module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.Real.Pi.Bounds | {
"line": 29,
"column": 2
} | {
"line": 34,
"column": 37
} | {
"line": 35,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ 2 ^ (n + 1) * √(2 - sqrtTwoAddSeries 0 n) < π",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"pow_pos",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"in... | [
"n : ℕ\nthis : √(2 - sqrtTwoAddSeries 0 n) / 2 * 2 ^ (n + 2) < π\n⊢ 2 ^ (n + 1) * √(2 - sqrtTwoAddSeries 0 n) < π"
] | have : √(2 - sqrtTwoAddSeries 0 n) / 2 * 2 ^ (n + 2) < π := by
rw [← lt_div_iff₀, ← sin_pi_over_two_pow_succ]
focus
apply sin_lt
apply div_pos pi_pos
all_goals apply pow_pos; norm_num | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.Real.Irrational | {
"line": 133,
"column": 79
} | {
"line": 135,
"column": 21
} | {
"line": 137,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ Irrational √↑n ↔ ¬IsSquare n",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Rat.instMul",
"Real",
"DivisionRing.toRatCast",
"congrArg",
"Nat.cast_nonneg",
"Iff.rfl",
... | [] | by
rw [← Rat.isSquare_natCast_iff, ← irrational_sqrt_ratCast_iff_of_nonneg n.cast_nonneg,
Rat.cast_natCast] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Real.Pi.Irrational | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 41
} | {
"line": 76,
"column": 2
} | [
{
"pp": "θ : ℝ\nn : ℕ\nf : ℝ → ℝ := fun x ↦ 1 - x ^ 2\nu₁ : ℝ → ℝ := fun x ↦ f x ^ (n + 1)\nu₁' : ℝ → ℝ := fun x ↦ -(2 * (↑n + 1) * x * f x ^ n)\nv₁ : ℝ → ℝ := fun x ↦ sin (x * θ)\nv₁' : ℝ → ℝ := fun x ↦ cos (x * θ) * θ\nu₂ : ℝ → ℝ := fun x ↦ x * f x ^ n\nu₂' : ℝ → ℝ := fun x ↦ f x ^ n - 2 * ↑n * x ^ 2 * f x ^ ... | [
"θ : ℝ\nn : ℕ\nf : ℝ → ℝ := fun x ↦ 1 - x ^ 2\nu₁ : ℝ → ℝ := fun x ↦ f x ^ (n + 1)\nu₁' : ℝ → ℝ := fun x ↦ -(2 * (↑n + 1) * x * f x ^ n)\nv₁ : ℝ → ℝ := fun x ↦ sin (x * θ)\nv₁' : ℝ → ℝ := fun x ↦ cos (x * θ) * θ\nu₂ : ℝ → ℝ := fun x ↦ x * f x ^ n\nu₂' : ℝ → ℝ := fun x ↦ f x ^ n - 2 * ↑n * x ^ 2 * f x ^ (n - 1)\nv₂ ... | let v₂' (x : ℝ) : ℝ := -sin (x * θ) * θ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.Real.Hyperreal | {
"line": 342,
"column": 2
} | {
"line": 344,
"column": 57
} | {
"line": 346,
"column": 0
} | [
{
"pp": "x : ℝ*\nhx : Germ.Tendsto x atTop\n⊢ ArchimedeanClass.mk x < 0",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"ArchimedeanOrder.of",
"Hyperreal.instField",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"instHSMul",
"Preorde... | [] | have : 0 < x := lt_of_tendsto_atTop 0 hx
intro n
simpa [abs_of_pos this] using! lt_of_tendsto_atTop n hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Real.Hyperreal | {
"line": 342,
"column": 2
} | {
"line": 344,
"column": 57
} | {
"line": 346,
"column": 0
} | [
{
"pp": "x : ℝ*\nhx : Germ.Tendsto x atTop\n⊢ ArchimedeanClass.mk x < 0",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"ArchimedeanOrder.of",
"Hyperreal.instField",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"instHSMul",
"Preorde... | [] | have : 0 < x := lt_of_tendsto_atTop 0 hx
intro n
simpa [abs_of_pos this] using! lt_of_tendsto_atTop n hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Complex.Arctan | {
"line": 62,
"column": 12
} | {
"line": 62,
"column": 26
} | {
"line": 62,
"column": 27
} | [
{
"pp": "case inl\nz : ℂ\nh₂ : z.re ≤ π / 2\nk : ℤ\nh₁ : -(π / 2) < ↑(2 * k + 1) * π / 2\nnr : z.re = ↑(2 * k + 1) * π / 2\n⊢ z.re = π / 2",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Int.cast",
"Real",
"instHDiv",
"Real.pi",
"HMul.hMul",
"Monoid.toMu... | [
"case inl\nz : ℂ\nh₂ : z.re ≤ π / 2\nk : ℤ\nh₁ : -(π / 2) < ↑(2 * k + 1) * (π / 2)\nnr : z.re = ↑(2 * k + 1) * π / 2\n⊢ z.re = π / 2"
] | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arctan | {
"line": 63,
"column": 31
} | {
"line": 63,
"column": 45
} | {
"line": 63,
"column": 46
} | [
{
"pp": "case inl\nz : ℂ\nk : ℤ\nh₂ : ↑(2 * k + 1) * π / 2 ≤ 1 * (π / 2)\nh₁ : -1 < ↑(2 * k + 1)\nnr : z.re = ↑(2 * k + 1) * π / 2\n⊢ z.re = π / 2",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Int.cast",
"MulOne.toOne",
"Real.instLE",
"Real",
"Preorder.toLT"... | [
"case inl\nz : ℂ\nk : ℤ\nh₂ : ↑(2 * k + 1) * (π / 2) ≤ 1 * (π / 2)\nh₁ : -1 < ↑(2 * k + 1)\nnr : z.re = ↑(2 * k + 1) * π / 2\n⊢ z.re = π / 2"
] | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arctan | {
"line": 91,
"column": 4
} | {
"line": 91,
"column": 41
} | {
"line": 92,
"column": 2
} | [
{
"pp": "case h₁\nx : ℝ\n⊢ -(π / 2) < Real.arctan x",
"ppTerm": "?h₁",
"assigned": true,
"usedConstants": [
"Real.neg_pi_div_two_lt_arctan"
],
"usedFVars": [
"x"
],
"usedGoals": []
}
] | [] | exact Real.neg_pi_div_two_lt_arctan _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Complex.Arctan | {
"line": 91,
"column": 4
} | {
"line": 91,
"column": 41
} | {
"line": 92,
"column": 2
} | [
{
"pp": "case h₁\nx : ℝ\n⊢ -(π / 2) < Real.arctan x",
"ppTerm": "?h₁",
"assigned": true,
"usedConstants": [
"Real.neg_pi_div_two_lt_arctan"
],
"usedFVars": [
"x"
],
"usedGoals": []
}
] | [] | exact Real.neg_pi_div_two_lt_arctan _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Complex.Arctan | {
"line": 91,
"column": 4
} | {
"line": 91,
"column": 41
} | {
"line": 92,
"column": 2
} | [
{
"pp": "case h₁\nx : ℝ\n⊢ -(π / 2) < Real.arctan x",
"ppTerm": "?h₁",
"assigned": true,
"usedConstants": [
"Real.neg_pi_div_two_lt_arctan"
],
"usedFVars": [
"x"
],
"usedGoals": []
}
] | [] | exact Real.neg_pi_div_two_lt_arctan _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Complex.Arctan | {
"line": 96,
"column": 29
} | {
"line": 96,
"column": 55
} | {
"line": 96,
"column": 56
} | [
{
"pp": "z : ℂ\nhz : ‖z‖ < 1\n⊢ |(1 + z).arg| < π / 2",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"instHDiv",
"Real.pi",
"Real.lattice",
"Real.instZero",
"abs",
"congrArg",
"Real.instDivInvMo... | [
"z : ℂ\nhz : ‖z‖ < 1\n⊢ 0 < (1 + z).re ∨ 1 + z = 0"
] | abs_arg_lt_pi_div_two_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Real.Pi.Wallis | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 42
} | {
"line": 99,
"column": 2
} | [
{
"pp": "this : 𝓝 (π / 2) = 𝓝 ((1 - 0) * (π / 2))\n⊢ Tendsto (fun i ↦ (2 * ↑i + 1) / (2 * ↑i + 2) * (π / 2)) atTop (𝓝 ((1 - 0) * (π / 2)))",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.p... | [
"this : 𝓝 (π / 2) = 𝓝 ((1 - 0) * (π / 2))\n⊢ Tendsto (fun i ↦ (2 * ↑i + 1) / (2 * ↑i + 2)) atTop (𝓝 (1 - 0))"
] | refine Tendsto.mul ?_ tendsto_const_nhds | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecialFunctions.ArithmeticGeometricMean | {
"line": 297,
"column": 38
} | {
"line": 297,
"column": 52
} | {
"line": 297,
"column": 53
} | [
{
"pp": "case succ\nk : ℝ≥0\nn : ℕ\nih : ∀ {x y : ℝ≥0}, ((k * x).agmSequences (k * y) n).2 = k * (x.agmSequences y n).2\nx y : ℝ≥0\n⊢ ((sqrt (k * x * (k * y))).agmSequences (k * (x + y) / 2) n).2 = k * (x.agmSequences y (n + 1)).2",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"NNRe... | [
"case succ\nk : ℝ≥0\nn : ℕ\nih : ∀ {x y : ℝ≥0}, ((k * x).agmSequences (k * y) n).2 = k * (x.agmSequences y n).2\nx y : ℝ≥0\n⊢ ((sqrt (k * x * (k * y))).agmSequences (k * ((x + y) / 2)) n).2 = k * (x.agmSequences y (n + 1)).2"
] | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.ConjSqrt | {
