module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 812,
"column": 4
} | {
"line": 812,
"column": 15
} | {
"line": 812,
"column": 16
} | [
{
"pp": "case refine_2\nL : PeriodPair\nl₀ x : ℂ\nr : NNReal\nhr0 : 0 < r\nhr : Metric.closedBall x ↑r ⊆ (↑L.lattice \\ {l₀})ᶜ\n⊢ Metric.eball x ↑r ⊆ Metric.closedBall x ↑r",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
... | [
"case refine_2\nL : PeriodPair\nl₀ x : ℂ\nr : NNReal\nhr0 : 0 < r\nhr : Metric.closedBall x ↑r ⊆ (↑L.lattice \\ {l₀})ᶜ\n⊢ Metric.ball x ↑r ⊆ Metric.closedBall x ↑r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 819,
"column": 2
} | {
"line": 819,
"column": 45
} | {
"line": 820,
"column": 4
} | [
{
"pp": "L : PeriodPair\nl : ℂ\nr : ℝ\nh₁ : 0 < r\nh₂ : Metric.closedBall l r ⊆ (↑L.lattice \\ {l})ᶜ\n⊢ HasFPowerSeriesAt ℘'[L - l] (FormalMultilinearSeries.ofScalars ℂ fun i ↦ (↑i + 1) * (↑i + 2) * L.sumInvPow l (i + 3))\n l",
"ppTerm": "?m.71",
"assigned": false,
"usedConstants": [],
"usedF... | [
"L : PeriodPair\nl : ℂ\nr : ℝ\nh₁ : 0 < r\nh₂ : Metric.closedBall l r ⊆ (↑L.lattice \\ {l})ᶜ\n⊢ HasFPowerSeriesAt ℘'[L - l] (FormalMultilinearSeries.ofScalars ℂ fun i ↦ (↑i + 1) * (↑i + 2) * L.sumInvPow l (i + 3))\n l"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 834,
"column": 4
} | {
"line": 834,
"column": 15
} | {
"line": 834,
"column": 16
} | [
{
"pp": "L : PeriodPair\nl : ℂ\nn : ℕ\n⊢ iteratedDeriv n ℘'[L - l] l / ↑n ! = (↑n + 1) * (↑n + 2) * L.sumInvPow l (n + 3)",
"ppTerm": "?m.64",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"L : PeriodPair\nl : ℂ\nn : ℕ\n⊢ iteratedDeriv n ℘'[L - l] l / ↑n ! = (↑n + 1) * (↑n + 2) * L.sumInvPow l (n + 3)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 138,
"column": 6
} | {
"line": 138,
"column": 71
} | {
"line": 138,
"column": 72
} | [
{
"pp": "case hbc.inl\nz : ℍ\nx : Fin 2 → ℤ\nhx : x ≠ 0\nhn0 : ‖x‖ ≠ 0\nh11 : ↑(x 0) * ↑z + ↑(x 1) = (↑(x 0) / ↑‖x‖ * ↑z + ↑(x 1) / ↑‖x‖) * ↑‖x‖\nH1 : 1 ≤ (↑(x 0) / ‖x‖) ^ 2\n⊢ r z ≤ ‖↑(x 0) / ↑↑(max (x 0).natAbs (x 1).natAbs) * ↑z + ↑(x 1) / ↑↑(max (x 0).natAbs (x 1).natAbs)‖",
"ppTerm": "?hbc.inl",
"a... | [
"case hbc.inl\nz : ℍ\nx : Fin 2 → ℤ\nhx : x ≠ 0\nhn0 : ‖x‖ ≠ 0\nh11 : ↑(x 0) * ↑z + ↑(x 1) = (↑(x 0) / ↑‖x‖ * ↑z + ↑(x 1) / ↑‖x‖) * ↑‖x‖\nH1 : 1 ≤ (↑(x 0) / ‖x‖) ^ 2\n⊢ r z ≤ ‖↑(x 0) / ↑↑(max (x 0).natAbs (x 1).natAbs) * ↑z + ↑(x 1) / ↑↑(max (x 0).natAbs (x 1).natAbs)‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema | {
"line": 112,
"column": 18
} | {
"line": 112,
"column": 29
} | {
"line": 112,
"column": 30
} | [
{
"pp": "n : ℕ\nhn : n ≠ 0\nx : ℝ\nhx : |x| ≤ 1\nk : ℕ\nhk : ↑k * π = ↑n * arccos x\nhk' : ↑k = ↑n * (arccos x / π)\nhkn : ↑k ≤ ↑n\n⊢ k ≤ n",
"ppTerm": "?m.185",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\nhn : n ≠ 0\nx : ℝ\nhx : |x| ≤ 1\nk : ℕ\nhk : ↑k * π = ↑n * arccos x\nhk' : ↑k = ↑n * (arccos x / π)\nhkn : ↑k ≤ ↑n\n⊢ k ≤ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 139,
"column": 6
} | {
"line": 139,
"column": 71
} | {
"line": 139,
"column": 72
} | [
{
"pp": "case hbc.inr\nz : ℍ\nx : Fin 2 → ℤ\nhx : x ≠ 0\nhn0 : ‖x‖ ≠ 0\nh11 : ↑(x 0) * ↑z + ↑(x 1) = (↑(x 0) / ↑‖x‖ * ↑z + ↑(x 1) / ↑‖x‖) * ↑‖x‖\nH2 : 1 ≤ (↑(x 1) / ‖x‖) ^ 2\n⊢ r z ≤ ‖↑(x 0) / ↑↑(max (x 0).natAbs (x 1).natAbs) * ↑z + ↑(x 1) / ↑↑(max (x 0).natAbs (x 1).natAbs)‖",
"ppTerm": "?hbc.inr",
"a... | [
"case hbc.inr\nz : ℍ\nx : Fin 2 → ℤ\nhx : x ≠ 0\nhn0 : ‖x‖ ≠ 0\nh11 : ↑(x 0) * ↑z + ↑(x 1) = (↑(x 0) / ↑‖x‖ * ↑z + ↑(x 1) / ↑‖x‖) * ↑‖x‖\nH2 : 1 ≤ (↑(x 1) / ‖x‖) ^ 2\n⊢ r z ≤ ‖↑(x 0) / ↑↑(max (x 0).natAbs (x 1).natAbs) * ↑z + ↑(x 1) / ↑↑(max (x 0).natAbs (x 1).natAbs)‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 929,
"column": 8
} | {
"line": 929,
"column": 25
} | {
"line": 929,
"column": 26
} | [
{
"pp": "case pos\nL : PeriodPair\nl₀ : ℂ\nh : l₀ ∈ L.lattice\nthis : AnalyticAt ℂ ℘[L - l₀] l₀\nhl₀ : l₀ = 0\n⊢ AnalyticAt ℂ (fun z ↦ (z - l₀) ^ 2 / l₀ ^ 2) l₀",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"InnerProductSpace.... | [
"case pos\nL : PeriodPair\nl₀ : ℂ\nh : l₀ ∈ L.lattice\nthis : AnalyticAt ℂ ℘[L - l₀] l₀\nhl₀ : l₀ = 0\n⊢ AnalyticAt ℂ (fun z ↦ 0) 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 32
} | {
"line": 167,
"column": 4
} | [
{
"pp": "case hΘ\nc e : ℤ\nz : ℂ\n⊢ (fun d ↦ ↑c * z) =o[cofinite] fun n ↦ ↑n",
"ppTerm": "?hΘ",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"HMul.hMul",... | [
"case hΘ\nc e : ℤ\nz : ℂ\n⊢ (c = 0 ∨ z = 0) ∨ Tendsto (norm ∘ fun n ↦ ↑n) cofinite atTop"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 32
} | {
"line": 167,
"column": 4
} | [
{
"pp": "case ho\nc e : ℤ\nz : ℂ\n⊢ (fun d ↦ ↑e) =o[cofinite] fun n ↦ ↑n",
"ppTerm": "?ho",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"Complex.instNormedAddCommGroup",
"congrArg",
... | [
"case ho\nc e : ℤ\nz : ℂ\n⊢ e = 0 ∨ Tendsto (norm ∘ fun n ↦ ↑n) cofinite atTop"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 254,
"column": 2
} | {
"line": 254,
"column": 23
} | {
"line": 254,
"column": 24
} | [
{
"pp": "z : ℂ\nc₁ c₂ : ℤ\n⊢ (fun n ↦ ((↑c₁ * z + ↑n + 1) * (↑c₂ * z + ↑n))⁻¹) =O[cofinite] fun n ↦ (|↑n| ^ 2)⁻¹",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instPow",
"Real",
"DivInvM... | [
"z : ℂ\nc₁ c₂ : ℤ\n⊢ ((fun n ↦ (↑c₂ * z + ↑n)⁻¹ * (↑c₁ * z + ↑n + 1)⁻¹) =O[atBot] fun n ↦ (↑n)⁻¹ * (↑n)⁻¹) ∧\n (fun n ↦ (↑c₂ * z + ↑n)⁻¹ * (↑c₁ * z + ↑n + 1)⁻¹) =O[atTop] fun n ↦ (↑n)⁻¹ * (↑n)⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 261,
"column": 2
} | {
"line": 261,
"column": 13
} | {
"line": 261,
"column": 14
} | [
{
"pp": "z : ℂ\nhz : z ≠ 0\nc₁ c₂ : ℤ\n⊢ (fun n ↦ ((↑n * z + ↑c₁) * (↑n * z + ↑c₂))⁻¹) =O[cofinite] fun n ↦ (↑n * ↑n)⁻¹",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"DivInvMonoid.toInv",
... | [
"z : ℂ\nhz : z ≠ 0\nc₁ c₂ : ℤ\n⊢ ((fun n ↦ (↑n * z + ↑c₂)⁻¹ * (↑n * z + ↑c₁)⁻¹) =O[atBot] fun n ↦ (↑n)⁻¹ * (↑n)⁻¹) ∧\n (fun n ↦ (↑n * z + ↑c₂)⁻¹ * (↑n * z + ↑c₁)⁻¹) =O[atTop] fun n ↦ (↑n)⁻¹ * (↑n)⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 270,
"column": 21
} | {
"line": 270,
"column": 63
} | {
"line": 271,
"column": 4
} | [
{
"pp": "z : ℍ\na b : ℤ\nh0 : z ∈ verticalStrip |z.re| z.im\nm : Fin 2 → ℤ\n⊢ ‖((↑(m 0) + ↑a) * ↑z + ↑(m 1) + ↑b)⁻¹‖ ≤ ?m.98",
"ppTerm": "?m.100",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Int.cast",
"Eq.mpr",
"Real",
"HMul.hMul",
"AddMonoid.toAddSemigrou... | [
"z : ℍ\na b : ℤ\nh0 : z ∈ verticalStrip |z.re| z.im\nm : Fin 2 → ℤ\n⊢ ‖(↑(m 0) + ↑a) * ↑z + (↑(m 1) + ↑b)‖⁻¹ ≤ ?m.98"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 277,
"column": 2
} | {
"line": 277,
"column": 55
} | {
"line": 278,
"column": 4
} | [
{
"pp": "α : Type u_1\na : α\ninst✝² : NormedAddCommGroup α\ninst✝¹ : DiscreteTopology α\ninst✝ : ProperSpace α\n⊢ (fun x ↦ a) =o[cofinite] fun x ↦ ‖x‖",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"congrArg",
"Function.comp",
... | [
"α : Type u_1\na : α\ninst✝² : NormedAddCommGroup α\ninst✝¹ : DiscreteTopology α\ninst✝ : ProperSpace α\n⊢ a = 0 ∨ Tendsto (fun x ↦ ‖x‖) cofinite atTop"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema | {
"line": 273,
"column": 4
} | {
"line": 273,
"column": 50
} | {
"line": 274,
"column": 2
} | [
{
