module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.MellinInversion | {
"line": 46,
"column": 43
} | {
"line": 46,
"column": 54
} | {
"line": 46,
"column": 55
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nx : ℝ\ns : ℂ\nf : E\n⊢ (-↑x).im ≤ π",
"ppTerm": "?m.94",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Real.pi",
"congrArg",
"Complex.im",
"id",
"Su... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nx : ℝ\ns : ℂ\nf : E\n⊢ 0 ≤ π"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.WeakOperatorTopology | {
"line": 397,
"column": 20
} | {
"line": 397,
"column": 31
} | {
"line": 397,
"column": 32
} | [
{
"pp": "𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁹ : NormedField 𝕜₁\ninst✝⁸ : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_3\nF : Type u_4\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : Module 𝕜₁ E\ninst✝⁴ : AddCommGroup F\ninst✝³ : TopologicalSpace F\ninst✝² : Module 𝕜₂ F\ninst✝¹ : IsTopologic... | [
"𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁹ : NormedField 𝕜₁\ninst✝⁸ : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_3\nF : Type u_4\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : Module 𝕜₁ E\ninst✝⁴ : AddCommGroup F\ninst✝³ : TopologicalSpace F\ninst✝² : Module 𝕜₂ F\ninst✝¹ : IsTopologicalAddGroup F... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MellinInversion | {
"line": 106,
"column": 6
} | {
"line": 106,
"column": 33
} | {
"line": 106,
"column": 34
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nσ : ℝ\nf : ℝ → E\nx : ℝ\nhx : 0 < x\nhf : IntegrableOn (fun x ↦ |(-rexp (-x))| • ↑((rexp ∘ Neg.neg) x) ^ (↑σ - 1) • f ((rexp ∘ Neg.neg) x)) univ volume\nhFf : VerticalIntegrable (mellin f) σ volume\nhfx : Co... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nσ : ℝ\nf : ℝ → E\nx : ℝ\nhx : 0 < x\nhf : IntegrableOn (fun x ↦ |(-rexp (-x))| • ↑((rexp ∘ Neg.neg) x) ^ (↑σ - 1) • f ((rexp ∘ Neg.neg) x)) univ volume\nhFf : VerticalIntegrable (mellin f) σ volume\nhfx : ContinuousAt f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MellinInversion | {
"line": 111,
"column": 6
} | {
"line": 111,
"column": 66
} | {
"line": 111,
"column": 67
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nσ : ℝ\nf : ℝ → E\nx : ℝ\nhx : 0 < x\nhFf : VerticalIntegrable (mellin f) σ volume\nhfx : ContinuousAt f x\ng : ℝ → E := fun u ↦ rexp (-σ * u) • f (rexp (-u))\nhf : Integrable g volume\nh2π : 2 * π ≠ 0\n⊢ Int... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nσ : ℝ\nf : ℝ → E\nx : ℝ\nhx : 0 < x\nhFf : VerticalIntegrable (mellin f) σ volume\nhfx : ContinuousAt f x\ng : ℝ → E := fun u ↦ rexp (-σ * u) • f (rexp (-u))\nhf : Integrable g volume\nh2π : 2 * π ≠ 0\n⊢ Integrable (𝓕 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MellinInversion | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 33
} | {
"line": 116,
"column": 34
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nσ : ℝ\nf : ℝ → E\nx : ℝ\nhx : 0 < x\nhfx : ContinuousAt f x\ng : ℝ → E := fun u ↦ rexp (-σ * u) • f (rexp (-u))\nhf : Integrable g volume\nhFf : Integrable (𝓕 g) volume\n⊢ ContinuousAt f (rexp (- -Real.log ... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nσ : ℝ\nf : ℝ → E\nx : ℝ\nhx : 0 < x\nhfx : ContinuousAt f x\ng : ℝ → E := fun u ↦ rexp (-σ * u) • f (rexp (-u))\nhf : Integrable g volume\nhFf : Integrable (𝓕 g) volume\n⊢ ContinuousAt f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MellinInversion | {
"line": 127,
"column": 22
} | {
"line": 127,
"column": 43
} | {
"line": 127,
"column": 44
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nσ : ℝ\nf : ℝ → E\nx : ℝ\nhx : 0 < x\ng : ℝ → E := fun u ↦ rexp (-σ * u) • f (rexp (-u))\nhf : Integrable g volume\nhFf : Integrable (𝓕 g) volume\nhfx : ContinuousAt g (-Real.log x)\n⊢ ↑x ^ ↑(-σ) • rexp (Rea... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nσ : ℝ\nf : ℝ → E\nx : ℝ\nhx : 0 < x\ng : ℝ → E := fun u ↦ rexp (-σ * u) • f (rexp (-u))\nhf : Integrable g volume\nhFf : Integrable (𝓕 g) volume\nhfx : ContinuousAt g (-Real.log x)\n⊢ ↑x ^ ↑(-σ) • x ^ σ • f (rexp (Real... | ← rpow_def_of_pos hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.LogBounds | {
"line": 76,
"column": 82
} | {
"line": 78,
"column": 56
} | {
"line": 80,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ logTaylor (n + 1) = logTaylor n + fun z ↦ (-1) ^ (n + 1) * z ^ n / ↑n",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"instHDiv",
"HMul.hMul",
"Complex.instDivInvMonoid",
"Complex.instMul",
"id",
"HDiv.hDiv",
"instOfNatNat",
... | [] | by
funext
simpa only [logTaylor] using! Finset.sum_range_succ .. | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Complex.LogBounds | {
"line": 83,
"column": 17
} | {
"line": 83,
"column": 49
} | {
"line": 83,
"column": 50
} | [
{
"pp": "case succ\nn : ℕ\nih : logTaylor n 0 = 0\n⊢ logTaylor (n + 1) 0 = 0",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NegZeroClass.toNeg",
"False",
"Nat.instMulZeroClass",... | [
"case succ\nn : ℕ\nih : logTaylor n 0 = 0\n⊢ ¬n = 0 ∨ n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.LogBounds | {
"line": 95,
"column": 4
} | {
"line": 98,
"column": 18
} | {
"line": 99,
"column": 4
} | [
{
"pp": "case succ\nz : ℂ\nn : ℕ\nih : HasDerivAt (logTaylor (n + 1)) (∑ j ∈ Finset.range n, (-1) ^ j * z ^ j) z\n⊢ HasDerivAt (fun z ↦ (-1) ^ (n + 1 + 1) * (z ^ (n + 1) / (↑n + 1))) ((-1) ^ n * z ^ n) z",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"IsModuleTopology.toContinuousSM... | [
"case succ\nz : ℂ\nn : ℕ\nih : HasDerivAt (logTaylor (n + 1)) (∑ j ∈ Finset.range n, (-1) ^ j * z ^ j) z\nthis : HasDerivAt (fun x ↦ x ^ (n + 1) / (↑n + 1)) (z ^ n) z\n⊢ HasDerivAt (fun z ↦ (-1) ^ (n + 1 + 1) * (z ^ (n + 1) / (↑n + 1))) ((-1) ^ n * z ^ n) z"
] | have : HasDerivAt (fun x : ℂ ↦ (x ^ (n + 1) / (n + 1))) (z ^ n) z := by
simp_rw [div_eq_mul_inv]
convert! HasDerivAt.mul_const (hasDerivAt_pow (n + 1) z) (((n : ℂ) + 1)⁻¹) using 1
simp [field] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.MellinTransform | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 29
} | {
"line": 152,
"column": 30
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℝ → E\ns : ℂ\na : ℝ\nha : 0 < a\n⊢ mellin (fun t ↦ f (t * a)) s = ↑a ^ (-s) • mellin f s",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℝ → E\ns : ℂ\na : ℝ\nha : 0 < a\n⊢ mellin (fun t ↦ f (a * t)) s = ↑a ^ (-s) • mellin f s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd | {
"line": 52,
"column": 2
} | {
"line": 57,
"column": 9
} | {
"line": 59,
"column": 0
} | [
{
"pp": "z : ℂ\nhz : z ≠ 0\nx : ℝ\n⊢ HasDerivAt (fun y ↦ -Complex.cos (2 * z * ↑y) / (2 * z)) (Complex.sin (2 * z * ↑x)) x",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"HasDerivAt.fun_neg",
"instInnerProductSpaceRealComplex",
"IsModuleTopology.toContinuousSMul",
... | [] | have a : HasDerivAt (fun y : ℂ => y * (2 * z)) _ x := hasDerivAt_mul_const _
have b : HasDerivAt (Complex.cos ∘ fun y : ℂ => (y * (2 * z))) _ x :=
HasDerivAt.comp (x : ℂ) (Complex.hasDerivAt_cos (x * (2 * z))) a
have c := (b.comp_ofReal.div_const (2 * z)).fun_neg
simp at c ⊢; field_simp at c ⊢; simp only [mul... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd | {
"line": 52,
"column": 2
} | {
"line": 57,
"column": 9
} | {
"line": 59,
"column": 0
} | [
{
"pp": "z : ℂ\nhz : z ≠ 0\nx : ℝ\n⊢ HasDerivAt (fun y ↦ -Complex.cos (2 * z * ↑y) / (2 * z)) (Complex.sin (2 * z * ↑x)) x",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"HasDerivAt.fun_neg",
"instInnerProductSpaceRealComplex",
"IsModuleTopology.toContinuousSMul",
... | [] | have a : HasDerivAt (fun y : ℂ => y * (2 * z)) _ x := hasDerivAt_mul_const _
have b : HasDerivAt (Complex.cos ∘ fun y : ℂ => (y * (2 * z))) _ x :=
HasDerivAt.comp (x : ℂ) (Complex.hasDerivAt_cos (x * (2 * z))) a
have c := (b.comp_ofReal.div_const (2 * z)).fun_neg
simp at c ⊢; field_simp at c ⊢; simp only [mul... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd | {
"line": 69,