"line": 40,
"column": 75
} | {
"line": 41,
"column": 46
} | {
"line": 43,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : PartialOrder A\ninst✝⁷ : Ring A\ninst✝⁶ : StarRing A\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : StarOrderedRing A\ninst✝³ : Algebra ℝ A\ninst✝² : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝¹ : NonnegSpectrumClass ℝ A\ninst✝ : SeparatelyContinuousMul A\nc a : A\nhc : ¬0 ≤ c\... | [] | by
simp [conjSqrt_apply, sqrt_of_not_nonneg hc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Real.Pi.Irrational | {
"line": 305,
"column": 4
} | {
"line": 305,
"column": 24
} | {
"line": 306,
"column": 2
} | [
{
"pp": "h' : ¬Irrational (π / 2)\na : ℤ\nb : ℕ\nhb : 0 < b\nh : π / 2 = ↑a / ↑b\nha : 0 < ↑a\nk : ∀ (n : ℕ), 0 < ↑a ^ (2 * n + 1) / ↑n !\nj : ∀ᶠ (n : ℕ) in atTop, ↑a ^ (2 * n + 1) / ↑n ! * I n (π / 2) < 1\nn : ℕ\nhn : ↑a ^ (2 * n + 1) / ↑n ! * I n (π / 2) < 1\nhn' : 0 < ↑a ^ (2 * n + 1) / ↑n ! * I n (π / 2)\nz... | [] | linear_combination e | Mathlib.Tactic.LinearCombination._aux_Mathlib_Tactic_LinearCombination___elabRules_Mathlib_Tactic_LinearCombination_linearCombination_1 | Mathlib.Tactic.LinearCombination.linearCombination |
Mathlib.Analysis.SpecialFunctions.Gamma.Deligne | {
"line": 197,
"column": 11
} | {
"line": 197,
"column": 25
} | {
"line": 197,
"column": 26
} | [
{
"pp": "case neg\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ -↑n\nh : s - 1 ≠ 0\nh' : ∀ (n : ℕ), s - 1 ≠ -↑n\n⊢ (s - 1).Gammaℂ * sin (↑π * s / 2) * (s - 1).Gammaℝ⁻¹ =\n (s - 1).Gammaℂ * (s - 1) / 2 / ↑π * sin (↑π * s / 2) * ((s - 1).Gammaℝ * (s - 1) / 2 / ↑π)⁻¹",
"ppTerm": "?neg✝",
"assigned": true,
"usedConsta... | [
"case neg\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ -↑n\nh : s - 1 ≠ 0\nh' : ∀ (n : ℕ), s - 1 ≠ -↑n\n⊢ (s - 1).Gammaℂ * sin (↑π * (s / 2)) * (s - 1).Gammaℝ⁻¹ =\n (s - 1).Gammaℂ * ((s - 1) / 2 / ↑π) * sin (↑π * (s / 2)) * ((s - 1).Gammaℝ * ((s - 1) / 2 / ↑π))⁻¹"
] | mul_div_assoc, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.RingInverseOrder | {
"line": 96,
"column": 46
} | {
"line": 96,
"column": 79
} | {
"line": 97,
"column": 6
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nx : A\nxpos : IsStrictlyPositive x\ny : A\nypos : IsStrictlyPositive y\na b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nz : A := (conjSqrt x⁻¹ʳ) y\nzpos : IsStrictlyPositive z\nxinvpos : IsStrictlyPositive x⁻¹ʳ... | [] | by rw [← inverse_eq_rpow_neg_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Harmonic.GammaDeriv | {
"line": 143,
"column": 4
} | {
"line": 143,
"column": 32
} | {
"line": 144,
"column": 4
} | [
{
"pp": "case e_a.e_a.e_a.refine_1\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⊢ HasDerivAt (fun s ↦ 1 - 2 * s) (-2) (1 / 2)",
"ppTerm": "?e_a.e_a.e_a.refine_1✝",
"assigned": true,
"usedConstants": [
"IsMod... | [
"case e_a.e_a.e_a.refine_1\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⊢ HasDerivAt (fun s ↦ 1 - s * 2) (-2) (1 / 2)"
] | simp_rw [mul_comm (2 : ℝ) _] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 127,
"column": 31
} | {
"line": 127,
"column": 45
} | {
"line": 127,
"column": 46
} | [
{
"pp": "p t x : ℝ\nhp : p ∈ Ioo 0 1\nht : 0 ≤ t\nhx : 0 ≤ x\nhxt : 0 ≤ x * t\nhx_zero : ¬x = 0\n⊢ (x * t) ^ (p - 1) * x / (x * t + x) = x ^ (p - 1) * t ^ (p - 1) * (x / (x * t + x))",
"ppTerm": "?m.215",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Semigroup.toM... | [
"p t x : ℝ\nhp : p ∈ Ioo 0 1\nht : 0 ≤ t\nhx : 0 ≤ x\nhxt : 0 ≤ x * t\nhx_zero : ¬x = 0\n⊢ (x * t) ^ (p - 1) * (x / (x * t + x)) = x ^ (p - 1) * t ^ (p - 1) * (x / (x * t + x))"
] | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 130,
"column": 18
} | {
"line": 130,
"column": 32
} | {
"line": 130,
"column": 33
} | [
{
"pp": "p t x : ℝ\nhp : p ∈ Ioo 0 1\nht : 0 ≤ t\nhx : 0 ≤ x\nhxt : 0 ≤ x * t\nhx_zero : ¬x = 0\nthis : x * t + x = x * (t + 1)\n⊢ x ^ (p - 1) * (t ^ (p - 1) * (x / (x * t + x))) = x ^ (p - 1) * (t ^ (p - 1) * 1 / (t + 1))",
"ppTerm": "?m.246",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"p t x : ℝ\nhp : p ∈ Ioo 0 1\nht : 0 ≤ t\nhx : 0 ≤ x\nhxt : 0 ≤ x * t\nhx_zero : ¬x = 0\nthis : x * t + x = x * (t + 1)\n⊢ x ^ (p - 1) * (t ^ (p - 1) * (x / (x * t + x))) = x ^ (p - 1) * (t ^ (p - 1) * (1 / (t + 1)))"
] | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation | {
"line": 102,
"column": 4
} | {
"line": 102,
"column": 79
} | {
"line": 103,
"column": 4
} | [
{
"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\n⊢ f =O[cocompact ℝ] fun x ↦ |x| ^ (-2)",
"ppTerm": "?m.339",
"assigned": true,
"... | [
"case convert_2\na : 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\n⊢ (-↑π * a).re < 0"
] | convert! (cexp_neg_quadratic_isLittleO_abs_rpow_cocompact ?_ _ (-2)).isBigO | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.NumberTheory.Harmonic.GammaDeriv | {
"line": 222,
"column": 4
} | {
"line": 222,
"column": 18
} | {
"line": 222,
"column": 19
} | [
{
"pp": "f : ℂ → ℂ := fun s ↦ ↑π ^ (-s / 2)\ng : ℂ → ℂ := fun s ↦ Gamma (s / 2)\naux : ↑π ^ (1 / 2) = ↑√π\naux2 : ↑√π ≠ 0\nhf : HasDerivAt f (-log ↑π / 2 / ↑√π) 1\nhg : HasDerivAt g (-↑√π * (↑γ + 2 * log 2) / 2) 1\n⊢ -log ↑π / 2 + (↑√π)⁻¹ * (-↑√π * (↑γ + 2 * log 2) / 2) = -(↑γ + log (4 * ↑π)) / 2",
"ppTerm"... | [
"f : ℂ → ℂ := fun s ↦ ↑π ^ (-s / 2)\ng : ℂ → ℂ := fun s ↦ Gamma (s / 2)\naux : ↑π ^ (1 / 2) = ↑√π\naux2 : ↑√π ≠ 0\nhf : HasDerivAt f (-log ↑π / 2 / ↑√π) 1\nhg : HasDerivAt g (-↑√π * (↑γ + 2 * log 2) / 2) 1\n⊢ -log ↑π / 2 + (↑√π)⁻¹ * (-↑√π * ((↑γ + 2 * log 2) / 2)) = -(↑γ + log (4 * ↑π)) / 2"
] | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 216,
"column": 16
} | {
"line": 216,
"column": 30
} | {
"line": 216,
"column": 31
} | [
{
"pp": "case hbc\np t : ℝ\nhp : p ∈ Ioo 0 1\nht : 1 ≤ t\n⊢ 1 / 2 * 1 / t ≤ 1 / (t + 1)",
"ppTerm": "?hbc",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Real.instDivInvMonoid",
"Na... | [
"case hbc\np t : ℝ\nhp : p ∈ Ioo 0 1\nht : 1 ≤ t\n⊢ 1 / 2 * (1 / t) ≤ 1 / (t + 1)"
] | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Int.Log | {
"line": 113,
"column": 4
} | {
"line": 114,
"column": 95
} | {
"line": 115,
"column": 4
} | [
{
"pp": "case inr.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\nhr1 : r < 1\nhcri : 1 < r⁻¹\n⊢ r < ↑b ^ (-↑(Nat.clog b ⌈r⁻¹⌉₊) + 1)",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstan... | [
"case inr.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\nhr1 : r < 1\nhcri : 1 < r⁻¹\nthis : 1 ≤ Nat.clog b ⌈r⁻¹⌉₊\n⊢ r < ↑b ^ (-↑(Nat.clog b ⌈r⁻¹⌉₊) + 1)"
] | have : 1 ≤ Nat.clog b ⌈r⁻¹⌉₊ :=
Nat.succ_le_of_lt (Nat.clog_pos hb <| Nat.one_lt_cast.1 <| hcri.trans_le (Nat.le_ceil _)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.Int.Log | {
"line": 235,
"column": 83
} | {
"line": 241,
"column": 52
} | {
"line": 243,
"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\nr : R\n⊢ r ≤ ↑b ^ clog b r",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"le_inv_comm₀",
"Iff.mpr",
"Eq.mpr",
"GroupWit... | [] | by
rcases le_or_gt r 0 with hr | hr
· rw [clog_of_right_le_zero _ hr, zpow_zero]
exact hr.trans zero_le_one
rw [← neg_log_inv_eq_clog, zpow_neg, le_inv_comm₀ hr (zpow_pos ..)]