"pp": "case mp.e_a\nn : ℕ\nhn : 2 ≤ n\nx : ℝ\nhx✝ : x ∈ Finset.image (fun k ↦ cos ((↑k + 1) * π / (↑(n - 1) + 1))) (Finset.range (n - 1))\nk : ℕ\nhk₁ : k ∈ Finset.range (n - 1)\nhx : cos ((↑k + 1) * π / (↑(n - 1) + 1)) = x\n⊢ n - 1 + 1 = n",
"ppTerm": "?mp.e_a",
"assigned": true,
"usedConstants": ... | [] | exact Nat.sub_add_cancel (Nat.one_le_of_lt hn) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 1044,
"column": 2
} | {
"line": 1044,
"column": 13
} | {
"line": 1044,
"column": 14
} | [
{
"pp": "case e_c\nL : PeriodPair\nx : ℂ\nl : ↥L.lattice\n⊢ (x + ↑l ∈ L.lattice) = (x ∈ L.lattice)",
"ppTerm": "?e_c",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"eq_iff_iff._simp_1",
"Membership.mem",
"id",
"Int",
"Complex.addCommGroup",
... | [
"case e_c\nL : PeriodPair\nx : ℂ\nl : ↥L.lattice\n⊢ x + ↑l ∈ L.lattice ↔ x ∈ L.lattice"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 1060,
"column": 6
} | {
"line": 1060,
"column": 52
} | {
"line": 1061,
"column": 6
} | [
{
"pp": "L : PeriodPair\nx : ℂ\nhx : x ∉ L.lattice\n⊢ AnalyticAt ℂ (fun z ↦ ℘'[L] z ^ 2 - 4 * ℘[L] z ^ 3 + L.g₂ * ℘[L] z + L.g₃) x",
"ppTerm": "?m.252",
"assigned": true,
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"Complex.instNormedAddCommGroup",
"Complex.instDenselyNor... | [
"L : PeriodPair\nx : ℂ\nhx : x ∉ L.lattice\nthis : AnalyticAt ℂ ℘'[L] x\n⊢ AnalyticAt ℂ (fun z ↦ ℘'[L] z ^ 2 - 4 * ℘[L] z ^ 3 + L.g₂ * ℘[L] z + L.g₃) x"
] | have := L.analyticOnNhd_derivWeierstrassP _ hx | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 1077,
"column": 2
} | {
"line": 1077,
"column": 50
} | {
"line": 1077,
"column": 51
} | [
{
"pp": "L : PeriodPair\nz : ℂ\nhz : z ∉ L.lattice\n⊢ ℘'[L] z ^ 2 = 4 * ℘[L] z ^ 3 - L.g₂ * ℘[L] z - L.g₃",
"ppTerm": "?m.45",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"L : PeriodPair\nz : ℂ\nhz : z ∉ L.lattice\n⊢ ℘'[L] z ^ 2 = 4 * ℘[L] z ^ 3 - L.g₂ * ℘[L] z - L.g₃"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.ArithmeticGeometric | {
"line": 90,
"column": 2
} | {
"line": 91,
"column": 57
} | {
"line": 93,
"column": 0
} | [
{
"pp": "R : Type u_1\na b : R\ninst✝ : Field R\nha : a ≠ 1\nn : ℕ\n⊢ arithGeom a b b n = b * (a ^ (n + 1) - 1) / (a - 1)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"neg_sub",
"neg_div",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing... | [] | rw [arithGeom_same_eq_mul_div' ha n, ← neg_sub _ a, div_neg,
← neg_sub _ (a ^ (n + 1)), mul_neg, neg_div, neg_neg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecificLimits.ArithmeticGeometric | {
"line": 90,
"column": 2
} | {
"line": 91,
"column": 57
} | {
"line": 93,
"column": 0
} | [
{
"pp": "R : Type u_1\na b : R\ninst✝ : Field R\nha : a ≠ 1\nn : ℕ\n⊢ arithGeom a b b n = b * (a ^ (n + 1) - 1) / (a - 1)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"neg_sub",
"neg_div",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing... | [] | rw [arithGeom_same_eq_mul_div' ha n, ← neg_sub _ a, div_neg,
← neg_sub _ (a ^ (n + 1)), mul_neg, neg_div, neg_neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecificLimits.ArithmeticGeometric | {
"line": 90,
"column": 2
} | {
"line": 91,
"column": 57
} | {
"line": 93,
"column": 0
} | [
{
"pp": "R : Type u_1\na b : R\ninst✝ : Field R\nha : a ≠ 1\nn : ℕ\n⊢ arithGeom a b b n = b * (a ^ (n + 1) - 1) / (a - 1)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"neg_sub",
"neg_div",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing... | [] | rw [arithGeom_same_eq_mul_div' ha n, ← neg_sub _ a, div_neg,
← neg_sub _ (a ^ (n + 1)), mul_neg, neg_div, neg_neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 37
} | {
"line": 87,
"column": 0
} | [
{
"pp": "x : ℂ\nhx : x ∈ ℂ_ℤ\nn : ℕ\n⊢ (↑n + 1) ^ 2 ≠ 0",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"GroupWithZero.toMonoidWithZero",
"False",
"eq_false",
"AddMonoid.toAddSemigroup",
"congrArg",
"AddMonoid.toAddZe... | [] | · simp [Nat.cast_add_one_ne_zero n] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 57
} | {
"line": 97,
"column": 2
} | [
{
"pp": "x : ℂ\n⊢ Summable fun i ↦ ‖sineTerm x i‖",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real",
"instHDiv",
"Nat.one_lt_two",
"AddGroupWithOne.toAddMonoidWithOne",
"PseudoMetricSpace.toUniformSpace",
"NormedField.toField",
... | [
"x : ℂ\nthis : Summable fun n ↦ ‖x ^ 2 / (↑n + ↑1) ^ 2‖\n⊢ Summable fun i ↦ ‖sineTerm x i‖"
] | have := summable_pow_div_add (x ^ 2) 2 1 Nat.one_lt_two | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 24
} | {
"line": 97,
"column": 25
} | [
{
"pp": "x : ℂ\nthis : Summable fun n ↦ ‖x ^ 2 / (↑n + ↑1) ^ 2‖\n⊢ Summable fun i ↦ ‖sineTerm x i‖",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr"... | [
"x : ℂ\nthis : Summable fun n ↦ ‖x ^ 2 / (↑n + ↑1) ^ 2‖\n⊢ Summable fun i ↦ ‖x‖ ^ 2 / ‖↑i + 1‖ ^ 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 111,
"column": 4
} | {
"line": 111,
"column": 15
} | {
"line": 111,
"column": 16
} | [
{
"pp": "case refine_1\nZ : Set ℂ\nhZ : IsCompact Z\nhf : ContinuousOn (fun x ↦ ‖-x ^ 2‖) Z\ns : ℝ\nhs : ∀ y ∈ (fun x ↦ ‖-x ^ 2‖) '' Z, y ≤ s\n⊢ Summable fun n ↦ ‖↑s / (↑n + 1) ^ 2‖",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing... | [
"case refine_1\nZ : Set ℂ\nhZ : IsCompact Z\nhf : ContinuousOn (fun x ↦ ‖-x ^ 2‖) Z\ns : ℝ\nhs : ∀ y ∈ (fun x ↦ ‖-x ^ 2‖) '' Z, y ≤ s\n⊢ Summable fun n ↦ |s| / ‖↑n + 1‖ ^ 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Int.Fib.Basic | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 25
} | {
"line": 57,
"column": 0
} | [
{
"pp": "case inr\nn : ℕ\nhn : Odd n\n⊢ fib (-↑n) = (-1) ^ (n + 1) * ↑(Nat.fib n)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"one_pow",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"NonUnitalCommRing.toNonUnitalNonAssocC... | [] | · simp [fib_of_odd, hn] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 51
} | {
"line": 132,
"column": 0
} | [
{
"pp": "Z : Set ℂ\nhZ2 : Z ⊆ ℂ_ℤ\nhZC : IsCompact Z\n⊢ Set.EqOn (fun x ↦ ∏' (i : ℕ), (1 + sineTerm x i)) (fun x ↦ Complex.sin (↑π * x) / (↑π * x)) Z",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Membership.mem",
"euler_sineTerm_tprod",
"Complex",
"Set.instMember... | [] | exact fun x hx => euler_sineTerm_tprod (by aesop) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 179,
"column": 4
} | {
"line": 179,
"column": 32
} | {
"line": 179,
"column": 33
} | [
{
"pp": "x : ℂ\nhx : x ∈ ℂ_ℤ\ni : ℕ\nh1 : x + ↑(↑i + 1) ≠ 0\n⊢ x - (↑i + 1) ≠ 0",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"neg_add_rev",
"Eq.mpr",
"AddGroupWithOne.toAddGroup",
"congrArg",
"AddMonoid.toAddZeroClass",
"sub_eq_add_neg",
"HSub.h... | [
"x : ℂ\nhx : x ∈ ℂ_ℤ\ni : ℕ\nh1 : x + ↑(↑i + 1) ≠ 0\n⊢ ¬x + (-1 + -↑i) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Int.Fib.Basic | {
"line": 153,
"column": 71
} | {
"line": 156,
"column": 62
} | {
"line": 158,
"column": 0
} | [
{
"pp": "m n : ℤ\n⊢ (fib m).gcd (fib n) = Nat.fib (m.gcd n)",
"ppTerm": "?m.2",
"assigned": true,
"usedConstants": [
"Nat.gcd",
"Int.gcd",
"congrArg",
"ite_self",
"Int.fib",
"Int.gcd_neg",
"Exists",
"Int.instDecidablePredEven",
"apply_ite",
... | [] | by
obtain ⟨m, (rfl | rfl)⟩ := m.eq_nat_or_neg
<;> obtain ⟨n, (rfl | rfl)⟩ := n.eq_nat_or_neg
<;> simp [fib_neg, Nat.fib_gcd, apply_ite, apply_ite_left] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 201,
"column": 2
} | {
"line": 201,
"column": 13
