [
... | [
"x : ℝ\nhx : 1 < x\n⊢ 1 < x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 342,
"column": 2
} | {
"line": 342,
"column": 45
} | {
"line": 342,
"column": 46
} | [
{
"pp": "x : ℝ\nhx : 1 < x\nhx' : 1 < (↑x).re\n⊢ abscissaOfAbsConv 1 < ↑x",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"LSeries.abscissaOfAbsConv_one",
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"EReal",
"id",
"Pi.in... | [
"x : ℝ\nhx : 1 < x\nhx' : 1 < (↑x).re\n⊢ 1 < ↑x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 374,
"column": 2
} | {
"line": 375,
"column": 9
} | {
"line": 375,
"column": 10
} | [
{
"pp": "n : ℕ\n⊢ ((fun n ↦ ↑(Λ n)) ⍟ fun n ↦ ↑(ζ n)) n = Complex.log ↑n",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"ArithmeticFunction.vonMangoldt",
"CharP.cast_eq_zero",
"Eq.mpr",
"Complex.log",
"Nat.instMulZeroClass",
"Real",
"HMul.hMul",
... | [
"n : ℕ\n⊢ (∑ x ∈ n.divisorsAntidiagonal, if x.2 = 0 then 0 else ↑(Λ x.1)) = Complex.log ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 387,
"column": 4
} | {
"line": 388,
"column": 11
} | {
"line": 388,
"column": 12
} | [
{
"pp": "s : ℂ\nhs : 1 < s.re\nhf : Summable fun x ↦ ‖term (logMul 1) s x‖\nn : ℕ\n⊢ ‖(fun n ↦ ↑(Λ n)) n‖ ≤ ‖Complex.log ↑n‖",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [
"ArithmeticFunction.vonMangoldt",
"Norm.norm",
"Eq.mpr",
"Complex.log",
"Real.instLE",
... | [
"s : ℂ\nhs : 1 < s.re\nhf : Summable fun x ↦ ‖term (logMul 1) s x‖\nn : ℕ\n⊢ Λ n ≤ Real.log ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 272,
"column": 2
} | {
"line": 273,
"column": 66
} | {
"line": 274,
"column": 4
} | [
{
"pp": "b p : ℝ\nhp : 0 < p\nhp' : HurwitzKernelBounds.F_int 1 ↑b =O[atTop] fun t ↦ rexp (-p * t)\nt : ℝ\nht : 0 < t\n⊢ ‖oddKernel (↑b) t‖ ≤ HurwitzKernelBounds.F_int 1 (↑b) t",
"ppTerm": "?m.97",
"assigned": true,
"usedConstants": [
"HurwitzKernelBounds.f_int",
"Norm.norm",
"Int.... | [
"b p : ℝ\nhp : 0 < p\nhp' : HurwitzKernelBounds.F_int 1 ↑b =O[atTop] fun t ↦ rexp (-p * t)\nt : ℝ\nht : 0 < t\n⊢ |∑' (b_1 : ℤ), (↑b_1 + b) * rexp (-(π * (↑b_1 + b) ^ 2 * t))| ≤ ∑' (n : ℤ), |↑n + b| * rexp (-(π * (↑n + b) ^ 2 * t))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 578,
"column": 42
} | {
"line": 578,
"column": 52
} | {
"line": 578,
"column": 53
} | [
{
"pp": "Λ : ℂ → ℂ\nhf : ∀ (s : ℂ), s ≠ 0 → s ≠ 1 → DifferentiableAt ℂ Λ s\nL : ℂ\nh_lim : Tendsto (fun s ↦ s * Λ s) (𝓝[≠] 0) (𝓝 L)\nclaim : ∀ (t : ℂ), t ≠ 0 → t ≠ 1 → DifferentiableAt ℂ (fun u ↦ Λ u / u.Gammaℝ) t\nclaim2 : Tendsto (fun s ↦ Λ s / s.Gammaℝ) (𝓝[≠] 0) (𝓝 (L / 2))\nhs' : 0 ≠ 1\nS_nhds : {1}ᶜ ∈ ... | [
"Λ : ℂ → ℂ\nhf : ∀ (s : ℂ), s ≠ 0 → s ≠ 1 → DifferentiableAt ℂ Λ s\nL : ℂ\nh_lim : Tendsto (fun s ↦ s * Λ s) (𝓝[≠] 0) (𝓝 L)\nclaim : ∀ (t : ℂ), t ≠ 0 → t ≠ 1 → DifferentiableAt ℂ (fun u ↦ Λ u / u.Gammaℝ) t\nclaim2 : Tendsto (fun s ↦ Λ s / s.Gammaℝ) (𝓝[≠] 0) (𝓝 (L / 2))\nhs' : 0 ≠ 1\nS_nhds : {1}ᶜ ∈ 𝓝 0\nx : ℂ\... | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 592,
"column": 4
} | {
"line": 592,
"column": 24
} | {
"line": 592,
"column": 25
} | [
{
"pp": "case inr\na : UnitAddCircle\nh : a ≠ 0 ∨ 0 ≠ 0\n⊢ Function.update (fun s ↦ completedHurwitzZetaEven a s / s.Gammaℝ) 0 (if a = 0 then -1 / 2 else 0) 0 =\n completedHurwitzZetaEven a 0 / Gammaℝ 0",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.... | [
"case inr\na : UnitAddCircle\nh : a ≠ 0 ∨ 0 ≠ 0\n⊢ ¬a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 318,
"column": 4
} | {
"line": 318,
"column": 15
} | {
"line": 318,
"column": 16
} | [
{
"pp": "a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : (fun x ↦ ‖oddKernel a x‖) =O[atTop] fun x ↦ rexp (-v * x)\n⊢ (fun x ↦ ‖(ofReal ∘ oddKernel a) x - 0‖) =O[atTop] fun x ↦ x ^ r",
"ppTerm": "?m.144",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toS... | [
"a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : (fun x ↦ ‖oddKernel a x‖) =O[atTop] fun x ↦ rexp (-v * x)\n⊢ oddKernel a =O[atTop] fun x ↦ x ^ r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 322,
"column": 4
} | {
"line": 322,
"column": 15
} | {
"line": 322,
"column": 16
} | [
{
"pp": "a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : (fun x ↦ ‖sinKernel a x‖) =O[atTop] fun x ↦ rexp (-v * x)\n⊢ (fun x ↦ ‖(ofReal ∘ sinKernel a) x - 0‖) =O[atTop] fun x ↦ x ^ r",
"ppTerm": "?m.223",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toS... | [
"a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : (fun x ↦ ‖sinKernel a x‖) =O[atTop] fun x ↦ rexp (-v * x)\n⊢ sinKernel a =O[atTop] fun x ↦ x ^ r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 650,
"column": 2
} | {
"line": 650,
"column": 13
} | {
"line": 650,
"column": 14
} | [
{
"pp": "a : UnitAddCircle\n⊢ Tendsto (fun s ↦ hurwitzZetaEven a s - 1 / (s - 1) / s.Gammaℝ) (𝓝 1) (𝓝 (hurwitzZetaEven a 1))",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"DivInvMonoid.toInv",
"instHDiv",
"c... | [
"a : UnitAddCircle\n⊢ Tendsto (fun s ↦ hurwitzZetaEven a s - (s - 1)⁻¹ / s.Gammaℝ) (𝓝 1) (𝓝 (hurwitzZetaEven a 1))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.EulerProduct.DirichletLSeries | {
"line": 40,
"column": 4
} | {
"line": 41,
"column": 11
} | {
"line": 41,
"column": 12
} | [
{