· exact zpow_log_le_self hb (inv_pos.mpr hr)
· exact Nat.cast_pos.mpr (zero_le_one.trans_lt hb) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Log.Monotone | {
"line": 71,
"column": 4
} | {
"line": 71,
"column": 39
} | {
"line": 71,
"column": 40
} | [
{
"pp": "a : ℝ\nha : 0 < a\nx : ℝ\nhex : x ∈ Ici (rexp a⁻¹)\ny : ℝ\nx✝ : y ∈ Ici (rexp a⁻¹)\nhxy : x ≤ y\nx_pos : 0 < x\ny_pos : 0 < y\n⊢ log ((y ^ a) ^ (1 / a)) / y ^ a ≤ log ((x ^ a) ^ (1 / a)) / x ^ a",
"ppTerm": "?m.99",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
... | [
"a : ℝ\nha : 0 < a\nx : ℝ\nhex : x ∈ Ici (rexp a⁻¹)\ny : ℝ\nx✝ : y ∈ Ici (rexp a⁻¹)\nhxy : x ≤ y\nx_pos : 0 < x\ny_pos : 0 < y\n⊢ 1 / a * log (y ^ a) / y ^ a ≤ log ((x ^ a) ^ (1 / a)) / x ^ a"
] | log_rpow (rpow_pos_of_pos y_pos a), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Log.Monotone | {
"line": 71,
"column": 76
} | {
"line": 71,
"column": 90
} | {
"line": 72,
"column": 4
} | [
{
"pp": "a : ℝ\nha : 0 < a\nx : ℝ\nhex : x ∈ Ici (rexp a⁻¹)\ny : ℝ\nx✝ : y ∈ Ici (rexp a⁻¹)\nhxy : x ≤ y\nx_pos : 0 < x\ny_pos : 0 < y\n⊢ 1 / a * log (y ^ a) / y ^ a ≤ 1 / a * log (x ^ a) / x ^ a",
"ppTerm": "?m.107",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"... | [
"a : ℝ\nha : 0 < a\nx : ℝ\nhex : x ∈ Ici (rexp a⁻¹)\ny : ℝ\nx✝ : y ∈ Ici (rexp a⁻¹)\nhxy : x ≤ y\nx_pos : 0 < x\ny_pos : 0 < y\n⊢ 1 / a * (log (y ^ a) / y ^ a) ≤ 1 / a * log (x ^ a) / x ^ a"
] | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Log.Monotone | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 18
} | {
"line": 72,
"column": 19
} | [
{
"pp": "a : ℝ\nha : 0 < a\nx : ℝ\nhex : x ∈ Ici (rexp a⁻¹)\ny : ℝ\nx✝ : y ∈ Ici (rexp a⁻¹)\nhxy : x ≤ y\nx_pos : 0 < x\ny_pos : 0 < y\n⊢ 1 / a * (log (y ^ a) / y ^ a) ≤ 1 / a * log (x ^ a) / x ^ a",
"ppTerm": "?m.113",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
... | [
"a : ℝ\nha : 0 < a\nx : ℝ\nhex : x ∈ Ici (rexp a⁻¹)\ny : ℝ\nx✝ : y ∈ Ici (rexp a⁻¹)\nhxy : x ≤ y\nx_pos : 0 < x\ny_pos : 0 < y\n⊢ 1 / a * (log (y ^ a) / y ^ a) ≤ 1 / a * (log (x ^ a) / x ^ a)"
] | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 495,
"column": 2
} | {
"line": 497,
"column": 9
} | {
"line": 498,
"column": 2
} | [
{
"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... | have hf : ContinuousOn (Function.uncurry f) (Ioi (0 : ℝ) ×ˢ quasispectrum ℝ a) := by
refine continuousOn_rpowIntegrand₀₁_uncurry hp (quasispectrum ℝ a) ?_
grind | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.BoundedContinuousFunction | {
"line": 120,
"column": 2
} | {
"line": 121,
"column": 6
} | {
"line": 123,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝⁸ : MeasurableSpace X\ninst✝⁷ : TopologicalSpace X\nμ : Measure X\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : OpensMeasurableSpace X\ninst✝⁴ : SecondCountableTopology E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : IsProbabilityMeasu... | [] | convert! f.norm_integral_le_mul_norm μ
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.BoundedContinuousFunction | {
"line": 120,
"column": 2
} | {
"line": 121,
"column": 6
} | {
"line": 123,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝⁸ : MeasurableSpace X\ninst✝⁷ : TopologicalSpace X\nμ : Measure X\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : OpensMeasurableSpace X\ninst✝⁴ : SecondCountableTopology E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : IsProbabilityMeasu... | [] | convert! f.norm_integral_le_mul_norm μ
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 588,
"column": 4
} | {
"line": 588,
"column": 63
} | {
"line": 589,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\np t : ℝ\nhp : p ∈ Ioo 0 1\nht : 0 < t\nx : A\nhx : x ∈ Ici 0\nhg : ContinuousOn (fun z ↦ (t + z)⁻¹) (spectrum ℝ x)\nhf : ContinuousOn (fun z ↦ 1 + z) (spectrum ℝ x)\n⊢ cfc (fun x ↦ t ^ (p - 1) - t ^ p * (t + x)⁻¹... | [
"A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\np t : ℝ\nhp : p ∈ Ioo 0 1\nht : 0 < t\nx : A\nhx : x ∈ Ici 0\nhg : ContinuousOn (fun z ↦ (t + z)⁻¹) (spectrum ℝ x)\nhf : ContinuousOn (fun z ↦ 1 + z) (spectrum ℝ x)\nhspectrum : ∀ r ∈ spectrum ℝ x, t + r ≠ 0\n⊢ cfc (fun x ↦ ... | have hspectrum : ∀ r ∈ spectrum ℝ x, t + r ≠ 0 := by grind | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SpecialFunctions.MulExpNegMulSqIntegral | {
"line": 136,
"column": 83
} | {
"line": 141,
"column": 34
} | {
"line": 143,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝³ : TopologicalSpace E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nP : Measure E\ninst✝ : IsFiniteMeasure P\nε : ℝ\nf : C(E, ℝ)\nK : Set E\nhK : MeasurableSet K\nhε : 0 < ε\nhKP : P Kᶜ < ↑ε.toNNReal\n⊢ |∫ (x : E), ε.mulExpNegMulSq (f x) ∂P - ∫ (x : E) in K, ε.mulExpNegMulSq (... | [] | by
apply lt_of_le_of_lt (norm_integral_sub_setIntegral_le
(Eventually.of_forall (fun _ => abs_mulExpNegMulSq_le hε)) hK
(integrable_mulExpNegMulSq_comp f hε))
rw [mul_inv_lt_iff₀ (sqrt_pos_of_pos hε), mul_self_sqrt (le_of_lt hε)]
exact toReal_lt_of_lt_ofReal hKP | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Pow.Integral | {
"line": 75,
"column": 2
} | {
"line": 76,
"column": 79
} | {
"line": 77,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nf_nn : 0 ≤ᵐ[μ] f\nf_mble : AEMeasurable f μ\np : ℝ\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : ℝ → ℝ := fun t ↦ t ^ (p - 1)\nobs : ∀ (x : ℝ), intervalIntegral g 0 x volume = x ^ p / p\ng_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t\... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nf_nn : 0 ≤ᵐ[μ] f\nf_mble : AEMeasurable f μ\np : ℝ\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : ℝ → ℝ := fun t ↦ t ^ (p - 1)\nobs : ∀ (x : ℝ), intervalIntegral g 0 x volume = x ^ p / p\ng_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t\ng_intble : ... | · congr with ω
rw [← ENNReal.ofReal_mul p_pos.le, mul_div_cancel₀ (f ω ^ p) p_pos.ne.symm] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Constructions.HaarToSphere | {
"line": 118,
"column": 2
} | {
"line": 125,
"column": 30
} | {
"line": 127,
"column": 0
} | [
{
"pp": "n : ℕ\nx : ↑(Ioi 0)\n⊢ (volumeIoiPow n) (Iio x) = ENNReal.ofReal (↑x ^ (n + 1) / (↑n + 1))",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Real.instIsOrderedRing",
"Eq.mpr",
"Set.Ioc",
"InnerProductSpace.toNormedSpace",
"Mea... | [] | have hr₀ : 0 ≤ x.1 := le_of_lt x.2
rw [volumeIoiPow, withDensity_apply _ measurableSet_Iio,
setLIntegral_subtype measurableSet_Ioi _ fun a : ℝ ↦ .ofReal (a ^ n),
image_subtype_val_Ioi_Iio, restrict_congr_set Ioo_ae_eq_Ioc,
← ofReal_integral_eq_lintegral_ofReal (intervalIntegrable_pow _).1, ← integral_of_l... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Constructions.HaarToSphere | {
"line": 118,
"column": 2
} | {
"line": 125,
"column": 30
} | {
"line": 127,
"column": 0
} | [
{
"pp": "n : ℕ\nx : ↑(Ioi 0)\n⊢ (volumeIoiPow n) (Iio x) = ENNReal.ofReal (↑x ^ (n + 1) / (↑n + 1))",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Real.instIsOrderedRing",
"Eq.mpr",
"Set.Ioc",
"InnerProductSpace.toNormedSpace",
"Mea... | [] | have hr₀ : 0 ≤ x.1 := le_of_lt x.2
rw [volumeIoiPow, withDensity_apply _ measurableSet_Iio,
setLIntegral_subtype measurableSet_Ioi _ fun a : ℝ ↦ .ofReal (a ^ n),
image_subtype_val_Ioi_Iio, restrict_congr_set Ioo_ae_eq_Ioc,
← ofReal_integral_eq_lintegral_ofReal (intervalIntegrable_pow _).1, ← integral_of_l... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Constructions.HaarToSphere | {
"line": 176,
"column": 4
} | {
"line": 176,
"column": 71
} | {