} | {
"line": 201,
"column": 14
} | [
{
"pp": "x : ℂ\nhx : x ∈ ℂ_ℤ\n⊢ Tendsto (fun n ↦ logDeriv (fun z ↦ ∏ j ∈ Finset.range n, (1 + sineTerm z j)) x) atTop\n (𝓝 (logDeriv (fun t ↦ Complex.sin (↑π * t) / (↑π * t)) x))",
"ppTerm": "?m.36",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℂ\nhx : x ∈ ℂ_ℤ\n⊢ Tendsto (fun n ↦ logDeriv (fun z ↦ ∏ j ∈ Finset.range n, (1 + sineTerm z j)) x) atTop\n (𝓝 (logDeriv (fun t ↦ Complex.sin (↑π * t) / (↑π * t)) x))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 208,
"column": 4
} | {
"line": 208,
"column": 32
} | {
"line": 208,
"column": 33
} | [
{
"pp": "case ha\nx : ℂ\nhz : x ∈ ℂ_ℤ\nn : ℕ\n⊢ x - (↑n + 1) ≠ 0",
"ppTerm": "?ha",
"assigned": true,
"usedConstants": [
"neg_add_rev",
"Eq.mpr",
"AddGroupWithOne.toAddGroup",
"congrArg",
"AddMonoid.toAddZeroClass",
"sub_eq_add_neg",
"HSub.hSub",
"AddZ... | [
"case ha\nx : ℂ\nhz : x ∈ ℂ_ℤ\nn : ℕ\n⊢ ¬x + (-1 + -↑n) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 209,
"column": 4
} | {
"line": 209,
"column": 15
} | {
"line": 209,
"column": 16
} | [
{
"pp": "case hb\nx : ℂ\nhz : x ∈ ℂ_ℤ\nn : ℕ\n⊢ x + (↑n + 1) ≠ 0",
"ppTerm": "?hb",
"assigned": true,
"usedConstants": [
"id",
"Ne",
"Complex.instNatCast",
"Nat.cast",
"Field.toSemifield",
"instHAdd",
"Semifield.toDivisionSemiring",
"HAdd.hAdd",
... | [
"case hb\nx : ℂ\nhz : x ∈ ℂ_ℤ\nn : ℕ\n⊢ ¬x + (↑n + 1) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 216,
"column": 4
} | {
"line": 216,
"column": 15
} | {
"line": 216,
"column": 16
} | [
{
"pp": "x : ℂ\nhz : x ∈ ℂ_ℤ\nthis : Summable fun n ↦ (x - ↑(n + 1))⁻¹ * (x + ↑(n + 1))⁻¹\n⊢ Summable fun i ↦ 1 / ((x + (↑i + 1)) * (x - (↑i + 1)))",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"MulOne.toOne",
"DivI... | [
"x : ℂ\nhz : x ∈ ℂ_ℤ\nthis : Summable fun n ↦ (x - ↑(n + 1))⁻¹ * (x + ↑(n + 1))⁻¹\n⊢ Summable fun i ↦ (x - (↑i + 1))⁻¹ * (x + (↑i + 1))⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 250,
"column": 2
} | {
"line": 250,
"column": 13
} | {
"line": 250,
"column": 14
} | [
{
"pp": "k : ℕ\nd : ℤ\n⊢ ContDiffOn ℂ (↑k) (fun z ↦ 1 / (z + ↑d)) ℂ_ℤ",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"MulOne.toOne",
"DivInvMonoid.toInv",
"instHDiv",
"Complex.instNormedAddC... | [
"k : ℕ\nd : ℤ\n⊢ ContDiffOn ℂ (↑k) (fun z ↦ (z + ↑d)⁻¹) ℂ_ℤ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 263,
"column": 4
} | {
"line": 263,
"column": 32
} | {
"line": 263,
"column": 33
} | [
{
"pp": "case hf\nk d : ℕ\nz : ℂ\nhz : z ∈ ℂ_ℤ\n⊢ ContDiffAt ℂ (↑k) (fun z ↦ 1 / (z - (↑d + 1))) z",
"ppTerm": "?hf",
"assigned": true,
"usedConstants": [
"neg_add_rev",
"ContDiffAt",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"DivInvMonoid.toInv",
"instHDiv",
... | [
"case hf\nk d : ℕ\nz : ℂ\nhz : z ∈ ℂ_ℤ\n⊢ ContDiffAt ℂ (↑k) (fun z ↦ (z + (-1 + -↑d))⁻¹) z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 265,
"column": 4
} | {
"line": 265,
"column": 15
} | {
"line": 265,
"column": 16
} | [
{
"pp": "case hg\nk d : ℕ\nz : ℂ\nhz : z ∈ ℂ_ℤ\n⊢ ContDiffAt ℂ (↑k) (fun z ↦ 1 / (z + (↑d + 1))) z",
"ppTerm": "?hg",
"assigned": true,
"usedConstants": [
"ContDiffAt",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"MulOne.toOne",
"DivInvMonoid.toInv",
"instHDiv",
... | [
"case hg\nk d : ℕ\nz : ℂ\nhz : z ∈ ℂ_ℤ\n⊢ ContDiffAt ℂ (↑k) (fun z ↦ (z + (↑d + 1))⁻¹) z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.NormNum.Prime | {
"line": 83,
"column": 4
} | {
"line": 83,
"column": 56
} | {
"line": 83,
"column": 57
} | [
{
"pp": "case refine_2.inr.inl\nn k k' : ℕ\ne : k + 2 = k'\nh : MinFacHelper n k\nnp : n.minFac ≠ k\nh2✝ : k < n.minFac\nh2 : k.succ = n.minFac\nh3 : 2 ∣ n.minFac\n⊢ 2 = n.minFac",
"ppTerm": "?refine_2.inr.inl",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case refine_2.inr.inl\nn k k' : ℕ\ne : k + 2 = k'\nh : MinFacHelper n k\nnp : n.minFac ≠ k\nh2✝ : k < n.minFac\nh2 : k.succ = n.minFac\nh3 : 2 ∣ n.minFac\n⊢ 2 = n.minFac"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 315,
"column": 4
} | {
"line": 316,
"column": 26
} | {
"line": 316,
"column": 27
} | [
{
"pp": "case hbc.h₁\nk : ℕ\nK : Set ℂ\nA B : ℝ\nhB : 0 < B\nhKAB : K ⊆ UpperHalfPlane.coe '' verticalStrip A B\nn : ℕ\na : ℍ\nhaAB : a ∈ verticalStrip A B\nha : ↑a ∈ K\nh1 :\n ‖↑(![1, ↑n + 1] 0) * ↑a + ↑(![1, ↑n + 1] 1)‖ ^ (-(↑k + 1)) ≤\n r { coe := { re := A, im := B }, coe_im_pos := hB } ^ (-(↑k + 1)) * ... | [
"case hbc.h₁\nk : ℕ\nK : Set ℂ\nA B : ℝ\nhB : 0 < B\nhKAB : K ⊆ UpperHalfPlane.coe '' verticalStrip A B\nn : ℕ\na : ℍ\nhaAB : a ∈ verticalStrip A B\nha : ↑a ∈ K\nh1 :\n ‖↑(![1, ↑n + 1] 0) * ↑a + ↑(![1, ↑n + 1] 1)‖ ^ (-(↑k + 1)) ≤\n r { coe := { re := A, im := B }, coe_im_pos := hB } ^ (-(↑k + 1)) * ‖![1, ↑n + 1... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 317,
"column": 4
} | {
"line": 318,
"column": 26
} | {
"line": 318,
"column": 27
} | [
{
"pp": "case h₂\nk : ℕ\nK : Set ℂ\nA B : ℝ\nhB : 0 < B\nhKAB : K ⊆ UpperHalfPlane.coe '' verticalStrip A B\nn : ℕ\na : ℍ\nhaAB : a ∈ verticalStrip A B\nha : ↑a ∈ K\nh1 :\n ‖↑(![1, ↑n + 1] 0) * ↑a + ↑(![1, ↑n + 1] 1)‖ ^ (-(↑k + 1)) ≤\n r { coe := { re := A, im := B }, coe_im_pos := hB } ^ (-(↑k + 1)) * ‖![1... | [
"case h₂\nk : ℕ\nK : Set ℂ\nA B : ℝ\nhB : 0 < B\nhKAB : K ⊆ UpperHalfPlane.coe '' verticalStrip A B\nn : ℕ\na : ℍ\nhaAB : a ∈ verticalStrip A B\nha : ↑a ∈ K\nh1 :\n ‖↑(![1, ↑n + 1] 0) * ↑a + ↑(![1, ↑n + 1] 1)‖ ^ (-(↑k + 1)) ≤\n r { coe := { re := A, im := B }, coe_im_pos := hB } ^ (-(↑k + 1)) * ‖![1, ↑n + 1]‖ ^... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 337,
"column": 4
} | {
"line": 337,
"column": 41
} | {
"line": 337,
"column": 42
} | [
{
"pp": "case hf\nn l : ℕ\nx : ℂ\nhx : x ∈ ℍₒ\n⊢ x + (↑n + 1) ≠ 0",
"ppTerm": "?hf✝",
"assigned": true,
"usedConstants": [
"neg_add_rev",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"AddGroupWithOne.toAddGroup",
"congrArg",
"AddMonoid.toA... | [
"case hf\nn l : ℕ\nx : ℂ\nhx : x ∈ ℍₒ\n⊢ ¬-1 + -↑n = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 338,
"column": 4
} | {
"line": 338,
"column": 29
} | {
"line": 338,
"column": 30
} | [
{
"pp": "case hg\nn l : ℕ\nx : ℂ\nhx : x ∈ ℍₒ\n⊢ x - (↑n + 1) ≠ 0",
"ppTerm": "?hg",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent.0.differentiableOn_iteratedDerivWithin_cotTerm._simp_1_2",
"AddGroupWithOne.toAddG... | [
"case hg\nn l : ℕ\nx : ℂ\nhx : x ∈ ℍₒ\n⊢ ¬x = ↑n + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 367,
"column": 41
} | {
"line": 367,
"column": 52
} | {
"line": 367,
"column": 53
} | [
{
"pp": "z : ℂ\nk : ℕ\nhk : 1 ≤ k\nhz : z ∈ ℍₒ\n⊢ Summable fun n ↦ (z + ↑(↑n + 1)) ^ (-1 - ↑k)",
"ppTerm": "?m.159",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Int.cast_natCast",
"AddMonoid.toAddSemigroup",
"... | [
"z : ℂ\nk : ℕ\nhk : 1 ≤ k\nhz : z ∈ ℍₒ\n⊢ Summable fun n ↦ (z + (↑n + 1)) ^ (-1 - ↑k)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 368,
"column": 8
} | {
"line": 368,
"column": 36
} | {
"line": 368,
"column": 37
} | [
{
"pp": "z : ℂ\nk : ℕ\nhk : 1 ≤ k\nhz : z ∈ ℍₒ\n⊢ Summable fun n ↦ (z + ↑(-(↑n + 1))) ^ (-1 - ↑k)",
"ppTerm": "?m.160",
"assigned": true,
"usedConstants": [
"neg_add_rev",
"Int.instAddCommGroup",
"AddGroup.toSubtractionMonoid",
"Int.cast_neg",
"Int.cast",
"Eq.mpr"... | [
