module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 154,
"column": 8
} | {
"line": 154,
"column": 84
} | [
{
"pp": "case refine_2.inr.inr.inl\nz τ : ℂ\nhτ✝ : τ.im ≤ 0\nhτ : τ.im = 0\nhz : z.im = 0\n⊢ (Summable fun x ↦ rexp (-(2 * π * ↑x * z.im))) → False",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"False",
"Real",
"Real.pi",
"HMul.hMul",
"Real.instZero",
"congrArg",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 200,
"column": 6
} | {
"line": 200,
"column": 32
} | [
{
"pp": "case inr.refine_2.h\na : ℝ\nha : 0 ≤ a\naux' : (fun t ↦ ((1 - rexp (-π * t)) ^ 2)⁻¹) =O[atTop] fun x ↦ 1\nha' : 0 < a\n⊢ (fun x ↦ rexp (-π * a ^ 2 * x) / (1 - rexp (-π * x))) =O[atTop] fun t ↦ rexp (-π * a ^ 2 * t)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 345,
"column": 37
} | {
"line": 345,
"column": 48
} | [
{
"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",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 346,
"column": 43
} | {
"line": 346,
"column": 54
} | [
{
"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",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"AddGroup.toSubtraction... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 405,
"column": 6
} | {
"line": 405,
"column": 17
} | [
{
"pp": "case refine_1.inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\ns : ℂ\nhs' : s ≠ ↑P.k ∨ P.g₀ = 0\nhs : s ≠ 0\n⊢ DifferentiableAt ℂ (fun s ↦ (1 / s) • P.f₀) s",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"DivInvMonoid.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.MellinEqDirichlet | {
"line": 76,
"column": 35
} | {
"line": 76,
"column": 46
} | [
{
"pp": "ι : Type u_1\ninst✝ : Countable ι\na : ι → ℂ\nq : ι → ℝ\nF : ℝ → ℂ\ns : ℂ\nhq : ∀ (i : ι), a i = 0 ∨ 0 < q i\nhs : 0 < s.re\nhF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a i * ↑(rexp (-π * q i * t))) (F t)\nh_sum : Summable fun i ↦ ‖a i‖ / q i ^ s.re\nhp : ∀ (i : ι), a i = 0 ∨ 0 < π * q i\n⊢ ∀ t ∈ Ioi 0, HasSum (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 434,
"column": 40
} | {
"line": 436,
"column": 94
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\ns : ℂ\n⊢ P.Λ (↑P.k - s) = P.ε • P.symm.Λ s",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"AddGroup.toSubtractionMonoid",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
... | by
linear_combination (norm := module) P.functional_equation₀ s - P.Λ₀_eq (P.k - s)
+ congr(P.ε • $(P.symm_Λ₀_eq s)) + congr(($(mul_inv_cancel₀ P.hε) / ((P.k:ℂ) - s)) • P.f₀) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 304,
"column": 4
} | {
"line": 304,
"column": 29
} | [
{
"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\nhVmem : (z, τ) ∈ V\n⊢ Convex ℝ {u | |u.im| < S}",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"AddGroup.toSubtractionMonoi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.MellinEqDirichlet | {
"line": 88,
"column": 4
} | {
"line": 88,
"column": 42
} | [
{
"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‖... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 466,
"column": 10
} | {
"line": 466,
"column": 21
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\n⊢ ↑P.k - 0 ≠ 0",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Real",
"Real.instZero",
"congrArg",
"sub_zero",
"HSub.hSub",
"Complex.instZero",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 170,
"column": 2
} | {
"line": 170,
"column": 20
} | [
{
"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))",
"usedConstants": [
"Int.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 180,
"column": 2
} | {
"line": 180,
"column": 20
} | [
{
"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))",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 319,
"column": 4
} | {
"line": 319,
