module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.SpecialFunctions.Gamma.Deligne | {
"line": 89,
"column": 28
} | {
"line": 89,
"column": 35
} | {
"line": 89,
"column": 36
} | [
{
"pp": "s : ℂ\n| s.Gammaℝ⁻¹",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"instHDiv",
"Real.pi",
"HMul.hMul",
"Complex.Gammaℝ.eq_1",
"congrArg",
"Complex.Gammaℝ",
"Complex.instPow",
"Complex.instDivInvMonoid",
"Complex.instMul",
... | [
"s : ℂ\n| (↑π ^ (-s / 2) * Gamma (s / 2))⁻¹"
] | Gammaℝ, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Deligne | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 9
} | {
"line": 110,
"column": 0
} | [
{
"pp": "h : Tendsto (fun z ↦ z / 2 * Gamma (z / 2)) (𝓝[≠] 0) (𝓝 1)\nh' : Tendsto (fun s ↦ 2 * ↑π ^ (-s / 2)) (𝓝[≠] 0) (𝓝 2)\nz : ℂ\n⊢ z * (↑π ^ (-z / 2) * Gamma (z / 2)) = 2 * ↑π ^ (-z / 2) * (z / 2 * Gamma (z / 2))",
"ppTerm": "?m.450",
"assigned": true,
"usedConstants": [
"Mathlib.Tacti... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.SpecialFunctions.Gamma.Deligne | {
"line": 118,
"column": 90
} | {
"line": 118,
"column": 97
} | {
"line": 119,
"column": 2
} | [
{
"pp": "s : ℂ\n⊢ ↑π ^ (-s / 2) * Gamma (s / 2) * (↑π ^ (-(s + 1) / 2) * Gamma ((s + 1) / 2)) =\n ↑π ^ (-s / 2) * ↑π ^ (-(s + 1) / 2) * (Gamma (s / 2) * Gamma (s / 2 + 1 / 2))",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.SpecialFunctions.Gamma.Deligne | {
"line": 118,
"column": 90
} | {
"line": 118,
"column": 97
} | {
"line": 119,
"column": 2
} | [
{
"pp": "s : ℂ\n⊢ ↑π ^ (-s / 2) * Gamma (s / 2) * (↑π ^ (-(s + 1) / 2) * Gamma ((s + 1) / 2)) =\n ↑π ^ (-s / 2) * ↑π ^ (-(s + 1) / 2) * (Gamma (s / 2) * Gamma (s / 2 + 1 / 2))",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.... | [] | ring_nf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Gamma.Deligne | {
"line": 118,
"column": 90
} | {
"line": 118,
"column": 97
} | {
"line": 119,
"column": 2
} | [
{
"pp": "s : ℂ\n⊢ ↑π ^ (-s / 2) * Gamma (s / 2) * (↑π ^ (-(s + 1) / 2) * Gamma ((s + 1) / 2)) =\n ↑π ^ (-s / 2) * ↑π ^ (-(s + 1) / 2) * (Gamma (s / 2) * Gamma (s / 2 + 1 / 2))",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.... | [] | ring_nf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Gamma.Deligne | {
"line": 122,
"column": 4
} | {
"line": 122,
"column": 11
} | {
"line": 123,
"column": 2
} | [
{
"pp": "s : ℂ\n⊢ ↑π ^ (-s / 2 + -(s + 1) / 2) * (Gamma (2 * (s / 2)) * 2 ^ (1 - 2 * (s / 2)) * ↑π ^ (1 / 2)) =\n 2 ^ (1 - s) * (↑π ^ (-1 / 2 - s) * ↑π ^ (1 / 2)) * Gamma s",
"ppTerm": "?m.227",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Ta... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.SpecialFunctions.Gamma.Deligne | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 11
} | {
"line": 127,
"column": 2
} | [
{
"pp": "s : ℂ\n⊢ 2 * 2 ^ (-s) * ↑π ^ (-1 / 2 - s + 1 / 2) * Gamma s = 2 * (2 ^ (-s) * ↑π ^ (-s)) * Gamma s",
"ppTerm": "?m.252",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNe... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.SpecialFunctions.Gamma.Deligne | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 11
} | {
"line": 138,
"column": 2
} | [
{
"pp": "s : ℂ\n⊢ ↑π ^ (-(1 - s) / 2) * Gamma ((1 - s) / 2) * (↑π ^ (-(1 + s) / 2) * Gamma ((1 + s) / 2)) =\n ↑π ^ ((s - 1) / 2) * ↑π ^ ((-1 - s) / 2) * (Gamma ((1 - s) / 2) * Gamma (1 - (1 - s) / 2))",
"ppTerm": "?m.216",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.SpecialFunctions.Gamma.Deligne | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 11
} | {
"line": 141,
"column": 2
} | [
{
"pp": "s : ℂ\n⊢ ↑π ^ ((s - 1) / 2) * ↑π ^ ((-1 - s) / 2) * (↑π / sin (↑π * ((1 - s) / 2))) =\n ↑π ^ ((s - 1) / 2) * ↑π ^ ((-1 - s) / 2) * ↑π / sin (↑π / 2 - ↑π * s / 2)",
"ppTerm": "?m.220",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tact... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.SpecialFunctions.Gamma.Deligne | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 11
} | {
"line": 145,
"column": 4
} | [
{
"pp": "s : ℂ\n⊢ ↑π ^ ((s - 1) / 2 + (-1 - s) / 2 + 1) / cos (↑π * s / 2) = (cos (↑π * s / 2))⁻¹",
"ppTerm": "?m.222",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.RingNF.nnrat_rawCast",
"Mathlib.Tactic.Ring.Common.neg_zero",
... | [
"s : ℂ\n⊢ ↑π ^ 0 * (cos (s * ↑π * (1 / 2)))⁻¹ = (cos (s * ↑π * (1 / 2)))⁻¹"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.SpecialFunctions.Gamma.Deligne | {
"line": 181,
"column": 52
} | {
"line": 181,
"column": 59
} | {
"line": 182,
"column": 4
} | [
{
"pp": "case pos\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ -↑n\nh : s = 1\n⊢ 1⁻¹ = Gammaℂ 1 * sin (↑π * 1 / 2) * (1 + 1).Gammaℝ⁻¹",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"Real.pi",
"HMul.hMul",
"Complex.Gammaℂ",
"Complex.Gammaℝ.eq_1",
... | [
"case pos\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ -↑n\nh : s = 1\n⊢ 1⁻¹ = Gammaℂ 1 * sin (↑π * 1 / 2) * (↑π ^ (-(1 + 1) / 2) * Gamma ((1 + 1) / 2))⁻¹"
] | Gammaℝ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Harmonic.GammaDeriv | {
"line": 134,
"column": 57
} | {
"line": 134,
"column": 64
} | {
"line": 134,
"column": 64
} | [
{
"pp": "case e_a.e_a.e_a\nh_diff : ∀ {s : ℝ}, 0 < s → DifferentiableAt ℝ Gamma s\nh_diff' : ∀ {s : ℝ}, 0 < s → DifferentiableAt ℝ (fun s ↦ Gamma (2 * s)) s\n⊢ -(2 * γ) = -2 * γ",
"ppTerm": "?e_a.e_a.e_a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"NonUnitalCommRing.t... | [
"case e_a.e_a.e_a\nh_diff : ∀ {s : ℝ}, 0 < s → DifferentiableAt ℝ Gamma s\nh_diff' : ∀ {s : ℝ}, 0 < s → DifferentiableAt ℝ (fun s ↦ Gamma (2 * s)) s\n⊢ -(2 * γ) = -(2 * γ)"
] | neg_mul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Digamma | {
"line": 48,
"column": 19
} | {
"line": 48,
"column": 34
} | {
"line": 48,
"column": 35
} | [
{
"pp": "⊢ logDeriv Gamma 1 = -↑Real.eulerMascheroniConstant",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"logDeriv",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"instHDiv",
"congrArg",
"deriv",
"NormedSpace.toModule",
"PseudoMetricSpa... | [
"⊢ deriv Gamma 1 / Gamma 1 = -↑Real.eulerMascheroniConstant"
] | logDeriv_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Digamma | {
"line": 51,
"column": 19
} | {
"line": 51,
"column": 34
} | {
"line": 51,
"column": 35
} | [
{
"pp": "⊢ logDeriv Gamma (1 / 2) = -2 * log 2 - ↑Real.eulerMascheroniConstant",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"logDeriv",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Complex.log",
"instHDiv",
"HMul.hMul",
"congrArg",
"d... | [
"⊢ deriv Gamma (1 / 2) / Gamma (1 / 2) = -2 * log 2 - ↑Real.eulerMascheroniConstant"
] | logDeriv_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Digamma | {
"line": 58,
"column": 19
} | {
"line": 58,
"column": 34
} | {
"line": 58,
"column": 35