"z : ℂ\nk : ℕ\nhk : 1 ≤ k\nhz : z ∈ ℍₒ\n⊢ Summable fun n ↦ (z + (-1 + -↑n)) ^ (-1 + -↑k)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Real.GoldenRatio | {
"line": 225,
"column": 2
} | {
"line": 233,
"column": 51
} | {
"line": 235,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ φ * ↑(Nat.fib (n + 1)) + ↑(Nat.fib n) = φ ^ (n + 1)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"CharP.cast_eq_zero",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic.Ring.Comm... | [] | induction n with
| zero => simp
| succ n ih =>
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... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.NumberTheory.Real.GoldenRatio | {
"line": 225,
"column": 2
} | {
"line": 233,
"column": 51
} | {
"line": 235,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ φ * ↑(Nat.fib (n + 1)) + ↑(Nat.fib n) = φ ^ (n + 1)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"CharP.cast_eq_zero",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic.Ring.Comm... | [] | induction n with
| zero => simp
| succ n ih =>
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... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Real.GoldenRatio | {
"line": 225,
"column": 2
} | {
"line": 233,
"column": 51
} | {
"line": 235,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ φ * ↑(Nat.fib (n + 1)) + ↑(Nat.fib n) = φ ^ (n + 1)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"CharP.cast_eq_zero",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic.Ring.Comm... | [] | induction n with
| zero => simp
| succ n ih =>
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... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 378,
"column": 2
} | {
"line": 378,
"column": 23
} | {
"line": 378,
"column": 24
} | [
{
"pp": "z✝ : ℂ\nk : ℕ\nhk : 1 ≤ k\nhz✝ : z✝ ∈ ℍₒ\nz : ℂ\nhz : z ∈ ℍₒ\n⊢ ↑π * (↑π * z).cot - z⁻¹ = ∑' (n : ℕ), cotTerm z n",
"ppTerm": "?m.117",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"DivInvMonoid.toInv",
"instHDiv",
"Real... | [
"z✝ : ℂ\nk : ℕ\nhk : 1 ≤ k\nhz✝ : z✝ ∈ ℍₒ\nz : ℂ\nhz : z ∈ ℍₒ\n⊢ ↑π * (↑π * z).cot - z⁻¹ = ∑' (n : ℕ), ((z - (↑n + 1))⁻¹ + (z + (↑n + 1))⁻¹)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SumIntegralExpDecay | {
"line": 50,
"column": 33
} | {
"line": 50,
"column": 44
} | {
"line": 50,
"column": 45
} | [
{
"pp": "k M : ℕ\nc : ℝ\nhc : 0 < c\nx : ℝ\nhx : x ∈ Icc ↑0 ↑M\ny : ℝ\nhy : y ∈ Icc ↑0 ↑M\nhxy : x ≤ y\n⊢ 0 ≤ x",
"ppTerm": "?m.179",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"k M : ℕ\nc : ℝ\nhc : 0 < c\nx : ℝ\nhx : x ∈ Icc ↑0 ↑M\ny : ℝ\nhy : y ∈ Icc ↑0 ↑M\nhxy : x ≤ y\n⊢ 0 ≤ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.FloorPow | {
"line": 63,
"column": 10
} | {
"line": 63,
"column": 21
} | {
"line": 63,
"column": 22
} | [
{
"pp": "case hbc\nu : 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 ... | [
"case hbc\nu : 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 atTo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.FloorPow | {
"line": 64,
"column": 4
} | {
"line": 66,
"column": 40
} | {
"line": 67,
"column": 4
} | [
{
"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... | [
"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 atTop, ↑(c (n ... | obtain ⟨a, ha⟩ :
∃ a : ℕ, ∀ b : ℕ, a ≤ b → (c (b + 1) : ℝ) ≤ (1 + ε) * c b ∧ u (c b) - c b * l ≤ ε * c b :=
eventually_atTop.1 (cgrowth.and L) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.SumIntegralComparisons | {
"line": 69,
"column": 18
} | {
"line": 69,
"column": 29
} | {
"line": 69,
"column": 30
} | [
{
"pp": "a b : ℕ\nf g : ℝ → ℝ\nhab : a ≤ b\nh : ∀ i ∈ Ico a b, ∀ x ∈ Ico ↑i ↑(i + 1), f ↑i ≤ g x\nhg : IntegrableOn g (Ico ↑a ↑b) volume\nA : ∀ i ∈ Finset.Ico a b, IntervalIntegrable g volume ↑i ↑(i + 1)\ni : ℕ\nhi : i ∈ Finset.Ico a b\nx : ℝ\nhx : x ∈ Ioo ↑i ↑(i + 1)\n⊢ i ∈ Ico a b",
"ppTerm": "?m.200",
... | [
"a b : ℕ\nf g : ℝ → ℝ\nhab : a ≤ b\nh : ∀ i ∈ Ico a b, ∀ x ∈ Ico ↑i ↑(i + 1), f ↑i ≤ g x\nhg : IntegrableOn g (Ico ↑a ↑b) volume\nA : ∀ i ∈ Finset.Ico a b, IntervalIntegrable g volume ↑i ↑(i + 1)\ni : ℕ\nhi : i ∈ Finset.Ico a b\nx : ℝ\nhx : x ∈ Ioo ↑i ↑(i + 1)\n⊢ a ≤ i ∧ i < b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.FloorPow | {
"line": 84,
"column": 6
} | {
"line": 84,
"column": 31
} | {
"line": 84,
"column": 32
} | [
{
"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... | [
"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 atTop, ↑(c (n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SumIntegralComparisons | {
"line": 100,
"column": 28
} | {
"line": 100,
"column": 49
} | {
"line": 100,
"column": 49
} | [
{
"pp": "a b : ℕ\nf : ℝ → ℝ\nhab : a ≤ b\nhf : AntitoneOn f (Icc ↑a ↑b)\n⊢ ∫ (x : ℝ) in ↑a..↑a + ↑(b - a), f x ≤ ∑ x ∈ Finset.Ico 0 (b - a), f ↑(a + x)",
"ppTerm": "?m.81",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real.instLE",
"Real"... | [
"a b : ℕ\nf : ℝ → ℝ\nhab : a ≤ b\nhf : AntitoneOn f (Icc ↑a ↑b)\n⊢ ∫ (x : ℝ) in ↑a..↑a + ↑(b - a), f x ≤ ∑ x ∈ Finset.range (b - a), f ↑(a + x)"
] | Nat.Ico_zero_eq_range | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Generator.Preadditive | {
"line": 33,
"column": 33
} | {
"line": 33,
"column": 68
} | {
"line": 33,
"column": 69
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nP : ObjectProperty C\nh𝒢 : P.IsSeparating\nX Y : C\nf : X ⟶ Y\nhf : ∀ (G : C), P G → ∀ (h : G ⟶ X), h ≫ f = 0\n⊢ ∀ (G : C), P G → ∀ (h : G ⟶ X), h ≫ f = h ≫ 0",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nP : ObjectProperty C\nh𝒢 : P.IsSeparating\nX Y : C\nf : X ⟶ Y\nhf : ∀ (G : C), P G → ∀ (h : G ⟶ X), h ≫ f = 0\n⊢ ∀ (G : C), P G → ∀ (h : G ⟶ X), h ≫ f = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Preadditive | {
"line": 34,
"column": 30
} | {
"line": 34,
"column": 82
} | {
"line": 34,
"column": 83
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nP : ObjectProperty C\nh𝒢 : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), (∀ (G : C), P G → ∀ (h : G ⟶ X), h ≫ f = 0) → f = 0\nX Y : C\nf g : X ⟶ Y\nhfg : ∀ (G : C), P G → ∀ (h : G ⟶ X), h ≫ f = h ≫ g\n⊢ ∀ (G : C), P G → ∀ (h : G ⟶ X), h ≫ (f - g) = 0",
"pp... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nP : ObjectProperty C\nh𝒢 : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), (∀ (G : C), P G → ∀ (h : G ⟶ X), h ≫ f = 0) → f = 0\nX Y : C\nf g : X ⟶ Y\nhfg : ∀ (G : C), P G → ∀ (h : G ⟶ X), h ≫ f = h ≫ g\n⊢ ∀ (G : C), P G → ∀ (h : G ⟶ X), h ≫ f = h ≫ g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Preadditive | {
"line": 39,
"column": 33
} | {
"line": 39,
"column": 68
} | {
"line": 39,
"column": 69
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nP : ObjectProperty C\nh𝒢 : P.IsCoseparating\nX Y : C\nf : X ⟶ Y\nhf : ∀ (G : C), P G → ∀ (h : Y ⟶ G), f ≫ h = 0\n⊢ ∀ (G : C), P G → ∀ (h : Y ⟶ G), f ≫ h = 0 ≫ h",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"E... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nP : ObjectProperty C\nh𝒢 : P.IsCoseparating\nX Y : C\nf : X ⟶ Y\nhf : ∀ (G : C), P G → ∀ (h : Y ⟶ G), f ≫ h = 0\n⊢ ∀ (G : C), P G → ∀ (h : Y ⟶ G), f ≫ h = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Preadditive | {
"line": 40,
"column": 30
} | {
"line": 40,
"column": 82
} | {
"line": 40,
"column": 83