"line": 177,
"column": 4
} | [
{
"pp": "E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nx : E\nhx : ‖x‖ = 1\nε : ℝ\nhε : 0 < ε\nhε2 : ε ≤ 2\nhabs : |1 - ε / 4| = 1 - ε / 4\nhy : dist 0 ((1 - ε / 4) • x) < ε / 4\n⊢ False",
"ppTerm": "?m.157",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionM... | [
"E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nx : E\nhx : ‖x‖ = 1\nε : ℝ\nhε : 0 < ε\nhε2 : ε ≤ 2\nhabs : |1 - ε / 4| = 1 - ε / 4\nhy : dist 0 ((1 - ε / 4) • x) < ε / 4\nthis : 1 - ε / 4 < ε / 4\n⊢ False"
] | have : 1 - ε / 4 < ε / 4 := by simpa [norm_smul, habs, hx] using hy | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Constructions.Polish.EmbeddingReal | {
"line": 29,
"column": 43
} | {
"line": 34,
"column": 58
} | {
"line": 36,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝³ : MeasurableSpace α\ninst✝² : StandardBorelSpace α\ninst✝¹ : Infinite α\ninst✝ : Countable α\n⊢ Nonempty (α ≃ᵐ ↑(range Nat.cast))",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Real.partialOrder",
"Real",
"FloorRing.toFloorSemiring",
"... | [] | by
have : PolishSpace (range ((↑) : ℕ → ℝ)) :=
Nat.isClosedEmbedding_coe_real.isClosedMap.isClosed_range.polishSpace
refine ⟨PolishSpace.Equiv.measurableEquiv ?_⟩
refine (nonempty_equiv_of_countable.some : α ≃ ℕ).trans ?_
exact Equiv.ofInjective ((↑) : ℕ → ℝ) Nat.cast_injective | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 93,
"column": 16
} | {
"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.87",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"AddGroup.toS... | [] | linear_combination e | Mathlib.Tactic.LinearCombination._aux_Mathlib_Tactic_LinearCombination___elabRules_Mathlib_Tactic_LinearCombination_linearCombination_1 | Mathlib.Tactic.LinearCombination.linearCombination |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 57
} | {
"line": 116,
"column": 2
} | [
{
"pp": "L : PeriodPair\nthis : Finset.univ = {0, 1}\n⊢ L.lattice = Submodule.span ℤ (Set.range ⇑L.basis)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCom... | [
"L : PeriodPair\nthis : Finset.univ = {0, 1}\n⊢ Submodule.span ℤ {L.ω₁, L.ω₂} = Submodule.span ℤ (⇑L.basis '' ↑{0, 1})"
] | rw [lattice, ← Set.image_univ, ← Finset.coe_univ, this] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 256,
"column": 4
} | {
"line": 256,
"column": 28
} | {
"line": 257,
"column": 2
} | [
{
"pp": "case h₁\nL : PeriodPair\nl₀ z : ℂ\nl : ↥L.lattice\n⊢ ↑((Equiv.neg ↥L.lattice) l) = l₀ ↔ ↑l = -l₀",
"ppTerm": "?h₁",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"Submodule",
"SubtractionMonoid.toInvolutiveNeg",
"Non... | [] | simp [neg_eq_iff_eq_neg] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 256,
"column": 4
} | {
"line": 256,
"column": 28
} | {
"line": 257,
"column": 2
} | [
{
"pp": "case h₁\nL : PeriodPair\nl₀ z : ℂ\nl : ↥L.lattice\n⊢ ↑((Equiv.neg ↥L.lattice) l) = l₀ ↔ ↑l = -l₀",
"ppTerm": "?h₁",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"Submodule",
"SubtractionMonoid.toInvolutiveNeg",
"Non... | [] | simp [neg_eq_iff_eq_neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 256,
"column": 4
} | {
"line": 256,
"column": 28
} | {
"line": 257,
"column": 2
} | [
{
"pp": "case h₁\nL : PeriodPair\nl₀ z : ℂ\nl : ↥L.lattice\n⊢ ↑((Equiv.neg ↥L.lattice) l) = l₀ ↔ ↑l = -l₀",
"ppTerm": "?h₁",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"Submodule",
"SubtractionMonoid.toInvolutiveNeg",
"Non... | [] | simp [neg_eq_iff_eq_neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.InverseDeriv | {
"line": 101,
"column": 4
} | {
"line": 101,
"column": 76
} | {
"line": 102,
"column": 4
} | [
{
"pp": "case neg\nx : ℝ\nh : ¬(x ≠ -1 ∧ x ≠ 1)\n⊢ deriv arcsin x = 1 / √(1 - x ^ 2)",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Real.differentiableAt_arcsin",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"instHDiv",
"Semiring.toModule",... | [
"case neg\nx : ℝ\nh : ¬(x ≠ -1 ∧ x ≠ 1)\n⊢ 0 = 1 / √(1 - x ^ 2)"
] | rw [deriv_zero_of_not_differentiableAt (mt differentiableAt_arcsin.1 h)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 423,
"column": 4
} | {
"line": 423,
"column": 28
} | {
"line": 424,
"column": 2
} | [
{
"pp": "case e'_5.h₁\nL : PeriodPair\nl₀ z : ℂ\nl : ↥L.lattice\n⊢ -↑l = l₀ ↔ ↑l = -l₀",
"ppTerm": "?e'_5.h₁",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"Submodule",
"_private.Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstr... | [] | simp [neg_eq_iff_eq_neg] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 423,
"column": 4
} | {
"line": 423,
"column": 28
} | {
"line": 424,
"column": 2
} | [
{
"pp": "case e'_5.h₁\nL : PeriodPair\nl₀ z : ℂ\nl : ↥L.lattice\n⊢ -↑l = l₀ ↔ ↑l = -l₀",
"ppTerm": "?e'_5.h₁",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"Submodule",
"_private.Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstr... | [] | simp [neg_eq_iff_eq_neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 423,
"column": 4
} | {
"line": 423,
"column": 28
} | {
"line": 424,
"column": 2
} | [
{
"pp": "case e'_5.h₁\nL : PeriodPair\nl₀ z : ℂ\nl : ↥L.lattice\n⊢ -↑l = l₀ ↔ ↑l = -l₀",
"ppTerm": "?e'_5.h₁",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"Submodule",
"_private.Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstr... | [] | simp [neg_eq_iff_eq_neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 639,
"column": 8
} | {
"line": 640,
"column": 44
} | {
"line": 641,
"column": 6
} | [
{
"pp": "L : PeriodPair\nl₀ x : ℂ\ni : ℕ\nhl₀ : l₀ ∈ L.lattice\n⊢ (FormalMultilinearSeries.ofScalars ℂ fun i ↦\n if i = 0 then ℘[L - l₀] x else (↑i + 1) * (L.sumInvPow x (i + 2) - ((l₀ - x) ^ (i + 2))⁻¹)).coeff\n (i + 1) =\n (↑i + 2) * (L.sumInvPow x (i + 3) - ∑' (l : ↥L.lattice), if l = ⟨l₀, h... | [] | rw [FormalMultilinearSeries.coeff_ofScalars, tsum_ite_eq, zpow_neg, zpow_natCast]
simp [add_assoc, one_add_one_eq_two] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 639,
"column": 8
} | {
"line": 640,
"column": 44
} | {
"line": 641,
"column": 6
} | [
{
"pp": "L : PeriodPair\nl₀ x : ℂ\ni : ℕ\nhl₀ : l₀ ∈ L.lattice\n⊢ (FormalMultilinearSeries.ofScalars ℂ fun i ↦\n if i = 0 then ℘[L - l₀] x else (↑i + 1) * (L.sumInvPow x (i + 2) - ((l₀ - x) ^ (i + 2))⁻¹)).coeff\n (i + 1) =\n (↑i + 2) * (L.sumInvPow x (i + 3) - ∑' (l : ↥L.lattice), if l = ⟨l₀, h... | [] | rw [FormalMultilinearSeries.coeff_ofScalars, tsum_ite_eq, zpow_neg, zpow_natCast]
simp [add_assoc, one_add_one_eq_two] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 639,
"column": 6
} | {
"line": 640,
"column": 44
} | {
"line": 641,
"column": 6
} | [
{
"pp": "L : PeriodPair\nl₀ x : ℂ\ni : ℕ\nhl₀ : l₀ ∈ L.lattice\n⊢ (FormalMultilinearSeries.ofScalars ℂ fun i ↦\n if i = 0 then ℘[L - l₀] x else (↑i + 1) * (L.sumInvPow x (i + 2) - ((l₀ - x) ^ (i + 2))⁻¹)).coeff\n (i + 1) =\n (↑i + 2) * (L.sumInvPow x (i + 3) - ∑' (l : ↥L.lattice), if l = ⟨l₀, h... | [
"L : PeriodPair\nl₀ x : ℂ\ni : ℕ\nhl₀ : l₀ ∈ L.lattice\n⊢ (↑i + 2) * (L.sumInvPow x (i + 3) - ∑' (l : ↥L.lattice), if l = ⟨l₀, hl₀⟩ then (l₀ - x) ^ (-↑(i + 3)) else 0) =\n ∑' (l : ↥L.lattice),\n if ↑l = l₀ then 0 else (↑(i + 1) + 1) * (↑l - x) ^ (-↑(i + 1 + 2)) - Nat.casesOn (i + 1) (↑l ^ (-2)) 0"
] | · rw [FormalMultilinearSeries.coeff_ofScalars, tsum_ite_eq, zpow_neg, zpow_natCast]
simp [add_assoc, one_add_one_eq_two] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 25
} | {
"line": 133,
"column": 4
} | [
{
"pp": "case h\nn : ℕ\nhn : n ≠ 0\nx : ℝ\nhx : eval x (T ℝ ↑n) = 1\nk : ℕ\nhk₁ : k ≤ n\nhk₂ : x = cos (↑k * π / ↑n)\n⊢ k ≤ n ∧ Even k ∧ x = cos (↑k * π / ↑n)",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"Real.pi",