"pp": "s : ℂ\nhs : s ≠ 0\nm n : ℕ\n⊢ ↑(m * n) ^ (-s) = ↑m ^ (-s) * ↑n ^ (-s)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.instMulZeroOneClass",
"HMul.hMul",
"congrArg",
"Complex.instPow",
... | [
"s : ℂ\nhs : s ≠ 0\nm n : ℕ\n⊢ (↑m * ↑n) ^ (-s) = ↑m ^ (-s) * ↑n ^ (-s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.EulerProduct.DirichletLSeries | {
"line": 52,
"column": 4
} | {
"line": 54,
"column": 11
} | {
"line": 54,
"column": 12
} | [
{
"pp": "s : ℂ\nn✝ : ℕ\nχ : DirichletCharacter ℂ n✝\nhs : s ≠ 0\nm n : ℕ\n⊢ χ ↑(m * n) * ↑↑(m * n) ^ (-s) = χ ↑m * ↑↑m ^ (-s) * (χ ↑n * ↑↑n ^ (-s))",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne... | [
"s : ℂ\nn✝ : ℕ\nχ : DirichletCharacter ℂ n✝\nhs : s ≠ 0\nm n : ℕ\n⊢ χ ↑m * χ ↑n * (↑↑m ^ (-s) * ↑↑n ^ (-s)) = χ ↑m * ↑↑m ^ (-s) * (χ ↑n * ↑↑n ^ (-s))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 765,
"column": 4
} | {
"line": 765,
"column": 47
} | {
"line": 765,
"column": 48
} | [
{
"pp": "a : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ -↑n\nhs' : a ≠ 0 ∨ s ≠ 1\n⊢ a ≠ 0 ∨ 1 - s ≠ 0",
"ppTerm": "?m.88",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"AddGroupWithOne.toAddGroup",
"congrArg",
"HSub.hSub",
"Complex.instZero",
"AddCo... | [
"a : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ -↑n\nhs' : a ≠ 0 ∨ s ≠ 1\n⊢ ¬a = 0 ∨ ¬1 = s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 767,
"column": 30
} | {
"line": 767,
"column": 41
} | {
"line": 767,
"column": 42
} | [
{
"pp": "a : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ -↑n\nhs' : a ≠ 0 ∨ s ≠ 1\nthis : hurwitzZetaEven a (1 - s) = completedHurwitzZetaEven a (1 - s) * (1 - s).Gammaℝ⁻¹\n⊢ s ≠ 0",
"ppTerm": "?m.118",
"assigned": true,
"usedConstants": [
"Complex.instZero",
"id",
"Ne",
"Zero.... | [
"a : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ -↑n\nhs' : a ≠ 0 ∨ s ≠ 1\nthis : hurwitzZetaEven a (1 - s) = completedHurwitzZetaEven a (1 - s) * (1 - s).Gammaℝ⁻¹\n⊢ ¬s = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 778,
"column": 4
} | {
"line": 778,
"column": 29
} | {
"line": 778,
"column": 30
} | [
{
"pp": "case h\na : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ 1 - ↑n\n⊢ 1 - s ≠ 0",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddGroupWithOne.toAddGroup",
"congrArg",
"_private.Mathlib.NumberTheory.LSeries.HurwitzZetaEven.0.HurwitzZeta.cosZeta_one_sub._... | [
"case h\na : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ 1 - ↑n\n⊢ ¬1 = s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.FLT.Basic | {
"line": 231,
"column": 30
} | {
"line": 231,
"column": 46
} | {
"line": 231,
"column": 46
} | [
{
"pp": "n : ℕ\nR : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : IsDomain R\ninst✝¹ : DecidableEq R\ninst✝ : NormalizedGCDMonoid R\nhn : ∀ (a b c : R), a ≠ 0 → b ≠ 0 → c ≠ 0 → {a, b, c}.gcd id = 1 → a ^ n + b ^ n ≠ c ^ n\na b c : R\ns : Finset R := {a, b, c}\nd : R := s.gcd id\nA : R\nhA : a = d * A\nB : R\nhB :... | [
"n : ℕ\nR : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : IsDomain R\ninst✝¹ : DecidableEq R\ninst✝ : NormalizedGCDMonoid R\nhn : ∀ (a b c : R), a ≠ 0 → b ≠ 0 → c ≠ 0 → {a, b, c}.gcd id = 1 → a ^ n + b ^ n ≠ c ^ n\na b c : R\ns : Finset R := {a, b, c}\nd : R := s.gcd id\nA : R\nhA : a = d * A\nB : R\nhB : b = d * B\n... | normalize_eq_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 779,
"column": 69
} | {
"line": 779,
"column": 80
} | {
"line": 779,
"column": 81
} | [
{
"pp": "a : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ 1 - ↑n\nthis : cosZeta a (1 - s) = completedCosZeta a (1 - s) * (1 - s).Gammaℝ⁻¹\nn : ℕ\n⊢ s ≠ -↑n",
"ppTerm": "?m.123",
"assigned": true,
"usedConstants": [
"id",
"Ne",
"Complex.instNatCast",
"Nat.cast",
"Complex",... | [
"a : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ 1 - ↑n\nthis : cosZeta a (1 - s) = completedCosZeta a (1 - s) * (1 - s).Gammaℝ⁻¹\nn : ℕ\n⊢ ¬s = -↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 780,
"column": 48
} | {
"line": 780,
"column": 59
} | {
"line": 780,
"column": 60
} | [
{
"pp": "a : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ 1 - ↑n\nthis : cosZeta a (1 - s) = completedCosZeta a (1 - s) * (1 - s).Gammaℝ⁻¹\n⊢ s ≠ 0",
"ppTerm": "?m.146",
"assigned": true,
"usedConstants": [
"Complex.instZero",
"id",
"Ne",
"Zero.toOfNat0",
"Complex",
... | [
"a : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ 1 - ↑n\nthis : cosZeta a (1 - s) = completedCosZeta a (1 - s) * (1 - s).Gammaℝ⁻¹\n⊢ ¬s = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 38,
"column": 2
} | {
"line": 38,
"column": 13
} | {
"line": 38,
"column": 14
} | [
{
"pp": "z : ℤ\n⊢ ¬↑(z * z) = ↑2",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"HMul.hMul",
"ZMod.commRing",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"AddGroupWithOne.toAddMonoidWithOne",
"id",
"NonUnitalNonAss... | [
"z : ℤ\n⊢ ¬↑z * ↑z = 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 148,
"column": 6
} | {
"line": 149,
"column": 26
} | {
"line": 149,
"column": 27
} | [
{
"pp": "x y z : ℤ\nh : PythagoreanTriple x y z\nh0 : x.gcd y = 0\nhx : x = 0\nhy : y = 0\n⊢ z = 0",