"column": 6
} | {
"line": 69,
"column": 17
} | {
"line": 69,
"column": 18
} | [
{
"pp": "z : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nx : ℝ\na✝ : x ∈ uIcc 0 (π / 2)\n⊢ HasDerivAt (fun y ↦ ↑(cos y)) (-↑(sin x)) x",
"ppTerm": "?m.158",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedComm... | [
"z : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nx : ℝ\na✝ : x ∈ uIcc 0 (π / 2)\n⊢ HasDerivAt (fun y ↦ Complex.cos ↑y) (-Complex.sin ↑x) x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MellinTransform | {
"line": 219,
"column": 6
} | {
"line": 220,
"column": 60
} | {
"line": 221,
"column": 6
} | [
{
"pp": "f : ℝ → ℝ\nhfc : AEStronglyMeasurable f (volume.restrict (Ioi 0))\na s : ℝ\nhf : f =O[atTop] fun x ↦ x ^ (-a)\nhs : s < a\nd e : ℝ\nhe : ∀ (b : ℝ), e ≤ b → ‖f b‖ ≤ d * ‖b ^ (-a)‖\nhe' : 0 < max e 1\nt : ℝ\nht : t ∈ Ioi (max e 1)\nht' : 0 < t\n⊢ ‖t ^ (s - 1) * f t‖ ≤ t ^ (s - 1 + -a) * d",
"ppTerm":... | [
"f : ℝ → ℝ\nhfc : AEStronglyMeasurable f (volume.restrict (Ioi 0))\na s : ℝ\nhf : f =O[atTop] fun x ↦ x ^ (-a)\nhs : s < a\nd e : ℝ\nhe : ∀ (b : ℝ), e ≤ b → ‖f b‖ ≤ d * ‖b ^ (-a)‖\nhe' : 0 < max e 1\nt : ℝ\nht : t ∈ Ioi (max e 1)\nht' : 0 < t\n⊢ t ^ (s - 1) * ‖f t‖ ≤ t ^ (s - 1) * (d * ‖t ^ (-a)‖)"
] | rw [norm_mul, rpow_add ht', ← norm_of_nonneg (rpow_nonneg ht'.le (-a)), mul_assoc,
mul_comm _ d, norm_of_nonneg (rpow_nonneg ht'.le _)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Gamma.Deriv | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 34
} | {
"line": 58,
"column": 35
} | [
{
"pp": "case convert_3\ns : ℂ\nhs : 0 < s.re\n⊢ (fun x ↦ rexp (-x)) =O[atTop] fun x ↦ x ^ (-(s.re + 1))",
"ppTerm": "?convert_3",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case convert_3\ns : ℂ\nhs : 0 < s.re\n⊢ (fun x ↦ rexp (-x)) =O[atTop] fun x ↦ x ^ (-(s.re + 1))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Deriv | {
"line": 73,
"column": 34
} | {
"line": 73,
"column": 45
} | {
"line": 73,
"column": 46
} | [
{
"pp": "s✝ : ℂ\nhs✝ : ∀ (m : ℕ), s✝ ≠ -↑m\ns : ℂ\nhsre : -↑0 < s.re\nhs : ∀ (m : ℕ), s ≠ -↑m\n⊢ 0 < s.re",
"ppTerm": "?m.75",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"s✝ : ℂ\nhs✝ : ∀ (m : ℕ), s✝ ≠ -↑m\ns : ℂ\nhsre : -↑0 < s.re\nhs : ∀ (m : ℕ), s ≠ -↑m\n⊢ 0 < s.re"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Deriv | {
"line": 75,
"column": 40
} | {
"line": 75,
"column": 51
} | {
"line": 75,
"column": 52
} | [
{
"pp": "s✝ : ℂ\nhs✝ : ∀ (m : ℕ), s✝ ≠ -↑m\ns : ℂ\nhs : ∀ (m : ℕ), s ≠ -↑m\nhsre : 0 < s.re\nthis : IsOpen {s | 0 < s.re}\n⊢ 0 < s.re",
"ppTerm": "?m.109",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"s✝ : ℂ\nhs✝ : ∀ (m : ℕ), s✝ ≠ -↑m\ns : ℂ\nhs : ∀ (m : ℕ), s ≠ -↑m\nhsre : 0 < s.re\nthis : IsOpen {s | 0 < s.re}\n⊢ 0 < s.re"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Deriv | {
"line": 115,
"column": 62
} | {
"line": 115,
"column": 73
} | {
"line": 115,
"column": 74
} | [
{
"pp": "n : ℕ\nih : ContinuousAt Gamma (-(↑n + 1))\nthis : ContinuousAt (fun s ↦ Gamma (s - 1 + 1)) (-↑n)\n⊢ ContinuousAt Gamma (-↑n)",
"ppTerm": "?m.65",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\nih : ContinuousAt Gamma (-(↑n + 1))\nthis : ContinuousAt (fun s ↦ Gamma (s - 1 + 1)) (-↑n)\n⊢ ContinuousAt Gamma (-↑n)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.LogBounds | {
"line": 319,
"column": 6
} | {
"line": 319,
"column": 17
} | {
"line": 319,
"column": 18
} | [
{
"pp": "g : ℝ → ℂ\nt : ℂ\nhg : Tendsto (fun x ↦ ↑x * g x) atTop (𝓝 t)\n⊢ (fun x ↦ (↑x * g x) ^ 2 * ↑x⁻¹) =O[atTop] fun x ↦ ↑x⁻¹",
"ppTerm": "?m.146",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"DivisionCommMonoid.toDivisionMonoid",
"Complex.in... | [
"g : ℝ → ℂ\nt : ℂ\nhg : Tendsto (fun x ↦ ↑x * g x) atTop (𝓝 t)\n⊢ (fun x ↦ (↑x * g x) ^ 2 * (↑x)⁻¹) =O[atTop] fun x ↦ (↑x)⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Beta | {
"line": 91,
"column": 2
} | {
"line": 92,
"column": 9
} | {
"line": 92,
"column": 10
} | [
{
"pp": "u v : ℂ\n⊢ v.betaIntegral u = u.betaIntegral v",
"ppTerm": "?m.2",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"MeasureTheory.Measure",
"HMul.hMul",
"Complex.instNorme... | [
"u v : ℂ\n⊢ ∫ (x : ℝ) in 0..1, ↑x ^ (v + -1) * (-↑x + 1) ^ (u + -1) = ∫ (x : ℝ) in 0..1, ↑x ^ (u + -1) * (-↑x + 1) ^ (v + -1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MellinTransform | {
"line": 361,
"column": 6
} | {
"line": 362,
"column": 97
} | {
"line": 363,
"column": 6
} | [
{
"pp": "case hbc.inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b : ℝ\nf : ℝ → E\ns : ℂ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x ↦ x ^ (-a)\nhs_top : s.re < a\nhf_bot : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)\nhs_bot : b < s.re\nF : ℂ → ℝ → E := fun z t ↦ ↑t ... | [
"case hbc.inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b : ℝ\nf : ℝ → E\ns : ℂ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x ↦ x ^ (-a)\nhs_top : s.re < a\nhf_bot : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)\nhs_bot : b < s.re\nF : ℂ → ℝ → E := fun z t ↦ ↑t ^ (z - 1) • ... | refine
le_add_of_nonneg_of_le (rpow_pos_of_pos ht _).le (rpow_le_rpow_of_exponent_ge ht h.le ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 108,
"column": 8
} | {
"line": 108,
"column": 24
} | {
"line": 108,
"column": 25
} | [
{
"pp": "case e'_4\ns t a b : ℝ\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : ℝ → ℝ → ℝ → ℝ := fun c u x ↦ rexp (-c * x) * x ^ (c * (u - 1))\ne : (1 / a).HolderConjugate (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : ∀ (c u x : ℝ), x ∈ Ioi 0 → 0 ≤ f c u x\nposf' : ∀ (c u : ℝ)... | [
"case e'_4\ns t a b : ℝ\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : ℝ → ℝ → ℝ → ℝ := fun c u x ↦ rexp (-c * x) * x ^ (c * (u - 1))\ne : (1 / a).HolderConjugate (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : ∀ (c u x : ℝ), x ∈ Ioi 0 → 0 ≤ f c u x\nposf' : ∀ (c u : ℝ), ∀ᵐ (x : ℝ)... | one_div_one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 174,
"column": 2
} | {
"line": 174,
"column": 32
} | {
"line": 175,
"column": 4
} | [
{
"pp": "f : ℝ → ℝ\nx : ℝ\nn : ℕ\nhf_conv : ConvexOn ℝ (Ioi 0) f\nhf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y\nhn : n ≠ 0\nhx : 0 < x\nhx' : x ≤ 1\nhn' : 0 < ↑n\nthis : f ↑n + x * log ↑n = (1 - x) * f ↑n + x * f (↑n + 1)\n⊢ f ((1 - x) * ↑n + x * (↑n + 1)) ≤ (1 - x) * f ↑n + x * f (↑n + 1)",
"ppTerm"... | [
"f : ℝ → ℝ\nx : ℝ\nn : ℕ\nhf_conv : ConvexOn ℝ (Ioi 0) f\nhf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y\nhn : n ≠ 0\nhx : 0 < x\nhx' : x ≤ 1\nhn' : 0 < ↑n\nthis : f ↑n + x * log ↑n = (1 - x) * f ↑n + x * f (↑n + 1)\n⊢ f ((1 - x) * ↑n + x * (↑n + 1)) ≤ (1 - x) * f ↑n + x * f (↑n + 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Affine.AsymptoticCone | {
"line": 51,
"column": 21
} | {
"line": 51,
"column": 53
} | {
"line": 51,
"column": 54
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : FiniteDimensional ℝ V\ns : Set P\np : P\nhs : ∀ i ∈ Metric.sphere 0 1, s ∈ asymptoticNhds ℝ P i\n⊢ s ∈ cobounded P",
"ppTerm": "?m.90",
"assigned": ... | [
"V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : FiniteDimensional ℝ V\ns : Set P\np : P\nhs : ∀ i ∈ Metric.sphere 0 1, s ∈ atTop • 𝓝 i +ᵥ pure p\n⊢ s ∈ cobounded P"
] | asymptoticNhds_eq_smul_vadd _ p, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Algebra.AsymptoticCone | {
"line": 132,
"column": 2
} | {
"line": 134,
"column": 53
} | {
"line": 136,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝¹⁰ : Field k\ninst✝⁹ : LinearOrder k\ninst✝⁸ : AddCommGroup V\ninst✝⁷ : Module k V\ninst✝⁶ : AddTorsor V P\ninst✝⁵ : TopologicalSpace V\ninst✝⁴ : TopologicalSpace k\ninst✝³ : OrderTopology k\ninst✝² : IsStrictOrderedRing k\ninst✝¹ : IsTopologicalAddGroup V... | [] | have ⟨p⟩ : Nonempty P := inferInstance
rw [← asymptoticNhds_vadd_pure 0 p, asymptoticNhds_zero', vadd_pure]
exact (Equiv.vaddConst p).surjective.filter_map_top | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.AsymptoticCone | {
"line": 132,