} | [
{
"pp": "s : ℂ\nhs : ∀ (m : ℕ), s ≠ -↑m\nhs0 : s ≠ 0\n⊢ logDeriv Gamma (s + 1) = logDeriv Gamma s + s⁻¹",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"logDeriv",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"instHDiv",
"congrArg",
"deriv",
"N... | [
"s : ℂ\nhs : ∀ (m : ℕ), s ≠ -↑m\nhs0 : s ≠ 0\n⊢ deriv Gamma (s + 1) / Gamma (s + 1) = logDeriv Gamma s + s⁻¹"
] | logDeriv_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Digamma | {
"line": 58,
"column": 35
} | {
"line": 58,
"column": 50
} | {
"line": 58,
"column": 51
} | [
{
"pp": "s : ℂ\nhs : ∀ (m : ℕ), s ≠ -↑m\nhs0 : s ≠ 0\n⊢ deriv Gamma (s + 1) / Gamma (s + 1) = logDeriv Gamma s + s⁻¹",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"logDeriv",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"instHDiv",
"congrArg",
"der... | [
"s : ℂ\nhs : ∀ (m : ℕ), s ≠ -↑m\nhs0 : s ≠ 0\n⊢ deriv Gamma (s + 1) / Gamma (s + 1) = deriv Gamma s / Gamma s + s⁻¹"
] | logDeriv_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Harmonic.GammaDeriv | {
"line": 108,
"column": 7
} | {
"line": 146,
"column": 38
} | {
"line": 148,
"column": 0
} | [] | [] | deriv Gamma (1 / 2)
_ = (deriv (fun s ↦ Gamma s * Gamma (s + 1 / 2)) (1 / 2)) + √π * γ := by
rw [deriv_fun_mul, Gamma_one_half_eq,
add_assoc, ← mul_add, deriv_comp_add_const,
(by norm_num : 1 / 2 + 1 / 2 = (1 : ℝ)), Gamma_one, mul_one,
eulerMascheroniConstant_eq_neg_deriv, add_neg_cancel, mul_ze... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation | {
"line": 45,
"column": 12
} | {
"line": 45,
"column": 19
} | {
"line": 46,
"column": 2
} | [
{
"pp": "a : ℝ\nha : a < 0\nb s x : ℝ\n⊢ -x - (a * x ^ 2 + b * x) = x * (-a * x - (b + 1))",
"ppTerm": "?m.212",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAss... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation | {
"line": 54,
"column": 11
} | {
"line": 54,
"column": 28
} | {
"line": 54,
"column": 29
} | [
{
"pp": "a : ℂ\nha : a.re < 0\nb : ℂ\ns x : ℝ\n⊢ ‖cexp (a * ↑x ^ 2 + b * ↑x)‖ = rexp (a.re * x ^ 2 + b.re * x)",
"ppTerm": "?m.128",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"HMul.hMul",
"congrArg",... | [
"a : ℂ\nha : a.re < 0\nb : ℂ\ns x : ℝ\n⊢ rexp (a * ↑x ^ 2 + b * ↑x).re = rexp (a.re * x ^ 2 + b.re * x)"
] | Complex.norm_exp, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation | {
"line": 59,
"column": 2
} | {
"line": 59,
"column": 46
} | {
"line": 60,
"column": 2
} | [
{
"pp": "a : ℂ\nha : a.re < 0\nb : ℂ\ns : ℝ\n⊢ (fun x ↦ cexp (a * ↑x ^ 2 + b * ↑x)) =o[cocompact ℝ] fun x ↦ |x| ^ s",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"Real.instPow",
"Real.partialOrder",
"ConditionallyCompleteL... | [
"a : ℂ\nha : a.re < 0\nb : ℂ\ns : ℝ\n⊢ ((fun x ↦ cexp (a * ↑x ^ 2 + b * ↑x)) =o[atBot] fun x ↦ |x| ^ s) ∧\n (fun x ↦ cexp (a * ↑x ^ 2 + b * ↑x)) =o[atTop] fun x ↦ |x| ^ s"
] | rw [cocompact_eq_atBot_atTop, isLittleO_sup] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation | {
"line": 108,
"column": 15
} | {
"line": 108,
"column": 22
} | {
"line": 108,
"column": 22
} | [
{
"pp": "a : ℂ\nha : 0 < a.re\nb : ℂ\nf : ℝ → ℂ := fun x ↦ cexp (-↑π * a * ↑x ^ 2 + 2 * ↑π * b * ↑x)\nhFf : 𝓕 f = fun x ↦ 1 / a ^ (1 / 2) * cexp (-↑π / a * (↑x + I * b) ^ 2)\nh1 : 0 < (↑π * a).re\nh2 : 0 < (↑π / a).re\nf_bd : f =O[cocompact ℝ] fun x ↦ |x| ^ (-2)\nx : ℝ\n⊢ -↑π / a * (↑x + I * b) ^ 2 = -↑π / a *... | [
"a : ℂ\nha : 0 < a.re\nb : ℂ\nf : ℝ → ℂ := fun x ↦ cexp (-↑π * a * ↑x ^ 2 + 2 * ↑π * b * ↑x)\nhFf : 𝓕 f = fun x ↦ 1 / a ^ (1 / 2) * cexp (-↑π / a * (↑x + I * b) ^ 2)\nh1 : 0 < (↑π * a).re\nh2 : 0 < (↑π / a).re\nf_bd : f =O[cocompact ℝ] fun x ↦ |x| ^ (-2)\nx : ℝ\n⊢ -(↑π * a⁻¹ * ↑x * I * b * 2) - ↑π * a⁻¹ * ↑x ^ 2 -... | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 243,
"column": 6
} | {
"line": 247,
"column": 25
} | {
"line": 248,
"column": 4
} | [
{
"pp": "p x : ℝ\nhp : p ∈ Ioo 0 1\nhx : 0 ≤ x\n⊢ HasFiniteIntegral (fun t ↦ t ^ (p - 1)) (volume.restrict (Ioo 0 1))",
"ppTerm": "?m.111",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
... | [] | apply Integrable.hasFiniteIntegral
rw [Set.mem_Ioo] at hp
rw [← IntegrableOn, intervalIntegral.integrableOn_Ioo_rpow_iff]
· linarith
· exact zero_lt_one | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 243,
"column": 6
} | {
"line": 247,
"column": 25
} | {
"line": 248,
"column": 4
} | [
{
"pp": "p x : ℝ\nhp : p ∈ Ioo 0 1\nhx : 0 ≤ x\n⊢ HasFiniteIntegral (fun t ↦ t ^ (p - 1)) (volume.restrict (Ioo 0 1))",
"ppTerm": "?m.111",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
... | [] | apply Integrable.hasFiniteIntegral
rw [Set.mem_Ioo] at hp
rw [← IntegrableOn, intervalIntegral.integrableOn_Ioo_rpow_iff]
· linarith
· exact zero_lt_one | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Int.Log | {
"line": 240,
"column": 4
} | {
"line": 240,
"column": 46
} | {
"line": 241,
"column": 2
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝³ : Semifield R\ninst✝² : LinearOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\nhr : 0 < r\n⊢ ↑b ^ log b r⁻¹ ≤ r⁻¹",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"GroupWithZero.toMonoid... | [] | exact zpow_log_le_self hb (inv_pos.mpr hr) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Int.Log | {
"line": 240,
"column": 4
} | {
"line": 240,
"column": 46
} | {
"line": 241,
"column": 2
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝³ : Semifield R\ninst✝² : LinearOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\nhr : 0 < r\n⊢ ↑b ^ log b r⁻¹ ≤ r⁻¹",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"GroupWithZero.toMonoid... | [] | exact zpow_log_le_self hb (inv_pos.mpr hr) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Int.Log | {
"line": 240,
"column": 4
} | {
"line": 240,
"column": 46
} | {
"line": 241,
"column": 2
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝³ : Semifield R\ninst✝² : LinearOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\nhr : 0 < r\n⊢ ↑b ^ log b r⁻¹ ≤ r⁻¹",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"GroupWithZero.toMonoid... | [] | exact zpow_log_le_self hb (inv_pos.mpr hr) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 219,
"column": 6
} | {
"line": 219,
"column": 19
} | {
"line": 219,
"column": 19
} | [
{
"pp": "b x : ℝ\nhb : 1 < b\nhx : 0 < x\n⊢ 0 < logb b x ↔ 1 < x",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.logb_one",
"Real",
"Real.instZero",
"congrArg",
"Real.instLT",
"id",
"Real.instOne",
"Real.logb",
"If... | [
"b x : ℝ\nhb : 1 < b\nhx : 0 < x\n⊢ logb b 1 < logb b x ↔ 1 < x"
] | ← @logb_one b | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Int.Log | {
"line": 270,
"column": 2
} | {
"line": 270,
"column": 62
} | {