"column": 40
} | [
{
"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\nhVmem : (z, τ) ∈ V\nhVp : IsPreconnected V\nf : ℤ → ℂ × ℂ → ℂ := fun n p ↦ jacobiTheta₂_term n p.1 p.2\nf' : ℤ → ℂ × ℂ → ℂ × ℂ →L[ℂ] ℂ := fun n p ↦ j... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 320,
"column": 2
} | {
"line": 320,
"column": 61
} | [
{
"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\nhVmem : (z, τ) ∈ V\nhVp : IsPreconnected V\nf : ℤ → ℂ × ℂ → ℂ := fun n p ↦ jacobiTheta₂_term n p.1 p.2\nf' : ℤ → ℂ × ℂ → ℂ × ℂ →L[ℂ] ℂ := fun n p ↦ j... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 190,
"column": 4
} | {
"line": 191,
"column": 11
} | [
{
"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)",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Eq.mpr",
"NegZeroClass.toNeg",
"R... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 192,
"column": 37
} | {
"line": 192,
"column": 55
} | [
{
"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)",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 192,
"column": 2
} | {
"line": 195,
"column": 49
} | [
{
"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)",
"usedConstants": [
"Mathlib.Tactic.Push.not_forall_eq",
"AddGroup.toSubtractionMonoid",
"Int.cast_n... | · 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
} | [
{
"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)",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"NonUnitalCommRing.toNonUnitalN... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 367,
"column": 29
} | {
"line": 368,
"column": 11
} | [
{
"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",
"usedConstants": [
"Int.cast",
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 13
} | [
{
"pp": "x : ℝ\n⊢ oddKernel 0 x = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 13
} | [
{
"pp": "x : ℝ\n⊢ sinKernel 0 x = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 478,
"column": 4
} | {
"line": 478,
"column": 96
} | [
{
"pp": "z τ : ℂ\nhτ : 0 < τ.im\n⊢ 0 < (-I * τ).re",
"usedConstants": [
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Complex.mul_re",
"HMul.hMul",
"Real.instZero",
"congrArg",
"Complex.im",
"Real.instSub",
"MulZeroClass.zero_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 275,
"column": 4
} | {
"line": 275,
"column": 15
} | [
{
"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",
"usedConstants": [
"Real.instPow",
"Real",
"Complex.instNormedAddCommGroup",
"Asymptotics.IsBigO",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 513,
"column": 4
} | {
"line": 513,
"column": 40
} | [
{
"pp": "z τ : ℂ\nhτ : 0 < τ.im\nhτ' : 0 < (-1 / τ).im\nthis : HasDerivAt (fun x ↦ jacobiTheta₂ x (-1 / τ)) (jacobiTheta₂' (z / τ) (-1 / τ)) (z / τ)\n⊢ HasDerivAt (fun w ↦ jacobiTheta₂ (w / τ) (-1 / τ)) (1 / τ * jacobiTheta₂' (z / τ) (-1 / τ)) z",
"usedConstants": [
"IsModuleTopology.toContinuousSMul"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 430,
"column": 2
} | {
"line": 430,
"column": 18
} | [
{
"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 ... | 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
} | [
{
"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)",
"usedConstants": [
"Eq.mpr",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"congrArg",
"Real.instDivInvMonoid",
"Nat.instAtLeastTwoHAddOfNat",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.RiemannZeta | {
"line": 101,
"column": 6
} | {
"line": 101,
"column": 39
} | [
{
"pp": "s : ℂ\n⊢ completedRiemannZeta₀ (1 - s) = completedRiemannZeta₀ s",
"usedConstants": [
"Eq.mpr",
"Real",
"HurwitzZeta.completedHurwitzZetaEven₀",
"congrArg",
"HSub.hSub",
"AddCommGroup.toAddGroup",
"id",
"SubtractionMonoid.toSubNegZeroMonoid",