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nP : ObjectProperty C\nh𝒢 : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), (∀ (G : C), P G → ∀ (h : Y ⟶ G), f ≫ h = 0) → f = 0\nX Y : C\nf g : X ⟶ Y\nhfg : ∀ (G : C), P G → ∀ (h : Y ⟶ G), f ≫ h = g ≫ h\n⊢ ∀ (G : C), P G → ∀ (h : Y ⟶ G), (f - g) ≫ h = 0",
"pp... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nP : ObjectProperty C\nh𝒢 : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), (∀ (G : C), P G → ∀ (h : Y ⟶ G), f ≫ h = 0) → f = 0\nX Y : C\nf g : X ⟶ Y\nhfg : ∀ (G : C), P G → ∀ (h : Y ⟶ G), f ≫ h = g ≫ h\n⊢ ∀ (G : C), P G → ∀ (h : Y ⟶ G), f ≫ h = g ≫ h"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Preadditive | {
"line": 44,
"column": 37
} | {
"line": 44,
"column": 72
} | {
"line": 44,
"column": 73
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nG : C\nhG : IsSeparator G\nX Y : C\nf : X ⟶ Y\nhf : ∀ (h : G ⟶ X), h ≫ f = 0\n⊢ ∀ (h : G ⟶ X), h ≫ f = h ≫ 0",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nG : C\nhG : IsSeparator G\nX Y : C\nf : X ⟶ Y\nhf : ∀ (h : G ⟶ X), h ≫ f = 0\n⊢ ∀ (h : G ⟶ X), h ≫ f = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Preadditive | {
"line": 46,
"column": 32
} | {
"line": 46,
"column": 84
} | {
"line": 46,
"column": 85
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nG : C\nhG : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), (∀ (h : G ⟶ X), h ≫ f = 0) → f = 0\nX Y : C\nf g : X ⟶ Y\nhfg : ∀ (h : G ⟶ X), h ≫ f = h ≫ g\n⊢ ∀ (h : G ⟶ X), h ≫ (f - g) = 0",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nG : C\nhG : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), (∀ (h : G ⟶ X), h ≫ f = 0) → f = 0\nX Y : C\nf g : X ⟶ Y\nhfg : ∀ (h : G ⟶ X), h ≫ f = h ≫ g\n⊢ ∀ (h : G ⟶ X), h ≫ f = h ≫ g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Preadditive | {
"line": 50,
"column": 37
} | {
"line": 50,
"column": 72
} | {
"line": 50,
"column": 73
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nG : C\nhG : IsCoseparator G\nX Y : C\nf : X ⟶ Y\nhf : ∀ (h : Y ⟶ G), f ≫ h = 0\n⊢ ∀ (h : Y ⟶ G), f ≫ h = 0 ≫ h",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiv... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nG : C\nhG : IsCoseparator G\nX Y : C\nf : X ⟶ Y\nhf : ∀ (h : Y ⟶ G), f ≫ h = 0\n⊢ ∀ (h : Y ⟶ G), f ≫ h = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Preadditive | {
"line": 52,
"column": 32
} | {
"line": 52,
"column": 84
} | {
"line": 52,
"column": 85
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nG : C\nhG : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), (∀ (h : Y ⟶ G), f ≫ h = 0) → f = 0\nX Y : C\nf g : X ⟶ Y\nhfg : ∀ (h : Y ⟶ G), f ≫ h = g ≫ h\n⊢ ∀ (h : Y ⟶ G), (f - g) ≫ h = 0",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nG : C\nhG : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), (∀ (h : Y ⟶ G), f ≫ h = 0) → f = 0\nX Y : C\nf g : X ⟶ Y\nhfg : ∀ (h : Y ⟶ G), f ≫ h = g ≫ h\n⊢ ∀ (h : Y ⟶ G), f ≫ h = g ≫ h"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.FloorPow | {
"line": 113,
"column": 10
} | {
"line": 113,
"column": 21
} | {
"line": 113,
"column": 22
} | [
{
"pp": "case hbc\nu : 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 ... | [
"case hbc\nu : 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 ≤ ε... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.FloorPow | {
"line": 136,
"column": 6
} | {
"line": 136,
"column": 31
} | {
"line": 136,
"column": 32
} | [
{
"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 ≤... | [
"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 ≤ ε * (1 + ε ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.FloorPow | {
"line": 144,
"column": 8
} | {
"line": 144,
"column": 23
} | {
"line": 144,
"column": 24
} | [
{
"pp": "case hbc\nu : 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 ... | [
"case hbc\nu : 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 ≤ ε... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SumIntegralComparisons | {
"line": 190,
"column": 4
} | {
"line": 190,
"column": 50
} | {
"line": 190,
"column": 51
} | [
{
"pp": "case pos\nf : ℝ → ℝ\na b : ℕ\nanti : AntitoneOn f (Icc ↑a ↑b)\nintegrable : IntegrableOn f (Ioi ↑a) volume\nnonneg : ∀ t ∈ Ioi ↑a, 0 ≤ f t\nhab : b < a\n⊢ ∑ n ∈ Finset.Ico a b, f ↑(n + 1) ≤ ∫ (x : ℝ) in Ioi ↑a, f x",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"case pos\nf : ℝ → ℝ\na b : ℕ\nanti : AntitoneOn f (Icc ↑a ↑b)\nintegrable : IntegrableOn f (Ioi ↑a) volume\nnonneg : ∀ t ∈ Ioi ↑a, 0 ≤ f t\nhab : b < a\n⊢ 0 ≤ ∫ (x : ℝ) in Ioi ↑a, f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.FloorPow | {
"line": 225,
"column": 4
} | {
"line": 225,
"column": 15
} | {
"line": 225,
"column": 16
} | [
{
"pp": "N : ℕ\nj : ℝ\nhj : 0 < j\nc : ℝ\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c⁻¹ ^ 2\nthis : c ^ 3 = c ^ 2 * c\n⊢ c ≤ c ^ 2",
"ppTerm": "?m.335",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"PartialOrder.toPreorder",
"Preorder.toLE",
... | [
"N : ℕ\nj : ℝ\nhj : 0 < j\nc : ℝ\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c⁻¹ ^ 2\nthis : c ^ 3 = c ^ 2 * c\n⊢ c ≤ c ^ 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.FloorPow | {
"line": 266,
"column": 6
} | {
"line": 266,
"column": 67
} | {
"line": 266,
"column": 68
} | [
{
"pp": "case h\nc : ℝ\nhc : 1 < c\ni : ℕ\ncpos : 0 < c\nhi : i ≠ 0\n⊢ 1 ≤ c ^ i * c⁻¹",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real.instLE",
"Real",
"instHDiv",
"HMul.h... | [
"case h\nc : ℝ\nhc : 1 < c\ni : ℕ\ncpos : 0 < c\nhi : i ≠ 0\n⊢ c ≤ c ^ i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.FloorPow | {
"line": 281,
"column": 6
} | {
"line": 290,
"column": 41
} | {
"line": 291,
"column": 4
} | [
{
"pp": "N : ℕ\nj : ℝ\nhj : 0 < j\nc : ℝ\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c⁻¹\n⊢ ∑ i ∈ range N with j < c ^ i, 1 / ↑⌊c ^ i⌋₊ ^ 2 ≤ ∑ i ∈ range N with j < c ^ i, (1 - c⁻¹)⁻¹ ^ 2 * (1 / (c ^ i) ^ 2)",
"ppTerm": "?m.317",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNon... | [] | gcongr with i
rw [mul_div_assoc', mul_one, div_le_div_iff₀]; rotate_left
· apply sq_pos_of_pos
refine zero_lt_one.trans_le ?_
simp only [Nat.le_floor, one_le_pow₀, hc.le, Nat.one_le_cast, Nat.cast_one]
· exact sq_pos_of_pos (pow_pos cpos _)
rw [one_mul, ← mul_pow]
gcongr
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecificLimits.FloorPow | {
"line": 281,
"column": 6
} | {
"line": 290,
"column": 41
} | {
"line": 291,
"column": 4
} | [
{
"pp": "N : ℕ\nj : ℝ\nhj : 0 < j\nc : ℝ\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c⁻¹\n⊢ ∑ i ∈ range N with j < c ^ i, 1 / ↑⌊c ^ i⌋₊ ^ 2 ≤ ∑ i ∈ range N with j < c ^ i, (1 - c⁻¹)⁻¹ ^ 2 * (1 / (c ^ i) ^ 2)",
"ppTerm": "?m.317",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNon... | [] | gcongr with i
rw [mul_div_assoc', mul_one, div_le_div_iff₀]; rotate_left
· apply sq_pos_of_pos
refine zero_lt_one.trans_le ?_
simp only [Nat.le_floor, one_le_pow₀, hc.le, Nat.one_le_cast, Nat.cast_one]
· exact sq_pos_of_pos (pow_pos cpos _)
rw [one_mul, ← mul_pow]
gcongr
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecificLimits.FloorPow | {
"line": 276,
"column": 2
} | {
"line": 297,
"column": 11
} | {
"line": 298,
"column": 0
} | [
{
"pp": "N : ℕ\nj : ℝ\nhj : 0 < j\nc : ℝ\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c⁻¹\n⊢ ∑ i ∈ range N with j < ↑⌊c ^ i⌋₊, 1 / ↑⌊c ^ i⌋₊ ^ 2 ≤ c ^ 5 * (c - 1)⁻¹ ^ 3 / j ^ 2",