"line": 272,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semifield R\ninst✝² : LinearOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nz : ℤ\n⊢ clog b (↑b ^ z) = z",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | rw [← neg_log_inv_eq_clog, ← zpow_neg, log_zpow hb, neg_neg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Int.Log | {
"line": 270,
"column": 2
} | {
"line": 270,
"column": 62
} | {
"line": 272,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semifield R\ninst✝² : LinearOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nz : ℤ\n⊢ clog b (↑b ^ z) = z",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | rw [← neg_log_inv_eq_clog, ← zpow_neg, log_zpow hb, neg_neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Int.Log | {
"line": 270,
"column": 2
} | {
"line": 270,
"column": 62
} | {
"line": 272,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semifield R\ninst✝² : LinearOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nz : ℤ\n⊢ clog b (↑b ^ z) = z",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | rw [← neg_log_inv_eq_clog, ← zpow_neg, log_zpow hb, neg_neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 312,
"column": 12
} | {
"line": 312,
"column": 31
} | {
"line": 312,
"column": 32
} | [
{
"pp": "p : ℝ\nhp : 0 < p ∧ p < 1\n⊢ 1 / 2 * -1 ^ (p - 1) / (p - 1) = ∫ (t : ℝ) in Ioi 1, 1 / 2 * t ^ (p - 2)",
"ppTerm": "?m.206",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"Real.instPow",
... | [
"p : ℝ\nhp : 0 < p ∧ p < 1\n⊢ 1 / 2 * -1 ^ (p - 1) / (p - 1) = 1 / 2 * ∫ (a : ℝ) in Ioi 1, a ^ (p - 2)"
] | integral_const_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 313,
"column": 8
} | {
"line": 313,
"column": 15
} | {
"line": 314,
"column": 2
} | [
{
"pp": "p : ℝ\nhp : 0 < p ∧ p < 1\n⊢ 1 / 2 * -1 ^ (p - 1) / (p - 1) = 1 / 2 * (-1 ^ (p - 2 + 1) / (p - 2 + 1))",
"ppTerm": "?m.233",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 303,
"column": 61
} | {
"line": 303,
"column": 99
} | {
"line": 303,
"column": 99
} | [
{
"pp": "b x y : ℝ\nb_pos : 0 < b\nb_lt_one : b < 1\nhy : 0 < y\n⊢ b ^ logb b y ≤ b ^ x ↔ y ≤ b ^ x",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real.instLE",
"Real",
"congrArg",
"id",
"LE.le",
"Real.rpow_logb",
... | [
"b x y : ℝ\nb_pos : 0 < b\nb_lt_one : b < 1\nhy : 0 < y\n⊢ y ≤ b ^ x ↔ y ≤ b ^ x"
] | rpow_logb b_pos (b_ne_one b_lt_one) hy | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 306,
"column": 61
} | {
"line": 306,
"column": 99
} | {
"line": 306,
"column": 99
} | [
{
"pp": "b x y : ℝ\nb_pos : 0 < b\nb_lt_one : b < 1\nhy : 0 < y\n⊢ b ^ logb b y < b ^ x ↔ y < b ^ x",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"congrArg",
"Real.instLT",
"id",
"Real.rpow_logb",
"Real.log... | [
"b x y : ℝ\nb_pos : 0 < b\nb_lt_one : b < 1\nhy : 0 < y\n⊢ y < b ^ x ↔ y < b ^ x"
] | rpow_logb b_pos (b_ne_one b_lt_one) hy | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.MulExpNegMulSq | {
"line": 143,
"column": 33
} | {
"line": 143,
"column": 91
} | {
"line": 143,
"column": 91
} | [
{
"pp": "ε : ℝ\nhε : 0 < ε\nx y : ℝ\nh : edist (ε.mulExpNegMulSq x) (ε.mulExpNegMulSq y) ≤ edist x y\n⊢ dist (ε.mulExpNegMulSq x) (ε.mulExpNegMulSq y) ≤ dist x y",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Real.instLE",
"Real",
"congrArg",
"Eq.mp",
"LE.le... | [
"ε : ℝ\nhε : 0 < ε\nx y : ℝ\nh : (edist (ε.mulExpNegMulSq x) (ε.mulExpNegMulSq y)).toReal ≤ (edist x y).toReal\n⊢ dist (ε.mulExpNegMulSq x) (ε.mulExpNegMulSq y) ≤ dist x y"
] | ← (toReal_le_toReal (edist_ne_top _ _) (edist_ne_top _ _)) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 504,
"column": 8
} | {
"line": 504,
"column": 84
} | {
"line": 504,
"column": 84
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : NonUnitalNormedRing A\ninst✝⁸ : StarRing A\ninst✝⁷ : NormedSpace ℝ A\ninst✝⁶ : SMulCommClass ℝ A A\ninst✝⁵ : IsScalarTower ℝ A A\ninst✝⁴ : PartialOrder A\ninst✝³ : StarOrderedRing A\ninst✝² : NonnegSpectrumClass ℝ A\ninst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoin... | [
"A : Type u_1\ninst✝⁹ : NonUnitalNormedRing A\ninst✝⁸ : StarRing A\ninst✝⁷ : NormedSpace ℝ A\ninst✝⁶ : SMulCommClass ℝ A A\ninst✝⁵ : IsScalarTower ℝ A A\ninst✝⁴ : PartialOrder A\ninst✝³ : StarOrderedRing A\ninst✝² : NonnegSpectrumClass ℝ A\ninst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : C... | Real.norm_of_nonneg (rpowIntegrand₀₁_nonneg p_pos (le_of_lt ht) maxr_nonneg) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas | {
"line": 128,
"column": 69
} | {
"line": 138,
"column": 14
} | {
"line": 140,
"column": 0
} | [
{
"pp": "n a b : ℕ\nhn : n ≠ 0\n⊢ a ≤ n.nthRoot b ↔ a ^ n ≤ b",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Preorder.toLT",
"eq_false",
"congrArg",
"Nat.instMonoid",
"PartialOrder.toPreorder",
"Preorder.toLE",
"tru... | [] | by
cases le_or_gt a (nthRoot n b) with
| inl hle =>
simp only [hle, true_iff]
refine le_trans ?_ (pow_nthRoot_le (.inl hn))
gcongr
| inr hlt =>
simp only [hlt.not_ge, false_iff, not_le]
refine (lt_pow_nthRoot_add_one hn b).trans_le ?_
gcongr
assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 93,
"column": 7
} | {
"line": 93,
"column": 36
} | {
"line": 93,
"column": 36
} | [
{
"pp": "L : PeriodPair\nα β : ℚ\nH : ↑α * L.ω₁ + ↑β * L.ω₂ ∈ L.lattice\nm n : ℤ\ne : ↑m * L.ω₁ + ↑n * L.ω₂ = ↑α * L.ω₁ + ↑β * L.ω₂\n⊢ (↑m - ↑α) • L.ω₁ + (↑n - ↑β) • L.ω₂ = 0",
"ppTerm": "?m.84",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Mathlib.Tactic.Rin... | [] | by simp; linear_combination e | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 202,
"column": 8
} | {
"line": 202,
"column": 22
} | {
"line": 202,
"column": 22
} | [
{
"pp": "case e_a\nr : ℝ\nhr : 0 < r\ns : ℂ\nhs : ‖s‖ < r\nl : ℂ\nh : 2 * r ≤ ‖l‖\nthis✝ : s ≠ l\nthis : 0 < ‖l‖\n⊢ ‖l ^ 2 - (s - l) ^ 2‖ = ‖s * (2 * l - s)‖",
"ppTerm": "?e_a✝",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"HMul.hMul",
"Complex.c... | [
"case e_a\nr : ℝ\nhr : 0 < r\ns : ℂ\nhs : ‖s‖ < r\nl : ℂ\nh : 2 * r ≤ ‖l‖\nthis✝ : s ≠ l\nthis : 0 < ‖l‖\n⊢ ‖(l + (s - l)) * (l - (s - l))‖ = ‖s * (2 * l - s)‖"
] | rw [sq_sub_sq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Constructions.HaarToSphere | {
"line": 293,
"column": 86
} | {
"line": 294,
"column": 76
} | {
"line": 296,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : Nontrivial E\nμ : Measure E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : BorelSpace E\ninst✝ : μ.IsAddHaarMeasure\nf : ℝ → F\nr... | [] | by
rw [← integrable_indicator_iff measurableSet_Ioo, ← integrableOn_univ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Basic | {
"line": 64,
"column": 4
} | {
"line": 64,
"column": 11
} | {
"line": 65,
"column": 2
} | [
{