"S... | ← completedHurwitzZetaEven₀_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.RiemannZeta | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 49
} | [
{
"pp": "case h\ns : ℂ\n⊢ hurwitzZeta 0 s = riemannZeta s",
"usedConstants": [
"Eq.mpr",
"Real",
"riemannZeta",
"AddLeftCancelSemigroup.toIsLeftCancelAdd",
"AddMonoid.toAddZeroClass",
"AddGroupWithOne.toAddMonoidWithOne",
"HurwitzZeta.hurwitzZetaOdd",
"AddComm... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 272,
"column": 2
} | {
"line": 273,
"column": 66
} | [
{
"pp": "case h\nb 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",
"usedConstants": [
"HurwitzKernelBounds.f_int",
"Norm.norm",
"Int.cast",
"Eq.mpr",
"Functio... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.RiemannZeta | {
"line": 192,
"column": 2
} | {
"line": 193,
"column": 21
} | [
{
"pp": "s : ℂ\nhs : 1 < s.re\n⊢ riemannZeta s = ∑' (n : ℕ), 1 / ↑n ^ s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 320,
"column": 4
} | {
"line": 320,
"column": 15
} | [
{
"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",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instPow",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 324,
"column": 4
} | {
"line": 324,
"column": 15
} | [
{
"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",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instPow",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.SumPrimeReciprocals | {
"line": 85,
"column": 17
} | {
"line": 85,
"column": 47
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 578,
"column": 42
} | {
"line": 578,
"column": 52
} | [
{
"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 ∈ ... | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 79,
"column": 2
} | {
"line": 80,
"column": 9
} | [
{
"pp": "⊢ (abscissaOfAbsConv fun n ↦ ↑(μ n)) = 1",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"instInfSetEReal",
"Real",
"Set.Ioi",
"ArithmeticFunction.instFunLikeNat",
"congrArg",
"_private.Mathlib.NumberTheory.LSeries.Dirichlet.0.ArithmeticFunction.abscissaOf... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 592,
"column": 4
} | {
"line": 592,
"column": 24
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NegZeroClass.toNeg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 57
} | [
{
"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",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"ArithmeticFunction.instFunLikeNat",
"ZMod.commRing",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 624,
"column": 4
} | {
"line": 624,
"column": 51
} | [
{
"pp": "a : UnitAddCircle\n⊢ Tendsto (fun s ↦ (s - 1) * completedHurwitzZetaEven a s / s.Gammaℝ) (𝓝[≠] 1) (𝓝 1)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 202,
"column": 2
} | {
"line": 203,
"column": 9
} | [
{
"pp": "N : ℕ\nhn : N ≠ 0\nχ : DirichletCharacter ℂ N\n⊢ (abscissaOfAbsConv fun n ↦ χ ↑n) = 1",
"usedConstants": [
"Eq.mpr",
"instInfSetEReal",
"Real",
"Set.Ioi",
"ZMod.commRing",
"congrArg",
"PartialOrder.toPreorder",
"setOf",
"AddGroupWithOne.toAddMon... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 211,
"column": 2
} | {
"line": 211,
"column": 13
} | [
{
"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‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"ZMod.commRing",
"congrArg",
"AddGroupWithOne.toAdd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 650,
"column": 2
} | {
"line": 650,
"column": 13
} | [
{
"pp": "a : UnitAddCircle\n⊢ Tendsto (fun s ↦ hurwitzZetaEven a s - 1 / (s - 1) / s.Gammaℝ) (𝓝 1) (𝓝 (hurwitzZetaEven a 1))",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"DivInvMonoid.toInv",
"instHDiv",
"congrArg",
"Complex.Gammaℝ",
"Comp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 230,
"column": 13
} | {