"ppTerm": "?m.33",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x y z : ℤ\nh : PythagoreanTriple x y z\nh0 : x.gcd y = 0\nhx : x = 0\nhy : y = 0\n⊢ z = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.FLT.Four | {
"line": 148,
"column": 17
} | {
"line": 148,
"column": 48
} | {
"line": 150,
"column": 0
} | [
{
"pp": "r s : ℤ\nh : IsCoprime r s\n⊢ IsCoprime (s ^ 2 + r ^ 2) s",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Int.isCoprime_of_sq_sum"
],
"usedFVars": [
"s",
"r",
"h"
],
"usedGoals": []
}
] | [] | apply Int.isCoprime_of_sq_sum h | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 163,
"column": 6
} | {
"line": 164,
"column": 26
} | {
"line": 164,
"column": 27
} | [
{
"pp": "x y z : ℤ\nh : PythagoreanTriple x y z\nh0 : x.gcd y = 0\nhx : x = 0\nhy : y = 0\n⊢ z = 0",
"ppTerm": "?m.43",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x y z : ℤ\nh : PythagoreanTriple x y z\nh0 : x.gcd y = 0\nhx : x = 0\nhy : y = 0\n⊢ z = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 32
} | {
"line": 165,
"column": 33
} | [
{
"pp": "case pos\nx y z : ℤ\nh : PythagoreanTriple x y z\nh0 : x.gcd y = 0\nhx : x = 0\nhy : y = 0\nhz : z = 0\n⊢ PythagoreanTriple (x / ↑(x.gcd y)) (y / ↑(x.gcd y)) (z / ↑(x.gcd y))",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"CharP.cast_eq_zero",
"Int.gcd",
"Eq.mpr... | [
"case pos\nx y z : ℤ\nh : PythagoreanTriple x y z\nh0 : x.gcd y = 0\nhx : x = 0\nhy : y = 0\nhz : z = 0\n⊢ PythagoreanTriple 0 0 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 249,
"column": 6
} | {
"line": 249,
"column": 56
} | {
"line": 249,
"column": 57
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nhk : ∀ (x : K), 1 + x ^ 2 ≠ 0\nx : K\n⊢ ¬1 = -1",
"ppTerm": "?m.267",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"AddGroupWithOne.toAddGroup",
"congrArg",
"AddMonoid.... | [
"K : Type u_1\ninst✝ : Field K\nhk : ∀ (x : K), 1 + x ^ 2 ≠ 0\nx : K\n⊢ ¬1 + 1 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.EulerProduct.DirichletLSeries | {
"line": 234,
"column": 47
} | {
"line": 246,
"column": 14
} | {
"line": 247,
"column": 4
} | [
{
"pp": "N : ℕ\nχ : DirichletCharacter ℂ N\ns : ℂ\nhs : 1 < s.re\nhpow_le : ∀ (p : Primes), ‖χ ↑↑p * ↑↑p ^ (-s)‖ < 1\nf : ℕ → ℂ := fun n ↦ χ ↑n * ↑(Λ n) / ↑(Real.log ↑n) * ↑n ^ (-s)\n⊢ ∑' (p : Primes) (k : ℕ),\n χ (↑↑p ^ (k + 1)) * ↑(↑p ^ (k + 1)) ^ (-s) * ↑(Λ (↑p ^ (k + 1))) / ↑(Real.log (↑↑p ^ (k + 1))) ... | [] | by
rw [← tsum_primes_pow_eq]
· exact tsum_congr fun p ↦ tsum_congr fun k ↦ (by unfold f; simp; ring)
· apply comp_injective _ Subtype.coe_injective (f := f)
apply of_norm_bounded_eventually_nat (g := (↑· ^ (-s.re)))
· simp [hs]
· filter_upwards [eventually_gt_atTop 1] with n hn... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 293,
"column": 2
} | {
"line": 295,
"column": 8
} | {
"line": 296,
"column": 2
} | [
{
"pp": "m n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (m ^ 2 + n ^ 2) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ m ^ 2 + n ^ 2\nh2m : ↑p ∣ 2 * m ^ 2\n⊢ False",
"ppTerm": "?m.146",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic... | [
"m n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (m ^ 2 + n ^ 2) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ m ^ 2 + n ^ 2\nh2m : ↑p ∣ 2 * m ^ 2\nh2n : ↑p ∣ 2 * n ^ 2\n⊢ False"
] | have h2n : (p : ℤ) ∣ 2 * n ^ 2 := by
convert! dvd_sub hp2 hp1 using 1
ring | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.EulerProduct.DirichletLSeries | {
"line": 273,
"column": 39
} | {
"line": 273,
"column": 50
} | {
"line": 273,
"column": 51
} | [
{
"pp": "s : ℝ\nhs : 1 < s\n⊢ 1 < (↑s).re",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Real",
"Real.instLT",
"id",
"Complex.ofReal",
"Complex.re",
"Real.instOne",
"LT.lt",
"One.toOfNat1",
"OfNat.ofNat"
],
"usedFVars": [
... | [
"s : ℝ\nhs : 1 < s\n⊢ 1 < s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Radical.Basic | {
"line": 200,
"column": 2
} | {
"line": 200,
"column": 27
} | {
"line": 202,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝² : CommMonoidWithZero M\ninst✝¹ : NormalizationMonoid M\ninst✝ : UniqueFactorizationMonoid M\na : M\nha : Prime a\nn : ℕ\nhn : n ≠ 0\n⊢ radical a = normalize a",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"UniqueFactorizationMonoid.radical_of_prime"
... | [] | exact radical_of_prime ha | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Radical.Basic | {
"line": 220,
"column": 54
} | {
"line": 220,
"column": 84
} | {
"line": 220,
"column": 85
} | [
{
"pp": "M : Type u_1\ninst✝² : CommMonoidWithZero M\ninst✝¹ : NormalizationMonoid M\ninst✝ : UniqueFactorizationMonoid M\na b : M\nha : Irreducible a\nhb : b ≠ 0\nha' : a ∣ b\nc : M\nhc : c ∈ normalizedFactors b\nhc' : Associated a c\n⊢ c ∈ primeFactors b",
"ppTerm": "?m.73",
"assigned": true,
"use... | [
"M : Type u_1\ninst✝² : CommMonoidWithZero M\ninst✝¹ : NormalizationMonoid M\ninst✝ : UniqueFactorizationMonoid M\na b : M\nha : Irreducible a\nhb : b ≠ 0\nha' : a ∣ b\nc : M\nhc : c ∈ normalizedFactors b\nhc' : Associated a c\n⊢ c ∈ normalizedFactors b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.FLT.MasonStothers | {
"line": 77,
"column": 4
} | {
"line": 77,
"column": 28
} | {
"line": 77,
"column": 29
} | [
{