"pp": "case add_two\nθ : ℂ\nn : ℕ\nih1 : eval (cos θ) (T ℂ (↑n + 1)) = cos (↑(↑n + 1) * θ)\nih2 : eval (cos θ) (T ℂ ↑n) = cos (↑↑n * θ)\n⊢ 2 * cos θ * cos ((↑n + 1) * θ) = 2 * cos (((↑n + 2) * θ + ↑n * θ) / 2) * cos (((↑n + 2) * θ - ↑n * θ) / 2)",
"ppTerm": "?add_two",
"assigned": true,
"usedConst... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Basic | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 11
} | {
"line": 71,
"column": 0
} | [
{
"pp": "case neg_add_one\nθ : ℂ\nn : ℕ\nih1 : eval (cos θ) (T ℂ (-↑n)) = cos (↑(-↑n) * θ)\nih2 : eval (cos θ) (T ℂ (-↑n + 1)) = cos (↑(-↑n + 1) * θ)\n⊢ 2 * cos θ * cos (-↑n * θ) = 2 * cos (((-↑n + 1) * θ + (-↑n - 1) * θ) / 2) * cos (((-↑n + 1) * θ - (-↑n - 1) * θ) / 2)",
"ppTerm": "?neg_add_one",
"assi... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Basic | {
"line": 82,
"column": 4
} | {
"line": 82,
"column": 11
} | {
"line": 83,
"column": 2
} | [
{
"pp": "case add_two\nθ : ℂ\nn : ℕ\nih1 : eval (cos θ) (U ℂ (↑n + 1)) * sin θ = sin ((↑(↑n + 1) + 1) * θ)\nih2 : eval (cos θ) (U ℂ ↑n) * sin θ = sin ((↑↑n + 1) * θ)\n⊢ 2 * (cos θ * sin ((↑n + 1 + 1) * θ)) =\n 2 * (sin (((↑n + 2 + 1) * θ + (↑n + 1) * θ) / 2) * cos (((↑n + 2 + 1) * θ - (↑n + 1) * θ) / 2))",
... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Basic | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 11
} | {
"line": 89,
"column": 0
} | [
{
"pp": "case neg_add_one\nθ : ℂ\nn : ℕ\nih1 : eval (cos θ) (U ℂ (-↑n)) * sin θ = sin ((↑(-↑n) + 1) * θ)\nih2 : eval (cos θ) (U ℂ (-↑n + 1)) * sin θ = sin ((↑(-↑n + 1) + 1) * θ)\n⊢ 2 * (cos θ * sin ((-↑n + 1) * θ)) =\n 2 * (sin (((-↑n + 1 + 1) * θ + (-↑n - 1 + 1) * θ) / 2) * cos (((-↑n + 1 + 1) * θ - (-↑n - ... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.SpecialFunctions.Stirling | {
"line": 222,
"column": 2
} | {
"line": 222,
"column": 9
} | {
"line": 224,
"column": 0
} | [
{
"pp": "n : ℕ\nhn : n ≠ 0\nthis : 4 = 2 * 2\n⊢ rexp (4 * ↑n) * ((2 * ↑n) ^ (2 * n)) ^ 2 = (↑n ^ n) ^ 4 * rexp (2 * (2 * ↑n)) * 2 ^ (4 * n)",
"ppTerm": "?m.96",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero"... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Orthogonality | {
"line": 111,
"column": 14
} | {
"line": 111,
"column": 91
} | {
"line": 111,
"column": 91
} | [
{
"pp": "n : ℤ\nhn : n ≠ 0\nhn' : ↑n ≠ 0\nh✝ : 0 ≤ n\n⊢ ∫ (θ : ℝ) in 0..↑n * π, deriv sin θ = 0",
"ppTerm": "?m.124",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.instIsOrderedRing",
"Int.cast",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real.partialOr... | [
"n : ℤ\nhn : n ≠ 0\nhn' : ↑n ≠ 0\nh✝ : 0 ≤ n\n⊢ sin (↑n * π) - sin 0 = 0"
] | rw [integral_deriv_of_contDiffOn_Icc contDiff_sin.contDiffOn (by positivity)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Extremal | {
"line": 90,
"column": 2
} | {
"line": 102,
"column": 21
} | {
"line": 104,
"column": 0
} | [
{
"pp": "n i : ℕ\nhi : i ≤ n\n⊢ 0 < (-1) ^ i * ∏ j ∈ (Finset.range (n + 1)).erase i, (node n i - node n j)",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"sub_neg",
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"Iff.mpr",
"sub_pos",
"AddGroup.toSubtr... | [] | rcases eq_or_ne n 0 with rfl | hn
· simp [Nat.le_zero.mp hi]
have h₁ : 0 < ∏ j ∈ Finset.range i, ((-1) * (node n i - node n j)) :=
Finset.prod_pos (fun j hj => mul_pos_of_neg_of_neg neg_one_lt_zero <| sub_neg.mpr <|
node_lt hi (Finset.mem_range.mp hj))
rw [Finset.prod_mul_distrib, Finset.prod_const, Finse... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Extremal | {
"line": 90,
"column": 2
} | {
"line": 102,
"column": 21
} | {
"line": 104,
"column": 0
} | [
{
"pp": "n i : ℕ\nhi : i ≤ n\n⊢ 0 < (-1) ^ i * ∏ j ∈ (Finset.range (n + 1)).erase i, (node n i - node n j)",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"sub_neg",
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"Iff.mpr",
"sub_pos",
"AddGroup.toSubtr... | [] | rcases eq_or_ne n 0 with rfl | hn
· simp [Nat.le_zero.mp hi]
have h₁ : 0 < ∏ j ∈ Finset.range i, ((-1) * (node n i - node n j)) :=
Finset.prod_pos (fun j hj => mul_pos_of_neg_of_neg neg_one_lt_zero <| sub_neg.mpr <|
node_lt hi (Finset.mem_range.mp hj))
rw [Finset.prod_mul_distrib, Finset.prod_const, Finse... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Niven | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 52
} | {
"line": 99,
"column": 0
} | [
{
"pp": "q : ℚ\n⊢ IsIntegral ℤ ((algebraMap ℝ ℂ) (2 * sin (↑q * π)))",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"Real.pi",
"RCLike.toNormedAlgebra",
"HMul.hMul",
"Algebra.algebraMap",
"congrArg",
... | [] | simp [Complex.isIntegral_two_mul_sin_rat_mul_pi] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.Niven | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 52
} | {
"line": 99,
"column": 0
} | [
{
"pp": "q : ℚ\n⊢ IsIntegral ℤ ((algebraMap ℝ ℂ) (2 * sin (↑q * π)))",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"Real.pi",
"RCLike.toNormedAlgebra",
"HMul.hMul",
"Algebra.algebraMap",
"congrArg",
... | [] | simp [Complex.isIntegral_two_mul_sin_rat_mul_pi] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Niven | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 52
} | {
"line": 99,
"column": 0
} | [
{
"pp": "q : ℚ\n⊢ IsIntegral ℤ ((algebraMap ℝ ℂ) (2 * sin (↑q * π)))",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"Real.pi",
"RCLike.toNormedAlgebra",
"HMul.hMul",
"Algebra.algebraMap",
"congrArg",
... | [] | simp [Complex.isIntegral_two_mul_sin_rat_mul_pi] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 639,
"column": 76
} | {
"line": 639,
"column": 88
} | {
"line": 639,
"column": 88
} | [
{
"pp": "L : PeriodPair\nl₀ x : ℂ\ni : ℕ\nhl₀ : l₀ ∈ L.lattice\n⊢ (if i + 1 = 0 then ℘[L - l₀] x else (↑(i + 1) + 1) * (L.sumInvPow x (i + 1 + 2) - ((l₀ - x) ^ (i + 1 + 2))⁻¹)) =\n (↑i + 2) * (L.sumInvPow x (i + 3) - ((l₀ - x) ^ ↑(i + 3))⁻¹)",
"ppTerm": "?m.131",
"assigned": true,
"usedConstants"... | [
"L : PeriodPair\nl₀ x : ℂ\ni : ℕ\nhl₀ : l₀ ∈ L.lattice\n⊢ (if i + 1 = 0 then ℘[L - l₀] x else (↑(i + 1) + 1) * (L.sumInvPow x (i + 1 + 2) - ((l₀ - x) ^ (i + 1 + 2))⁻¹)) =\n (↑i + 2) * (L.sumInvPow x (i + 3) - ((l₀ - x) ^ (i + 3))⁻¹)"
] | zpow_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Extremal | {
"line": 192,
"column": 2
} | {
"line": 192,
"column": 47
} | {
"line": 194,
"column": 0
} | [
{
"pp": "n i : ℕ\nhi : i ≤ n\nthis : 0 < ((-1) ^ i * ∏ j ∈ (Finset.range (n + 1)).erase i, (node n i - node n j))⁻¹\n⊢ 0 < (-1) ^ i * leadingCoeffC n i",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"Real.partialOrder",
"Real",
"Pr... | [] | rwa [mul_inv, ← inv_pow, inv_neg_one] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 668,
"column": 4
} | {
"line": 671,
"column": 75
} | {
"line": 672,
"column": 4
} | [
{