"column": 2
} | {
"line": 134,
"column": 53
} | {
"line": 136,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝¹⁰ : Field k\ninst✝⁹ : LinearOrder k\ninst✝⁸ : AddCommGroup V\ninst✝⁷ : Module k V\ninst✝⁶ : AddTorsor V P\ninst✝⁵ : TopologicalSpace V\ninst✝⁴ : TopologicalSpace k\ninst✝³ : OrderTopology k\ninst✝² : IsStrictOrderedRing k\ninst✝¹ : IsTopologicalAddGroup V... | [] | have ⟨p⟩ : Nonempty P := inferInstance
rw [← asymptoticNhds_vadd_pure 0 p, asymptoticNhds_zero', vadd_pure]
exact (Equiv.vaddConst p).surjective.filter_map_top | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.AsymptoticCone | {
"line": 150,
"column": 11
} | {
"line": 150,
"column": 43
} | {
"line": 151,
"column": 4
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝¹⁰ : Field k\ninst✝⁹ : LinearOrder k\ninst✝⁸ : AddCommGroup V\ninst✝⁷ : Module k V\ninst✝⁶ : AddTorsor V P\ninst✝⁵ : TopologicalSpace V\ninst✝⁴ : TopologicalSpace k\ninst✝³ : OrderTopology k\ninst✝² : IsStrictOrderedRing k\ninst✝¹ : IsTopologicalAddGroup V... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝¹⁰ : Field k\ninst✝⁹ : LinearOrder k\ninst✝⁸ : AddCommGroup V\ninst✝⁷ : Module k V\ninst✝⁶ : AddTorsor V P\ninst✝⁵ : TopologicalSpace V\ninst✝⁴ : TopologicalSpace k\ninst✝³ : OrderTopology k\ninst✝² : IsStrictOrderedRing k\ninst✝¹ : IsTopologicalAddGroup V\ninst✝ : Co... | asymptoticNhds_eq_smul_vadd _ p, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Algebra.AsymptoticCone | {
"line": 162,
"column": 13
} | {
"line": 162,
"column": 45
} | {
"line": 162,
"column": 46
} | [
{
"pp": "case a\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝¹⁰ : Field k\ninst✝⁹ : LinearOrder k\ninst✝⁸ : AddCommGroup V\ninst✝⁷ : Module k V\ninst✝⁶ : AddTorsor V P\ninst✝⁵ : TopologicalSpace V\ninst✝⁴ : TopologicalSpace k\ninst✝³ : OrderTopology k\ninst✝² : IsStrictOrderedRing k\ninst✝¹ : IsTopologicalAd... | [
"case a\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝¹⁰ : Field k\ninst✝⁹ : LinearOrder k\ninst✝⁸ : AddCommGroup V\ninst✝⁷ : Module k V\ninst✝⁶ : AddTorsor V P\ninst✝⁵ : TopologicalSpace V\ninst✝⁴ : TopologicalSpace k\ninst✝³ : OrderTopology k\ninst✝² : IsStrictOrderedRing k\ninst✝¹ : IsTopologicalAddGroup V\nin... | asymptoticNhds_eq_smul_vadd _ p, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Algebra.AsymptoticCone | {
"line": 174,
"column": 6
} | {
"line": 174,
"column": 38
} | {
"line": 174,
"column": 39
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝¹² : Field k\ninst✝¹¹ : LinearOrder k\ninst✝¹⁰ : AddCommGroup V\ninst✝⁹ : Module k V\ninst✝⁸ : AddTorsor V P\ninst✝⁷ : TopologicalSpace V\ninst✝⁶ : TopologicalSpace k\ninst✝⁵ : OrderTopology k\ninst✝⁴ : IsStrictOrderedRing k\ninst✝³ : IsTopologicalAddGroup... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝¹² : Field k\ninst✝¹¹ : LinearOrder k\ninst✝¹⁰ : AddCommGroup V\ninst✝⁹ : Module k V\ninst✝⁸ : AddTorsor V P\ninst✝⁷ : TopologicalSpace V\ninst✝⁶ : TopologicalSpace k\ninst✝⁵ : OrderTopology k\ninst✝⁴ : IsStrictOrderedRing k\ninst✝³ : IsTopologicalAddGroup V\ninst✝² :... | asymptoticNhds_eq_smul_vadd _ p, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.AsymptoticCone | {
"line": 192,
"column": 2
} | {
"line": 192,
"column": 32
} | {
"line": 193,
"column": 2
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝¹⁰ : Field k\ninst✝⁹ : LinearOrder k\ninst✝⁸ : AddCommGroup V\ninst✝⁷ : Module k V\ninst✝⁶ : AddTorsor V P\ninst✝⁵ : TopologicalSpace V\ninst✝⁴ : TopologicalSpace k\ninst✝³ : OrderTopology k\ninst✝² : IsStrictOrderedRing k\ninst✝¹ : IsTopologicalAddGroup V... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝¹⁰ : Field k\ninst✝⁹ : LinearOrder k\ninst✝⁸ : AddCommGroup V\ninst✝⁷ : Module k V\ninst✝⁶ : AddTorsor V P\ninst✝⁵ : TopologicalSpace V\ninst✝⁴ : TopologicalSpace k\ninst✝³ : OrderTopology k\ninst✝² : IsStrictOrderedRing k\ninst✝¹ : IsTopologicalAddGroup V\ninst✝ : Co... | refine Filter.ext' fun p => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 366,
"column": 4
} | {
"line": 366,
"column": 44
} | {
"line": 366,
"column": 45
} | [
{
"pp": "x : ℝ\nhx : x ∈ Icc 1 2\nhmin : IsMinOn Γ (Icc 1 2) x\n⊢ Γ (3 / 2) < Γ 1 ∧ Γ (3 / 2) < Γ 2 ∧ Γ x ≤ Γ (3 / 2)",
"ppTerm": "?m.134",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"instHDiv",
"congrArg",
"Real.instDivInvMonoid",
... | [
"x : ℝ\nhx : x ∈ Icc 1 2\nhmin : IsMinOn Γ (Icc 1 2) x\n⊢ Γ x ≤ Γ (3 / 2)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Affine.Ceva | {
"line": 50,
"column": 57
} | {
"line": 50,
"column": 68
} | {
"line": 50,
"column": 69
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : SeminormedAddCommGroup V\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nt : Triangle 𝕜 P\np : Fin 3 → P\np' : P\nhp0 : ∀ (i : Fin 3), p i ≠ t.points (i + 2)\nhp : ∀ (i : Fin 3), p i ∈ line[𝕜... | [
"𝕜 : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : SeminormedAddCommGroup V\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nt : Triangle 𝕜 P\np : Fin 3 → P\np' : P\nhp0 : ∀ (i : Fin 3), p i ≠ t.points (i + 2)\nhp : ∀ (i : Fin 3), p i ∈ line[𝕜, t.points (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Ceva | {
"line": 86,
"column": 37
} | {
"line": 86,
"column": 53
} | {
"line": 86,
"column": 54
} | [
{
"pp": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : ι → P\nhp : AffineIndependent k p\ns : Set ι\nhs : s.Nonempty\nfs : ↑s → Finset ι\nw : ↑s → ι → k\nhw : ∀ (i : ↑s), ∑ j ∈ fs i, w i j = 1\np' : P... | [
"case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : ι → P\nhp : AffineIndependent k p\ns : Set ι\nhs : s.Nonempty\nfs : ↑s → Finset ι\nw : ↑s → ι → k\nhw : ∀ (i : ↑s), ∑ j ∈ fs i, w i j = 1\np' : P\nhp' : ∀ (i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Beta | {
"line": 352,
"column": 33
} | {
"line": 352,
"column": 44
} | {
"line": 352,
"column": 45
} | [
{
"pp": "m : ℕ\nIH : ∀ (s : ℂ), ⌊1 - s.re⌋₊ = m → Tendsto s.GammaSeq atTop (𝓝 (Gamma s))\ns : ℂ\nhs : ↑(m + 1) ≤ 1 - s.re ∧ 1 - s.re < ↑(m + 1) + 1\nhsne : s ≠ 0\nthis : s.re ≤ -↑m\n⊢ 0 ≤ 1 - (s + 1).re",
"ppTerm": "?m.213",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",... | [
"m : ℕ\nIH : ∀ (s : ℂ), ⌊1 - s.re⌋₊ = m → Tendsto s.GammaSeq atTop (𝓝 (Gamma s))\ns : ℂ\nhs : ↑(m + 1) ≤ 1 - s.re ∧ 1 - s.re < ↑(m + 1) + 1\nhsne : s ≠ 0\nthis : s.re ≤ -↑m\n⊢ s.re ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Ceva | {
"line": 86,
"column": 37
} | {
"line": 86,
"column": 53
} | {
"line": 86,
"column": 54
} | [
{
"pp": "case neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : ι → P\nhp : AffineIndependent k p\ns : Set ι\nhs : s.Nonempty\nfs : ↑s → Finset ι\nw : ↑s → ι → k\nhw : ∀ (i : ↑s), ∑ j ∈ fs i, w i j = 1\np' : P... | [
"case neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : ι → P\nhp : AffineIndependent k p\ns : Set ι\nhs : s.Nonempty\nfs : ↑s → Finset ι\nw : ↑s → ι → k\nhw : ∀ (i : ↑s), ∑ j ∈ fs i, w i j = 1\np' : P\nhp' : ∀ (i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.AsymptoticCone | {
"line": 328,
"column": 22
} | {
"line": 340,
"column": 85
} | {
"line": 342,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\ninst✝⁹ : Field k\ninst✝⁸ : LinearOrder k\ninst✝⁷ : IsStrictOrderedRing k\ninst✝⁶ : TopologicalSpace k\ninst✝⁵ : OrderTopology k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : TopologicalSpace V\ninst✝¹ : IsTopologicalAddGroup V\ninst✝ : ContinuousSMul k V\ns : Set V... | [] | by
refine isClosed_iff_frequently.mp hs₂ _ <|
tendsto_snd (f := atTop (α := k)) |>.const_smul _ |>.vadd_const _ |>.frequently ?_
rw [mem_asymptoticCone_iff, asymptoticNhds_eq_smul_vadd v p, vadd_pure, frequently_map,
← map₂_smul, ← map_prod_eq_map₂, frequently_map] at hv
apply hv.mp
filter_upwards [tend... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.AffineSpace.Ceva | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 19
} | {
"line": 146,
"column": 20