"HMul.hMul",
"Real.cos",
... | [
"case h\nn : ℕ\nhn : n ≠ 0\nx : ℝ\nhx : eval x (T ℝ ↑n) = 1\nk : ℕ\nhk₁ : k ≤ n\nhk₂ : x = cos (↑k * π / ↑n)\n⊢ Even k"
] | refine ⟨hk₁, ?_, hk₂⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema | {
"line": 136,
"column": 8
} | {
"line": 136,
"column": 11
} | {
"line": 136,
"column": 12
} | [
{
"pp": "case mpr\nn : ℕ\nhn : n ≠ 0\nx : ℝ\nk : ℕ\nhk₁ : k ≤ n\nhk₂ : Even k\nhx : x = cos (↑k * π / ↑n)\n⊢ eval x (T ℝ ↑n) = 1",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"Real",
"instHDiv",
"Real.pi",
"HMul.hMul",
... | [
"case mpr\nn : ℕ\nhn : n ≠ 0\nx : ℝ\nk : ℕ\nhk₁ : k ≤ n\nhk₂ : Even k\nhx : x = cos (↑k * π / ↑n)\n⊢ eval (cos (↑k * π / ↑n)) (T ℝ ↑n) = 1"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema | {
"line": 143,
"column": 4
} | {
"line": 144,
"column": 68
} | {
"line": 145,
"column": 4
} | [
{
"pp": "case mp\nn : ℕ\nhn : n ≠ 0\nx : ℝ\nhx : eval x (T ℝ ↑n) = -1\n⊢ ∃ k ≤ n, Odd k ∧ x = cos (↑k * π / ↑n)",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Real.instIsOrderedRing",
"Polynomial.eval",
"NegZeroClass.... | [
"case mp\nn : ℕ\nhn : n ≠ 0\nx : ℝ\nhx : eval x (T ℝ ↑n) = -1\nk : ℕ\nhk₁ : k ≤ n\nhk₂ : x = cos (↑k * π / ↑n)\n⊢ ∃ k ≤ n, Odd k ∧ x = cos (↑k * π / ↑n)"
] | obtain ⟨k, hk₁, hk₂⟩ := (abs_eval_T_real_eq_one_iff hn x).mp
((abs_eq_abs.mpr (.inl hx)).trans ((abs_neg 1).trans abs_one)) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 25
} | {
"line": 147,
"column": 4
} | [
{
"pp": "case h\nn : ℕ\nhn : n ≠ 0\nx : ℝ\nhx : eval x (T ℝ ↑n) = -1\nk : ℕ\nhk₁ : k ≤ n\nhk₂ : x = cos (↑k * π / ↑n)\n⊢ k ≤ n ∧ Odd k ∧ x = cos (↑k * π / ↑n)",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"Real.pi",
"HMul.hMul",
"Real.cos",
... | [
"case h\nn : ℕ\nhn : n ≠ 0\nx : ℝ\nhx : eval x (T ℝ ↑n) = -1\nk : ℕ\nhk₁ : k ≤ n\nhk₂ : x = cos (↑k * π / ↑n)\n⊢ Odd k"
] | refine ⟨hk₁, ?_, hk₂⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema | {
"line": 150,
"column": 8
} | {
"line": 150,
"column": 11
} | {
"line": 150,
"column": 12
} | [
{
"pp": "case mpr\nn : ℕ\nhn : n ≠ 0\nx : ℝ\nk : ℕ\nhk₁ : k ≤ n\nhk₂ : Odd k\nhx : x = cos (↑k * π / ↑n)\n⊢ eval x (T ℝ ↑n) = -1",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"Real",
"instHDiv",
"Real.pi",
"HMul.hMul",
... | [
"case mpr\nn : ℕ\nhn : n ≠ 0\nx : ℝ\nk : ℕ\nhk₁ : k ≤ n\nhk₂ : Odd k\nhx : x = cos (↑k * π / ↑n)\n⊢ eval (cos (↑k * π / ↑n)) (T ℝ ↑n) = -1"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 104,
"column": 2
} | {
"line": 109,
"column": 71
} | {
"line": 111,
"column": 0
} | [
{
"pp": "z : ℍ\nc d : ℝ\nhc : 1 ≤ c ^ 2\n⊢ r z ≤ ‖↑c * ↑z + ↑d‖",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Complex.mul_im",
"Real.instIsOrderedRing",
"Norm.norm",
"Eq.mpr",
"Real.partialOrder",
"Real.instLE",
"Real",
"Lattice.toSemilatt... | [] | rcases z with ⟨z, hz⟩
have H1 : z.im ≤ √((c * z.re + d) ^ 2 + (c * z).im ^ 2) := by
rw [Real.le_sqrt' hz, im_ofReal_mul, mul_pow]
exact (le_mul_of_one_le_left (sq_nonneg _) hc).trans <| le_add_of_nonneg_left (sq_nonneg _)
simpa only [r, norm_def, normSq_apply, add_re, re_ofReal_mul, coe_re, ← pow_two, add_i... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 104,
"column": 2
} | {
"line": 109,
"column": 71
} | {
"line": 111,
"column": 0
} | [
{
"pp": "z : ℍ\nc d : ℝ\nhc : 1 ≤ c ^ 2\n⊢ r z ≤ ‖↑c * ↑z + ↑d‖",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Complex.mul_im",
"Real.instIsOrderedRing",
"Norm.norm",
"Eq.mpr",
"Real.partialOrder",
"Real.instLE",
"Real",
"Lattice.toSemilatt... | [] | rcases z with ⟨z, hz⟩
have H1 : z.im ≤ √((c * z.re + d) ^ 2 + (c * z).im ^ 2) := by
rw [Real.le_sqrt' hz, im_ofReal_mul, mul_pow]
exact (le_mul_of_one_le_left (sq_nonneg _) hc).trans <| le_add_of_nonneg_left (sq_nonneg _)
simpa only [r, norm_def, normSq_apply, add_re, re_ofReal_mul, coe_re, ← pow_two, add_i... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Real.GoldenRatio | {
"line": 182,
"column": 68
} | {
"line": 195,
"column": 82
} | {
"line": 197,
"column": 0
} | [
{
"pp": "⊢ (fun n ↦ ↑(Nat.fib n)) = fun n ↦ (φ ^ n - ψ ^ n) / √5",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"CharP.cast_eq_zero",
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"P... | [] | by
rw [fibRec.eq_iff_eqOn_range_order]
· intro i hi
norm_cast at hi
fin_cases hi <;> simp
· exact fib_isSol_fibRec
· suffices LinearRecurrence.IsSolution fibRec
((fun n ↦ (√5)⁻¹ * φ ^ n) - (fun n ↦ (√5)⁻¹ * ψ ^ n)) by
convert! this
rw [Pi.sub_apply]
ring
apply (@fibRec ℝ _)... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SumIntegralExpDecay | {
"line": 28,
"column": 33
} | {
"line": 28,
"column": 36
} | {
"line": 28,
"column": 36
} | [
{
"pp": "k : ℕ\nM c : ℝ\nhM : 0 ≤ M\nhc : 0 < c\nhk : 0 < ↑k + 1\nkey : ∫ (t : ℝ) in Ioi 0, t ^ (↑k + 1 - 1) * rexp (-(c * t)) = (1 / c) ^ (↑k + 1) * Gamma (↑k + 1)\n⊢ ∫ (a : ℝ) in Ioi 0, a ^ (↑k + 1 - 1) * rexp (-(c * a)) ≠ 0",
"ppTerm": "?m.116",
"assigned": true,
"usedConstants": [
"Eq.mpr"... | [
"k : ℕ\nM c : ℝ\nhM : 0 ≤ M\nhc : 0 < c\nhk : 0 < ↑k + 1\nkey : ∫ (t : ℝ) in Ioi 0, t ^ (↑k + 1 - 1) * rexp (-(c * t)) = (1 / c) ^ (↑k + 1) * Gamma (↑k + 1)\n⊢ (1 / c) ^ (↑k + 1) * Gamma (↑k + 1) ≠ 0"
] | key | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SumIntegralComparisons | {
"line": 87,
"column": 4
} | {
"line": 88,
"column": 8
} | {
"line": 89,
"column": 2
} | [
{
"pp": "x₀ : ℝ\na : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\n⊢ ∫ (x : ℝ) in x₀..x₀ + ↑a, f x = ∑ i ∈ Finset.range a, ∫ (x : ℝ) in x₀ + ↑i..x₀ + ↑(i + 1), f x",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"CharP.cast_eq_zero",
"Eq.mpr",
"InnerProductSpace.toN... | [] | convert! (sum_integral_adjacent_intervals hf.intervalIntegrable_subset).symm
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SumIntegralComparisons | {
"line": 87,
"column": 4
} | {
"line": 88,
"column": 8
} | {
"line": 89,
"column": 2
} | [
{
"pp": "x₀ : ℝ\na : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\n⊢ ∫ (x : ℝ) in x₀..x₀ + ↑a, f x = ∑ i ∈ Finset.range a, ∫ (x : ℝ) in x₀ + ↑i..x₀ + ↑(i + 1), f x",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"CharP.cast_eq_zero",
"Eq.mpr",
"InnerProductSpace.toN... | [] | convert! (sum_integral_adjacent_intervals hf.intervalIntegrable_subset).symm
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.Injective.Dimension | {
"line": 282,
"column": 4
} | {
"line": 282,
"column": 61
} | {
"line": 283,
"column": 4
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nn : ℕ\nh : injectiveDimension X < ↑n\n⊢ HasInjectiveDimensionLT X n",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"CategoryTheory.HasInjectiveDimensionLT",
"WithBot.instPreorder",
... | [
"case refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nn : ℕ\nh : injectiveDimension X < ↑n\nthis : injectiveDimension X ∈ {n | ∀ (i : ℕ), n < ↑i → HasInjectiveDimensionLT X i}\n⊢ HasInjectiveDimensionLT X n"
] | have : injectiveDimension X ∈ _ := csInf_mem ⟨⊤, by simp⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Abelian.Injective.Dimension | {
"line": 315,
"column": 6
} | {
"line": 315,
"column": 28
} | {
"line": 316,
"column": 6
} | [
{
"pp": "case coe.top\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nhd : injectiveDimension X = ↑⊤\nthis : ∃ n, HasInjectiveDimensionLE X n\n⊢ False",
"ppTerm": "?coe.top",
"assigned": true,
"usedConstants": [
"False",
"Exists",
"CategoryTheory.HasInjectiveDime... | [