"pp": "L : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\n⊢ ∃ κ, 1 < κ ∧ ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ * κ < ‖↑l - x‖",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"not_e... | [
"L : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\nκ : ℝ\nhκ : κ > 0\nhκ' : Metric.ball 1 κ ⊆ (fun x_1 ↦ x_1 * ‖z - x‖) ⁻¹' (↑(upperClosure (dist x '' (↑L.lattice \\ {l₀}))))ᶜ\n⊢ ∃ κ, 1 < κ ∧ ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ * κ < ‖↑l - x‖"
] | obtain ⟨κ, hκ, hκ'⟩ := Metric.isOpen_iff.mp ((continuous_mul_const ‖z - x‖).isOpen_preimage _
(isClosedMap_dist x _
(L.isClosed_of_subset_lattice (Set.sdiff_subset (t := {l₀})))).upperClosure.isOpen_compl) 1
(by simpa [Complex.dist_eq, @forall_comm ℝ, norm_sub_rev x] using hx) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 698,
"column": 6
} | {
"line": 698,
"column": 52
} | {
"line": 699,
"column": 4
} | [
{
"pp": "L : PeriodPair\nl₀ z : ℂ\nκ : ℝ\nhκ : 1 < κ\ne : ℕ × ↥L.lattice ≃ ↥L.lattice ⊕ ℕ × ↥L.lattice :=\n (Equiv.prodCongrLeft fun x ↦ (Denumerable.eqv (Option ℕ)).symm).trans optionProdEquiv\nH₁ : Summable fun i ↦ (↑i + 2) * κ ^ (-↑i)\np : ℕ × ↥L.lattice\nhp :\n ¬↑((Equiv.prodCongrLeft fun x ↦\n ... | [] | simpa [(norm_nonneg _).not_gt] using hx p.2 hp | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 703,
"column": 18
} | {
"line": 703,
"column": 25
} | {
"line": 703,
"column": 25
} | [
{
"pp": "L : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\nκ : ℝ\nhκ : 1 < κ\nhκ' : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ * κ < ‖↑l - x‖\ne : ℕ × ↥L.lattice ≃ ↥L.lattice ⊕ ℕ × ↥L.lattice :=\n (Equiv.prodCongrLeft fun x ↦ (Denumerable.eqv (Option ℕ)).symm).trans optionProdEquiv\... | [
"L : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\nκ : ℝ\nhκ : 1 < κ\nhκ' : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ * κ < ‖↑l - x‖\ne : ℕ × ↥L.lattice ≃ ↥L.lattice ⊕ ℕ × ↥L.lattice :=\n (Equiv.prodCongrLeft fun x ↦ (Denumerable.eqv (Option ℕ)).symm).trans optionProdEquiv\nH₁ : Summab... | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 171,
"column": 64
} | {
"line": 172,
"column": 44
} | {
"line": 174,
"column": 0
} | [
{
"pp": "c : ℤ\nz : ℂ\n⊢ (fun d ↦ ↑c * z + ↑d) =Θ[cofinite] fun n ↦ ↑n",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"HMul.hMul",
"AddGroupWithOn... | [] | by
simpa using linear_isTheta_right_add c 0 z | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 246,
"column": 2
} | {
"line": 246,
"column": 28
} | {
"line": 247,
"column": 2
} | [
{
"pp": "z : ℂ\nhz : z ≠ 0\nd k : ℤ\nhk : 2 ≤ k\n⊢ (fun n ↦ ((↑n * z + ↑d) ^ k)⁻¹) =O[cofinite] fun n ↦ (|↑n| ^ ↑k)⁻¹",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Int.cast",
"Real.instPow",
"Real",
"HMul.hMul",
"Real.lattice",
"DivisionCommMonoid.toD... | [
"z : ℂ\nhz : z ≠ 0\nd : ℤ\nk : ℕ\nhk : 2 ≤ ↑k\n⊢ (fun n ↦ ((↑n * z + ↑d) ^ ↑k)⁻¹) =O[cofinite] fun n ↦ (|↑n| ^ ↑↑k)⁻¹"
] | lift k to ℕ using (by lia) | Mathlib.Tactic._aux_Mathlib_Tactic_Lift___elabRules_Mathlib_Tactic_lift_1 | Mathlib.Tactic.lift |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 43,
"column": 4
} | {
"line": 43,
"column": 11
} | {
"line": 44,
"column": 2
} | [
{
"pp": "z : ℂ\n⊢ cexp (z * I) + cexp (-z * I) = cexp (-(z * I) + 2 * I * z) + cexp (-(z * I)) * 1",
"ppTerm": "?m.103",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 45,
"column": 4
} | {
"line": 45,
"column": 11
} | {
"line": 46,
"column": 4
} | [
{
"pp": "z : ℂ\nh1 : cexp (z * I) + cexp (-z * I) = cexp (-(z * I)) * (cexp (2 * I * z) + 1)\n⊢ cexp (-z * I) - cexp (z * I) = cexp (-(z * I)) * (1 - cexp (2 * I * z))",
"ppTerm": "?m.145",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Rin... | [
"z : ℂ\nh1 : cexp (z * I) + cexp (-z * I) = cexp (-(z * I)) * (cexp (2 * I * z) + 1)\n⊢ cexp (-(z * I)) - cexp (z * I) = cexp (-(z * I)) - cexp (-(z * I)) * cexp (z * I * 2)"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 47,
"column": 4
} | {
"line": 47,
"column": 11
} | {
"line": 48,
"column": 2
} | [
{
"pp": "z : ℂ\nh1 : cexp (z * I) + cexp (-z * I) = cexp (-(z * I)) * (cexp (2 * I * z) + 1)\n⊢ cexp (-(z * I)) - cexp (z * I) = cexp (-(z * I)) - cexp (-(z * I) + z * (I * 2))",
"ppTerm": "?m.157",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.T... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 9
} | {
"line": 57,
"column": 0
} | [
{
"pp": "z : ℂ\n⊢ (cexp (2 * I * (↑π * z)) + 1) / (I * (1 - cexp (2 * I * (↑π * z)))) =\n (cexp (2 * ↑π * I * z) + 1) / (I * (1 - cexp (2 * ↑π * I * z)))",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.SpecificLimits.ArithmeticGeometric | {
"line": 76,
"column": 17
} | {
"line": 76,
"column": 40
} | {
"line": 78,
"column": 0
} | [
{
"pp": "case succ\nR : Type u_1\na b u₀ : R\ninst✝ : Field R\nha : a ≠ 1\nn : ℕ\nhn : arithGeom a b u₀ n = a ^ n * (u₀ - b / (1 - a)) + b / (1 - a)\n⊢ arithGeom a b u₀ (n + 1) = a ^ (n + 1) * (u₀ - b / (1 - a)) + b / (1 - a)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | unfold arithGeom; grind | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecificLimits.ArithmeticGeometric | {
"line": 76,
"column": 17
} | {
"line": 76,
"column": 40
} | {
"line": 78,
"column": 0
} | [
{
"pp": "case succ\nR : Type u_1\na b u₀ : R\ninst✝ : Field R\nha : a ≠ 1\nn : ℕ\nhn : arithGeom a b u₀ n = a ^ n * (u₀ - b / (1 - a)) + b / (1 - a)\n⊢ arithGeom a b u₀ (n + 1) = a ^ (n + 1) * (u₀ - b / (1 - a)) + b / (1 - a)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | unfold arithGeom; grind | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 910,
"column": 4
} | {
"line": 910,
"column": 58
} | {
"line": 911,
"column": 4
} | [
{
"pp": "case pos\nL : PeriodPair\nx : ℂ\nhx : x ∈ L.lattice\n⊢ MeromorphicAt ℘[L] x",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Submodule",
"instHDiv",
"Complex.instNormedAddCommGroup",
"congrArg",
... | [
"case pos\nL : PeriodPair\nx : ℂ\nhx : x ∈ L.lattice\n⊢ MeromorphicAt (fun x_1 ↦ ℘[L - x] x_1 + (1 / (x_1 - x) ^ 2 - 1 / x ^ 2)) x"
] | simp_rw [← funext <| L.weierstrassPExcept_add ⟨x, hx⟩] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 63
} | {
"line": 165,
"column": 64
} | [
{
"pp": "x : ℂ\nhz : x ∈ ℂ_ℤ\nthis : (fun t ↦ Complex.sin (↑π * t) / (↑π * t)) = fun z ↦ (Complex.sin ∘ fun t ↦ ↑π * t) z / (↑π * z)\n⊢ logDeriv (Complex.sin ∘ fun t ↦ ↑π * t) x - logDeriv (HMul.hMul ↑π) x = ↑π * (↑π * x).cot - 1 / x",
"ppTerm": "?m.143",
"assigned": true,
"usedConstants": [
"... | [
"x : ℂ\nhz : x ∈ ℂ_ℤ\nthis : (fun t ↦ Complex.sin (↑π * t) / (↑π * t)) = fun z ↦ (Complex.sin ∘ fun t ↦ ↑π * t) z / (↑π * z)\n⊢ logDeriv Complex.sin (↑π * x) * deriv (fun t ↦ ↑π * t) x - logDeriv (HMul.hMul ↑π) x = ↑π * (↑π * x).cot - 1 / x",
"x : ℂ\nhz : x ∈ ℂ_ℤ\nthis : (fun t ↦ Complex.sin (↑π * t) / (↑π * t)) ... | logDeriv_comp (Complex.differentiableAt_sin) (by fun_prop), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema | {