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : CommRing k\ninst✝³ : NoZeroDivisors k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nt : Triangle k P\nr : Fin 3 → k\np' : P\nhp' : ∀ (i : Fin 3), p' ∈ line[k, t.points i, (AffineMap.lineMap (t.points (i + 1)) (t.points (i + 2)... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : CommRing k\ninst✝³ : NoZeroDivisors k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nt : Triangle k P\nr : Fin 3 → k\np' : P\nhp' : ∀ (i : Fin 3), p' ∈ line[k, t.points i, (AffineMap.lineMap (t.points (i + 1)) (t.points (i + 2))) (r i)]\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Ceva | {
"line": 153,
"column": 4
} | {
"line": 153,
"column": 20
} | {
"line": 153,
"column": 21
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : CommRing k\ninst✝³ : NoZeroDivisors k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nt : Triangle k P\nr : Fin 3 → k\nh✝ : Nontrivial k\nw : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i + 1) (↑i +... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : CommRing k\ninst✝³ : NoZeroDivisors k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nt : Triangle k P\nr : Fin 3 → k\nh✝ : Nontrivial k\nw : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i + 1) (↑i + 2) (r ↑i)\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Beta | {
"line": 431,
"column": 10
} | {
"line": 431,
"column": 15
} | {
"line": 431,
"column": 16
} | [
{
"pp": "s : ℂ\nhs : ∀ (m : ℕ), s ≠ -↑m\nh_im : s.im = 0\n⊢ ↑s.re + ↑s.im * I = ↑s.re",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"Real.instZero",
"congrArg",
"Complex.im",
"Complex.instMul",
"id",
"Co... | [
"s : ℂ\nhs : ∀ (m : ℕ), s ≠ -↑m\nh_im : s.im = 0\n⊢ ↑s.re + ↑0 * I = ↑s.re"
] | h_im, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.Ceva | {
"line": 169,
"column": 6
} | {
"line": 169,
"column": 17
} | {
"line": 169,
"column": 18
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : CommRing k\ninst✝³ : NoZeroDivisors k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nt : Triangle k P\nr : Fin 3 → k\nh✝ : Nontrivial k\nw : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i + 1) (↑i +... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : CommRing k\ninst✝³ : NoZeroDivisors k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nt : Triangle k P\nr : Fin 3 → k\nh✝ : Nontrivial k\nw : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i + 1) (↑i + 2) (r ↑i)\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Beta | {
"line": 456,
"column": 2
} | {
"line": 456,
"column": 72
} | {
"line": 456,
"column": 73
} | [
{
"pp": "s : ℂ\nm : ℕ\nhs : s = -↑m\n⊢ s.re ≤ 0",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Real",
"Real.instZero",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"id",
"Real.instA... | [
"s : ℂ\nm : ℕ\nhs : s = -↑m\n⊢ 0 ≤ ↑m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Beta | {
"line": 560,
"column": 4
} | {
"line": 560,
"column": 76
} | {
"line": 561,
"column": 4
} | [
{
"pp": "s : ℂ\nh1 : AnalyticOnNhd ℂ (fun z ↦ (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ\n⊢ DifferentiableOn ℂ (fun z ↦ (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑√π) univ",
"ppTerm": "?m.420",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"InnerProductSpace.toNormedSpac... | [
"s : ℂ\nh1 : AnalyticOnNhd ℂ (fun z ↦ (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ\n⊢ Differentiable ℂ fun z ↦ (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1)"
] | refine (Differentiable.mul ?_ (differentiable_const _)).differentiableOn | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.AffineSpace.Ceva | {
"line": 175,
"column": 38
} | {
"line": 175,
"column": 54
} | {
"line": 175,
"column": 55
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : CommRing k\ninst✝³ : NoZeroDivisors k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nt : Triangle k P\nr : Fin 3 → k\nh✝ : Nontrivial k\nw : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i + 1) (↑i +... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : CommRing k\ninst✝³ : NoZeroDivisors k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nt : Triangle k P\nr : Fin 3 → k\nh✝ : Nontrivial k\nw : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i + 1) (↑i + 2) (r ↑i)\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Ceva | {
"line": 176,
"column": 38
} | {
"line": 176,
"column": 54
} | {
"line": 176,
"column": 55
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : CommRing k\ninst✝³ : NoZeroDivisors k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nt : Triangle k P\nr : Fin 3 → k\nh✝ : Nontrivial k\nw : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i + 1) (↑i +... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : CommRing k\ninst✝³ : NoZeroDivisors k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nt : Triangle k P\nr : Fin 3 → k\nh✝ : Nontrivial k\nw : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i + 1) (↑i + 2) (r ↑i)\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Ceva | {
"line": 180,
"column": 47
} | {
"line": 180,
"column": 77
} | {
"line": 180,
"column": 78
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : CommRing k\ninst✝³ : NoZeroDivisors k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nt : Triangle k P\nr : Fin 3 → k\nh✝ : Nontrivial k\nw : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i + 1) (↑i +... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : CommRing k\ninst✝³ : NoZeroDivisors k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nt : Triangle k P\nr : Fin 3 → k\nh✝ : Nontrivial k\nw : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i + 1) (↑i + 2) (r ↑i)\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Ceva | {
"line": 182,
"column": 5
} | {
"line": 182,
"column": 35
} | {
"line": 182,
"column": 36
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : CommRing k\ninst✝³ : NoZeroDivisors k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nt : Triangle k P\nr : Fin 3 → k\nh✝ : Nontrivial k\nw : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i + 1) (↑i +... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : CommRing k\ninst✝³ : NoZeroDivisors k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nt : Triangle k P\nr : Fin 3 → k\nh✝ : Nontrivial k\nw : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i + 1) (↑i + 2) (r ↑i)\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Ceva | {
"line": 184,
"column": 68
} | {
"line": 184,
"column": 79
} | {
"line": 184,
"column": 80
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : CommRing k\ninst✝³ : NoZeroDivisors k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nt : Triangle k P\nr : Fin 3 → k\nh✝ : Nontrivial k\nw : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i + 1) (↑i +... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : CommRing k\ninst✝³ : NoZeroDivisors k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nt : Triangle k P\nr : Fin 3 → k\nh✝ : Nontrivial k\nw : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i + 1) (↑i + 2) (r ↑i)\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Affine.Simplex | {
"line": 108,
"column": 4
} | {
"line": 108,
"column": 15
} | {
"line": 108,
"column": 16
} | [
{
"pp": "case refine_1\nR : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : Module R V\ninst✝ : NormedAddTorsor V P\nm n : ℕ\ns : Simplex R P m\ne : Fin (m + 1) ≃ Fin (n + 1)\nh : (s.reindex e).Regular\nσ : Equiv.Perm (Fin (m + 1))... | [
"case refine_1\nR : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : Module R V\ninst✝ : NormedAddTorsor V P\nm n : ℕ\ns : Simplex R P m\ne : Fin (m + 1) ≃ Fin (n + 1)\nh : (s.reindex e).Regular\nσ : Equiv.Perm (Fin (m + 1))\nx : P ≃ᵢ P... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Affine.Simplex | {
"line": 112,
"column": 4
} | {
"line": 112,
"column": 15
} | {
"line": 112,
"column": 16
} | [
{
"pp": "case refine_2\nR : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : Module R V\ninst✝ : NormedAddTorsor V P\nm n : ℕ\ns : Simplex R P m\ne : Fin (m + 1) ≃ Fin (n + 1)\nh : s.Regular\nσ : Equiv.Perm (Fin (n + 1))\nx : P ≃ᵢ P... | [