"line": 230,
"column": 28
} | [
{
"pp": "N : ℕ\nχ : DirichletCharacter ℂ N\ns : ℂ\nhs : 1 < s.re\nh : L (fun n ↦ χ ↑n) s = 0\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 299,
"column": 13
} | {
"line": 299,
"column": 41
} | [
{
"pp": "s : ℂ\nhs : 1 < s.re\nh : L (fun n ↦ ↑(ζ n)) s = 0\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 337,
"column": 34
} | {
"line": 337,
"column": 45
} | [
{
"pp": "x : ℝ\nhx : 1 < x\n⊢ 1 < (↑x).re",
"usedConstants": [
"Real",
"Real.instLT",
"id",
"Complex.ofReal",
"Complex.re",
"Real.instOne",
"LT.lt",
"One.toOfNat1",
"OfNat.ofNat"
]
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 340,
"column": 2
} | {
"line": 340,
"column": 45
} | [
{
"pp": "x : ℝ\nhx : 1 < x\nhx' : 1 < (↑x).re\n⊢ abscissaOfAbsConv 1 < ↑x",
"usedConstants": [
"LSeries.abscissaOfAbsConv_one",
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"EReal",
"id",
"Pi.instOne",
"Nat",
"LT.lt",
"in... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 372,
"column": 2
} | {
"line": 373,
"column": 9
} | [
{
"pp": "case h\nn : ℕ\n⊢ ((fun n ↦ ↑(Λ n)) ⍟ fun n ↦ ↑(ζ n)) n = Complex.log ↑n",
"usedConstants": [
"ArithmeticFunction.vonMangoldt",
"CharP.cast_eq_zero",
"Eq.mpr",
"Complex.log",
"Nat.instMulZeroClass",
"Real",
"HMul.hMul",
"Nat.divisorsAntidiagonal",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 385,
"column": 4
} | {
"line": 386,
"column": 11
} | [
{
"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‖",
"usedConstants": [
"ArithmeticFunction.vonMangoldt",
"Norm.norm",
"Eq.mpr",
"Complex.log",
"Real.instLE",
"Real",
"_private.Mathlib.NumberT... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 765,
"column": 4
} | {
"line": 765,
"column": 47
} | [
{
"pp": "a : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ -↑n\nhs' : a ≠ 0 ∨ s ≠ 1\n⊢ a ≠ 0 ∨ 1 - s ≠ 0",
"usedConstants": [
"Eq.mpr",
"Real",
"AddGroupWithOne.toAddGroup",
"congrArg",
"HSub.hSub",
"Complex.instZero",
"AddCommGroup.toAddGroup",
"Complex.addGroupW... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 767,
"column": 30
} | {
"line": 767,
"column": 41
} | [
{
"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",
"usedConstants": [
"Complex.instZero",
"id",
"Ne",
"Zero.toOfNat0",
"Complex",
"OfNat.ofNat... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 778,
"column": 4
} | {
"line": 778,
"column": 29
} | [
{
"pp": "case h\na : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ 1 - ↑n\n⊢ 1 - s ≠ 0",
"usedConstants": [
"Eq.mpr",
"AddGroupWithOne.toAddGroup",
"congrArg",
"HSub.hSub",
"Complex.instZero",
"Complex.addGroupWithOne",
"id",
"Ne",
"instHSub",
"One.toO... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 779,
"column": 69
} | {
"line": 779,
"column": 80
} | [
{
"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",
"usedConstants": [
"id",
"Ne",
"Complex.instNatCast",
"Nat.cast",
"Complex",
"Complex.instNeg",
"Neg.neg"
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 780,
"column": 48
} | {
"line": 780,
"column": 59
} | [
{
"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",
"usedConstants": [
"Complex.instZero",
"id",
"Ne",
"Zero.toOfNat0",
"Complex",
"OfNat.ofNat"
]
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.EulerProduct.DirichletLSeries | {
"line": 40,
"column": 4
} | {
"line": 41,
"column": 11
} | [
{
"pp": "s : ℂ\nhs : s ≠ 0\nm n : ℕ\n⊢ ↑(m * n) ^ (-s) = ↑m ^ (-s) * ↑n ^ (-s)",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.instMulZeroOneClass",
"HMul.hMul",
"congrArg",
"Complex.instPow",
"id",
"MulOne.toMul",
"AddMono... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.EulerProduct.DirichletLSeries | {
"line": 52,
"column": 4
} | {
"line": 54,
"column": 11
} | [
{
