module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.Real.Pi.Leibniz | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 37
} | {
"line": 45,
"column": 2
} | [
{
"pp": "l : ℝ\nh : Tendsto (fun n ↦ ∑ i ∈ range n, (-1) ^ i / (2 * ↑i + 1)) atTop (𝓝 l)\nabel : Tendsto (fun x ↦ ∑' (n : ℕ), (-1) ^ n / (2 * ↑n + 1) * x ^ n) (𝓝[<] 1) (𝓝 l)\nm : 𝓝[<] 1 ≤ 𝓝 1\nq : Tendsto (fun x ↦ x ^ 2) (𝓝[<] 1) (𝓝[<] 1)\n⊢ Tendsto (fun k ↦ ∑ i ∈ range k, (-1) ^ i / (2 * ↑i + 1)) atTop ... | [
"l : ℝ\nh : Tendsto (fun n ↦ ∑ i ∈ range n, (-1) ^ i / (2 * ↑i + 1)) atTop (𝓝 l)\nm : 𝓝[<] 1 ≤ 𝓝 1\nq : Tendsto (fun x ↦ x ^ 2) (𝓝[<] 1) (𝓝[<] 1)\nabel :\n Tendsto (fun x ↦ ((fun x ↦ ∑' (n : ℕ), (-1) ^ n / (2 * ↑n + 1) * x ^ n) ∘ fun x ↦ x ^ 2) x * x) (𝓝[<] 1) (𝓝 (l * 1))\n⊢ Tendsto (fun k ↦ ∑ i ∈ range k, ... | replace abel := (abel.comp q).mul m | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.Analysis.SpecialFunctions.Complex.Arctan | {
"line": 83,
"column": 2
} | {
"line": 83,
"column": 32
} | {
"line": 85,
"column": 0
} | [
{
"pp": "case hx₂\nz : ℂ\nh₀ : z ≠ ↑π / 2\nh₁ : -(π / 2) < z.re\nh₂ : z.re ≤ π / 2\nh : cos z ≠ 0\n⊢ 2 * z.re ≤ π",
"ppTerm": "?hx₂",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Real.partialOrder",
"Real.instLE",
"Real",
"le_d... | [] | · rwa [← le_div_iff₀' two_pos] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Real.Hyperreal | {
"line": 927,
"column": 49
} | {
"line": 927,
"column": 64
} | {
"line": 929,
"column": 0
} | [
{
"pp": "x : ℝ*\n⊢ x⁻¹⁻¹.Infinitesimal ∧ 0 < x⁻¹⁻¹ ↔ x.Infinitesimal ∧ 0 < x",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Hyperreal.instField",
"Eq.mpr",
"Preorder.toLT",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"congrA... | [] | by rw [inv_inv] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Real.Hyperreal | {
"line": 932,
"column": 49
} | {
"line": 932,
"column": 64
} | {
"line": 934,
"column": 0
} | [
{
"pp": "x : ℝ*\n⊢ x⁻¹⁻¹.Infinitesimal ∧ x⁻¹⁻¹ < 0 ↔ x.Infinitesimal ∧ x < 0",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Hyperreal.instField",
"Eq.mpr",
"Preorder.toLT",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"congrA... | [] | by rw [inv_inv] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Real.Hyperreal | {
"line": 936,
"column": 13
} | {
"line": 936,
"column": 28
} | {
"line": 936,
"column": 28
} | [
{
"pp": "x : ℝ*\nh : x ≠ 0\n⊢ x.Infinitesimal ↔ x⁻¹⁻¹.Infinitesimal",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Hyperreal.instField",
"Eq.mpr",
"DivInvMonoid.toInv",
"GroupWithZero.toDivInvMonoid",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMon... | [] | by rw [inv_inv] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Real.Hyperreal | {
"line": 940,
"column": 48
} | {
"line": 940,
"column": 59
} | {
"line": 942,
"column": 0
} | [
{
"pp": "x : ℝ*\nhi : ¬x.Infinitesimal\nhr : x.IsSt 0\n⊢ False",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"hi",
"hr"
],
"usedGoals": []
}
] | [] | exact hi hr | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Artanh | {
"line": 76,
"column": 75
} | {
"line": 78,
"column": 81
} | {
"line": 80,
"column": 0
} | [
{
"pp": "x : ℝ\nhx : x ∈ Ioo (-1) 1\n⊢ tanh (artanh x) = x",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Real",
"HMul.hMul",
"AddGroupWithOne.toAddGroup",
"CommSemiring.toSemiring",
"HSub.hSub",
"Distrib.toAdd",
"instOfNatNat",
"Real.commR... | [] | by
have := sq_sub_sq 1 x