"case coe.top\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nhd : injectiveDimension X = ↑⊤\nn : ℕ\nhn : HasInjectiveDimensionLE X n\n⊢ False"
] | obtain ⟨n, hn⟩ := this | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Abelian.Pseudoelements | {
"line": 261,
"column": 85
} | {
"line": 263,
"column": 55
} | {
"line": 265,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nh : ∀ (a : Pseudoelement P), pseudoApply f a = 0\n⊢ f = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"Eq.mpr",
"CategoryTheory.Over",
... | [] | by
rw [← Category.id_comp f]
exact (pseudoZero_iff (𝟙 P ≫ f : Over Q)).1 (h (𝟙 P)) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Abelian.Pseudoelements | {
"line": 419,
"column": 47
} | {
"line": 419,
"column": 82
} | {
"line": 419,
"column": 82
} | [
{
"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 ... | [] | by simp [a'', sub_eq_add_neg, this] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Abelian.RightDerived | {
"line": 107,
"column": 6
} | {
"line": 107,
"column": 21
} | {
"line": 107,
"column": 22
} | [
{
"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\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... | [
"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\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\nF : C ⥤ D\... | Iso.hom_inv_id, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.ObjectProperty.Kernels | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 32
} | {
"line": 142,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nP : ObjectProperty C\ninst✝ : P.IsClosedUnderQuotients\ni✝ X₁✝ X₂✝ : C\nf✝ : X₁✝ ⟶ X₂✝\nk : CokernelCofork f✝\nhk : IsColimit k\nhf : MorphismProperty.ofObjectProperty P P f✝\nthis : Epi (Cofork.π k) := Cofork.IsColimit.epi hk\n... | [] | exact P.prop_of_epi k.π hf.2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Abelian.SerreClass.Localization | {
"line": 309,
"column": 2
} | {
"line": 309,
"column": 31
} | {
"line": 310,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : Abelian C\nD : Type u'\ninst✝⁴ : Category.{v', u'} D\nL : C ⥤ D\nP : ObjectProperty C\ninst✝³ : P.IsSerreClass\ninst✝² : L.IsLocalization P.isoModSerre\ninst✝¹ : Preadditive D\ninst✝ : L.Additive\nX Y : C\nf : X ⟶ Y\nthis✝ : L.PreservesMonomorphisms\nthi... | [
"C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : Abelian C\nD : Type u'\ninst✝⁴ : Category.{v', u'} D\nL : C ⥤ D\nP : ObjectProperty C\ninst✝³ : P.IsSerreClass\ninst✝² : L.IsLocalization P.isoModSerre\ninst✝¹ : Preadditive D\ninst✝ : L.Additive\nX Y : C\nf : X ⟶ Y\nthis✝ : L.PreservesMonomorphisms\nthis : L.EssSur... | obtain ⟨Z', t, ht, fac⟩ := hw | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Action.Basic | {
"line": 138,
"column": 18
} | {
"line": 138,
"column": 33
} | {
"line": 138,
"column": 34
} | [
{
"pp": "V : Type u_1\ninst✝¹ : Category.{v_1, u_1} V\nG : Type u_2\ninst✝ : Monoid G\nM N : Action V G\nf : M ≅ N\n⊢ (f.hom ≫ f.inv).hom = 𝟙 M.V",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"cong... | [
"V : Type u_1\ninst✝¹ : Category.{v_1, u_1} V\nG : Type u_2\ninst✝ : Monoid G\nM N : Action V G\nf : M ≅ N\n⊢ (𝟙 M).hom = 𝟙 M.V"
] | Iso.hom_inv_id, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.SerreClass.Localization | {
"line": 339,
"column": 2
} | {
"line": 339,
"column": 31
} | {
"line": 340,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : Abelian C\nD : Type u'\ninst✝⁴ : Category.{v', u'} D\nL : C ⥤ D\nP : ObjectProperty C\ninst✝³ : P.IsSerreClass\ninst✝² : L.IsLocalization P.isoModSerre\ninst✝¹ : Preadditive D\ninst✝ : L.Additive\nX Y : C\nf : X ⟶ Y\nthis✝ : L.PreservesEpimorphisms\nthis... | [
"C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : Abelian C\nD : Type u'\ninst✝⁴ : Category.{v', u'} D\nL : C ⥤ D\nP : ObjectProperty C\ninst✝³ : P.IsSerreClass\ninst✝² : L.IsLocalization P.isoModSerre\ninst✝¹ : Preadditive D\ninst✝ : L.Additive\nX Y : C\nf : X ⟶ Y\nthis✝ : L.PreservesEpimorphisms\nthis : L.EssSurj... | obtain ⟨Z', t, ht, fac⟩ := hw | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Action.Continuous | {
"line": 103,
"column": 4
} | {
"line": 103,
"column": 38
} | {
"line": 104,
"column": 4
} | [
{
"pp": "V : Type u_1\ninst✝⁷ : Category.{v_1, u_1} V\nFV : V → V → Type u_2\nCV : V → Type u_3\ninst✝⁶ : (X Y : V) → FunLike (FV X Y) (CV X) (CV Y)\ninst✝⁵ : ConcreteCategory V FV\ninst✝⁴ : HasForget₂ V TopCat\nG : Type u_4\ninst✝³ : Monoid G\ninst✝² : TopologicalSpace G\nH : Type u_5\ninst✝¹ : Monoid H\ninst✝... | [
"V : Type u_1\ninst✝⁷ : Category.{v_1, u_1} V\nFV : V → V → Type u_2\nCV : V → Type u_3\ninst✝⁶ : (X Y : V) → FunLike (FV X Y) (CV X) (CV Y)\ninst✝⁵ : ConcreteCategory V FV\ninst✝⁴ : HasForget₂ V TopCat\nG : Type u_4\ninst✝³ : Monoid G\ninst✝² : TopologicalSpace G\nH : Type u_5\ninst✝¹ : Monoid H\ninst✝ : Topologic... | have : Continuous v := by fun_prop | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Adjunction.Triple | {
"line": 306,
"column": 2
} | {
"line": 306,
"column": 75
} | {
"line": 307,
"column": 2
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\nF : C ⥤ D\nG : D ⥤ C\nH : C ⥤ D\nt : Triple F G H\ninst✝¹ : F.Full\ninst✝ : F.Faithful\n⊢ (∀ (X : C), Mono (t.leftToRight.app X)) ↔ ∀ (X : D), Mono (t.adj₁.counit.app X ≫ t.adj₂.unit.app X)",
"ppTerm": "?m.9... | [
"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\nF : C ⥤ D\nG : D ⥤ C\nH : C ⥤ D\nt : Triple F G H\ninst✝¹ : F.Full\ninst✝ : F.Faithful\nh : ∀ (X : D), Mono (t.adj₁.counit.app X ≫ t.adj₂.unit.app X)\nX : C\n⊢ Mono (t.leftToRight.app X)"
] | refine ⟨fun h X ↦ by rw [← leftToRight_app_obj]; exact h _, fun h X ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Oplax | {
"line": 163,
"column": 13
} | {
"line": 165,
"column": 16
} | {
"line": 167,
"column": 0
} | [
{
"pp": "B : Type u₁\ninst✝¹ : Bicategory B\nC : Type u₂\ninst✝ : Bicategory C\nF G H : B ⥤ᵒᵖᴸ C\nη : LaxTrans F G\nθ : LaxTrans G H\na b c : B\nf : a ⟶ b\ng : b ⟶ c\n⊢ 𝟙 ((η.app a ≫ θ.app a) ≫ H.map (f ≫ g)) ⊗≫\n η.app a ◁ (θ.app a ◁ H.mapComp f g ⊗≫ θ.naturality f ▷ H.map g) ⊗≫\n ((η.app a ≫ G.ma... | [] | by
rw [whisker_exchange]
bicategory | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Lax | {
"line": 167,
"column": 13
} | {
"line": 169,
"column": 16
} | {
"line": 171,
"column": 0
} | [
{
"pp": "B : Type u₁\ninst✝¹ : Bicategory B\nC : Type u₂\ninst✝ : Bicategory C\nF G H : B ⥤ᴸ C\nη : LaxTrans F G\nθ : LaxTrans G H\na b c : B\nf : a ⟶ b\ng : b ⟶ c\n⊢ 𝟙 (η.vCompApp θ a ≫ H.map f ≫ H.map g) ⊗≫\n η.app a ◁ θ.naturality f ▷ H.map g ⊗≫\n ((η.app a ≫ G.map f) ◁ θ.naturality g ≫ η.natura... | [] | by
rw [whisker_exchange]
bicategory | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Enriched.EnrichedCat | {
"line": 101,
"column": 2
} | {
"line": 102,
"column": 79
} | {
"line": 103,
"column": 2
} | [
{
"pp": "V : Type v\ninst✝⁴ : Category.{w, v} V\ninst✝³ : MonoidalCategory V\nC : Type u\ninst✝² : EnrichedCategory V C\nD : Type u₁\ninst✝¹ : EnrichedCategory V D\nE : Type u₂\ninst✝ : EnrichedCategory V E\nF G H : EnrichedFunctor V C D\nα : F ⟶ G\nβ : G ⟶ H\nI : EnrichedFunctor V D E\nX : C\n⊢ (whiskerRight {... | [
"V : Type v\ninst✝⁴ : Category.{w, v} V\ninst✝³ : MonoidalCategory V\nC : Type u\ninst✝² : EnrichedCategory V C\nD : Type u₁\ninst✝¹ : EnrichedCategory V D\nE : Type u₂\ninst✝ : EnrichedCategory V E\nF G H : EnrichedFunctor V C D\nα : F ⟶ G\nβ : G ⟶ H\nI : EnrichedFunctor V D E\nX : C\n⊢ homOf V (homTo V (α.out.app... | simp only [whiskerRight_out_app, NatTrans.comp_app, EnrichedFunctor.category_comp_out,