"pp": "k : Type u_1\ninst✝¹ : Field k\ninst✝ : DecidableEq k\na b c : k[X]\nha : a ≠ 0\nhb : b ≠ 0\nhc : c ≠ 0\nhab : IsCoprime a b\nhsum : b + c + a = 0\nw : k[X] := a.wronskian b\nwab : w = a.wronskian b\nhbc : IsCoprime b c\nhsum' : b + c + a = 0\nhca : IsCoprime c a\nwbc : w = b.wronskian c\n⊢ w = c.wrons... | [
"k : Type u_1\ninst✝¹ : Field k\ninst✝ : DecidableEq k\na b c : k[X]\nha : a ≠ 0\nhb : b ≠ 0\nhc : c ≠ 0\nhab : IsCoprime a b\nhsum : b + c + a = 0\nw : k[X] := a.wronskian b\nwab : w = a.wronskian b\nhbc : IsCoprime b c\nhsum' : b + c + a = 0\nhca : IsCoprime c a\nwbc : w = b.wronskian c\n⊢ w = c.wronskian a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 468,
"column": 2
} | {
"line": 472,
"column": 7
} | {
"line": 473,
"column": 2
} | [
{
"pp": "case neg.inl.inl\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhyo : y % 2 = 1\nhzpos : 0 < z\nh0 : ¬x = 0\nv : ℚ := ↑x / ↑z\nw : ℚ := ↑y / ↑z\nhq : v ^ 2 + w ^ 2 = 1\nhvz : v ≠ 0\nhw1 : w ≠ -1\nhQ : ∀ (x : ℚ), 1 + x ^ 2 ≠ 0\nhp : (v, w) ∈ {p | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1}\nq : ℚ := (... | [
"case neg.inl.inr\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhyo : y % 2 = 1\nhzpos : 0 < z\nh0 : ¬x = 0\nv : ℚ := ↑x / ↑z\nw : ℚ := ↑y / ↑z\nhq : v ^ 2 + w ^ 2 = 1\nhvz : v ≠ 0\nhw1 : w ≠ -1\nhQ : ∀ (x : ℚ), 1 + x ^ 2 ≠ 0\nhp : (v, w) ∈ {p | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1}\nq : ℚ := (circleEquivG... | · -- m even, n even
exfalso
have h1 : 2 ∣ (Int.gcd n m : ℤ) :=
Int.dvd_coe_gcd (Int.dvd_of_emod_eq_zero hn2) (Int.dvd_of_emod_eq_zero hm2)
lia | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.NumberField.FractionalIdeal | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 13
} | {
"line": 114,
"column": 14
} | [
{
"pp": "case e_6\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\ne : Free.ChooseBasisIndex ℤ (𝓞 K) ≃ Free.ChooseBasisIndex ℤ ↥↑↑I\nx✝ : Free.ChooseBasisIndex ℤ (𝓞 K)\n⊢ ((basisOfFractionalIdeal K I).reindex e.symm) x✝ = (Subtype.val ∘ ⇑((fractionalIdealBasis K ↑I).re... | [
"case e_6\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\ne : Free.ChooseBasisIndex ℤ (𝓞 K) ≃ Free.ChooseBasisIndex ℤ ↥↑↑I\nx✝ : Free.ChooseBasisIndex ℤ (𝓞 K)\n⊢ (basisOfFractionalIdeal K I) (e x✝) = ↑((fractionalIdealBasis K ↑I) (e x✝))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 510,
"column": 45
} | {
"line": 510,
"column": 89
} | {
"line": 510,
"column": 90
} | [
{
"pp": "x y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhz : z ≤ 0\n⊢ PythagoreanTriple x y (-z)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
... | [
"x y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhz : z ≤ 0\n⊢ x * x + y * y = z * z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 539,
"column": 6
} | {
"line": 539,
"column": 17
} | {
"line": 539,
"column": 18
} | [
{
"pp": "case h.inl\nz m n : ℤ\nh : PythagoreanTriple (m ^ 2 - n ^ 2) (2 * m * n) z ∧ (m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\nco : m.gcd n = 1\npp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0\nthis : z ^ 2 = (m ^ 2 + n ^ 2) ^ 2\n⊢ z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)",
"ppTerm": "?h.inl",
"assigned"... | [
"case h.inl\nz m n : ℤ\nh : PythagoreanTriple (m ^ 2 - n ^ 2) (2 * m * n) z ∧ (m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\nco : m.gcd n = 1\npp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0\nthis : z ^ 2 = (m ^ 2 + n ^ 2) ^ 2\n⊢ z = m ^ 2 + n ^ 2 ∨ z = -n ^ 2 + -m ^ 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 544,
"column": 6
} | {
"line": 544,
"column": 17
} | {
"line": 544,
"column": 18
} | [
{
"pp": "case h.inr\nz m n : ℤ\nh : PythagoreanTriple (2 * m * n) (m ^ 2 - n ^ 2) z ∧ (2 * m * n).gcd (m ^ 2 - n ^ 2) = 1\nco : m.gcd n = 1\npp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0\nthis : z ^ 2 = (m ^ 2 + n ^ 2) ^ 2\n⊢ z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)",
"ppTerm": "?h.inr",
"assigned"... | [
"case h.inr\nz m n : ℤ\nh : PythagoreanTriple (2 * m * n) (m ^ 2 - n ^ 2) z ∧ (2 * m * n).gcd (m ^ 2 - n ^ 2) = 1\nco : m.gcd n = 1\npp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0\nthis : z ^ 2 = (m ^ 2 + n ^ 2) ^ 2\n⊢ z = m ^ 2 + n ^ 2 ∨ z = -n ^ 2 + -m ^ 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 119,
"column": 17
} | {
"line": 119,
"column": 28
} | {
"line": 119,
"column": 29
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nx✝¹ x✝ : InfinitePlace K\nh : x✝¹.embedding = x✝.embedding\n⊢ x✝¹ = x✝",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"K : Type u_1\ninst✝ : Field K\nx✝¹ x✝ : InfinitePlace K\nh : x✝¹.embedding = x✝.embedding\n⊢ x✝¹ = x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 374,
"column": 2
} | {
"line": 379,
"column": 82
} | {
"line": 380,
"column": 2
} | [
{
"pp": "case e'_2\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : K\n⊢ ∏ w, w x ^ w.mult = ‖∏ σ, σ x‖",
"ppTerm": "?e'_2",
"assigned": true,
"usedConstants": [
"Norm.norm",
"NumberField.InfinitePlace.instFunLikeReal",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRin... | [
"case e'_3\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : K\n⊢ ↑|(Algebra.norm ℚ) x| = ‖(algebraMap ℚ ℂ) ((Algebra.norm ℚ) x)‖"