"line": 345,
"column": 4
} | {
"line": 345,
"column": 34
} | {
"line": 346,
"column": 2
} | [
{
"pp": "k : ℕ\nx : ℝ\nhx : |x| ≤ 1\nn : ℕ\nthis : (⇑derivative)^[k] (T ℝ ↑n) ∈ Submodule.span ℕ ((fun m ↦ T ℝ ↑m) '' Set.Icc 0 (n - k))\nf : ℝ[X] →₀ ℕ\nhfsupp : ↑f.support ⊆ (fun m ↦ T ℝ ↑m) '' Set.Icc 0 (n - k)\nhfderiv : ∑ p ∈ f.support, f p • p = (⇑derivative)^[k] (T ℝ ↑n)\ny : ℝ\n⊢ ∑ p ∈ f.support, f p • e... | [] | simp_rw [Polynomial.eval_smul] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Data.Int.Fib.Basic | {
"line": 124,
"column": 15
} | {
"line": 124,
"column": 46
} | {
"line": 125,
"column": 2
} | [
{
"pp": "x✝ : ℕ\n⊢ fib (-↑x✝ + -↑1) = fib (-↑x✝ - 1) * fib (-↑1) + fib (-↑x✝) * fib (-↑1 + 1)",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"add_neg_cancel",
"NegZeroClass.toNeg",
"HMul.hMul",
"congrArg",
"AddMonoid.toAddZeroCla... | [] | simp [sub_eq_neg_add, add_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Int.Fib.Basic | {
"line": 124,
"column": 15
} | {
"line": 124,
"column": 46
} | {
"line": 125,
"column": 2
} | [
{
"pp": "x✝ : ℕ\n⊢ fib (-↑x✝ + -↑1) = fib (-↑x✝ - 1) * fib (-↑1) + fib (-↑x✝) * fib (-↑1 + 1)",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"add_neg_cancel",
"NegZeroClass.toNeg",
"HMul.hMul",
"congrArg",
"AddMonoid.toAddZeroCla... | [] | simp [sub_eq_neg_add, add_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Int.Fib.Basic | {
"line": 124,
"column": 15
} | {
"line": 124,
"column": 46
} | {
"line": 125,
"column": 2
} | [
{
"pp": "x✝ : ℕ\n⊢ fib (-↑x✝ + -↑1) = fib (-↑x✝ - 1) * fib (-↑1) + fib (-↑x✝) * fib (-↑1 + 1)",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"add_neg_cancel",
"NegZeroClass.toNeg",
"HMul.hMul",
"congrArg",
"AddMonoid.toAddZeroCla... | [] | simp [sub_eq_neg_add, add_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 296,
"column": 77
} | {
"line": 296,
"column": 89
} | {
"line": 296,
"column": 89
} | [
{
"pp": "A B : ℝ\nhB : 0 < B\nk : ℕ\nhk : 1 ≤ k\nn : ℕ\n| ‖((↑n + 1) ^ ↑(1 + k))⁻¹‖",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Norm.norm",
"Real",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
... | [
"A B : ℝ\nhB : 0 < B\nk : ℕ\nhk : 1 ≤ k\nn : ℕ\n| ‖((↑n + 1) ^ (1 + k))⁻¹‖"
] | zpow_natCast | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Analysis.SpecificLimits.Fibonacci | {
"line": 30,
"column": 4
} | {
"line": 30,
"column": 11
} | {
"line": 31,
"column": 4
} | [
{
"pp": "h₁ : ∀ (n : ℕ), ↑(fib (n + 1)) / ↑(fib n) = (φ - ψ * (ψ / φ) ^ n) / (1 - (ψ / φ) ^ n)\n⊢ -φ < ψ ∧ ψ < φ",
"ppTerm": "?m.107",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.RingNF.nnrat_rawCast",
"AddGroup.toSubtractionMonoid... | [
"h₁ : ∀ (n : ℕ), ↑(fib (n + 1)) / ↑(fib n) = (φ - ψ * (ψ / φ) ^ n) / (1 - (ψ / φ) ^ n)\n⊢ -1 / 2 + √5 * (-1 / 2) < 1 / 2 + √5 * (-1 / 2) ∧ 1 / 2 + √5 * (-1 / 2) < 1 / 2 + √5 * (1 / 2)"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.SpecificLimits.Fibonacci | {
"line": 29,
"column": 4
} | {
"line": 31,
"column": 9
} | {
"line": 32,
"column": 2
} | [
{
"pp": "h₁ : ∀ (n : ℕ), ↑(fib (n + 1)) / ↑(fib n) = (φ - ψ * (ψ / φ) ^ n) / (1 - (ψ / φ) ^ n)\n⊢ |ψ / φ| < 1",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"Mathlib.Tactic.R... | [] | rw [abs_div, div_lt_one <| by positivity, abs_of_pos goldenRatio_pos, abs_lt]
ring_nf
bound | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecificLimits.Fibonacci | {
"line": 29,
"column": 4
} | {
"line": 31,
"column": 9
} | {
"line": 32,
"column": 2
} | [
{
"pp": "h₁ : ∀ (n : ℕ), ↑(fib (n + 1)) / ↑(fib n) = (φ - ψ * (ψ / φ) ^ n) / (1 - (ψ / φ) ^ n)\n⊢ |ψ / φ| < 1",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"Mathlib.Tactic.R... | [] | rw [abs_div, div_lt_one <| by positivity, abs_of_pos goldenRatio_pos, abs_lt]
ring_nf
bound | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Real.GoldenRatio | {
"line": 214,
"column": 2
} | {
"line": 214,
"column": 9
} | {
"line": 215,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ (φ ^ (n + 1) - ψ ^ (n + 1) - (φ ^ (n + 1) - φ * ψ ^ n)) / √5 = ψ ^ n",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.RingNF.nnrat_rawCast",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
... | [
"n : ℕ\n⊢ √5 * (√5)⁻¹ * (1 / 2 + √5 * (-1 / 2)) ^ n = (1 / 2 + √5 * (-1 / 2)) ^ n"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.Real.GoldenRatio | {
"line": 216,
"column": 2
} | {
"line": 216,
"column": 43
} | {
"line": 218,
"column": 0
} | [
{
"pp": "n : ℕ\nnz : √5 ≠ 0\n⊢ √5 * (√5)⁻¹ * (1 / 2 + √5 * (-1 / 2)) ^ n = (1 / 2 + √5 * (-1 / 2)) ^ n",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | rw [← (mul_inv_cancel₀ nz).symm, one_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Real.GoldenRatio | {
"line": 228,
"column": 4
} | {
"line": 233,
"column": 51
} | {
"line": 235,
"column": 0
} | [
{
"pp": "case succ\nn : ℕ\nih : φ * ↑(Nat.fib (n + 1)) + ↑(Nat.fib n) = φ ^ (n + 1)\n⊢ φ * ↑(Nat.fib (n + 1 + 1)) + ↑(Nat.fib (n + 1)) = φ ^ (n + 1 + 1)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"NonAssocSemiring.t... | [] | calc
_ = φ * (Nat.fib n) + φ ^ 2 * (Nat.fib (n + 1)) := by
simp only [Nat.fib_add_one (Nat.succ_ne_zero n), Nat.succ_sub_succ_eq_sub,
Nat.cast_add, goldenRatio_sq, Nat.sub_zero]; ring
_ = φ * ((Nat.fib n) + φ * (Nat.fib (n + 1))) := by ring
_ = φ ^ (n + 2) := by rw [add_comm, ih]; ri... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.NumberTheory.Real.GoldenRatio | {
"line": 228,
"column": 4
} | {
"line": 233,
"column": 51
} | {
"line": 235,
"column": 0
} | [
{
"pp": "case succ\nn : ℕ\nih : φ * ↑(Nat.fib (n + 1)) + ↑(Nat.fib n) = φ ^ (n + 1)\n⊢ φ * ↑(Nat.fib (n + 1 + 1)) + ↑(Nat.fib (n + 1)) = φ ^ (n + 1 + 1)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"NonAssocSemiring.t... | [] | calc
_ = φ * (Nat.fib n) + φ ^ 2 * (Nat.fib (n + 1)) := by
simp only [Nat.fib_add_one (Nat.succ_ne_zero n), Nat.succ_sub_succ_eq_sub,
Nat.cast_add, goldenRatio_sq, Nat.sub_zero]; ring
_ = φ * ((Nat.fib n) + φ * (Nat.fib (n + 1))) := by ring
_ = φ ^ (n + 2) := by rw [add_comm, ih]; ri... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Real.GoldenRatio | {
"line": 228,
"column": 4
} | {
"line": 233,
"column": 51
} | {
"line": 235,
"column": 0
} | [
{
"pp": "case succ\nn : ℕ\nih : φ * ↑(Nat.fib (n + 1)) + ↑(Nat.fib n) = φ ^ (n + 1)\n⊢ φ * ↑(Nat.fib (n + 1 + 1)) + ↑(Nat.fib (n + 1)) = φ ^ (n + 1 + 1)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"NonAssocSemiring.t... | [] | calc
_ = φ * (Nat.fib n) + φ ^ 2 * (Nat.fib (n + 1)) := by
simp only [Nat.fib_add_one (Nat.succ_ne_zero n), Nat.succ_sub_succ_eq_sub,
Nat.cast_add, goldenRatio_sq, Nat.sub_zero]; ring
_ = φ * ((Nat.fib n) + φ * (Nat.fib (n + 1))) := by ring