"case refine_2\nR : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : Module R V\ninst✝ : NormedAddTorsor V P\nm n : ℕ\ns : Simplex R P m\ne : Fin (m + 1) ≃ Fin (n + 1)\nh : s.Regular\nσ : Equiv.Perm (Fin (n + 1))\nx : P ≃ᵢ P\nhx : s.poi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Ceva | {
"line": 186,
"column": 36
} | {
"line": 186,
"column": 54
} | {
"line": 186,
"column": 55
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : CommRing k\ninst✝³ : NoZeroDivisors k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nt : Triangle k P\nr : Fin 3 → k\nh✝ : Nontrivial k\nw : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i + 1) (↑i +... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : CommRing k\ninst✝³ : NoZeroDivisors k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nt : Triangle k P\nr : Fin 3 → k\nh✝ : Nontrivial k\nw : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i + 1) (↑i + 2) (r ↑i)\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Ceva | {
"line": 187,
"column": 53
} | {
"line": 187,
"column": 83
} | {
"line": 187,
"column": 84
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : CommRing k\ninst✝³ : NoZeroDivisors k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nt : Triangle k P\nr : Fin 3 → k\nh✝ : Nontrivial k\nw : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i + 1) (↑i +... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : CommRing k\ninst✝³ : NoZeroDivisors k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nt : Triangle k P\nr : Fin 3 → k\nh✝ : Nontrivial k\nw : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i + 1) (↑i + 2) (r ↑i)\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Ceva | {
"line": 188,
"column": 48
} | {
"line": 188,
"column": 78
} | {
"line": 188,
"column": 79
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : CommRing k\ninst✝³ : NoZeroDivisors k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nt : Triangle k P\nr : Fin 3 → k\nh✝ : Nontrivial k\nw : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i + 1) (↑i +... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : CommRing k\ninst✝³ : NoZeroDivisors k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nt : Triangle k P\nr : Fin 3 → k\nh✝ : Nontrivial k\nw : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i + 1) (↑i + 2) (r ↑i)\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Affine.Simplex | {
"line": 114,
"column": 82
} | {
"line": 132,
"column": 10
} | {
"line": 134,
"column": 0
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : Module R V\ninst✝ : NormedAddTorsor V P\nn : ℕ\ns : Simplex R P n\nhr : s.Regular\n⊢ s.Equilateral",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
... | [] | by
refine ⟨dist (s.points 0) (s.points 1), fun i j hij ↦ ?_⟩
have hn : n ≠ 0 := by lia
by_cases hi : i = 1
· rw [hi, dist_comm]
rcases hr (Equiv.swap 0 j) with ⟨x, hx⟩
nth_rw 2 [← x.dist_eq]
simp_rw [← Function.comp_apply (f := x), ← hx]
simp only [comp_apply, Equiv.swap_apply_left]
convert!... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.AffineSpace.Ceva | {
"line": 190,
"column": 68
} | {
"line": 190,
"column": 79
} | {
"line": 190,
"column": 80
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : CommRing k\ninst✝³ : NoZeroDivisors k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nt : Triangle k P\nr : Fin 3 → k\nh✝ : Nontrivial k\nw : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i + 1) (↑i +... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : CommRing k\ninst✝³ : NoZeroDivisors k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nt : Triangle k P\nr : Fin 3 → k\nh✝ : Nontrivial k\nw : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i + 1) (↑i + 2) (r ↑i)\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Polynomial.Factorization | {
"line": 49,
"column": 83
} | {
"line": 56,
"column": 82
} | {
"line": 58,
"column": 0
} | [
{
"pp": "f : ℝ[X]\nn : ℕ\nhf : f.IsMonicOfDegree (n + 1)\n⊢ ∃ f₁ f₂, (f₁.IsMonicOfDegree 1 ∨ f₁.IsMonicOfDegree 2) ∧ f = f₁ * f₂",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"NormedCommRing.toNormedRing",
"Irreducible.natDegree_le_two",
"Semigroup.toMu... | [] | by
obtain ⟨f₁, hm, hirr, f₂, hf₂⟩ :=
exists_monic_irreducible_factor f <| not_isUnit_of_natDegree_pos f <|
by grind [IsMonicOfDegree.natDegree_eq]
refine ⟨f₁, f₂, ?_, hf₂⟩
have help {P : ℕ → Prop} {m : ℕ} (hm₀ : 0 < m) (hm₂ : m ≤ 2) (h : P m) : P 1 ∨ P 2 := by
interval_cases m <;> tauto
exact help... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.AffineSpace.Ceva | {
"line": 192,
"column": 36
} | {
"line": 192,
"column": 54
} | {
"line": 192,
"column": 55
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : CommRing k\ninst✝³ : NoZeroDivisors k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nt : Triangle k P\nr : Fin 3 → k\nh✝ : Nontrivial k\nw : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i + 1) (↑i +... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : CommRing k\ninst✝³ : NoZeroDivisors k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nt : Triangle k P\nr : Fin 3 → k\nh✝ : Nontrivial k\nw : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i + 1) (↑i + 2) (r ↑i)\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Ceva | {
"line": 173,
"column": 2
} | {
"line": 194,
"column": 66
} | {
"line": 195,
"column": 2
} | [
{
"pp": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : CommRing k\ninst✝³ : NoZeroDivisors k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nt : Triangle k P\nr : Fin 3 → k\nh✝ : Nontrivial k\nw : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i ... | [
"case neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : CommRing k\ninst✝³ : NoZeroDivisors k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nt : Triangle k P\nr : Fin 3 → k\nh✝ : Nontrivial k\nw : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i + 1) (↑i + 2... | · rw [Finset.prod_eq_zero_iff] at hc
obtain ⟨i, -, hi⟩ := hc
have hw'i1 : w' (i + 1) = 0 := by simpa [hi] using (hc1 i).symm
have hw'i2 : w' (i + 2) = 0 := by simpa [hi] using (hc2 i).symm
have hw'i0 : w' i = 1 := by
rw [← hw', Fin.sum_univ_three]
fin_cases i <;> grind
have hi1 : c (i + ... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Algebra.MatrixExponential | {
"line": 190,
"column": 10
} | {
"line": 190,
"column": 49
} | {
"line": 190,
"column": 50
} | [
{
"pp": "m : Type u_1\n𝔸 : Type u_5\ninst✝⁴ : Fintype m\ninst✝³ : DecidableEq m\ninst✝² : NormedCommRing 𝔸\ninst✝¹ : NormedAlgebra ℚ 𝔸\ninst✝ : CompleteSpace 𝔸\nU A : Matrix m m 𝔸\nhy : IsUnit U\nu : (Matrix m m 𝔸)ˣ\nhu : ↑u = U\n⊢ exp (↑u * A * (↑u)⁻¹) = ↑u * exp A * (↑u)⁻¹",
"ppTerm": "?m.39",
"... | [
"m : Type u_1\n𝔸 : Type u_5\ninst✝⁴ : Fintype m\ninst✝³ : DecidableEq m\ninst✝² : NormedCommRing 𝔸\ninst✝¹ : NormedAlgebra ℚ 𝔸\ninst✝ : CompleteSpace 𝔸\nU A : Matrix m m 𝔸\nhy : IsUnit U\nu : (Matrix m m 𝔸)ˣ\nhu : ↑u = U\n⊢ exp (↑u * A * (↑u)⁻¹) = ↑u * exp A * (↑u)⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.MatrixExponential | {
"line": 195,
"column": 10
} | {
"line": 195,
"column": 49
} | {
"line": 195,
"column": 50
} | [
{
"pp": "m : Type u_1\n𝔸 : Type u_5\ninst✝⁴ : Fintype m\ninst✝³ : DecidableEq m\ninst✝² : NormedCommRing 𝔸\ninst✝¹ : NormedAlgebra ℚ 𝔸\ninst✝ : CompleteSpace 𝔸\nU A : Matrix m m 𝔸\nhy : IsUnit U\nu : (Matrix m m 𝔸)ˣ\nhu : ↑u = U\n⊢ exp ((↑u)⁻¹ * A * ↑u) = (↑u)⁻¹ * exp A * ↑u",
"ppTerm": "?m.39",
"... | [
"m : Type u_1\n𝔸 : Type u_5\ninst✝⁴ : Fintype m\ninst✝³ : DecidableEq m\ninst✝² : NormedCommRing 𝔸\ninst✝¹ : NormedAlgebra ℚ 𝔸\ninst✝ : CompleteSpace 𝔸\nU A : Matrix m m 𝔸\nhy : IsUnit U\nu : (Matrix m m 𝔸)ˣ\nhu : ↑u = U\n⊢ exp ((↑u)⁻¹ * A * ↑u) = (↑u)⁻¹ * exp A * ↑u"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.GelfandMazur | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 47