] | · rw [norm_prod, ← Fintype.prod_equiv (RingHom.equivRatAlgHom K ℂ) (fun f => ‖f x‖)
(fun φ => ‖φ x‖) fun _ => by simp [RingHom.equivRatAlgHom_apply]]
rw [← Finset.prod_fiberwise Finset.univ mk (fun φ => ‖φ x‖)]
have (w : InfinitePlace K) (φ) (hφ : φ ∈ ({φ | mk φ = w} : Finset _)) :
‖φ x‖ = w x := ... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.NumberField.Units.Basic | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 55
} | {
"line": 141,
"column": 4
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (𝓞 K)ˣ\n⊢ ∑ w, ↑w.mult * Real.log (w ((algebraMap (𝓞 K) K) ↑x)) = 0",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (𝓞 K)ˣ\n⊢ ∑ w, ↑w.mult * Real.log (w ((algebraMap (𝓞 K) K) ↑x)) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 619,
"column": 6
} | {
"line": 619,
"column": 17
} | {
"line": 619,
"column": 18
} | [
{
"pp": "case h.inl\nz k m n : ℤ\nright✝ : m.gcd n = 1\nh : PythagoreanTriple (k * (m ^ 2 - n ^ 2)) (k * (2 * m * n)) z\nthis : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2\n⊢ z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2)",
"ppTerm": "?h.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"No... | [
"case h.inl\nz k m n : ℤ\nright✝ : m.gcd n = 1\nh : PythagoreanTriple (k * (m ^ 2 - n ^ 2)) (k * (2 * m * n)) z\nthis : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2\n⊢ z = k * (m ^ 2 + n ^ 2) ∨ z = -(k * (m ^ 2 + n ^ 2))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 624,
"column": 6
} | {
"line": 624,
"column": 17
} | {
"line": 624,
"column": 18
} | [
{
"pp": "case h.inr\nz k m n : ℤ\nright✝ : m.gcd n = 1\nh : PythagoreanTriple (k * (2 * m * n)) (k * (m ^ 2 - n ^ 2)) z\nthis : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2\n⊢ z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2)",
"ppTerm": "?h.inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"No... | [
"case h.inr\nz k m n : ℤ\nright✝ : m.gcd n = 1\nh : PythagoreanTriple (k * (2 * m * n)) (k * (m ^ 2 - n ^ 2)) z\nthis : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2\n⊢ z = k * (m ^ 2 + n ^ 2) ∨ z = -(k * (m ^ 2 + n ^ 2))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Units.Basic | {
"line": 208,
"column": 4
} | {
"line": 208,
"column": 59
} | {
"line": 209,
"column": 4
} | [
{
"pp": "case refine_1\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nζ : (𝓞 K)ˣ\nh : ζ ^ torsionOrder K = 1\n⊢ ζ ∈ CommGroup.torsion (𝓞 K)ˣ",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"NumberField.instCommRingRingOfIntegers",
... | [
"case refine_1\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nζ : (𝓞 K)ˣ\nh : ζ ^ torsionOrder K = 1\n⊢ ∃ n, 0 < n ∧ ζ ^ n = 1"
] | rw [CommGroup.mem_torsion, isOfFinOrder_iff_pow_eq_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 580,
"column": 2
} | {
"line": 580,
"column": 30
} | {
"line": 582,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nx : ℚ\nval✝ : AbsoluteValue K ℝ\nproperty✝ : ∃ φ, place φ = val✝\n⊢ ⟨val✝, property✝⟩ ↑x = ‖x‖",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Norm.norm",
"NumberField.InfinitePlace.instFunLikeReal",
"Eq.mpr",
"RingHom.instRi... | [] | aesop (add simp [coe_apply]) | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 579,
"column": 2
} | {
"line": 580,
"column": 30
} | {
"line": 582,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nv : InfinitePlace K\nx : ℚ\n⊢ v ↑x = ‖x‖",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Norm.norm",
"NumberField.InfinitePlace.instFunLikeReal",
"Eq.mpr",
"RingHom.instRingHomClass",
"Real.partialOrder",
"Real",
... | [] | rcases v with ⟨_, _⟩
aesop (add simp [coe_apply]) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 579,
"column": 2
} | {
"line": 580,
"column": 30
} | {
"line": 582,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nv : InfinitePlace K\nx : ℚ\n⊢ v ↑x = ‖x‖",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Norm.norm",
"NumberField.InfinitePlace.instFunLikeReal",
"Eq.mpr",
"RingHom.instRingHomClass",
"Real.partialOrder",
"Real",
... | [] | rcases v with ⟨_, _⟩
aesop (add simp [coe_apply]) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 586,
"column": 2
} | {
"line": 586,
"column": 30
} | {
"line": 588,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nn : ℕ\nval✝ : AbsoluteValue K ℝ\nproperty✝ : ∃ φ, place φ = val✝\n⊢ ⟨val✝, property✝⟩ ↑n = ↑n",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Norm.norm",
"NumberField.InfinitePlace.instFunLikeReal",
"NonAssocSemiring.toAddCommMonoi... | [] | aesop (add simp [coe_apply]) | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 585,
"column": 2
} | {
"line": 586,
"column": 30
} | {
"line": 588,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nv : InfinitePlace K\nn : ℕ\n⊢ v ↑n = ↑n",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Norm.norm",
"NumberField.InfinitePlace.instFunLikeReal",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"Rea... | [] | rcases v with ⟨_, _⟩
aesop (add simp [coe_apply]) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 585,
"column": 2
} | {
"line": 586,
"column": 30
} | {
"line": 588,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nv : InfinitePlace K\nn : ℕ\n⊢ v ↑n = ↑n",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Norm.norm",