"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))",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.instMulZeroOneClass",
"Re... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.EulerProduct.DirichletLSeries | {
"line": 146,
"column": 2
} | {
"line": 147,
"column": 9
} | [
{
"pp": "N : ℕ\nχ : DirichletCharacter ℂ N\ns : ℂ\nhs : 1 < s.re\nf : ℕ →*₀ ℂ := dirichletSummandHom χ ⋯\nh : ∀ (n : ℕ), term (fun n ↦ χ ↑n) s n = f n\n⊢ cexp (∑' (p : Primes), -Complex.log (1 - χ ↑↑p * ↑↑p ^ (-s))) = L (fun n ↦ χ ↑n) s",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormed... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.FLT.Basic | {
"line": 231,
"column": 30
} | {
"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 :... | normalize_eq_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.FLT.Four | {
"line": 147,
"column": 17
} | {
"line": 147,
"column": 48
} | [
{
"pp": "r s : ℤ\nh : IsCoprime r s\n⊢ IsCoprime (s ^ 2 + r ^ 2) s",
"usedConstants": [
"Int.isCoprime_of_sq_sum"
]
}
] | apply Int.isCoprime_of_sq_sum h | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Radical.Basic | {
"line": 194,
"column": 2
} | {
"line": 194,
"column": 27
} | [
{
"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",
"usedConstants": [
"UniqueFactorizationMonoid.radical_of_prime"
]
}
] | exact radical_of_prime ha | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Radical.Basic | {
"line": 214,
"column": 54
} | {
"line": 214,
"column": 84
} | [
{
"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",
"usedConstants": [
"UniqueFactorizationMono... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.FLT.MasonStothers | {
"line": 77,
"column": 4
} | {
"line": 77,
"column": 28
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 38,
"column": 2
} | {
"line": 38,
"column": 13
} | [
{
"pp": "z : ℤ\n⊢ ¬↑(z * z) = ↑2",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"HMul.hMul",
"ZMod.commRing",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"AddGroupWithOne.toAddMonoidWithOne",
"id",
"NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring",
"A... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 148,
"column": 6
} | {
"line": 149,
"column": 26
} | [
{
"pp": "x y z : ℤ\nh : PythagoreanTriple x y z\nh0 : x.gcd y = 0\nhx : x = 0\nhy : y = 0\n⊢ z = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 163,
"column": 6
} | {
"line": 164,
"column": 26
} | [
{
"pp": "x y z : ℤ\nh : PythagoreanTriple x y z\nh0 : x.gcd y = 0\nhx : x = 0\nhy : y = 0\n⊢ z = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 32
} | [
{
"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))",
"usedConstants": [
"CharP.cast_eq_zero",
"Int.gcd",
"Eq.mpr",
"Int.instDiv",
"instHDiv",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 251,
"column": 6
} | {
"line": 251,
"column": 56
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nhk : ∀ (x : K), 1 + x ^ 2 ≠ 0\nx : K\n⊢ ¬1 = -1",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"AddGroupWithOne.toAddGroup",
"congrArg",
"AddMonoid.toAddZeroClass",
"AddGroupWithOne.toAddM... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 297,
"column": 2
} | {
"line": 299,
"column": 8
} | [
{
"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",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tact... | 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.PythagoreanTriples | {
"line": 472,
"column": 2
} | {
"line": 476,
"column": 7
} | [
{
"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 := (... | · -- 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.PythagoreanTriples | {
"line": 514,
"column": 45
} | {
"line": 514,
"column": 89
} | [
{
"pp": "x y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhz : z ≤ 0\n⊢ PythagoreanTriple x y (-z)",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"congrArg",