_ = φ ^ (n + 2) := by rw [add_comm, ih]; ri... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 371,
"column": 2
} | {
"line": 371,
"column": 9
} | {
"line": 373,
"column": 0
} | [
{
"pp": "z : ℂ\nk : ℕ\nhk : 1 ≤ k\nhz : z ∈ ℍₒ\n⊢ (-1) ^ k * ↑k ! * z ^ (-1 - ↑k) +\n (-1) ^ k * ↑k ! * (∑' (b : ℕ), (z + (↑b + 1)) ^ (-1 - ↑k) + ∑' (b : ℕ), (z - (↑b + 1)) ^ (-1 - ↑k)) =\n (-1) ^ k * ↑k ! *\n (∑' (n : ℕ), (z + (↑n + 1)) ^ (-1 - ↑k) + (z + 0) ^ (-1 - ↑k) + ∑' (n : ℕ), (z + -(↑n + 1... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.CategoryTheory.Preadditive.Yoneda.Projective | {
"line": 36,
"column": 2
} | {
"line": 36,
"column": 56
} | {
"line": 37,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nP : C\n⊢ Projective P ↔ (preadditiveCoyoneda.obj (op P)).PreservesEpimorphisms",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor",
"Opposite",
"CategoryTheory.coy... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nP : C\n⊢ (coyoneda.obj (op P)).PreservesEpimorphisms ↔ (preadditiveCoyoneda.obj (op P)).PreservesEpimorphisms"
] | rw [projective_iff_preservesEpimorphisms_coyoneda_obj] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Preadditive.Yoneda.Projective | {
"line": 46,
"column": 2
} | {
"line": 46,
"column": 56
} | {
"line": 47,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nP : C\n⊢ Projective P ↔ (preadditiveCoyonedaObj P).PreservesEpimorphisms",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor",
"Opposite",
"CategoryTheory.coyoneda"... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nP : C\n⊢ (coyoneda.obj (op P)).PreservesEpimorphisms ↔ (preadditiveCoyonedaObj P).PreservesEpimorphisms"
] | rw [projective_iff_preservesEpimorphisms_coyoneda_obj] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecificLimits.FloorPow | {
"line": 79,
"column": 8
} | {
"line": 79,
"column": 27
} | {
"line": 80,
"column": 6
} | [
{
"pp": "u : ℕ → ℝ\nl : ℝ\nhmono : Monotone u\nhlim :\n ∀ (a : ℝ),\n 1 < a →\n ∃ c,\n (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧\n Tendsto c atTop atTop ∧ Tendsto (fun n ↦ u (c n) / ↑(c n)) atTop (𝓝 l)\nlnonneg : 0 ≤ l\nε : ℝ\nεpos : 0 < ε\nc : ℕ → ℕ\ncgrowth : ∀ᶠ (n : ℕ) in at... | [] | exact mem_range.2 h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SumIntegralComparisons | {
"line": 119,
"column": 2
} | {
"line": 122,
"column": 54
} | {
"line": 124,
"column": 0
} | [
{
"pp": "a b : ℕ\nf : ℝ → ℝ\nhab : a ≤ b\nhf : AntitoneOn f (Icc ↑a ↑b)\n⊢ ∑ i ∈ Finset.Ico a b, f ↑(i + 1) ≤ ∫ (x : ℝ) in ↑a..↑b, f x",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real.instLE",
"Real",
"MeasureT... | [] | suffices ∑ i ∈ .Ico (0 + a) (b - a + a), f ↑(i + 1) ≤ ∫ x in a..a + ↑(b - a), f x by simp_all
simp_rw [← Finset.sum_Ico_add, Nat.Ico_zero_eq_range, add_assoc]
suffices ∑ x ∈ .range (b - a), f (a + ↑(x + 1)) ≤ ∫ x in a..a + ↑(b - a), f x by simp_all
exact AntitoneOn.sum_le_integral (by simp [hf, hab]) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SumIntegralComparisons | {
"line": 119,
"column": 2
} | {
"line": 122,
"column": 54
} | {
"line": 124,
"column": 0
} | [
{
"pp": "a b : ℕ\nf : ℝ → ℝ\nhab : a ≤ b\nhf : AntitoneOn f (Icc ↑a ↑b)\n⊢ ∑ i ∈ Finset.Ico a b, f ↑(i + 1) ≤ ∫ (x : ℝ) in ↑a..↑b, f x",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real.instLE",
"Real",
"MeasureT... | [] | suffices ∑ i ∈ .Ico (0 + a) (b - a + a), f ↑(i + 1) ≤ ∫ x in a..a + ↑(b - a), f x by simp_all
simp_rw [← Finset.sum_Ico_add, Nat.Ico_zero_eq_range, add_assoc]
suffices ∑ x ∈ .range (b - a), f (a + ↑(x + 1)) ≤ ∫ x in a..a + ↑(b - a), f x by simp_all
exact AntitoneOn.sum_le_integral (by simp [hf, hab]) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp | {
"line": 119,
"column": 6
} | {
"line": 124,
"column": 11
} | {
"line": 124,
"column": 11
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nA : C\nk : A ⟶ X ⊞ Y\nhk : k ≫ φ f g = 0\n⊢ (k ≫ biprod.fst) ≫ f ≫ g = 0",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
... | [] | obtain ⟨k₁, k₂, rfl⟩ := biprod.decomp_hom_to k
simp only [biprod.ext_to_iff, add_comp, assoc, inl_φ, BinaryBicone.inl_fst,
comp_id, inr_φ_fst, comp_neg, zero_comp, BinaryBicone.inl_snd, comp_zero, φ_snd,
BinaryBicone.inr_snd_assoc, zero_add, add_neg_eq_zero] at hk
obtain ⟨rfl, hk⟩ := hk
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp | {
"line": 119,
"column": 6
} | {
"line": 124,
"column": 11
} | {
"line": 124,
"column": 11
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nA : C\nk : A ⟶ X ⊞ Y\nhk : k ≫ φ f g = 0\n⊢ (k ≫ biprod.fst) ≫ f ≫ g = 0",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
... | [] | obtain ⟨k₁, k₂, rfl⟩ := biprod.decomp_hom_to k
simp only [biprod.ext_to_iff, add_comp, assoc, inl_φ, BinaryBicone.inl_fst,
comp_id, inr_φ_fst, comp_neg, zero_comp, BinaryBicone.inl_snd, comp_zero, φ_snd,
BinaryBicone.inr_snd_assoc, zero_add, add_neg_eq_zero] at hk
obtain ⟨rfl, hk⟩ := hk
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp | {
"line": 119,
"column": 6
} | {
"line": 124,
"column": 11
} | {
"line": 124,
"column": 11
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nA : C\nk : A ⟶ X ⊞ Y\nhk : k ≫ φ f g = 0\n⊢ kernel.lift (f ≫ g) (k ≫ biprod.fst) ⋯ ≫ ι f g = k",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abeli... | [] | obtain ⟨k₁, k₂, rfl⟩ := biprod.decomp_hom_to k
simp only [biprod.ext_to_iff, add_comp, assoc, inl_φ, BinaryBicone.inl_fst,
comp_id, inr_φ_fst, comp_neg, zero_comp, BinaryBicone.inl_snd, comp_zero, φ_snd,
BinaryBicone.inr_snd_assoc, zero_add, add_neg_eq_zero] at hk
obtain ⟨rfl, hk⟩ := hk
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp | {
"line": 119,
"column": 6
} | {
"line": 124,
"column": 11
} | {
"line": 124,
"column": 11
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nA : C\nk : A ⟶ X ⊞ Y\nhk : k ≫ φ f g = 0\n⊢ kernel.lift (f ≫ g) (k ≫ biprod.fst) ⋯ ≫ ι f g = k",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abeli... | [] | obtain ⟨k₁, k₂, rfl⟩ := biprod.decomp_hom_to k
simp only [biprod.ext_to_iff, add_comp, assoc, inl_φ, BinaryBicone.inl_fst,
comp_id, inr_φ_fst, comp_neg, zero_comp, BinaryBicone.inl_snd, comp_zero, φ_snd,
BinaryBicone.inr_snd_assoc, zero_add, add_neg_eq_zero] at hk
obtain ⟨rfl, hk⟩ := hk
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecificLimits.FloorPow | {
"line": 130,
"column": 8
} | {
"line": 130,
"column": 27
} | {
"line": 131,
"column": 6
} | [
{
"pp": "u : ℕ → ℝ\nl : ℝ\nhmono : Monotone u\nhlim :\n ∀ (a : ℝ),\n 1 < a →\n ∃ c,\n (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧\n Tendsto c atTop atTop ∧ Tendsto (fun n ↦ u (c n) / ↑(c n)) atTop (𝓝 l)\nlnonneg : 0 ≤ l\nA : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤... | [] | exact mem_range.2 h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Generator.Abelian | {