"NumberField.InfinitePlace.instFunLikeReal",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"Rea... | [] | rcases v with ⟨_, _⟩
aesop (add simp [coe_apply]) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 592,
"column": 2
} | {
"line": 592,
"column": 30
} | {
"line": 594,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nz : ℤ\nval✝ : AbsoluteValue K ℝ\nproperty✝ : ∃ φ, place φ = val✝\n⊢ ⟨val✝, property✝⟩ ↑z = ‖z‖",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Int.cast",
"NumberField.InfinitePlace.... | [] | aesop (add simp [coe_apply]) | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 591,
"column": 2
} | {
"line": 592,
"column": 30
} | {
"line": 594,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nv : InfinitePlace K\nz : ℤ\n⊢ v ↑z = ‖z‖",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Int.cast",
"NumberField.InfinitePlace.instFunLikeReal",
"Eq.mpr",
"RingHom.instRi... | [] | rcases v with ⟨_, _⟩
aesop (add simp [coe_apply]) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 591,
"column": 2
} | {
"line": 592,
"column": 30
} | {
"line": 594,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nv : InfinitePlace K\nz : ℤ\n⊢ v ↑z = ‖z‖",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Int.cast",
"NumberField.InfinitePlace.instFunLikeReal",
"Eq.mpr",
"RingHom.instRi... | [] | rcases v with ⟨_, _⟩
aesop (add simp [coe_apply]) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 598,
"column": 7
} | {
"line": 598,
"column": 18
} | {
"line": 598,
"column": 19
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nv w : InfinitePlace K\nt : ℝ\nh : (fun x ↦ w x) ^ t = ⇑v\nn : ℕ\nhn : 1 < n\n⊢ ↑n ^ t = ↑n ^ 1",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"congrArg",
"id",
"Nat.cast",
"... | [
"K : Type u_1\ninst✝ : Field K\nv w : InfinitePlace K\nt : ℝ\nh : (fun x ↦ w x) ^ t = ⇑v\nn : ℕ\nhn : 1 < n\n⊢ ↑n ^ t = ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 605,
"column": 27
} | {
"line": 605,
"column": 76
} | {
"line": 605,
"column": 77
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nv w : InfinitePlace K\nh✝ : (↑w).IsEquiv ↑v\nt : ℝ\nleft✝ : 0 < t\nh : (fun x ↦ ↑w x ^ t) = ⇑↑v\nk : K\n⊢ w k = v k",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"NumberField.InfinitePlace.instFunLikeReal",
"Real",
"id",
"Nu... | [
"K : Type u_1\ninst✝ : Field K\nv w : InfinitePlace K\nh✝ : (↑w).IsEquiv ↑v\nt : ℝ\nleft✝ : 0 < t\nh : (fun x ↦ ↑w x ^ t) = ⇑↑v\nk : K\n⊢ ↑w k = ↑v k"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 171,
"column": 4
} | {
"line": 171,
"column": 25
} | {
"line": 171,
"column": 26
} | [
{
"pp": "case refine_2\nK : Type u_1\ninst✝ : Field K\nf : InfinitePlace K → ℝ≥0\nw₀ : { w // w.IsComplex }\nx : K\nx✝ :\n (∀ (a : InfinitePlace K), a.IsReal → a x < ↑(f a)) ∧\n ∀ (a : InfinitePlace K) (b : a.IsComplex),\n a.embedding x ∈ if ⟨a, b⟩ = w₀ then {x | |x.re| < 1 ∧ |x.im| < ↑(f a) ^ 2} else ... | [
"case refine_2\nK : Type u_1\ninst✝ : Field K\nf : InfinitePlace K → ℝ≥0\nw₀ : { w // w.IsComplex }\nx : K\nx✝ :\n (∀ (a : InfinitePlace K), a.IsReal → a x < ↑(f a)) ∧\n ∀ (a : InfinitePlace K) (b : a.IsComplex),\n a.embedding x ∈ if ⟨a, b⟩ = w₀ then {x | |x.re| < 1 ∧ |x.im| < ↑(f a) ^ 2} else ball 0 ↑(f a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex | {
"line": 190,
"column": 4
} | {
"line": 191,
"column": 77
} | {
"line": 192,
"column": 2
} | [
{
"pp": "K : Type u_2\ninst✝² : Field K\ninst✝¹ : CharZero K\ninst✝ : Algebra.IsAlgebraic ℚ K\nh : maximalRealSubfield K = ⊤\n⊢ IsTotallyReal K",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Algebra.IsIntegral.tower_top",
"Eq.mpr",
"Subfield.toDivisionRing",
"le_r... | [] | have : Algebra.IsIntegral (⊤ : Subfield K) K := Algebra.IsIntegral.tower_top ℚ
rw [← isTotallyReal_top_iff, isTotallyReal_iff_le_maximalRealSubfield, h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex | {
"line": 190,
"column": 4
} | {
"line": 191,
"column": 77
} | {
"line": 192,
"column": 2
} | [
{
"pp": "K : Type u_2\ninst✝² : Field K\ninst✝¹ : CharZero K\ninst✝ : Algebra.IsAlgebraic ℚ K\nh : maximalRealSubfield K = ⊤\n⊢ IsTotallyReal K",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Algebra.IsIntegral.tower_top",
"Eq.mpr",
"Subfield.toDivisionRing",
"le_r... | [] | have : Algebra.IsIntegral (⊤ : Subfield K) K := Algebra.IsIntegral.tower_top ℚ
rw [← isTotallyReal_top_iff, isTotallyReal_iff_le_maximalRealSubfield, h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 118,
"column": 4
} | {
"line": 118,
"column": 36
} | {
"line": 119,
"column": 4
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ Basis (Free.ChooseBasisIndex ℤ (𝓞 K)) ℂ ((K →+* ℂ) → ℂ)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Pi.Function.module",
"Semiring.toModule",
"Pi.addCommMonoid",
"RingHom",
"Finite.of_fint... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : Basis (K →+* ℂ) ℂ ((K →+* ℂ) → ℂ) := Pi.basisFun ℂ (K →+* ℂ)\n⊢ Basis (Free.ChooseBasisIndex ℤ (𝓞 K)) ℂ ((K →+* ℂ) → ℂ)"
] | let B := Pi.basisFun ℂ (K →+* ℂ) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 233,
"column": 8
} | {
"line": 233,
"column": 35