"neg_neg",
"id",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 543,
"column": 6
} | {
"line": 543,
"column": 17
} | [
{
"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)",
"usedConstants": [
"neg_add_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.FractionalIdeal | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 13
} | [
{
"pp": "case e_a.h.e_6.h.h\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 ∘ ⇑((fractionalIdealBasi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 548,
"column": 6
} | {
"line": 548,
"column": 17
} | [
{
"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)",
"usedConstants": [
"neg_add_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.Complex | {
"line": 75,
"column": 8
} | {
"line": 75,
"column": 43
} | [
{
"pp": "K : Subfield ℂ\nψ : ↥K →+* ℂ\nhc : UniformContinuous ⇑ψ\nthis✝ : IsTopologicalDivisionRing ℂ :=\n { toIsTopologicalRing := NormedDivisionRing.to_isTopologicalDivisionRing.toIsTopologicalRing,\n toContinuousInv₀ := NormedDivisionRing.to_isTopologicalDivisionRing.toContinuousInv₀ }\nthis : IsTopologi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.Complex | {
"line": 100,
"column": 8
} | {
"line": 100,
"column": 43
} | [
{
"pp": "K : Subfield ℂ\nψ : ↥K →+* ℂ\nhc : UniformContinuous ⇑ψ\nthis✝ : IsTopologicalDivisionRing ℂ :=\n { toIsTopologicalRing := NormedDivisionRing.to_isTopologicalDivisionRing.toIsTopologicalRing,\n toContinuousInv₀ := NormedDivisionRing.to_isTopologicalDivisionRing.toContinuousInv₀ }\nthis : IsTopologi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 623,
"column": 6
} | {
"line": 623,
"column": 17
} | [
{
"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)",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 628,
"column": 6
} | {
"line": 628,
"column": 17
} | [
{
"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)",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.FLT.Polynomial | {
"line": 252,
"column": 4
} | {
"line": 252,
"column": 90
} | [
{
"pp": "k : Type u_1\ninst✝ : Field k\nn : ℕ\nhn✝ : 3 ≤ n\nchn : ↑n ≠ 0\na b c : k[X]\nha : a ≠ 0\nhb : b ≠ 0\nhc : c ≠ 0\na' b' : k[X]\nd : k[X] := gcd a b\nheq : d ^ n * (a' ^ n + b' ^ n) = c ^ n\neq_a : a = d * a'\neq_b : b = d * b'\nhd : d ≠ 0\nhn : 0 < n\nhdncn : d ^ n ∣ c ^ n\n⊢ d ∣ c",
"usedConstant... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 119,
"column": 17
} | {
"line": 119,
"column": 28
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nx✝¹ x✝ : InfinitePlace K\nh : x✝¹.embedding = x✝.embedding\n⊢ x✝¹ = x✝",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Units.Basic | {
"line": 135,
"column": 2
} | {
"line": 135,
"column": 55
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Units.Basic | {
"line": 203,
"column": 4
} | {
"line": 203,
"column": 59
} | [
{
"pp": "case h.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",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"NumberField.instCommRingRingOfIntegers",
"Monoid.toMulOneClass",
"congrArg",... | 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": 564,
"column": 2
} | {
"line": 564,
"column": 30
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nx : ℚ\nval✝ : AbsoluteValue K ℝ\nproperty✝ : ∃ φ, place φ = val✝\n⊢ ⟨val✝, property✝⟩ ↑x = ‖x‖",
"usedConstants": [
"Norm.norm",
"NumberField.InfinitePlace.instFunLikeReal",
"Eq.mpr",
"RingHom.instRingHomClass",
"Real.partialOrder",
... | aesop (add simp [coe_apply]) | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 563,
"column": 2
} | {
"line": 564,
"column": 30
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nv : InfinitePlace K\nx : ℚ\n⊢ v ↑x = ‖x‖",
"usedConstants": [
"Norm.norm",