} | {
"line": 156,
"column": 48
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_2\ninst✝⁴ : NormedField 𝕜\ninst✝³ : ProperSpace 𝕜\ninst✝² : SeminormedRing F\ninst✝¹ : NormedAlgebra 𝕜 F\ninst✝ : NormOneClass F\nx : F\nthis : Tendsto (fun x_1 ↦ ‖x - (algebraMap 𝕜 F) x_1‖) (cobounded 𝕜) atTop\n⊢ Bornology.IsBounded {x_1 | ‖x - (algebraMap 𝕜 F) x_1‖ ≤ ‖... | [
"𝕜 : Type u_1\nF : Type u_2\ninst✝⁴ : NormedField 𝕜\ninst✝³ : ProperSpace 𝕜\ninst✝² : SeminormedRing F\ninst✝¹ : NormedAlgebra 𝕜 F\ninst✝ : NormOneClass F\nx : F\nthis : Tendsto (fun x_1 ↦ ‖x - (algebraMap 𝕜 F) x_1‖) (cobounded 𝕜) atTop\n⊢ {a | ‖x‖ < ‖x - (algebraMap 𝕜 F) a‖} ∈ cobounded 𝕜"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.GelfandMazur | {
"line": 180,
"column": 4
} | {
"line": 181,
"column": 11
} | {
"line": 181,
"column": 12
} | [
{
"pp": "case succ\nF : Type u_1\ninst✝³ : NormedRing F\ninst✝² : NormOneClass F\ninst✝¹ : NormMulClass F\ninst✝ : NormedAlgebra ℂ F\nx : F\nM : ℝ\nhM : 0 ≤ M\nh : ∀ (z' : ℂ), M ≤ ‖x - (algebraMap ℂ F) z'‖\nc : ℂ\nn : ℕ\nih : ∀ {p : ℂ[X]}, p.IsMonicOfDegree n → M ^ n ≤ ‖(aeval (x - (algebraMap ℂ F) c)) p‖\np : ... | [
"case succ\nF : Type u_1\ninst✝³ : NormedRing F\ninst✝² : NormOneClass F\ninst✝¹ : NormMulClass F\ninst✝ : NormedAlgebra ℂ F\nx : F\nM : ℝ\nhM : 0 ≤ M\nh : ∀ (z' : ℂ), M ≤ ‖x - (algebraMap ℂ F) z'‖\nc : ℂ\nn : ℕ\nih : ∀ {p : ℂ[X]}, p.IsMonicOfDegree n → M ^ n ≤ ‖(aeval (x - (algebraMap ℂ F) c)) p‖\np : ℂ[X]\nhp : p... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Quaternion | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 41
} | {
"line": 176,
"column": 4
} | [
{
"pp": "⊢ Continuous ⇑normSq",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ Continuous ⇑normSq"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 31
} | {
"line": 113,
"column": 32
} | [
{
"pp": "𝕜 : Type u_1\nR : Type u_3\nM : Type u_4\ninst✝¹⁶ : Field 𝕜\ninst✝¹⁵ : CharZero 𝕜\ninst✝¹⁴ : Ring R\ninst✝¹³ : AddCommGroup M\ninst✝¹² : Algebra 𝕜 R\ninst✝¹¹ : Module 𝕜 M\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module Rᵐᵒᵖ M\ninst✝⁸ : SMulCommClass R Rᵐᵒᵖ M\ninst✝⁷ : IsScalarTower 𝕜 R M\ninst✝⁶ : IsScala... | [
"𝕜 : Type u_1\nR : Type u_3\nM : Type u_4\ninst✝¹⁶ : Field 𝕜\ninst✝¹⁵ : CharZero 𝕜\ninst✝¹⁴ : Ring R\ninst✝¹³ : AddCommGroup M\ninst✝¹² : Algebra 𝕜 R\ninst✝¹¹ : Module 𝕜 M\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module Rᵐᵒᵖ M\ninst✝⁸ : SMulCommClass R Rᵐᵒᵖ M\ninst✝⁷ : IsScalarTower 𝕜 R M\ninst✝⁶ : IsScalarTower 𝕜 Rᵐ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 43
} | {
"line": 115,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nR : Type u_3\nM : Type u_4\ninst✝¹⁶ : Field 𝕜\ninst✝¹⁵ : CharZero 𝕜\ninst✝¹⁴ : Ring R\ninst✝¹³ : AddCommGroup M\ninst✝¹² : Algebra 𝕜 R\ninst✝¹¹ : Module 𝕜 M\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module Rᵐᵒᵖ M\ninst✝⁸ : SMulCommClass R Rᵐᵒᵖ M\ninst✝⁷ : IsScalarTower 𝕜 R M\ninst✝⁶ : IsScala... | [
"𝕜 : Type u_1\nR : Type u_3\nM : Type u_4\ninst✝¹⁶ : Field 𝕜\ninst✝¹⁵ : CharZero 𝕜\ninst✝¹⁴ : Ring R\ninst✝¹³ : AddCommGroup M\ninst✝¹² : Algebra 𝕜 R\ninst✝¹¹ : Module 𝕜 M\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module Rᵐᵒᵖ M\ninst✝⁸ : SMulCommClass R Rᵐᵒᵖ M\ninst✝⁷ : IsScalarTower 𝕜 R M\ninst✝⁶ : IsScalarTower 𝕜 Rᵐ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Series | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 36
} | {
"line": 125,
"column": 37
} | [
{
"pp": "z : ℂ\n⊢ HasSum (fun n ↦ z ^ (2 * n) / ↑(2 * n)!) (cosh z)",
"ppTerm": "?m.42",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"z : ℂ\n⊢ HasSum (fun n ↦ z ^ (2 * n) / ↑(2 * n)!) (cosh z)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Series | {
"line": 129,
"column": 2
} | {
"line": 130,
"column": 9
} | {
"line": 130,
"column": 10
} | [
{
"pp": "z : ℂ\n⊢ HasSum (fun n ↦ z ^ (2 * n + 1) / ↑(2 * n + 1)!) (sinh z)",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Complex.sinh",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg"... | [
"z : ℂ\n⊢ HasSum (fun n ↦ (z ^ 2) ^ n * z / ↑(2 * n + 1)!) (sinh z)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Ultra | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 41
} | {
"line": 52,
"column": 42
} | [
{
"pp": "R : Type u_1\ninst✝² : SeminormedRing R\ninst✝¹ : NormOneClass R\ninst✝ : IsUltrametricDist R\nx : R\n⊢ ‖x + 1‖ ≤ max ‖x‖ 1",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"SeminormedRing.toNorm",
"Real.instLE",
"Real",
"P... | [
"R : Type u_1\ninst✝² : SeminormedRing R\ninst✝¹ : NormOneClass R\ninst✝ : IsUltrametricDist R\nx : R\n⊢ ‖x + 1‖ ≤ ‖x‖ ∨ ‖x + 1‖ ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Ultra | {
"line": 64,
"column": 17
} | {
"line": 64,
"column": 80
} | {
"line": 65,
"column": 4
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝² : SeminormedRing R\ninst✝¹ : NormOneClass R\ninst✝ : IsUltrametricDist R\nn : ℕ\nhn : ‖↑n‖₊ ≤ 1\n⊢ ‖↑(n + 1)‖₊ ≤ 1",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddMonoid.toAddSemigroup",
"congrArg",
"SeminormedA... | [
"case succ\nR : Type u_1\ninst✝² : SeminormedRing R\ninst✝¹ : NormOneClass R\ninst✝ : IsUltrametricDist R\nn : ℕ\nhn : ‖↑n‖₊ ≤ 1\n⊢ ‖↑n + 1‖₊ ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Ultra | {
"line": 74,
"column": 2
} | {
"line": 75,
"column": 9
} | {
"line": 75,
"column": 10
} | [
{
"pp": "case ofNat\nR : Type u_1\ninst✝² : SeminormedRing R\ninst✝¹ : NormOneClass R\ninst✝ : IsUltrametricDist R\na✝ : ℕ\n⊢ ‖↑(Int.ofNat a✝)‖₊ ≤ 1",
"ppTerm": "?ofNat",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Int.cast_natCast",
"congrArg",
"Seminor... | [
"case ofNat\nR : Type u_1\ninst✝² : SeminormedRing R\ninst✝¹ : NormOneClass R\ninst✝ : IsUltrametricDist R\na✝ : ℕ\n⊢ ‖↑a✝‖₊ ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Ultra | {
"line": 74,
"column": 2
} | {
"line": 75,
"column": 9
} | {
"line": 75,
"column": 10
} | [
{
"pp": "case negSucc\nR : Type u_1\ninst✝² : SeminormedRing R\ninst✝¹ : NormOneClass R\ninst✝ : IsUltrametricDist R\na✝ : ℕ\n⊢ ‖↑(Int.negSucc a✝)‖₊ ≤ 1",
"ppTerm": "?negSucc",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Eq.mpr",
"NegZero... | [
"case negSucc\nR : Type u_1\ninst✝² : SeminormedRing R\ninst✝¹ : NormOneClass R\ninst✝ : IsUltrametricDist R\na✝ : ℕ\n⊢ ‖↑(a✝ + 1)‖₊ ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 31
} | {
"line": 49,
"column": 32
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (x : R), ‖x + 1‖ ≤ max ‖x‖ 1\nx : R\n⊢ ‖x + 0‖ ≤ max ‖x‖ ‖0‖",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"Real.instLE",
"Real",
"S... | [
"case inl\nR : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (x : R), ‖x + 1‖ ≤ max ‖x‖ 1\nx : R\n⊢ ‖x‖ ≤ max ‖x‖ ‖0‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 51,
"column": 4
} | {
"line": 52,
"column": 36
} | {
"line": 52,
"column": 37
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (x : R), ‖x + 1‖ ≤ max ‖x‖ 1\nx y : R\nhy : y ≠ 0\np : 0 < ‖y‖\n⊢ ‖x + y‖ ≤ max ‖x‖ ‖y‖",
"ppTerm": "?inr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case inr\nR : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (x : R), ‖x + 1‖ ≤ max ‖x‖ 1\nx y : R\nhy : y ≠ 0\np : 0 < ‖y‖\n⊢ ‖x + y‖ ≤ max ‖x‖ ‖y‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 61,
"column": 19
} | {
"line": 61,
"column": 30
} | {
"line": 61,
"column": 31
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (x : R), ‖x‖ ≤ 1 → ‖x + 1‖ ≤ 1\nx : R\nH : 1 < ‖x‖\n⊢ x ≠ 0",
"ppTerm": "?m.100",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"DivisionSemiring.toGroupWithZero",
"NormedDivisionRing.toDivisionRing... | [
"R : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (x : R), ‖x‖ ≤ 1 → ‖x + 1‖ ≤ 1\nx : R\nH : 1 < ‖x‖\n⊢ ¬x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 68,
"column": 4
} | {
"line": 68,
"column": 43
} | {
"line": 68,
"column": 44
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (x : R), ‖x‖ ≤ 1 → ‖x - 1‖ ≤ 1\nx : R\nhx : ‖x‖ ≤ 1\n⊢ ‖x + 1‖ ≤ 1",