} | {
"line": 233,
"column": 36
} | [
{
"pp": "case e_a.e_a\nK : Type u_1\ninst✝¹ : Field K\nf : InfinitePlace K → ℝ≥0\nw₀ : { w // w.IsComplex }\ninst✝ : NumberField K\nvol_box : ∀ (B : ℝ≥0), volume {x | |x.re| < 1 ∧ |x.im| < ↑B ^ 2} = 4 * ↑B ^ 2\n⊢ volume (if w₀ = w₀ then {x | |x.re| < 1 ∧ |x.im| < ↑(f ↑w₀) ^ 2} else ball 0 ↑(f ↑w₀)) = 4 * ↑(f ↑w... | [
"case e_a.e_a\nK : Type u_1\ninst✝¹ : Field K\nf : InfinitePlace K → ℝ≥0\nw₀ : { w // w.IsComplex }\ninst✝ : NumberField K\nvol_box : ∀ (B : ℝ≥0), volume {x | |x.re| < 1 ∧ |x.im| < ↑B ^ 2} = 4 * ↑B ^ 2\n⊢ volume {x | |x.re| < 1 ∧ |x.im| < ↑(f ↑w₀) ^ 2} = 4 * ↑(f ↑w₀) ^ 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 255,
"column": 2
} | {
"line": 255,
"column": 17
} | {
"line": 255,
"column": 18
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw : { w // w.IsReal }\nA : AffineSubspace ℝ (mixedSpace K) := ↑{ carrier := {x | x.1 w = 0}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }\nh : A = ⊤\n⊢ False",
"ppTerm": "?m.66",
"assigned": false,
"usedConstants": [],
"usedFVars... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw : { w // w.IsReal }\nA : AffineSubspace ℝ (mixedSpace K) := ↑{ carrier := {x | x.1 w = 0}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }\nh : A = ⊤\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 277,
"column": 7
} | {
"line": 277,
"column": 18
} | {
"line": 277,
"column": 19
} | [
{
"pp": "k : Type u_1\ninst✝² : Field k\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra k K\nw : InfinitePlace K\nh : IsRamified k w\n⊢ ¬ComplexEmbedding.IsReal (conjugate w.embedding)",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsSelfAdjoint",
"NumberFiel... | [
"k : Type u_1\ninst✝² : Field k\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra k K\nw : InfinitePlace K\nh : IsRamified k w\n⊢ ¬IsSelfAdjoint w.embedding"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 284,
"column": 26
} | {
"line": 284,
"column": 63
} | {
"line": 284,
"column": 64
} | [
{
"pp": "case refine_1.inr\nk : Type u_1\ninst✝² : Field k\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra k K\nφ : K →+* ℂ\nh : IsRamified k (mk φ)\nhr : (mk φ).embedding = conjugate φ\n⊢ IsMixed k (star (star φ))",
"ppTerm": "?refine_1.inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"case refine_1.inr\nk : Type u_1\ninst✝² : Field k\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra k K\nφ : K →+* ℂ\nh : IsRamified k (mk φ)\nhr : (mk φ).embedding = conjugate φ\n⊢ IsMixed k (conjugate (mk φ).embedding)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 300,
"column": 2
} | {
"line": 300,
"column": 13
} | {
"line": 300,
"column": 14
} | [
{
"pp": "k : Type u_1\ninst✝² : Field k\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra k K\nw : InfinitePlace K\nh : IsUnramified k w\nhw : (w.comap (algebraMap k K)).IsReal\n⊢ ComplexEmbedding.IsReal (conjugate w.embedding)",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [
"k : Type u_1\ninst✝² : Field k\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra k K\nw : InfinitePlace K\nh : IsUnramified k w\nhw : (w.comap (algebraMap k K)).IsReal\n⊢ IsSelfAdjoint w.embedding"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 307,
"column": 26
} | {
"line": 307,
"column": 63
} | {
"line": 307,
"column": 64
} | [
{
"pp": "case refine_1.inr\nk : Type u_1\ninst✝² : Field k\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra k K\nφ : K →+* ℂ\nh : IsUnramified k (mk φ)\nhr : (mk φ).embedding = conjugate φ\n⊢ IsUnmixed k (star (star φ))",
"ppTerm": "?refine_1.inr",
"assigned": true,
"usedConstants": [
"Eq.mpr"... | [
"case refine_1.inr\nk : Type u_1\ninst✝² : Field k\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra k K\nφ : K →+* ℂ\nh : IsUnramified k (mk φ)\nhr : (mk φ).embedding = conjugate φ\n⊢ IsUnmixed k (conjugate (mk φ).embedding)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 302,
"column": 6
} | {
"line": 302,
"column": 30
} | {
"line": 302,
"column": 30
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : mixedSpace K\n⊢ ‖x‖ ≤ convexBodySumFun x",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",
"NumberField.mixedEmbedding.norm_eq_sup'_normAtPlace",
... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : mixedSpace K\n⊢ (Finset.univ.sup' ⋯ fun w ↦ (normAtPlace w) x) ≤ convexBodySumFun x"
] | norm_eq_sup'_normAtPlace | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 311,
"column": 23
} | {
"line": 311,
"column": 34
} | {
"line": 311,
"column": 35
} | [
{
"pp": "k : Type u_1\ninst✝² : Field k\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra k K\nφ : K →+* ℂ\nh : IsUnmixed k φ\nhv : ¬ComplexEmbedding.IsReal (φ.comp (algebraMap k K))\n⊢ ((mk φ).comap (algebraMap k K)).IsComplex",
"ppTerm": "?m.110",
"assigned": true,
"usedConstants": [
"Algebra... | [
"k : Type u_1\ninst✝² : Field k\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra k K\nφ : K →+* ℂ\nh : IsUnmixed k φ\nhv : ¬ComplexEmbedding.IsReal (φ.comp (algebraMap k K))\n⊢ (mk (φ.comp (algebraMap k K))).IsComplex"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 351,
"column": 2
} | {
"line": 356,
"column": 50
} | {