"NumberField.InfinitePlace.instFunLikeReal",
"Eq.mpr",
"RingHom.instRingHomClass",
"Real.partialOrder",
"Real",
"Real.lattice",
"map_ratCast",
... | rcases v with ⟨_, _⟩
aesop (add simp [coe_apply]) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 563,
"column": 2
} | {
"line": 564,
"column": 30
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nv : InfinitePlace K\nx : ℚ\n⊢ v ↑x = ‖x‖",
"usedConstants": [
"Norm.norm",
"NumberField.InfinitePlace.instFunLikeReal",
"Eq.mpr",
"RingHom.instRingHomClass",
"Real.partialOrder",
"Real",
"Real.lattice",
"map_ratCast",
... | rcases v with ⟨_, _⟩
aesop (add simp [coe_apply]) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 569,
"column": 2
} | {
"line": 569,
"column": 30
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nn : ℕ\nval✝ : AbsoluteValue K ℝ\nproperty✝ : ∃ φ, place φ = val✝\n⊢ ⟨val✝, property✝⟩ ↑n = ↑n",
"usedConstants": [
"Norm.norm",
"NumberField.InfinitePlace.instFunLikeReal",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
... | aesop (add simp [coe_apply]) | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 568,
"column": 2
} | {
"line": 569,
"column": 30
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nv : InfinitePlace K\nn : ℕ\n⊢ v ↑n = ↑n",
"usedConstants": [
"Norm.norm",
"NumberField.InfinitePlace.instFunLikeReal",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"Real.partialOrder",
"Real",
"congrA... | rcases v with ⟨_, _⟩
aesop (add simp [coe_apply]) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 568,
"column": 2
} | {
"line": 569,
"column": 30
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nv : InfinitePlace K\nn : ℕ\n⊢ v ↑n = ↑n",
"usedConstants": [
"Norm.norm",
"NumberField.InfinitePlace.instFunLikeReal",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"Real.partialOrder",
"Real",
"congrA... | rcases v with ⟨_, _⟩
aesop (add simp [coe_apply]) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 574,
"column": 2
} | {
"line": 574,
"column": 30
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nz : ℤ\nval✝ : AbsoluteValue K ℝ\nproperty✝ : ∃ φ, place φ = val✝\n⊢ ⟨val✝, property✝⟩ ↑z = ‖z‖",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Int.cast",
"NumberField.InfinitePlace.instFunLikeReal",
"Eq.mpr",
"Ring... | aesop (add simp [coe_apply]) | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 573,
"column": 2
} | {
"line": 574,
"column": 30
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nv : InfinitePlace K\nz : ℤ\n⊢ v ↑z = ‖z‖",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Int.cast",
"NumberField.InfinitePlace.instFunLikeReal",
"Eq.mpr",
"RingHom.instRingHomClass",
"Real.partialOrder",
... | rcases v with ⟨_, _⟩
aesop (add simp [coe_apply]) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 573,
"column": 2
} | {
"line": 574,
"column": 30
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nv : InfinitePlace K\nz : ℤ\n⊢ v ↑z = ‖z‖",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Int.cast",
"NumberField.InfinitePlace.instFunLikeReal",
"Eq.mpr",
"RingHom.instRingHomClass",
"Real.partialOrder",
... | rcases v with ⟨_, _⟩
aesop (add simp [coe_apply]) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 580,
"column": 7
} | {
"line": 580,
"column": 18
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"congrArg",
"id",
"Nat.cast",
"Real.rpow_one",
"Real.instOne",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 587,
"column": 27
} | {
"line": 587,
"column": 76
} | [
{
"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",
"usedConstants": [
"NumberField.InfinitePlace.instFunLikeReal",
"Real",
"id",
"NumberField.InfinitePlace",
"Eq",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 627,
"column": 33
} | {
"line": 627,