grind [tanh_eq_sinh_div_cosh, sinh_artanh, cosh_artanh, sqrt_ne_zero', mul_pos] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.BinaryEntropy | {
"line": 279,
"column": 2
} | {
"line": 286,
"column": 41
} | {
"line": 287,
"column": 2
} | [
{
"pp": "case hf\n⊢ Tendsto (fun x ↦ log (1 - x)) (𝓝[<] 1) atBot",
"ppTerm": "?hf",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Real.partialOrder",
"Real",
"Set.Ioi",
... | [
"case hg\n⊢ ∀ᶠ (x : ℝ) in 𝓝[<] 1, -log x ≤ -log (1 - 2⁻¹)"
] | · have : Tendsto log (𝓝[>] 0) atBot := Real.tendsto_log_nhdsGT_zero
apply Tendsto.comp (f := (1 - ·)) (g := log) this
have contF : Continuous ((1 : ℝ) - ·) := continuous_sub_left 1
have : MapsTo ((1 : ℝ) - ·) (Iio 1) (Ioi 0) := by
intro p hx
simp_all only [mem_Iio, mem_Ioi, sub_pos]
convert... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.ConjSqrt | {
"line": 60,
"column": 2
} | {
"line": 62,
"column": 41
} | {
"line": 64,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝¹⁰ : PartialOrder A\ninst✝⁹ : Ring A\ninst✝⁸ : StarRing A\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : StarOrderedRing A\ninst✝⁵ : Algebra ℝ A\ninst✝⁴ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝³ : NonnegSpectrumClass ℝ A\ninst✝² : SeparatelyContinuousMul A\ninst✝¹ : IsSemitopo... | [] | have hc' : IsSelfAdjoint (sqrt c) := by cfc_tac
rw [conjSqrt_apply]
by_cases ha : IsSelfAdjoint a <;> grind | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.ConjSqrt | {
"line": 60,
"column": 2
} | {
"line": 62,
"column": 41
} | {
"line": 64,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝¹⁰ : PartialOrder A\ninst✝⁹ : Ring A\ninst✝⁸ : StarRing A\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : StarOrderedRing A\ninst✝⁵ : Algebra ℝ A\ninst✝⁴ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝³ : NonnegSpectrumClass ℝ A\ninst✝² : SeparatelyContinuousMul A\ninst✝¹ : IsSemitopo... | [] | have hc' : IsSelfAdjoint (sqrt c) := by cfc_tac
rw [conjSqrt_apply]
by_cases ha : IsSelfAdjoint a <;> grind | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.ExpLog.Order | {
"line": 87,
"column": 2
} | {
"line": 88,
"column": 43
} | {
"line": 89,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\ns : Set A := {a | IsStrictlyPositive a}\nf : ℝ → A → A := fun p a ↦ if a ∈ s then cfc (fun x ↦ p⁻¹ * (x ^ p - 1)) a else 0\ng : A → A := fun a ↦ if a ∈ s then log a else 0\nhg : Set.EqOn g log s\n⊢ MonotoneOn g {... | [
"A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\ns : Set A := {a | IsStrictlyPositive a}\nf : ℝ → A → A := fun p a ↦ if a ∈ s then cfc (fun x ↦ p⁻¹ * (x ^ p - 1)) a else 0\ng : A → A := fun a ↦ if a ∈ s then log a else 0\nhg : Set.EqOn g log s\n⊢ ∀ᶠ (x : ℝ) in 𝓝[>] 0, f x... | refine isClosed_monotoneOn.mem_of_tendsto (f := f) (b := (𝓝[>] 0))
tendsto_ite_cfc_rpow_sub_one_ite_log ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.Harmonic.GammaDeriv | {
"line": 60,
"column": 59
} | {
"line": 60,
"column": 72
} | {
"line": 60,
"column": 72
} | [
{
"pp": "case succ\nn✝ : ℕ\nf : ℝ → ℝ := log ∘ Gamma\nhc : ConvexOn ℝ (Ioi 0) f\nh_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x\nhder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x\nhder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x\nn : ℕ\nhn : deriv f (↑n + 1) = deriv f 1 + ↑(harmonic n)\n⊢ de... | [