"ppTerm": "?m.39",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (x : R), ‖x‖ ≤ 1 → ‖x - 1‖ ≤ 1\nx : R\nhx : ‖x‖ ≤ 1\n⊢ ‖x + 1‖ ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 51
} | {
"line": 159,
"column": 0
} | [
{
"pp": "R : Type u_3\nM : Type u_4\ninst✝¹⁴ : CommRing R\ninst✝¹³ : AddCommGroup M\ninst✝¹² : Algebra ℚ R\ninst✝¹¹ : Module ℚ M\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module Rᵐᵒᵖ M\ninst✝⁸ : IsCentralScalar R M\ninst✝⁷ : TopologicalSpace R\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : IsTopologicalRing R\ninst✝⁴ : IsTopolog... | [] | rw [exp_def, fst_add, fst_inl, fst_inr, add_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 51
} | {
"line": 159,
"column": 0
} | [
{
"pp": "R : Type u_3\nM : Type u_4\ninst✝¹⁴ : CommRing R\ninst✝¹³ : AddCommGroup M\ninst✝¹² : Algebra ℚ R\ninst✝¹¹ : Module ℚ M\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module Rᵐᵒᵖ M\ninst✝⁸ : IsCentralScalar R M\ninst✝⁷ : TopologicalSpace R\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : IsTopologicalRing R\ninst✝⁴ : IsTopolog... | [] | rw [exp_def, fst_add, fst_inl, fst_inr, add_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 51
} | {
"line": 159,
"column": 0
} | [
{
"pp": "R : Type u_3\nM : Type u_4\ninst✝¹⁴ : CommRing R\ninst✝¹³ : AddCommGroup M\ninst✝¹² : Algebra ℚ R\ninst✝¹¹ : Module ℚ M\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module Rᵐᵒᵖ M\ninst✝⁸ : IsCentralScalar R M\ninst✝⁷ : TopologicalSpace R\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : IsTopologicalRing R\ninst✝⁴ : IsTopolog... | [] | rw [exp_def, fst_add, fst_inl, fst_inr, add_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 114,
"column": 6
} | {
"line": 114,
"column": 50
} | {
"line": 114,
"column": 51
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (n : ℕ), ‖↑n‖ ≤ 1\nx✝ : R\nm : ℕ\nx : R\nn : ℕ\nhx : 0 < ‖x‖\n⊢ ‖↑n‖ * ‖x‖ ≤ ‖x‖",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"... | [
"case inr\nR : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (n : ℕ), ‖↑n‖ ≤ 1\nx✝ : R\nm : ℕ\nx : R\nn : ℕ\nhx : 0 < ‖x‖\n⊢ ‖↑n‖ ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 118,
"column": 4
} | {
"line": 119,
"column": 26
} | {
"line": 119,
"column": 27
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedDivisionRing R\nx : R\nm : ℕ\nh : ∀ (x : R) (n : ℕ), ‖n • x‖ ≤ ‖x‖\n⊢ ‖x + 1‖ ^ m ≤ ∑ k ∈ Finset.range (m + 1), ‖x‖ ^ k",
"ppTerm": "?m.86",
"assigned": true,
"usedConstants": [
"one_pow",
"Norm.norm",
"Eq.mpr",
"NonAssocSemiring.toAddComm... | [
"R : Type u_1\ninst✝ : NormedDivisionRing R\nx : R\nm : ℕ\nh : ∀ (x : R) (n : ℕ), ‖n • x‖ ≤ ‖x‖\n⊢ ‖∑ x_1 ∈ Finset.range (m + 1), x ^ x_1 * ↑(m.choose x_1)‖ ≤ ∑ x_1 ∈ Finset.range (m + 1), ‖x ^ x_1‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 129,
"column": 22
} | {
"line": 129,
"column": 89
} | {
"line": 129,
"column": 90
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedDivisionRing R\nx : R\nh : ∀ (x : R) (n : ℕ), ‖n • x‖ ≤ ‖x‖\ni : ℕ\nhm : max 1 (‖x‖ ^ 0) = 1\nhx : ‖x‖ ^ 0 ≤ 1\nhi : i ∈ Finset.range (0 + 1)\n⊢ i = 0",
"ppTerm": "?m.244",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : NormedDivisionRing R\nx : R\nh : ∀ (x : R) (n : ℕ), ‖n • x‖ ≤ ‖x‖\ni : ℕ\nhm : max 1 (‖x‖ ^ 0) = 1\nhx : ‖x‖ ^ 0 ≤ 1\nhi : i ∈ Finset.range (0 + 1)\n⊢ i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Ultra | {
"line": 31,
"column": 4
} | {
"line": 31,
"column": 15
} | {
"line": 31,
"column": 16
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝³ : NormedField K\ninst✝² : SeminormedRing L\ninst✝¹ : NormOneClass L\ninst✝ : NormedAlgebra K L\nh : IsUltrametricDist L\nx y z : K\n⊢ dist x z ≤ max (dist x y) (dist y z)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Norm... | [
"K : Type u_1\nL : Type u_2\ninst✝³ : NormedField K\ninst✝² : SeminormedRing L\ninst✝¹ : NormOneClass L\ninst✝ : NormedAlgebra K L\nh : IsUltrametricDist L\nx y z : K\n⊢ dist x z ≤ dist x y ∨ dist x z ≤ dist y z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 134,
"column": 4
} | {
"line": 134,
"column": 33
} | {
"line": 134,
"column": 34
} | [
{
"pp": "case hmn\nR : Type u_1\ninst✝ : NormedDivisionRing R\nx : R\nm : ℕ\nh : ∀ (x : R) (n : ℕ), ‖n • x‖ ≤ ‖x‖\nhm : max 1 (‖x‖ ^ m) = ‖x‖ ^ m\nhx : 1 < ‖x‖ ^ m\ni : ℕ\nhi : i ∈ Finset.range (m + 1)\n⊢ i ≤ m",
"ppTerm": "?hmn",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"use... | [
"case hmn\nR : Type u_1\ninst✝ : NormedDivisionRing R\nx : R\nm : ℕ\nh : ∀ (x : R) (n : ℕ), ‖n • x‖ ≤ ‖x‖\nhm : max 1 (‖x‖ ^ m) = ‖x‖ ^ m\nhx : 1 < ‖x‖ ^ m\ni : ℕ\nhi : i ∈ Finset.range (m + 1)\n⊢ i ≤ m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.GelfandMazur | {
"line": 377,
"column": 2
} | {
"line": 378,
"column": 9
} | {
"line": 378,
"column": 10
} | [
{
"pp": "case inr\nF : Type u_1\ninst✝² : NormedRing F\ninst✝¹ : NormedAlgebra ℝ F\ninst✝ : NormOneClass F\nx : F\nu : ℝ\nhc₀ : 0 < ‖x - (algebraMap ℝ F) u‖\nhu : ∀ (x_1 : ℝ), ‖x - (algebraMap ℝ F) u‖ ≤ ‖x - (algebraMap ℝ F) x_1‖\n⊢ Bornology.IsBounded {x_1 | ‖φ x x_1‖ ≤ ‖φ x (0, 0)‖}",
"ppTerm": "?inr",
... | [
"case inr\nF : Type u_1\ninst✝² : NormedRing F\ninst✝¹ : NormedAlgebra ℝ F\ninst✝ : NormOneClass F\nx : F\nu : ℝ\nhc₀ : 0 < ‖x - (algebraMap ℝ F) u‖\nhu : ∀ (x_1 : ℝ), ‖x - (algebraMap ℝ F) u‖ ≤ ‖x - (algebraMap ℝ F) x_1‖\n⊢ {a | ‖φ x (0, 0)‖ < ‖φ x a‖} ∈ cobounded (ℝ × ℝ)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.RingSeminorm | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 45
} | {
"line": 152,
"column": 46
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝ : Ring R\np : RingSeminorm R\nhp : p 1 ≤ 1\nh : p ≠ 0\nhp0 : p 1 = 0\nx : R\n⊢ p x ≤ 0",
"ppTerm": "?inl",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case inl\nR : Type u_1\ninst✝ : Ring R\np : RingSeminorm R\nhp : p 1 ≤ 1\nh : p ≠ 0\nhp0 : p 1 = 0\nx : R\n⊢ p x ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.RingSeminorm | {
"line": 154,
"column": 4
} | {
"line": 154,
"column": 30
} | {
"line": 154,
"column": 31
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝ : Ring R\np : RingSeminorm R\nhp : p 1 ≤ 1\nh : p ≠ 0\nhp0 : 0 < p 1\n⊢ p 1 ≤ p 1 * p 1",
"ppTerm": "?inr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case inr\nR : Type u_1\ninst✝ : Ring R\np : RingSeminorm R\nhp : p 1 ≤ 1\nh : p ≠ 0\nhp0 : 0 < p 1\n⊢ p 1 ≤ p 1 * p 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 78,
"column": 23
} | {
"line": 78,
"column": 51
} | {
"line": 78,
"column": 52
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nx : R\nhx : IsUnit x\nhfx : f x ≠ 0\nn : ℕ\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nh1 : f 1 ≠ 0\nhxn : f (x ^ n) = 0\n⊢ f 1 ≤ 0",
"ppTerm": "?m.59",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nx : R\nhx : IsUnit x\nhfx : f x ≠ 0\nn : ℕ\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nh1 : f 1 ≠ 0\nhxn : f (x ^ n) = 0\n⊢ f 1 ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 86,
"column": 38
} | {
"line": 86,
"column": 54
} | {
"line": 86,
"column": 55
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx : R\nhx : f x = 0\ny : R\n⊢ f (x * y) ≤ 0",
"ppTerm": "?m.38",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx : R\nhx : f x = 0\ny : R\n⊢ f (x * y) ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 103,
"column": 38
} | {
"line": 103,
"column": 54
} | {
"line": 103,
"column": 55
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nx : R\nf_mul : 1 ≤ c * f 1\nhx : 0 < f x\nf_nonneg : 0 ≤ f 1\nh1 : f 1 = 0\n⊢ 1 ≤ 0",
"ppTerm": "?m.161",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nx : R\nf_mul : 1 ≤ c * f 1\nhx : 0 < f x\nf_nonneg : 0 ≤ f 1\nh1 : f 1 = 0\n⊢ 1 ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 21
} | {
"line": 116,
"column": 22
} | [
{
"pp": "case h.inl\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx y : R\nhy0 : f y = 0 y\n⊢ (fun y ↦ f (x * y) / f y) y ≤ c * f x",
"ppTerm": "?h.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero... | [