"line": 45,
"column": 4
} | {
"line": 45,
"column": 50
} | {
"line": 46,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasLimits C\ninst✝ : EnoughInjectives C\nG : C\nhG : IsSeparator G\nthis✝ : WellPowered.{v, v, u} C\nthis : HasProductsOfShape (Subobject (op G)) C\nT : C := Injective.under (∏ᶜ fun P ↦ unop (Subobject.underlying.obj P))\nX Y : C\nf :... | [] | rw [← Limits.image.fac (h ≫ f), hh, zero_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Generator.Abelian | {
"line": 45,
"column": 4
} | {
"line": 45,
"column": 50
} | {
"line": 46,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasLimits C\ninst✝ : EnoughInjectives C\nG : C\nhG : IsSeparator G\nthis✝ : WellPowered.{v, v, u} C\nthis : HasProductsOfShape (Subobject (op G)) C\nT : C := Injective.under (∏ᶜ fun P ↦ unop (Subobject.underlying.obj P))\nX Y : C\nf :... | [] | rw [← Limits.image.fac (h ≫ f), hh, zero_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Generator.Abelian | {
"line": 45,
"column": 4
} | {
"line": 45,
"column": 50
} | {
"line": 46,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasLimits C\ninst✝ : EnoughInjectives C\nG : C\nhG : IsSeparator G\nthis✝ : WellPowered.{v, v, u} C\nthis : HasProductsOfShape (Subobject (op G)) C\nT : C := Injective.under (∏ᶜ fun P ↦ unop (Subobject.underlying.obj P))\nX Y : C\nf :... | [] | rw [← Limits.image.fac (h ≫ f), hh, zero_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.Yoneda | {
"line": 63,
"column": 4
} | {
"line": 63,
"column": 26
} | {
"line": 64,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\nD : Type u'\ninst✝² : Category.{v', u'} D\nF : D ⥤ C\ninst✝¹ : F.Full\nG : C\ninst✝ : Projective G\nhG : IsSeparator G\nhG₂ : ∀ (X : D), ∃ p, Epi p\nX Y : D\nf : (F ⋙ preadditiveCoyonedaObj G).obj X ⟶ (F ⋙ preadditiveCoyonedaObj G).obj Y\n⊢ ∃ ... | [
"C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\nD : Type u'\ninst✝² : Category.{v', u'} D\nF : D ⥤ C\ninst✝¹ : F.Full\nG : C\ninst✝ : Projective G\nhG : IsSeparator G\nhG₂ : ∀ (X : D), ∃ p, Epi p\nX Y : D\nf : (F ⋙ preadditiveCoyonedaObj G).obj X ⟶ (F ⋙ preadditiveCoyonedaObj G).obj Y\np : G ⟶ F.obj X\... | obtain ⟨p, _⟩ := hG₂ X | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Limits.Indization.FilteredColimits | {
"line": 86,
"column": 2
} | {
"line": 88,
"column": 79
} | {
"line": 90,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\nI : Type v\ninst✝⁴ : SmallCategory I\nF : I ⥤ Cᵒᵖ ⥤ Type v\nJ : Type v\ninst✝³ : SmallCategory J\ninst✝² : FinCategory J\nG : J ⥤ CostructuredArrow yoneda (colimit F)\nK : Type v\ninst✝¹ : SmallCategory K\nH : K ⥤ Over (colimit F)\ninst✝ : IsFiltered K\nc : Cocon... | [] | refine exists_nonempty_limit_obj_of_colimit F G H ?_
suffices T ≅ colimit H from Nonempty.map (lim.map (whiskerLeft 𝒢 (yoneda.map this.hom))) h
refine hT.symm ≪≫ IsColimit.coconePointUniqueUpToIso hc (colimit.isColimit _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Indization.FilteredColimits | {
"line": 86,
"column": 2
} | {
"line": 88,
"column": 79
} | {
"line": 90,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\nI : Type v\ninst✝⁴ : SmallCategory I\nF : I ⥤ Cᵒᵖ ⥤ Type v\nJ : Type v\ninst✝³ : SmallCategory J\ninst✝² : FinCategory J\nG : J ⥤ CostructuredArrow yoneda (colimit F)\nK : Type v\ninst✝¹ : SmallCategory K\nH : K ⥤ Over (colimit F)\ninst✝ : IsFiltered K\nc : Cocon... | [] | refine exists_nonempty_limit_obj_of_colimit F G H ?_
suffices T ≅ colimit H from Nonempty.map (lim.map (whiskerLeft 𝒢 (yoneda.map this.hom))) h
refine hT.symm ≪≫ IsColimit.coconePointUniqueUpToIso hc (colimit.isColimit _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Indization.LocallySmall | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 42
} | {
"line": 73,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nI : Type u₁\ninst✝³ : Category.{v₁, u₁} I\ninst✝² : HasColimitsOfShape I (Type v)\ninst✝¹ : HasLimitsOfShape Iᵒᵖ (Type v)\ninst✝ : HasLimitsOfShape Iᵒᵖ (Type (max u v))\nF : I ⥤ C\nG : Cᵒᵖ ⥤ Type v\nη : colimit (F ⋙ yoneda) ⟶ G\ni : Iᵒᵖ\nthis :\n ∀ (a : limit ((... | [
"C : Type u\ninst✝⁴ : Category.{v, u} C\nI : Type u₁\ninst✝³ : Category.{v₁, u₁} I\ninst✝² : HasColimitsOfShape I (Type v)\ninst✝¹ : HasLimitsOfShape Iᵒᵖ (Type v)\ninst✝ : HasLimitsOfShape Iᵒᵖ (Type (max u v))\nF : I ⥤ C\nG : Cᵒᵖ ⥤ Type v\nη : colimit (F ⋙ yoneda) ⟶ G\ni : Iᵒᵖ\nthis :\n ∀ (a : limit ((F.op ⋙ G) ⋙ ... | erw [colimitYonedaHomIsoLimitOp_π_apply] | Lean.Parser.Tactic._aux_Init_Meta___macroRules_Lean_Parser_Tactic_tacticErw____1 | Lean.Parser.Tactic.tacticErw___ |
Mathlib.CategoryTheory.Preadditive.LiftToFinset | {
"line": 67,
"column": 4
} | {
"line": 68,
"column": 38
} | {
"line": 69,
"column": 2
} | [
{
"pp": "case h₀\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Preadditive C\ninst✝² : HasFiniteProducts C\nα : Type w\ninst✝¹ : DecidableEq α\nf : α → C\ninst✝ : HasProduct f\nS : (Finset (Discrete α))ᵒᵖ\nv : ↥(Opposite.unop S)\n⊢ ∀ b ∈ (Opposite.unop S).attach, b ≠ v → Pi.π f (↑b).as ≫ Pi.ι (fun a ↦ f (↑a... | [] | intro b hb hb₁
rw [Pi.ι_π_of_ne _ hb₁, comp_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Preadditive.LiftToFinset | {
"line": 67,
"column": 4
} | {
"line": 68,
"column": 38
} | {
"line": 69,
"column": 2
} | [
{
"pp": "case h₀\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Preadditive C\ninst✝² : HasFiniteProducts C\nα : Type w\ninst✝¹ : DecidableEq α\nf : α → C\ninst✝ : HasProduct f\nS : (Finset (Discrete α))ᵒᵖ\nv : ↥(Opposite.unop S)\n⊢ ∀ b ∈ (Opposite.unop S).attach, b ≠ v → Pi.π f (↑b).as ≫ Pi.ι (fun a ↦ f (↑a... | [] | intro b hb hb₁
rw [Pi.ι_π_of_ne _ hb₁, comp_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.FilteredColimitCommutesProduct | {
"line": 265,
"column": 6
} | {
"line": 265,
"column": 48
} | {
"line": 266,
"column": 6
} | [
{
"pp": "α : Type u\nI : α → Type u\ninst✝¹ : (i : α) → SmallCategory (I i)\ninst✝ : ∀ (i : α), IsFiltered (I i)\nF : (i : α) → I i ⥤ Type u\nky : (i : α) → I i\nyk₀ : (pointwiseProduct F).obj ky\nky' : (i : α) → I i\nyk₀' : (pointwiseProduct F).obj ky'\nk : (i : α) → I i := IsFiltered.max ky ky'\nyk : ∏ᶜ (Func... | [
"α : Type u\nI : α → Type u\ninst✝¹ : (i : α) → SmallCategory (I i)\ninst✝ : ∀ (i : α), IsFiltered (I i)\nF : (i : α) → I i ⥤ Type u\nky : (i : α) → I i\nyk₀ : (pointwiseProduct F).obj ky\nky' : (i : α) → I i\nyk₀' : (pointwiseProduct F).obj ky'\nk : (i : α) → I i := IsFiltered.max ky ky'\nyk : ∏ᶜ (Functor.pi F).ob... | conv_rhs => rw [IsFiltered.coeq_condition] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convRHS_1 | Mathlib.Tactic.Conv.convRHS |
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