"case succ\nn✝ : ℕ\nf : ℝ → ℝ := log ∘ Gamma\nhc : ConvexOn ℝ (Ioi 0) f\nh_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x\nhder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x\nhder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x\nn : ℕ\nhn : deriv f (↑n + 1) = deriv f 1 + ↑(harmonic n)\n⊢ deriv f 1 + ↑(... | harmonic_succ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Harmonic.GammaDeriv | {
"line": 218,
"column": 4
} | {
"line": 218,
"column": 41
} | {
"line": 219,
"column": 2
} | [
{
"pp": "case refine_1\nf : 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\n⊢ HasDerivAt (fun s ↦ s / 2) (1 / 2) 1",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Nat.instAtLea... | [] | · exact (hasDerivAt_id _).div_const _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Int.Log | {
"line": 276,
"column": 2
} | {
"line": 276,
"column": 67
} | {
"line": 277,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semifield R\ninst✝² : LinearOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : FloorSemiring R\nb : ℕ\nr₁ r₂ : R\nh₀ : 0 < r₁\nh : r₁ ≤ r₂\nh₀' : 0 < r₂\n⊢ clog b r₁ ≤ clog b r₂",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoi... | [
"R : Type u_1\ninst✝³ : Semifield R\ninst✝² : LinearOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : FloorSemiring R\nb : ℕ\nr₁ r₂ : R\nh₀ : 0 < r₁\nh : r₁ ≤ r₂\nh₀' : 0 < r₂\n⊢ log b r₂⁻¹ ≤ log b r₁⁻¹"
] | rw [← neg_log_inv_eq_clog, ← neg_log_inv_eq_clog, neg_le_neg_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 311,
"column": 8
} | {
"line": 313,
"column": 15
} | {
"line": 314,
"column": 2
} | [
{
"pp": "p : ℝ\nhp : p ∈ Ioo 0 1\n⊢ 1 / 2 * -1 ^ (p - 1) / (p - 1) = ∫ (t : ℝ) in Ioi 1, 1 / 2 * t ^ (p - 2)",
"ppTerm": "?m.137",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"... | [] | push _ ∈ _ at hp
rw [integral_const_mul, integral_Ioi_rpow_of_lt (by linarith) zero_lt_one]
ring_nf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 311,
"column": 8
} | {
"line": 313,
"column": 15
} | {
"line": 314,
"column": 2
} | [
{
"pp": "p : ℝ\nhp : p ∈ Ioo 0 1\n⊢ 1 / 2 * -1 ^ (p - 1) / (p - 1) = ∫ (t : ℝ) in Ioi 1, 1 / 2 * t ^ (p - 2)",
"ppTerm": "?m.137",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"... | [] | push _ ∈ _ at hp
rw [integral_const_mul, integral_Ioi_rpow_of_lt (by linarith) zero_lt_one]
ring_nf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 382,
"column": 4
} | {
"line": 382,
"column": 57
} | {
"line": 383,
"column": 2
} | [
{
"pp": "case pos\np t x : ℝ\nhp : 1 < p\nht : 0 ≤ t\nhx : 0 ≤ x\nht' : 0 < t\n⊢ 0 ≤ (p - 1).rpowIntegrand₀₁ t x",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Real",
"Real.instSub",
"HSub.hSub",
"Real.rpowIntegrand₀₁_nonneg",
"Real.instOne",
"_private... | [] | exact rpowIntegrand₀₁_nonneg (by grind) (by grind) hx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.MulExpNegMulSq | {
"line": 88,
"column": 12
} | {
"line": 88,
"column": 42
} | {
"line": 88,
"column": 42
} | [
{
"pp": "ε y : ℝ\n⊢ HasDerivAt ε.mulExpNegMulSq (rexp (-(ε * y * y)) + y * (rexp (-(ε * y * y)) * (-2 * ε * y))) y",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"MulOne.toOne",
"Real",
"NonUnitalCommRing.toNon... | [
"ε y : ℝ\n⊢ HasDerivAt ε.mulExpNegMulSq (1 * rexp (-(ε * y * y)) + y * (rexp (-(ε * y * y)) * (-2 * ε * y))) y"
] | ← one_mul (exp (-(ε * y * y))) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 536,
"column": 2
} | {
"line": 540,
"column": 75
} | {
"line": 542,
"column": 0
} | [
{
"pp": "case inr\nc : ℝ\nhc : c ≠ 0\n⊢ (fun x ↦ log (c * x)) =O[atTop] log",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"NormedCommRing.toSeminormedCommRing",