"ppTerm": "?m.159",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"NonUnitalNonAss... | [] | calc
(∑ i ∈ range N with j < ⌊c ^ i⌋₊, (1 : ℝ) / (⌊c ^ i⌋₊ : ℝ) ^ 2) ≤
∑ i ∈ range N with j < c ^ i, (1 : ℝ) / (⌊c ^ i⌋₊ : ℝ) ^ 2 := by
gcongr with k hk; exact Nat.floor_le (by positivity)
_ ≤ ∑ i ∈ range N with j < c ^ i, (1 - c⁻¹)⁻¹ ^ 2 * ((1 : ℝ) / (c ^ i) ^ 2) := by
gcongr with i
r... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.CategoryTheory.Limits.Indization.FilteredColimits | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 17
} | {
"line": 74,
"column": 2
} | [
{
"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\nh : Nonem... | [
"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\nt : limit ((G.op ⋙ (C... | obtain ⟨t⟩ := h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Comma.Final | {
"line": 71,
"column": 6
} | {
"line": 72,
"column": 82
} | {
"line": 74,
"column": 0
} | [
{
"pp": "case refine_2\nA : Type u₁\ninst✝⁵ : Category.{v₁, u₁} A\nB : Type u₂\ninst✝⁴ : Category.{v₂, u₂} B\nT : Type u₃\ninst✝³ : Category.{v₃, u₃} T\nL : A ⥤ T\nR : B ⥤ T\ninst✝² : IsCofiltered A\ninst✝¹ : IsCofiltered B\ninst✝ : ∀ (b : B), IsCofiltered (CostructuredArrow L (R.obj b))\nj₁ j₂ : Comma L R\nu v... | [] | exact ⟨⟨i₀, IsCofiltered.eq u.right v.right, L.map (β ≫ va₁) ≫ Q.hom⟩,
⟨β ≫ va₂, IsCofiltered.eqHom u.right v.right, by cat_disch⟩, by cat_disch⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Comma.StructuredArrow.CommaMap | {
"line": 41,
"column": 10
} | {
"line": 44,
"column": 49
} | {
"line": 45,
"column": 8
} | [
{
"pp": "C : Type u₁\ninst✝⁶ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\nT : Type u₃\ninst✝⁴ : Category.{v₃, u₃} T\nL : C ⥤ T\nR : D ⥤ T\nC' : Type u₄\ninst✝³ : Category.{v₄, u₄} C'\nD' : Type u₅\ninst✝² : Category.{v₅, u₅} D'\nT' : Type u₆\ninst✝¹ : Category.{v₆, u₆} T'\nL' : C' ⥤ T'\nR' ... | [
"C : Type u₁\ninst✝⁶ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\nT : Type u₃\ninst✝⁴ : Category.{v₃, u₃} T\nL : C ⥤ T\nR : D ⥤ T\nC' : Type u₄\ninst✝³ : Category.{v₄, u₄} C'\nD' : Type u₅\ninst✝² : Category.{v₅, u₅} D'\nT' : Type u₆\ninst✝¹ : Category.{v₆, u₆} T'\nL' : C' ⥤ T'\nR' : D' ⥤ T'\nF... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Comma.StructuredArrow.CommaMap | {
"line": 59,
"column": 6
} | {
"line": 59,
"column": 17
} | {
"line": 59,
"column": 18
} | [
{
"pp": "C : Type u₁\ninst✝⁶ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\nT : Type u₃\ninst✝⁴ : Category.{v₃, u₃} T\nL : C ⥤ T\nR : D ⥤ T\nC' : Type u₄\ninst✝³ : Category.{v₄, u₄} C'\nD' : Type u₅\ninst✝² : Category.{v₅, u₅} D'\nT' : Type u₆\ninst✝¹ : Category.{v₆, u₆} T'\nL' : C' ⥤ T'\nR' ... | [
"C : Type u₁\ninst✝⁶ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\nT : Type u₃\ninst✝⁴ : Category.{v₃, u₃} T\nL : C ⥤ T\nR : D ⥤ T\nC' : Type u₄\ninst✝³ : Category.{v₄, u₄} C'\nD' : Type u₅\ninst✝² : Category.{v₅, u₅} D'\nT' : Type u₆\ninst✝¹ : Category.{v₆, u₆} T'\nL' : C' ⥤ T'\nR' : D' ⥤ T'\nF... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.FilteredColimitCommutesProduct | {
"line": 252,
"column": 4
} | {
"line": 253,
"column": 66
} | {
"line": 254,
"column": 4
} | [
{
"pp": "case refine_1\nα : 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\ny y' : (fun X ↦ X) (colimit (pointwiseProduct F))\nhy : (hom (colimitPointwiseProductToProductColimit F)) y = (hom (colimitPointwiseProductToProductColimit... | [
"case refine_1\nα : 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\ny y' : (fun X ↦ X) (colimit (pointwiseProduct F))\nhy : (hom (colimitPointwiseProductToProductColimit F)) y = (hom (colimitPointwiseProductToProductColimit F)) y'\nky ... | let yk' : (pointwiseProduct F).obj k :=
(pointwiseProduct F).map (IsFiltered.rightToMax ky ky') yk₀' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Connected | {
"line": 73,
"column": 32
} | {
"line": 73,
"column": 43
} | {
"line": 73,
"column": 44
} | [
{
"pp": "J : Type w\ninst✝⁶ : Category.{w', w} J\ninst✝⁵ : IsConnected J\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasPullbacks C\ninst✝¹ : HasColimitsOfShape J C\ninst✝ : HasExactColimitsOfShape J C\nF : J ⥤ C\nc : Cocone F\nhc : IsColimit c\nX Y : C\nf : X ⟶ c.pt\ng : c.pt... | [
"J : Type w\ninst✝⁶ : Category.{w', w} J\ninst✝⁵ : IsConnected J\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasPullbacks C\ninst✝¹ : HasColimitsOfShape J C\ninst✝ : HasExactColimitsOfShape J C\nF : J ⥤ C\nc : Cocone F\nhc : IsColimit c\nX Y : C\nf : X ⟶ c.pt\ng : c.pt ⟶ Y\nhf : ∀... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Connected | {
"line": 111,
"column": 31
} | {
"line": 111,
"column": 42
} | {
"line": 111,
"column": 43
} | [
{
"pp": "J : Type w\ninst✝⁶ : Category.{w', w} J\ninst✝⁵ : IsConnected J\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasPushouts C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : HasExactLimitsOfShape J C\nF : J ⥤ C\nc : Cone F\nhc : IsLimit c\nX Y : C\ng : Y ⟶ c.pt\nf : c.pt ⟶ X\nhf ... | [
"J : Type w\ninst✝⁶ : Category.{w', w} J\ninst✝⁵ : IsConnected J\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasPushouts C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : HasExactLimitsOfShape J C\nF : J ⥤ C\nc : Cone F\nhc : IsLimit c\nX Y : C\ng : Y ⟶ c.pt\nf : c.pt ⟶ X\nhf : ∀ (j : J),... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.FilteredColimitCommutesProduct | {
"line": 274,
"column": 48
} | {
"line": 274,
"column": 88
} | {
"line": 274,
"column": 89
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.FilteredColimitCommutesProduct | {
"line": 282,
"column": 25
} | {
"line": 282,
"column": 49
} | {
"line": 282,
"column": 50
} | [
{
"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\nx : (fun X ↦ X) (∏ᶜ fun s ↦ colimit (F s))\nk : (s : α) → I s\np : (s : α) → (F s).obj (k s)\nhk : ∀ (s : α), (hom (colimit.ι (F s) (k s))) (p s) = (hom (Pi.π (fun s ↦ co... | [
"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\nx : (fun X ↦ X) (∏ᶜ fun s ↦ colimit (F s))\nk : (s : α) → I s\np : (s : α) → (F s).obj (k s)\nhk : ∀ (s : α), (hom (colimit.ι (F s) (k s))) (p s) = (hom (Pi.π (fun s ↦ colimit (F s))... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.ModuleEmbedding.GabrielPopescu | {
"line": 93,
"column": 53
} | {
"line": 93,
"column": 83
} | {
"line": 93,
"column": 84
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : IsGrothendieckAbelian.{v, v, u} C\nG : C\nhG : IsSeparator G\nA B : C\nM : ModuleCat (End G)ᵐᵒᵖ\ng : M ⟶ ModuleCat.of (End G)ᵐᵒᵖ (G ⟶ A)\nhg : Mono g\nf : M ⟶ ModuleCat.of (End G)ᵐᵒᵖ (G ⟶ B)\nF : Finset (Discrete ↑M)\nh : G ⟶ pullback ... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : IsGrothendieckAbelian.{v, v, u} C\nG : C\nhG : IsSeparator G\nA B : C\nM : ModuleCat (End G)ᵐᵒᵖ\ng : M ⟶ ModuleCat.of (End G)ᵐᵒᵖ (G ⟶ A)\nhg : Mono g\nf : M ⟶ ModuleCat.of (End G)ᵐᵒᵖ (G ⟶ B)\nF : Finset (Discrete ↑M)\nh : G ⟶ pullback (∑ a ∈ F.att... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.ModuleEmbedding.GabrielPopescu | {
"line": 132,
"column": 4
} | {
"line": 143,
"column": 15
} | {