EnrichedFunctor.forget, EnrichedFunctor.comp_obj, EnrichedFunctor.comp_map] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Localization.Monoidal.Basic | {
"line": 412,
"column": 49
} | {
"line": 412,
"column": 64
} | {
"line": 412,
"column": 65
} | [
{
"pp": "case e_f\nC : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\nL : C ⥤ D\nW : MorphismProperty C\ninst✝² : MonoidalCategory C\ninst✝¹ : W.IsMonoidal\ninst✝ : L.IsLocalization W\nunit : D\nε : L.obj (𝟙_ C) ≅ unit\nX Y : LocalizedMonoidal L W ε\nX' Y' : C\ne₁ : L'.... | [
"case e_f\nC : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\nL : C ⥤ D\nW : MorphismProperty C\ninst✝² : MonoidalCategory C\ninst✝¹ : W.IsMonoidal\ninst✝ : L.IsLocalization W\nunit : D\nε : L.obj (𝟙_ C) ≅ unit\nX Y : LocalizedMonoidal L W ε\nX' Y' : C\ne₁ : L'.obj X' ≅ X\n... | Iso.hom_inv_id, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Localization.Monoidal.Basic | {
"line": 433,
"column": 4
} | {
"line": 433,
"column": 19
} | {
"line": 433,
"column": 20
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\nL : C ⥤ D\nW : MorphismProperty C\ninst✝² : MonoidalCategory C\ninst✝¹ : W.IsMonoidal\ninst✝ : L.IsLocalization W\nunit : D\nε : L.obj (𝟙_ C) ≅ unit\nX Y : LocalizedMonoidal L W ε\nX' : C\ne₁ : L'.obj X' ≅ X\nY... | [
"C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\nL : C ⥤ D\nW : MorphismProperty C\ninst✝² : MonoidalCategory C\ninst✝¹ : W.IsMonoidal\ninst✝ : L.IsLocalization W\nunit : D\nε : L.obj (𝟙_ C) ≅ unit\nX Y : LocalizedMonoidal L W ε\nX' : C\ne₁ : L'.obj X' ≅ X\nY' : C\ne₂ : ... | Iso.hom_inv_id, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.FiberedCategory.HomLift | {
"line": 87,
"column": 90
} | {
"line": 88,
"column": 28
} | {
"line": 90,
"column": 0
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₂} 𝒳\ninst✝¹ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ⟶ b\ninst✝ : p.IsHomLift f φ\n⊢ f = eqToHom ⋯ ≫ p.map φ ≫ eqToHom ⋯",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"CategoryTheory.Fun... | [] | by
subst_hom_lift p f φ; simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.FiberedCategory.HomLift | {
"line": 90,
"column": 91
} | {
"line": 91,
"column": 28
} | {
"line": 93,
"column": 0
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₂} 𝒳\ninst✝¹ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ⟶ b\ninst✝ : p.IsHomLift f φ\n⊢ p.map φ = eqToHom ⋯ ≫ f ≫ eqToHom ⋯",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"CategoryTheory.Fun... | [] | by
subst_hom_lift p f φ; simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.FiberedCategory.Cocartesian | {
"line": 117,
"column": 52
} | {
"line": 120,
"column": 21
} | {
"line": 122,
"column": 0
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₁} 𝒮\ninst✝¹ : Category.{v₂, u₂} 𝒳\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ⟶ b\ninst✝ : p.IsCocartesian f φ\n⊢ IsCocartesian.map p f φ φ = 𝟙 b",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"CategoryTheory.Fu... | [] | by
subst_hom_lift p f φ; symm
apply map_uniq
simp only [comp_id] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.FiberedCategory.Cocartesian | {
"line": 236,
"column": 55
} | {
"line": 239,
"column": 21
} | {
"line": 241,
"column": 0
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₁} 𝒮\ninst✝¹ : Category.{v₂, u₂} 𝒳\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ⟶ b\ninst✝ : p.IsStronglyCocartesian f φ\n⊢ map p f φ ⋯ φ = 𝟙 b",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functo... | [] | by
subst_hom_lift p f φ; symm
apply map_uniq
simp only [comp_id] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Functor.Derived.LeftDerived | {
"line": 170,
"column": 2
} | {
"line": 171,
"column": 40
} | {
"line": 173,
"column": 0
} | [
{
"pp": "C : Type u_1\nH : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Category.{v_5, u_2} H\nF F' : C ⥤ H\ne : F ≅ F'\nW : MorphismProperty C\n⊢ F.HasLeftDerivedFunctor W ↔ F'.HasLeftDerivedFunctor W",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheo... | [] | rw [hasLeftDerivedFunctor_iff F W.Q W, hasLeftDerivedFunctor_iff F' W.Q W,
hasRightExtension_iff_of_iso₂ W.Q e] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Functor.Derived.LeftDerived | {
"line": 170,
"column": 2
} | {
"line": 171,
"column": 40
} | {
"line": 173,
"column": 0
} | [
{
"pp": "C : Type u_1\nH : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Category.{v_5, u_2} H\nF F' : C ⥤ H\ne : F ≅ F'\nW : MorphismProperty C\n⊢ F.HasLeftDerivedFunctor W ↔ F'.HasLeftDerivedFunctor W",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheo... | [] | rw [hasLeftDerivedFunctor_iff F W.Q W, hasLeftDerivedFunctor_iff F' W.Q W,
hasRightExtension_iff_of_iso₂ W.Q e] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Functor.Derived.LeftDerived | {
"line": 170,
"column": 2
} | {
"line": 171,
"column": 40
} | {
"line": 173,
"column": 0
} | [
{
"pp": "C : Type u_1\nH : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Category.{v_5, u_2} H\nF F' : C ⥤ H\ne : F ≅ F'\nW : MorphismProperty C\n⊢ F.HasLeftDerivedFunctor W ↔ F'.HasLeftDerivedFunctor W",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheo... | [] | rw [hasLeftDerivedFunctor_iff F W.Q W, hasLeftDerivedFunctor_iff F' W.Q W,
hasRightExtension_iff_of_iso₂ W.Q e] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Functor.Derived.PointwiseLeftDerived | {
"line": 71,
"column": 48
} | {
"line": 73,
"column": 91
} | {
"line": 75,
"column": 0
} | [
{
"pp": "C : Type u₁\nH : Type u₃\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : Category.{v₃, u₃} H\nF : C ⥤ H\nW : MorphismProperty C\nX Y : C\nw : X ⟶ Y\nhw : W w\n⊢ F.HasPointwiseLeftDerivedFunctorAt W X ↔ F.HasPointwiseLeftDerivedFunctorAt W Y",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
... | [] | by
simp only [F.hasPointwiseLeftDerivedFunctorAt_iff W.Q W]
exact hasPointwiseRightKanExtensionAt_iff_of_iso W.Q F (Localization.isoOfHom W.Q W w hw) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.CombinedProducts | {
"line": 52,
"column": 43
} | {
"line": 55,
"column": 59
} | {
"line": 55,
"column": 59
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{u₂, u₁} C\nι₁ : Type u_1\nι₂ : Type u_2\nX : C\nf₁ : ι₁ → C\nf₂ : ι₂ → C\nc₁ : Fan f₁\nc₂ : Fan f₂\nbc : BinaryFan c₁.pt c₂.pt\nh₁ : IsLimit c₁\nh₂ : IsLimit c₂\nh : IsLimit bc\ns : Fan (Sum.elim f₁ f₂)\ni : WalkingPair\n⊢ s.pt ⟶ WalkingPair.casesOn i c₁.pt c₂.pt",
"p... | [] | by
cases i
· exact Fan.IsLimit.lift h₁ (fun a ↦ s.proj (.inl a))
· exact Fan.IsLimit.lift h₂ (fun a ↦ s.proj (.inr a)) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Galois.Equivalence | {
"line": 80,
"column": 15
} | {
"line": 82,
"column": 76
} | {
"line": 82,
"column": 76
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{u₂, u₁} C\nF : C ⥤ FintypeCat\ninst✝¹ : GaloisCategory C\ninst✝ : FiberFunctor F\nF' : C ⥤ FintypeCat := F ⋙ FintypeCat.uSwitch\nthis✝ : FiberFunctor F' := FiberFunctor.comp_right FintypeCat.uSwitch\nthis : (functorToContAction F').EssSurj\nf✝ : Aut F ≃ₜ* Aut F' := autEq... | [] | by
ext : 2
exact FintypeCat.uSwitchEquivalence.unitIso.hom.naturality (F.map f) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Groupoid.Subgroupoid | {
"line": 308,
"column": 16
} | {
"line": 308,
"column": 61
} | {
"line": 310,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Groupoid C\nS : Subgroupoid C\nSn : S.IsNormal\nc✝ d✝ : C\np : d✝ ⟶ c✝\nγ : c✝ ⟶ c✝\nhs : γ ∈ S.arrows c✝ c✝\n⊢ p ≫ γ ≫ Groupoid.inv p ∈ S.arrows d✝ d✝",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"outParam",