"line": 358,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : ℝ\n⊢ Bornology.IsBounded (convexBodySum K B)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing"... | [] | classical
refine Metric.isBounded_iff.mpr ⟨B + B, fun x hx y hy => ?_⟩
simp_rw [dist_eq_norm]
refine le_trans (norm_sub_le x y) (add_le_add ?_ ?_)
· exact le_trans (norm_le_convexBodySumFun x) hx
· exact le_trans (norm_le_convexBodySumFun y) hy | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 351,
"column": 2
} | {
"line": 356,
"column": 50
} | {
"line": 358,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : ℝ\n⊢ Bornology.IsBounded (convexBodySum K B)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing"... | [] | classical
refine Metric.isBounded_iff.mpr ⟨B + B, fun x hx y hy => ?_⟩
simp_rw [dist_eq_norm]
refine le_trans (norm_sub_le x y) (add_le_add ?_ ?_)
· exact le_trans (norm_le_convexBodySumFun x) hx
· exact le_trans (norm_le_convexBodySumFun y) hy | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 351,
"column": 2
} | {
"line": 356,
"column": 50
} | {
"line": 358,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : ℝ\n⊢ Bornology.IsBounded (convexBodySum K B)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing"... | [] | classical
refine Metric.isBounded_iff.mpr ⟨B + B, fun x hx y hy => ?_⟩
simp_rw [dist_eq_norm]
refine le_trans (norm_sub_le x y) (add_le_add ?_ ?_)
· exact le_trans (norm_le_convexBodySumFun x) hx
· exact le_trans (norm_le_convexBodySumFun y) hy | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 349,
"column": 61
} | {
"line": 354,
"column": 83
} | {
"line": 356,
"column": 0
} | [
{
"pp": "k : Type u_1\ninst✝² : Field k\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra k K\nφ : K →+* ℂ\nσ : Gal(K/k)\n⊢ σ ∈ Stab (mk φ) ↔ σ = 1 ∨ ComplexEmbedding.IsConj φ σ",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"or_congr",
"Eq.mpr",
"AlgEquiv.instEquivLike",... | [] | by
simp only [MulAction.mem_stabilizer_iff, smul_mk, mk_eq_iff]
rw [← ComplexEmbedding.isConj_symm, ComplexEmbedding.conjugate, star_eq_iff_star_eq]
refine or_congr ⟨fun H ↦ ?_, fun H ↦ H ▸ rfl⟩ Iff.rfl
exact congr_arg AlgEquiv.symm
(AlgEquiv.ext (g := AlgEquiv.refl) fun x ↦ φ.injective (RingHom.congr_fun H... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 595,
"column": 52
} | {
"line": 595,
"column": 73
} | {
"line": 596,
"column": 4
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (K →+* ℂ) → ℂ\nhx : ∀ (φ : K →+* ℂ), (starRingEnd ℂ) (x φ) = x (ComplexEmbedding.conjugate φ)\nc : index K\n⊢ ↑(((stdBasis K).repr (fun w ↦ (x (↑w).embedding).re, fun w ↦ x (↑w).embedding)) c) =\n ∑ x_1,\n fromBlocks (diagonal fun x ↦ 1... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (K →+* ℂ) → ℂ\nhx : ∀ (φ : K →+* ℂ), (starRingEnd ℂ) (x φ) = x (ComplexEmbedding.conjugate φ)\nc : index K\n⊢ ↑(((stdBasis K).repr (fun w ↦ (x (↑w).embedding).re, fun w ↦ x (↑w).embedding)) c) =\n ∑ a₁,\n fromBlocks (diagonal fun x ↦ 1) 0 0\n ... | Fintype.sum_sum_type, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 489,
"column": 4
} | {
"line": 493,
"column": 34
} | {
"line": 494,
"column": 2
} | [
{
"pp": "k : Type u_1\ninst✝⁵ : Field k\nK : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Algebra k K\ninst✝² : NumberField K\ninst✝¹ : NumberField k\ninst✝ : IsGalois k K\nw : InfinitePlace K\nhw : IsUnramifiedIn K ((fun x ↦ x.comap (algebraMap k K)) w)\n⊢ #(MulAction.orbit Gal(K/k) w).toFinset = Nat.card Gal(K/k)",
... | [] | · rw [Nat.card_eq_fintype_card,
← MulAction.card_orbit_mul_card_stabilizer_eq_card_group _ w,
← Nat.card_eq_fintype_card (α := Stab w), card_stabilizer, if_pos,
mul_one, Set.toFinset_card]
rwa [← isUnramifiedIn_comap] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 599,
"column": 32
} | {
"line": 599,
"column": 52
} | {
"line": 599,
"column": 53
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (K →+* ℂ) → ℂ\nhx : ∀ (φ : K →+* ℂ), (starRingEnd ℂ) (x φ) = x (ComplexEmbedding.conjugate φ)\nc : index K\n⊢ ↑(((stdBasis K).repr (fun w ↦ (x (↑w).embedding).re, fun w ↦ x (↑w).embedding)) c) =\n ∑ x_1,\n fromBlocks 1 0 0\n ... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (K →+* ℂ) → ℂ\nhx : ∀ (φ : K →+* ℂ), (starRingEnd ℂ) (x φ) = x (ComplexEmbedding.conjugate φ)\nc : index K\n⊢ ↑(((stdBasis K).repr (fun w ↦ (x (↑w).embedding).re, fun w ↦ x (↑w).embedding)) c) =\n ∑ x_1,\n fromBlocks 1 0 0\n ((blockDia... | Equiv.prodComm_symm, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 492,
"column": 19
} | {
"line": 492,
"column": 30
} | {
"line": 492,
"column": 31
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nf : InfinitePlace K → ℝ≥0\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nh : minkowskiBound K I < volume (convexBodyLT K f)\nh_fund :\n IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup)\n (fundamentalDomain (fr... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nf : InfinitePlace K → ℝ≥0\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nh : minkowskiBound K I < volume (convexBodyLT K f)\nh_fund :\n IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup)\n (fundamentalDomain (fractionalIdea... | 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.