"line": 145,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : IsGrothendieckAbelian.{v, v, u} C\nG : C\nhG : IsSeparator G\nB : C\nhB : Injective B\n⊢ Injective ((preadditiveCoyonedaObj G).obj B)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"CategoryTheory.IsGrothen... | [] | rw [← Module.injective_iff_injective_object]
simp only [preadditiveCoyonedaObj_obj_carrier]
refine Module.Baer.injective (fun M g => ?_)
have h := exists_d_comp_eq_d hG B (ModuleCat.ofHom
⟨⟨fun i => i.1.unop, by cat_disch⟩, by cat_disch⟩) ?_ (ModuleCat.ofHom g)
· obtain ⟨l, hl⟩ := h
refine ⟨... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.ModuleEmbedding.GabrielPopescu | {
"line": 132,
"column": 4
} | {
"line": 143,
"column": 15
} | {
"line": 145,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : IsGrothendieckAbelian.{v, v, u} C\nG : C\nhG : IsSeparator G\nB : C\nhB : Injective B\n⊢ Injective ((preadditiveCoyonedaObj G).obj B)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"CategoryTheory.IsGrothen... | [] | rw [← Module.injective_iff_injective_object]
simp only [preadditiveCoyonedaObj_obj_carrier]
refine Module.Baer.injective (fun M g => ?_)
have h := exists_d_comp_eq_d hG B (ModuleCat.ofHom
⟨⟨fun i => i.1.unop, by cat_disch⟩, by cat_disch⟩) ?_ (ModuleCat.ofHom g)
· obtain ⟨l, hl⟩ := h
refine ⟨... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Preserves.Shapes.Square | {
"line": 107,
"column": 4
} | {
"line": 107,
"column": 46
} | {
"line": 107,
"column": 47
} | [
{
"pp": "case refine_1\nsq₁ : Square (Type v)\nsq₂ : Square (Type u)\ne₁ : sq₁.X₁ ≃ sq₂.X₁\ne₂ : sq₁.X₂ ≃ sq₂.X₂\ne₃ : sq₁.X₃ ≃ sq₂.X₃\ne₄ : sq₁.X₄ ≃ sq₂.X₄\ncomm₁₂ : ⇑e₂ ∘ ⇑(ConcreteCategory.hom sq₁.f₁₂) = ⇑(ConcreteCategory.hom sq₂.f₁₂) ∘ ⇑e₁\ncomm₁₃ : ⇑e₃ ∘ ⇑(ConcreteCategory.hom sq₁.f₁₃) = ⇑(ConcreteCategor... | [
"case refine_1\nsq₁ : Square (Type v)\nsq₂ : Square (Type u)\ne₁ : sq₁.X₁ ≃ sq₂.X₁\ne₂ : sq₁.X₂ ≃ sq₂.X₂\ne₃ : sq₁.X₃ ≃ sq₂.X₃\ne₄ : sq₁.X₄ ≃ sq₂.X₄\ncomm₁₂ : ⇑e₂ ∘ ⇑(ConcreteCategory.hom sq₁.f₁₂) = ⇑(ConcreteCategory.hom sq₂.f₁₂) ∘ ⇑e₁\ncomm₁₃ : ⇑e₃ ∘ ⇑(ConcreteCategory.hom sq₁.f₁₃) = ⇑(ConcreteCategory.hom sq₂.f₁... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Preserves.Shapes.Square | {
"line": 108,
"column": 4
} | {
"line": 108,
"column": 46
} | {
"line": 108,
"column": 47
} | [
{
"pp": "case refine_2\nsq₁ : Square (Type v)\nsq₂ : Square (Type u)\ne₁ : sq₁.X₁ ≃ sq₂.X₁\ne₂ : sq₁.X₂ ≃ sq₂.X₂\ne₃ : sq₁.X₃ ≃ sq₂.X₃\ne₄ : sq₁.X₄ ≃ sq₂.X₄\ncomm₁₂ : ⇑e₂ ∘ ⇑(ConcreteCategory.hom sq₁.f₁₂) = ⇑(ConcreteCategory.hom sq₂.f₁₂) ∘ ⇑e₁\ncomm₁₃ : ⇑e₃ ∘ ⇑(ConcreteCategory.hom sq₁.f₁₃) = ⇑(ConcreteCategor... | [
"case refine_2\nsq₁ : Square (Type v)\nsq₂ : Square (Type u)\ne₁ : sq₁.X₁ ≃ sq₂.X₁\ne₂ : sq₁.X₂ ≃ sq₂.X₂\ne₃ : sq₁.X₃ ≃ sq₂.X₃\ne₄ : sq₁.X₄ ≃ sq₂.X₄\ncomm₁₂ : ⇑e₂ ∘ ⇑(ConcreteCategory.hom sq₁.f₁₂) = ⇑(ConcreteCategory.hom sq₂.f₁₂) ∘ ⇑e₁\ncomm₁₃ : ⇑e₃ ∘ ⇑(ConcreteCategory.hom sq₁.f₁₃) = ⇑(ConcreteCategory.hom sq₂.f₁... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Preserves.Shapes.Square | {
"line": 109,
"column": 4
} | {
"line": 109,
"column": 46
} | {
"line": 109,
"column": 47
} | [
{
"pp": "case refine_3\nsq₁ : Square (Type v)\nsq₂ : Square (Type u)\ne₁ : sq₁.X₁ ≃ sq₂.X₁\ne₂ : sq₁.X₂ ≃ sq₂.X₂\ne₃ : sq₁.X₃ ≃ sq₂.X₃\ne₄ : sq₁.X₄ ≃ sq₂.X₄\ncomm₁₂ : ⇑e₂ ∘ ⇑(ConcreteCategory.hom sq₁.f₁₂) = ⇑(ConcreteCategory.hom sq₂.f₁₂) ∘ ⇑e₁\ncomm₁₃ : ⇑e₃ ∘ ⇑(ConcreteCategory.hom sq₁.f₁₃) = ⇑(ConcreteCategor... | [
"case refine_3\nsq₁ : Square (Type v)\nsq₂ : Square (Type u)\ne₁ : sq₁.X₁ ≃ sq₂.X₁\ne₂ : sq₁.X₂ ≃ sq₂.X₂\ne₃ : sq₁.X₃ ≃ sq₂.X₃\ne₄ : sq₁.X₄ ≃ sq₂.X₄\ncomm₁₂ : ⇑e₂ ∘ ⇑(ConcreteCategory.hom sq₁.f₁₂) = ⇑(ConcreteCategory.hom sq₂.f₁₂) ∘ ⇑e₁\ncomm₁₃ : ⇑e₃ ∘ ⇑(ConcreteCategory.hom sq₁.f₁₃) = ⇑(ConcreteCategory.hom sq₂.f₁... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Preserves.Shapes.Square | {
"line": 110,
"column": 4
} | {
"line": 110,
"column": 46
} | {
"line": 110,
"column": 47
} | [
{
"pp": "case refine_4\nsq₁ : Square (Type v)\nsq₂ : Square (Type u)\ne₁ : sq₁.X₁ ≃ sq₂.X₁\ne₂ : sq₁.X₂ ≃ sq₂.X₂\ne₃ : sq₁.X₃ ≃ sq₂.X₃\ne₄ : sq₁.X₄ ≃ sq₂.X₄\ncomm₁₂ : ⇑e₂ ∘ ⇑(ConcreteCategory.hom sq₁.f₁₂) = ⇑(ConcreteCategory.hom sq₂.f₁₂) ∘ ⇑e₁\ncomm₁₃ : ⇑e₃ ∘ ⇑(ConcreteCategory.hom sq₁.f₁₃) = ⇑(ConcreteCategor... | [
"case refine_4\nsq₁ : Square (Type v)\nsq₂ : Square (Type u)\ne₁ : sq₁.X₁ ≃ sq₂.X₁\ne₂ : sq₁.X₂ ≃ sq₂.X₂\ne₃ : sq₁.X₃ ≃ sq₂.X₃\ne₄ : sq₁.X₄ ≃ sq₂.X₄\ncomm₁₂ : ⇑e₂ ∘ ⇑(ConcreteCategory.hom sq₁.f₁₂) = ⇑(ConcreteCategory.hom sq₂.f₁₂) ∘ ⇑e₁\ncomm₁₃ : ⇑e₃ ∘ ⇑(ConcreteCategory.hom sq₁.f₁₃) = ⇑(ConcreteCategory.hom sq₂.f₁... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.Preradical.Colon | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 13
} | {
"line": 146,
"column": 14
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : Abelian C\nΦ Ψ : Preradical C\nX : C\n⊢ IsPullback ((Φ.colon Ψ).ι.app X) ((Φ.colonπ Ψ).app X) (Φ.π.app X) (Ψ.ι.app (Φ.quotient.obj X))",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.Preradical.col... | [
"C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : Abelian C\nΦ Ψ : Preradical C\nX : C\n⊢ IsPullback ((Φ.colon Ψ).ι.app X) ((Φ.colonπ Ψ).app X) (Φ.π.app X) (Ψ.ι.app (Φ.quotient.obj X))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.Preradical.Colon | {
"line": 194,
"column": 2
} | {
"line": 195,
"column": 39
} | {
"line": 195,
"column": 40
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : Abelian C\nΦ Ψ : Preradical C\n⊢ IsIso (Φ.toColon Ψ) ↔ IsZero (Φ.quotient ⋙ Ψ.r)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Over",
"CategoryTheory.Functor",
"_private.Mathl... | [
"C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : Abelian C\nΦ Ψ : Preradical C\n⊢ (∀ (X : C), IsIso ((Over.Hom.left (Φ.toColon Ψ).hom).app X)) ↔ ∀ (X : C), IsZero (Ψ.r.obj (Φ.quotient.obj X))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.Injective.Resolution | {
"line": 192,
"column": 4
} | {
"line": 192,
"column": 25
} | {
"line": 192,
"column": 26
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nI J : InjectiveResolution X\n⊢ Homotopy (desc (𝟙 X ≫ 𝟙 X) I I) (𝟙 I.cocomplex)",
"ppTerm": "?m.75",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"Eq.mpr",
"HomologicalCompl... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nI J : InjectiveResolution X\n⊢ Homotopy (desc (𝟙 X) I I) (𝟙 I.cocomplex)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.Injective.Resolution | {
"line": 194,
"column": 4
} | {
"line": 194,
"column": 25
} | {
"line": 194,
"column": 26
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nI J : InjectiveResolution X\n⊢ Homotopy (desc (𝟙 X ≫ 𝟙 X) J J) (𝟙 J.cocomplex)",
"ppTerm": "?m.105",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"Eq.mpr",
"HomologicalComp... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nI J : InjectiveResolution X\n⊢ Homotopy (desc (𝟙 X) J J) (𝟙 J.cocomplex)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.Injective.Resolution | {
"line": 252,
"column": 2
} | {
"line": 252,
"column": 17
} | {
"line": 254,
"column": 0
} | [
{
"pp": "case g_comm\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasInjectiveResolutions C\nX Y : C\nf : X ⟶ Y\nI : InjectiveResolution X\nJ : InjectiveResolution Y\nφ : I.cocomplex ⟶ J.cocomplex\ncomm : I.ι.f 0 ≫ φ.f 0 = f ≫ J.ι.f 0\n⊢ (injectiveResolution X).ι ≫\n desc f (injecti... | [] | all_goals aesop | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.CategoryTheory.Abelian.Injective.Ext | {