"CategoryTheory.Categ... | [] | by convert! Sn.conj (Groupoid.inv p) hs; simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Galois.EssSurj | {
"line": 272,
"column": 2
} | {
"line": 273,
"column": 45
} | {
"line": 275,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝⁵ : Category.{u₂, u₁} C\nF : C ⥤ FintypeCat\ninst✝⁴ : GaloisCategory C\ninst✝³ : FiberFunctor F\nX : Action FintypeCat (Aut F)\ninst✝² : TopologicalSpace X.V.obj\ninst✝¹ : DiscreteTopology X.V.obj\ninst✝ : ContinuousSMul (Aut F) X.V.obj\nι : Type\nhfin : Finite ι\nf : ι → OpenSubgroup... | [] | exact ⟨∐ g, ⟨PreservesCoproduct.iso (functorToAction F) g ≪≫
Sigma.mapIso (fun i ↦ (gu i).some) ≪≫ u⟩⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Groupoid.Subgroupoid | {
"line": 395,
"column": 15
} | {
"line": 395,
"column": 67
} | {
"line": 396,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Groupoid C\nD : Type u_1\ninst✝ : Groupoid D\nφ : C ⥤ D\nS : Subgroupoid D\nSn : S.IsNormal\nc : C\n⊢ 𝟙 c ∈ (comap φ S).arrows c c",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"CategoryTheory.Subgroupoid.comap._proof_1",
"Eq.mpr",
"Catego... | [] | rw [comap, mem_setOf, Functor.map_id]; apply Sn.wide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Groupoid.Subgroupoid | {
"line": 395,
"column": 15
} | {
"line": 395,
"column": 67
} | {
"line": 396,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Groupoid C\nD : Type u_1\ninst✝ : Groupoid D\nφ : C ⥤ D\nS : Subgroupoid D\nSn : S.IsNormal\nc : C\n⊢ 𝟙 c ∈ (comap φ S).arrows c c",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"CategoryTheory.Subgroupoid.comap._proof_1",
"Eq.mpr",
"Catego... | [] | rw [comap, mem_setOf, Functor.map_id]; apply Sn.wide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Groupoid.Subgroupoid | {
"line": 397,
"column": 4
} | {
"line": 397,
"column": 93
} | {
"line": 398,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝¹ : Groupoid C\nD : Type u_1\ninst✝ : Groupoid D\nφ : C ⥤ D\nS : Subgroupoid D\nSn : S.IsNormal\nc✝ d✝ : C\nf : c✝ ⟶ d✝\nγ : c✝ ⟶ c✝\nhγ : γ ∈ (comap φ S).arrows c✝ c✝\n⊢ Groupoid.inv f ≫ γ ≫ f ∈ (comap φ S).arrows d✝ d✝",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants... | [
"C : Type u\ninst✝¹ : Groupoid C\nD : Type u_1\ninst✝ : Groupoid D\nφ : C ⥤ D\nS : Subgroupoid D\nSn : S.IsNormal\nc✝ d✝ : C\nf : c✝ ⟶ d✝\nγ : c✝ ⟶ c✝\nhγ : γ ∈ (comap φ S).arrows c✝ c✝\n⊢ Groupoid.inv (φ.map f) ≫ φ.map γ ≫ φ.map f ∈ S.arrows (φ.obj d✝) (φ.obj d✝)"
] | simp_rw [inv_eq_inv f, comap, mem_setOf, Functor.map_comp, Functor.map_inv, ← inv_eq_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.CategoryTheory.Limits.FormalCoproducts.Basic | {
"line": 375,
"column": 49
} | {
"line": 375,
"column": 71
} | {
"line": 375,
"column": 71
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nA : Type u₁\ninst✝¹ : Category.{v₁, u₁} A\ninst✝ : HasCoproducts A\nJ : Type w\nf : J → FormalCoproduct C\nF✝ F : C ⥤ A\nX✝ Y✝ Z✝ : FormalCoproduct C\nx✝² : X✝ ⟶ Y✝\nx✝¹ : Y✝ ⟶ Z✝\nx✝ : X✝.I\n⊢ (Sigma.ι (fun i ↦ F.obj (X✝.obj i)) x✝ ≫\n Sigma.desc fun i ↦ F.... | [] | by simp [Sigma.ι_desc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.FormalCoproducts.Basic | {
"line": 385,
"column": 32
} | {
"line": 385,
"column": 54
} | {
"line": 385,
"column": 54
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nA : Type u₁\ninst✝¹ : Category.{v₁, u₁} A\ninst✝ : HasCoproducts A\nJ : Type w\nf✝ : J → FormalCoproduct C\nF✝ F : C ⥤ A\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\n⊢ ∀ (b : ((incl C).obj X✝).I),\n Sigma.ι (fun i ↦ F.obj (((incl C).obj X✝).obj i)) b ≫\n ((eval C A ⋙ (whis... | [] | by simp [Sigma.ι_desc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.ChosenPullback | {
"line": 105,
"column": 14
} | {
"line": 105,
"column": 49
} | {
"line": 105,
"column": 50
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX₁ X₂ S : C\nf₁ : X₁ ⟶ S\nf₂ : X₂ ⟶ S\nh : ChosenPullback f₁ f₂\nY : C\ng₁ : Y ⟶ X₁\ng₂ : Y ⟶ X₂\nb : Y ⟶ S\nw : g₁ ≫ f₁ = g₂ ≫ f₂\nhf₁ : g₁ ≫ f₁ = b\nl : Y ⟶ h.pullback\nh₁ : l ≫ h.p₁ = g₁\nh₂ : l ≫ h.p₂ = g₂\n⊢ l ≫ h.p = b",
"ppTerm": "?m.90",
"assigned"... | [] | rw [← h.hp₁, ← hf₁, reassoc_of% h₁] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.ChosenPullback | {
"line": 105,
"column": 14
} | {
"line": 105,
"column": 49
} | {
"line": 105,
"column": 50
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX₁ X₂ S : C\nf₁ : X₁ ⟶ S\nf₂ : X₂ ⟶ S\nh : ChosenPullback f₁ f₂\nY : C\ng₁ : Y ⟶ X₁\ng₂ : Y ⟶ X₂\nb : Y ⟶ S\nw : g₁ ≫ f₁ = g₂ ≫ f₂\nhf₁ : g₁ ≫ f₁ = b\nl : Y ⟶ h.pullback\nh₁ : l ≫ h.p₁ = g₁\nh₂ : l ≫ h.p₂ = g₂\n⊢ l ≫ h.p = b",
"ppTerm": "?m.90",
"assigned"... | [] | rw [← h.hp₁, ← hf₁, reassoc_of% h₁] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.ChosenPullback | {
"line": 105,
"column": 14
} | {
"line": 105,
"column": 49
} | {
"line": 105,
"column": 50
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX₁ X₂ S : C\nf₁ : X₁ ⟶ S\nf₂ : X₂ ⟶ S\nh : ChosenPullback f₁ f₂\nY : C\ng₁ : Y ⟶ X₁\ng₂ : Y ⟶ X₂\nb : Y ⟶ S\nw : g₁ ≫ f₁ = g₂ ≫ f₂\nhf₁ : g₁ ≫ f₁ = b\nl : Y ⟶ h.pullback\nh₁ : l ≫ h.p₁ = g₁\nh₂ : l ≫ h.p₂ = g₂\n⊢ l ≫ h.p = b",
"ppTerm": "?m.90",
"assigned"... | [] | rw [← h.hp₁, ← hf₁, reassoc_of% h₁] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.ChosenPullback | {
"line": 200,
"column": 37
} | {
"line": 201,
"column": 34
} | {
"line": 203,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX₁ X₂ X₃ S : C\nf₁ : X₁ ⟶ S\nf₂ : X₂ ⟶ S\nf₃ : X₃ ⟶ S\nh₁₂ : ChosenPullback f₁ f₂\nh₂₃ : ChosenPullback f₂ f₃\nh₁₃ : ChosenPullback f₁ f₃\nh : ChosenPullback₃ h₁₂ h₂₃ h₁₃\n⊢ h.p₁₃ ≫ h₁₃.p = h.p",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
... | [] | by
rw [← h₁₃.hp₁, p₁₃_p₁_assoc, w₁] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Types.End | {
"line": 129,
"column": 75
} | {
"line": 131,
"column": 13
} | {
"line": 133,
"column": 0
} | [
{
"pp": "J : Type u\ninst✝ : Category.{v, u} J\nF : Jᵒᵖ ⥤ J ⥤ Type (max w u)\ni j : J\nf : i ⟶ j\n⊢ π F i ≫ (F.obj (op i)).map f = π F j ≫ (F.map f.op).app j",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor",
"Opposite",
"CategoryTheory.CategoryStru... | [] | by
ext x
exact x.2 f | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Localization.Pi | {
"line": 38,
"column": 2
} | {
"line": 54,
"column": 52
} | {
"line": 55,
"column": 2
} | [
{
"pp": "case of_equiv\n⊢ ∀ {α β : Type w} (a : α ≃ β),\n (∀ {C : α → Type u₁} {D : α → Type u₂} [inst : (j : α) → Category.{v₁, u₁} (C j)]\n [inst_1 : (j : α) → Category.{v₂, u₂} (D j)] (L : (j : α) → C j ⥤ D j) (W : (j : α) → MorphismProperty (C j))\n [∀ (j : α), (W j).ContainsIdentities] [∀ ... | [
"case h_empty\n⊢ ∀ {C : PEmpty.{w + 1} → Type u₁} {D : PEmpty.{w + 1} → Type u₂}\n [inst : (j : PEmpty.{w + 1}) → Category.{v₁, u₁} (C j)] [inst_1 : (j : PEmpty.{w + 1}) → Category.{v₂, u₂} (D j)]\n (L : (j : PEmpty.{w + 1}) → C j ⥤ D j) (W : (j : PEmpty.{w + 1}) → MorphismProperty (C j))\n [∀ (j : PEmpty.... | · intro J₁ J₂ e hJ₁ C₂ D₂ _ _ L₂ W₂ _ _
let L₁ := fun j => (L₂ (e j))
let E := Pi.equivalenceOfEquiv C₂ e
let E' := Pi.equivalenceOfEquiv D₂ e
haveI : CatCommSq E.functor (Functor.pi L₁) (Functor.pi L₂) E'.functor :=
(CatCommSq.hInvEquiv E (Functor.pi L₁) (Functor.pi L₂) E').symm ⟨Iso.refl _⟩
... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
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