"column": 73
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nz : (v : InfinitePlace K) → WithAbs ↑v\nr : ℝ\nhr : r > 0\na : InfinitePlace K → K\nhx : ∀ (i : InfinitePlace K), 1 < ↑i (a i) ∧ ∀ (j : InfinitePlace K), j ≠ i → ↑j (a i) < 1\ny : ℕ → K := fun n ↦ ∑ v, 1 / (1 + (a v)⁻¹ ^ n) * (WithAbs.equiv ↑v) (z ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 628,
"column": 6
} | {
"line": 628,
"column": 17
} | [
{
"pp": "case pos\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nz : (v : InfinitePlace K) → WithAbs ↑v\nr : ℝ\nhr : r > 0\na : InfinitePlace K → K\nhx : ∀ (i : InfinitePlace K), 1 < ↑i (a i) ∧ ∀ (j : InfinitePlace K), j ≠ i → ↑j (a i) < 1\ny : ℕ → K := fun n ↦ ∑ v, 1 / (1 + (a v)⁻¹ ^ n) * (WithAbs.equ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 631,
"column": 36
} | {
"line": 631,
"column": 65
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nz : (v : InfinitePlace K) → WithAbs ↑v\nr : ℝ\nhr : r > 0\na : InfinitePlace K → K\nhx : ∀ (i : InfinitePlace K), 1 < ↑i (a i) ∧ ∀ (j : InfinitePlace K), j ≠ i → ↑j (a i) < 1\ny : ℕ → K := fun n ↦ ∑ v, 1 / (1 + (a v)⁻¹ ^ n) * (WithAbs.equiv ↑v) (z ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 635,
"column": 6
} | {
"line": 635,
"column": 17
} | [
{
"pp": "case neg\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nz : (v : InfinitePlace K) → WithAbs ↑v\nr : ℝ\nhr : r > 0\na : InfinitePlace K → K\nhx : ∀ (i : InfinitePlace K), 1 < ↑i (a i) ∧ ∀ (j : InfinitePlace K), j ≠ i → ↑j (a i) < 1\ny : ℕ → K := fun n ↦ ∑ v, 1 / (1 + (a v)⁻¹ ^ n) * (WithAbs.equ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex | {
"line": 190,
"column": 4
} | {
"line": 191,
"column": 77
} | [
{
"pp": "K : Type u_2\ninst✝² : Field K\ninst✝¹ : CharZero K\ninst✝ : Algebra.IsAlgebraic ℚ K\nh : maximalRealSubfield K = ⊤\n⊢ IsTotallyReal K",
"usedConstants": [
"Algebra.IsIntegral.tower_top",
"Eq.mpr",
"Subfield.toDivisionRing",
"le_refl",
"Subfield.toAlgebra",
"Semi... | 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
} | [
{
"pp": "K : Type u_2\ninst✝² : Field K\ninst✝¹ : CharZero K\ninst✝ : Algebra.IsAlgebraic ℚ K\nh : maximalRealSubfield K = ⊤\n⊢ IsTotallyReal K",
"usedConstants": [
"Algebra.IsIntegral.tower_top",
"Eq.mpr",
"Subfield.toDivisionRing",
"le_refl",
"Subfield.toAlgebra",
"Semi... | 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.ConvexBody | {
"line": 167,
"column": 4
} | {
"line": 167,
"column": 25
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 229,
"column": 8
} | {
"line": 229,
"column": 35
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 206,
"column": 85
} | {
"line": 245,
"column": 63
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\nf : InfinitePlace K → ℝ≥0\nw₀ : { w // w.IsComplex }\ninst✝ : NumberField K\n⊢ volume (convexBodyLT' K f w₀) = ↑(convexBodyLT'Factor K) * ↑(∏ w, f w ^ w.mult)",
"usedConstants": [
"instWeaklyLocallyCompactSpaceOfLocallyCompactSpace",
"NumberField.Infinite... | by
have vol_box : ∀ B : ℝ≥0, volume {x : ℂ | |x.re| < 1 ∧ |x.im| < B ^ 2} = 4 * B ^ 2 := by
intro B
rw [← (Complex.volume_preserving_equiv_real_prod.symm).measure_preimage]
· simp_rw [Set.preimage_setOf_eq, Complex.measurableEquivRealProd_symm_apply]
rw [show {a : ℝ × ℝ | |a.1| < 1 ∧ |a.2| < B ^ 2} ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 298,
"column": 6
} | {
"line": 298,
"column": 30
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : mixedSpace K\n⊢ ‖x‖ ≤ convexBodySumFun x",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",
"NumberField.mixedEmbedding.norm_eq_sup'_normAtPlace",
"NumberField.mixedEmbedding.convexBodyS... | norm_eq_sup'_normAtPlace | Lean.Elab.Tactic.evalRewriteSeq | null |
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