"line": 204,
"column": 9
} | {
"line": 204,
"column": 89
} | {
"line": 204,
"column": 89
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX Y : C\nR : InjectiveResolution Y\nn : ℕ\nf : X ⟶ R.cocomplex.X n\nm : ℕ\nhm : n + 1 = m\nhf : f ≫ R.cocomplex.d n m = 0\np : ℕ\nhp : p + 1 = n\nx✝ :\n ∃ g,\n g ≫ (R.cochainComplexXIso (↑p) p ⋯).hom ≫ R.cocomplex.d p n ≫... | [] | simp only [← cancel_mono (R.cochainComplexXIso n n rfl).inv, Category.assoc, hg] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Abelian.Injective.Ext | {
"line": 204,
"column": 9
} | {
"line": 204,
"column": 89
} | {
"line": 204,
"column": 89
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX Y : C\nR : InjectiveResolution Y\nn : ℕ\nf : X ⟶ R.cocomplex.X n\nm : ℕ\nhm : n + 1 = m\nhf : f ≫ R.cocomplex.d n m = 0\np : ℕ\nhp : p + 1 = n\nx✝ :\n ∃ g,\n g ≫ (R.cochainComplexXIso (↑p) p ⋯).hom ≫ R.cocomplex.d p n ≫... | [] | simp only [← cancel_mono (R.cochainComplexXIso n n rfl).inv, Category.assoc, hg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.Injective.Ext | {
"line": 204,
"column": 9
} | {
"line": 204,
"column": 89
} | {
"line": 204,
"column": 89
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX Y : C\nR : InjectiveResolution Y\nn : ℕ\nf : X ⟶ R.cocomplex.X n\nm : ℕ\nhm : n + 1 = m\nhf : f ≫ R.cocomplex.d n m = 0\np : ℕ\nhp : p + 1 = n\nx✝ :\n ∃ g,\n g ≫ (R.cochainComplexXIso (↑p) p ⋯).hom ≫ R.cocomplex.d p n ≫... | [] | simp only [← cancel_mono (R.cochainComplexXIso n n rfl).inv, Category.assoc, hg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.Injective.Ext | {
"line": 214,
"column": 7
} | {
"line": 215,
"column": 61
} | {
"line": 215,
"column": 62
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX Y : C\nR : InjectiveResolution Y\nn m : ℕ\nhm : n + 1 = m\nf : X ⟶ R.cochainComplex.X ↑n\nhf : f ≫ R.cochainComplex.d ↑n ↑m = 0\n⊢ (f ≫ (R.cochainComplexXIso (↑n) n ⋯).hom) ≫ R.cocomplex.d n m = 0",
"ppTerm": "?m.163",
... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX Y : C\nR : InjectiveResolution Y\nn m : ℕ\nhm : n + 1 = m\nf : X ⟶ R.cochainComplex.X ↑n\nhf : f ≫ R.cochainComplex.d ↑n ↑m = 0\n⊢ f ≫ (R.cochainComplexXIso (↑n) n ⋯).hom ≫ R.cocomplex.d n m ≫ (R.cochainComplexXIso (↑m) m ⋯).inv = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.Pseudoelements | {
"line": 415,
"column": 12
} | {
"line": 415,
"column": 23
} | {
"line": 415,
"column": 24
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nx y : Pseudoelement P\na a' : Over P\nh : pseudoApply f ⟦a⟧ = pseudoApply f ⟦a'⟧\nR : C\np : R ⟶ ((fun g ↦ app f g) a).left\nq : R ⟶ ((fun g ↦ app f g) a').left\nep : Epi p\nw✝¹ : Epi q\ncomm : p ≫ ((fun g ↦ app f g) a).hom ... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nx y : Pseudoelement P\na a' : Over P\nh : pseudoApply f ⟦a⟧ = pseudoApply f ⟦a'⟧\nR : C\np : R ⟶ ((fun g ↦ app f g) a).left\nq : R ⟶ ((fun g ↦ app f g) a').left\nep : Epi p\nw✝¹ : Epi q\ncomm : p ≫ ((fun g ↦ app f g) a).hom = q ≫ ((fun ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.Projective.Ext | {
"line": 212,
"column": 2
} | {
"line": 212,
"column": 30
} | {
"line": 212,
"column": 31
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX Y : C\nR : ProjectiveResolution X\nn : ℕ\nf : R.complex.X n ⟶ Y\nm : ℕ\nhm : n + 1 = m\nhf : R.complex.d m n ≫ f = 0\np : ℕ\nhp : p + 1 = n\nx✝ :\n ∃ g,\n ((R.cochainComplexXIso (-↑n) n ⋯).hom ≫ R.complex.d n p ≫ (R.coc... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX Y : C\nR : ProjectiveResolution X\nn : ℕ\nf : R.complex.X n ⟶ Y\nm : ℕ\nhm : n + 1 = m\nhf : R.complex.d m n ≫ f = 0\np : ℕ\nhp : p + 1 = n\nx✝ :\n ∃ g,\n ((R.cochainComplexXIso (-↑n) n ⋯).hom ≫ R.complex.d n p ≫ (R.cochainComplexX... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.Projective.Ext | {
"line": 222,
"column": 2
} | {
"line": 222,
"column": 50
} | {
"line": 222,
"column": 51
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX Y : C\nR : ProjectiveResolution X\nn m : ℕ\nhm : n + 1 = m\nf : R.cochainComplex.X (-↑n) ⟶ Y\nhf : R.cochainComplex.d (-↑m) (-↑n) ≫ f = 0\n⊢ (R.cochainComplexXIso (-↑m) m ⋯).hom ≫ R.complex.d m n ≫ (R.cochainComplexXIso (-↑... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX Y : C\nR : ProjectiveResolution X\nn m : ℕ\nhm : n + 1 = m\nf : R.cochainComplex.X (-↑n) ⟶ Y\nhf : R.cochainComplex.d (-↑m) (-↑n) ≫ f = 0\n⊢ R.complex.d m n ≫ (R.cochainComplexXIso (-↑n) n ⋯).inv ≫ f = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.Pseudoelements | {
"line": 448,
"column": 2
} | {
"line": 453,
"column": 58
} | {
"line": 454,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝ : Ring R\nG : ModuleCat R\nx y : Over G\nP : ModuleCat R\np : P ⟶ x.left\nq : P ⟶ y.left\nhp : Epi p\nhq : Epi q\nH : p ≫ x.hom = q ≫ y.hom\na : ↑G\nha : a ∈ (ModuleCat.Hom.hom x.hom).range\n⊢ a ∈ (ModuleCat.Hom.hom y.hom).range",
"ppTerm": "?refine_1",
"assig... | [
"case refine_2\nR : Type u_1\ninst✝ : Ring R\nG : ModuleCat R\nx y : Over G\nP : ModuleCat R\np : P ⟶ x.left\nq : P ⟶ y.left\nhp : Epi p\nhq : Epi q\nH : p ≫ x.hom = q ≫ y.hom\na : ↑G\nha : a ∈ (ModuleCat.Hom.hom y.hom).range\n⊢ a ∈ (ModuleCat.Hom.hom x.hom).range"
] | · obtain ⟨a', ha'⟩ := ha
obtain ⟨a'', ha''⟩ := (ModuleCat.epi_iff_surjective p).1 hp a'
refine ⟨q a'', ?_⟩
dsimp at ha' ⊢
rw [← LinearMap.comp_apply, ← ModuleCat.hom_comp, ← H,
ModuleCat.hom_comp, LinearMap.comp_apply, ha'', ha'] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Abelian.RightDerived | {
"line": 318,
"column": 22
} | {
"line": 320,
"column": 49
} | {
"line": 321,
"column": 6
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\nD : Type u_1\ninst✝⁴ : Category.{v_1, u_1} D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : F.Additive\nX Y : C\nf : X ⟶ Y\n⊢ F.map f ≫\n (injectiveResolution Y).toRightDerivedZero' F ≫\n ((F.mapHomolog... | [
"C : Type u\ninst✝⁵ : Category.{v, u} C\nD : Type u_1\ninst✝⁴ : Category.{v_1, u_1} D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : F.Additive\nX Y : C\nf : X ⟶ Y\n⊢ (injectiveResolution X).toRightDerivedZero' F ≫\n HomologicalComplex.cyclesMap\n ((F.m... | InjectiveResolution.toRightDerivedZero'_naturality_assoc f
(injectiveResolution X) (injectiveResolution Y)
(InjectiveResolution.desc f _ _) (by simp), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.SerreClass.Localization | {
"line": 111,
"column": 6
} | {
"line": 111,
"column": 58
} | {
"line": 111,
"column": 59
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Abelian C\nD : Type u'\ninst✝³ : Category.{v', u'} D\nL : C ⥤ D\nP : ObjectProperty C\ninst✝² : P.IsSerreClass\nE : Type u''\ninst✝¹ : Category.{v'', u''} E\ninst✝ : Abelian E\nX' X Y : C\nf₁ f₂ : X ⟶ Y\ns : X' ⟶ X\nhs : P.isoModSerre s\ne... | [
"case refine_2\nC : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Abelian C\nD : Type u'\ninst✝³ : Category.{v', u'} D\nL : C ⥤ D\nP : ObjectProperty C\ninst✝² : P.IsSerreClass\nE : Type u''\ninst✝¹ : Category.{v'', u''} E\ninst✝ : Abelian E\nX' X Y : C\nf₁ f₂ : X ⟶ Y\ns : X' ⟶ X\nhs : P.isoModSerre s\neq : s ≫ f₁ =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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