"case h.inl\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx y : R\nhy0 : f y = 0 y\n⊢ 0 ≤ c * f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 33
} | {
"line": 117,
"column": 34
} | [
{
"pp": "case h.inr\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx y : R\nhy0 : 0 y < f y\n⊢ (fun y ↦ f (x * y) / f y) y ≤ c * f x",
"ppTerm": "?h.inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero... | [
"case h.inr\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx y : R\nhy0 : 0 y < f y\n⊢ f (x * y) ≤ c * f x * f y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 20
} | {
"line": 126,
"column": 21
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx y : R\nhy : f y = 0 y\n⊢ f (x * y) / f y ≤ c * f x",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero... | [
"case inl\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx y : R\nhy : f y = 0 y\n⊢ 0 ≤ c * f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.RingSeminorm | {
"line": 412,
"column": 6
} | {
"line": 419,
"column": 30
} | {
"line": 420,
"column": 6
} | [
{
"pp": "R : Type u_1\nK : Type u_2\ninst✝ : Field K\nf : RingSeminorm K\nhnt : f ≠ 0\nx : K\nhx : f.toFun x = 0\nc : K\nhc : f c ≠ 0\nhn0 : ¬x = 0\n⊢ False",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
... | [
"R : Type u_1\nK : Type u_2\ninst✝ : Field K\nf : RingSeminorm K\nhnt : f ≠ 0\nx : K\nhx : f.toFun x = 0\nc : K\nhc : f c ≠ 0\nhn0 : ¬x = 0\nhc0 : f c = 0\n⊢ False"
] | have hc0 : f c = 0 := by
rw [← mul_one c, ← mul_inv_cancel₀ hn0, ← mul_assoc, mul_comm c, mul_assoc]
exact
le_antisymm
(le_trans (map_mul_le_mul f _ _)
(by rw [← RingSeminorm.toFun_eq_coe, ← AddGroupSeminorm.toFun_eq_coe, hx,
zero_mul]))
(a... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Field.Approximation | {
"line": 89,
"column": 6
} | {
"line": 89,
"column": 41
} | {
"line": 89,
"column": 42
} | [
{
"pp": "K : Type u_1\ninst✝ : NormedField K\nf g : K[X]\nε : ℝ\nhε : 0 < ε\na : K\nha : eval a f = 0\nhfm : f.Monic\nhgm : g.Monic\nhdeg : g.natDegree = f.natDegree\nhcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε\nhg : g.Splits\n⊢ (g - f).natDegree < g.natDegree + 1",
"ppTerm": "?m.596",
"assigned": t... | [
"K : Type u_1\ninst✝ : NormedField K\nf g : K[X]\nε : ℝ\nhε : 0 < ε\na : K\nha : eval a f = 0\nhfm : f.Monic\nhgm : g.Monic\nhdeg : g.natDegree = f.natDegree\nhcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε\nhg : g.Splits\n⊢ (g - f).natDegree ≤ f.natDegree"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Approximation | {
"line": 93,
"column": 4
} | {
"line": 93,
"column": 42
} | {
"line": 93,
"column": 43
} | [
{
"pp": "K : Type u_1\ninst✝ : NormedField K\nf g : K[X]\nε : ℝ\nhε : 0 < ε\na : K\nha : eval a f = 0\nhfm : f.Monic\nhgm : g.Monic\nhdeg : g.natDegree = f.natDegree\nhcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε\nhg : g.Splits\nthis :\n ‖∑ i ∈ Finset.range (g.natDegree + 1), eval a (C (g.coeff i - f.coeff i... | [
"K : Type u_1\ninst✝ : NormedField K\nf g : K[X]\nε : ℝ\nhε : 0 < ε\na : K\nha : eval a f = 0\nhfm : f.Monic\nhgm : g.Monic\nhdeg : g.natDegree = f.natDegree\nhcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε\nhg : g.Splits\nthis :\n ‖∑ i ∈ Finset.range (g.natDegree + 1), eval a (C (g.coeff i - f.coeff i) * X ^ i)‖ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Approximation | {
"line": 106,
"column": 8
} | {
"line": 106,
"column": 31
} | {
"line": 106,
"column": 32
} | [
{
"pp": "case convert_3.h₁\nK : Type u_1\ninst✝ : NormedField K\nf g : K[X]\nε : ℝ\nhε : 0 < ε\na : K\nha : eval a f = 0\nhfm : f.Monic\nhgm : g.Monic\nhdeg : g.natDegree = f.natDegree\nhcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε\nhg : g.Splits\ni : ℕ\nhi : i < f.natDegree + 1\n⊢ ‖g.coeff i - f.coeff i‖ < ε... | [
"case convert_3.h₁\nK : Type u_1\ninst✝ : NormedField K\nf g : K[X]\nε : ℝ\nhε : 0 < ε\na : K\nha : eval a f = 0\nhfm : f.Monic\nhgm : g.Monic\nhdeg : g.natDegree = f.natDegree\nhcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε\nhg : g.Splits\ni : ℕ\nhi : i < f.natDegree + 1\n⊢ ‖g.coeff i - f.coeff i‖ < ε"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Approximation | {
"line": 119,
"column": 40
} | {
"line": 119,
"column": 51
} | {
"line": 119,
"column": 52
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝² : NormedField K\ninst✝¹ : NormedField L\ninst✝ : NormedAlgebra K L\nf g : K[X]\nε : ℝ\nhε : 0 < ε\na : L\nha : (aeval a) f = 0\nhfm : f.Monic\nhgm : g.Monic\nhdeg : g.natDegree = f.natDegree\nhcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε\nhg : (map (algebraMap K L) ... | [
"K : Type u_1\nL : Type u_2\ninst✝² : NormedField K\ninst✝¹ : NormedField L\ninst✝ : NormedAlgebra K L\nf g : K[X]\nε : ℝ\nhε : 0 < ε\na : L\nha : (aeval a) f = 0\nhfm : f.Monic\nhgm : g.Monic\nhdeg : g.natDegree = f.natDegree\nhcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε\nhg : (map (algebraMap K L) g).Splits\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Approximation | {
"line": 120,
"column": 10
} | {
"line": 120,
"column": 21
} | {
"line": 120,
"column": 22
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝² : NormedField K\ninst✝¹ : NormedField L\ninst✝ : NormedAlgebra K L\nf g : K[X]\nε : ℝ\nhε : 0 < ε\na : L\nha : (aeval a) f = 0\nhfm : f.Monic\nhgm : g.Monic\nhdeg : g.natDegree = f.natDegree\nhcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε\nhg : (map (algebraMap K L) ... | [
"K : Type u_1\nL : Type u_2\ninst✝² : NormedField K\ninst✝¹ : NormedField L\ninst✝ : NormedAlgebra K L\nf g : K[X]\nε : ℝ\nhε : 0 < ε\na : L\nha : (aeval a) f = 0\nhfm : f.Monic\nhgm : g.Monic\nhdeg : g.natDegree = f.natDegree\nhcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε\nhg : (map (algebraMap K L) g).Splits\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Approximation | {
"line": 120,
"column": 32
} | {
"line": 120,
"column": 55
} | {
"line": 120,
"column": 56
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝² : NormedField K\ninst✝¹ : NormedField L\ninst✝ : NormedAlgebra K L\nf g : K[X]\nε : ℝ\nhε : 0 < ε\na : L\nha : (aeval a) f = 0\nhfm : f.Monic\nhgm : g.Monic\nhdeg : g.natDegree = f.natDegree\nhcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε\nhg : (map (algebraMap K L) ... | [
"K : Type u_1\nL : Type u_2\ninst✝² : NormedField K\ninst✝¹ : NormedField L\ninst✝ : NormedAlgebra K L\nf g : K[X]\nε : ℝ\nhε : 0 < ε\na : L\nha : (aeval a) f = 0\nhfm : f.Monic\nhgm : g.Monic\nhdeg : g.natDegree = f.natDegree\nhcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε\nhg : (map (algebraMap K L) g).Splits\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Approximation | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 13
} | {
"line": 122,
"column": 14
} | [
{
"pp": "case right\nK : Type u_1\nL : Type u_2\ninst✝² : NormedField K\ninst✝¹ : NormedField L\ninst✝ : NormedAlgebra K L\nf g : K[X]\nε : ℝ\nhε : 0 < ε\na : L\nha : (aeval a) f = 0\nhfm : f.Monic\nhgm : g.Monic\nhdeg : g.natDegree = f.natDegree\nhcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε\nhg : (map (alge... | [
"case right\nK : Type u_1\nL : Type u_2\ninst✝² : NormedField K\ninst✝¹ : NormedField L\ninst✝ : NormedAlgebra K L\nf g : K[X]\nε : ℝ\nhε : 0 < ε\na : L\nha : (aeval a) f = 0\nhfm : f.Monic\nhgm : g.Monic\nhdeg : g.natDegree = f.natDegree\nhcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε\nhg : (map (algebraMap K L) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Approximation | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 16
} | {
"line": 124,
"column": 0
} | [
{
"pp": "case right\nK : Type u_1\nL : Type u_2\ninst✝² : NormedField K\ninst✝¹ : NormedField L\ninst✝ : NormedAlgebra K L\nf g : K[X]\nε : ℝ\nhε : 0 < ε\na : L\nha : (aeval a) f = 0\nhfm : f.Monic\nhgm : g.Monic\nhdeg : g.natDegree = f.natDegree\nhcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε\nhg : (map (alge... | [] | simpa using h2 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.