"Real.partialOrder",
"Real",
"Trans.trans",
"Preorder.toLT",
... | [] | · calc (fun x ↦ log (c * x))
=ᶠ[atTop] (fun x => log c + log x) := by
filter_upwards [eventually_gt_atTop 0] with a ha using log_mul hc ha.ne'
_ =O[atTop] log :=
isLittleO_const_log_atTop.isBigO.add (Asymptotics.isBigO_refl ..) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Constructions.HaarToSphere | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 30
} | {
"line": 121,
"column": 31
} | [
{
"pp": "n : ℕ\nx : ↑(Ioi 0)\nhr₀ : 0 ≤ ↑x\n⊢ ∫⁻ (x : ℝ) in Subtype.val '' Iio x, ENNReal.ofReal (x ^ n) ∂volume = ENNReal.ofReal (↑x ^ (n + 1) / (↑n + 1))",
"ppTerm": "?m.90",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Set.Ioi",
"instHDiv",
"Real.instZero"... | [
"n : ℕ\nx : ↑(Ioi 0)\nhr₀ : 0 ≤ ↑x\n⊢ ∫⁻ (x : ℝ) in Ioo 0 ↑x, ENNReal.ofReal (x ^ n) ∂volume = ENNReal.ofReal (↑x ^ (n + 1) / (↑n + 1))"
] | image_subtype_val_Ioi_Iio, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Stirling | {
"line": 190,
"column": 2
} | {
"line": 190,
"column": 43
} | {
"line": 191,
"column": 2
} | [
{
"pp": "x : ℝ\nx_pos : 0 < x\nhx : ∀ (n : ℕ), x ≤ stirlingSeq (n + 1)\nhx' : x ∈ lowerBounds (Set.range (stirlingSeq ∘ succ))\n⊢ Tendsto stirlingSeq atTop (𝓝 (sInf (Set.range (stirlingSeq ∘ succ))))",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ConditionallyCompl... | [
"x : ℝ\nx_pos : 0 < x\nhx : ∀ (n : ℕ), x ≤ stirlingSeq (n + 1)\nhx' : x ∈ lowerBounds (Set.range (stirlingSeq ∘ succ))\n⊢ Tendsto (fun n ↦ stirlingSeq (n + 1)) atTop (𝓝 (sInf (Set.range (stirlingSeq ∘ succ))))"
] | rw [← Filter.tendsto_add_atTop_iff_nat 1] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.InverseDeriv | {
"line": 37,
"column": 2
} | {
"line": 37,
"column": 33
} | {
"line": 38,
"column": 2
} | [
{
"pp": "case inr\nx : ℝ\nh₁✝ : x ≠ -1\nh₂ : x ≠ 1\nh₁ : -1 < x\n⊢ HasStrictDerivAt arcsin (1 / √(1 - x ^ 2)) x ∧ ContDiffAt ℝ ω arcsin x",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"ContDiffAt",
"NormedCommRing.toSeminormedCommRing",
"Real",
"Preorder.toLT",
... | [
"case inr.inl\nx : ℝ\nh₁✝ : x ≠ -1\nh₂✝ : x ≠ 1\nh₁ : -1 < x\nh₂ : x < 1\n⊢ HasStrictDerivAt arcsin (1 / √(1 - x ^ 2)) x ∧ ContDiffAt ℝ ω arcsin x",
"case inr.inr\nx : ℝ\nh₁✝ : x ≠ -1\nh₂✝ : x ≠ 1\nh₁ : -1 < x\nh₂ : 1 < x\n⊢ HasStrictDerivAt arcsin (1 / √(1 - x ^ 2)) x ∧ ContDiffAt ℝ ω arcsin x"
] | rcases h₂.lt_or_gt with h₂ | h₂ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 511,
"column": 6
} | {
"line": 511,
"column": 35
} | {
"line": 511,
"column": 36
} | [
{
"pp": "L : PeriodPair\nz : ℂ\nl : ↥L.lattice\n⊢ ℘[L] (z - ↑l) = ℘[L] z",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"congrArg",
"HSub.hSub",
"Membership.mem",
"id",
"Int",
"PeriodPair.weierstrassP_add_coe",
"C... | [
"L : PeriodPair\nz : ℂ\nl : ↥L.lattice\n⊢ ℘[L] (z - ↑l + ↑l) = ℘[L] z"
] | ← L.weierstrassP_add_coe _ l, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Niven | {
"line": 141,
"column": 51
} | {
"line": 145,
"column": 30
} | {
"line": 147,
"column": 0
} | [
{
"pp": "θ : ℝ\nhθ : ∃ r, θ = ↑r * π\nhcos : ∃ q, sin θ = ↑q\n⊢ sin θ ∈ {-1, -1 / 2, 0, 1 / 2, 1}",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Iff.mpr",
"Rat.instOfNat",
"Real.instIsOrderedRing",
"Not.intro",
... | [] | by
convert! ← niven (θ := θ - π / 2) ?_ ?_ using 1
· exact cos_sub_pi_div_two θ
· exact hθ.imp' (· - 1 / 2) (by intros; push_cast; linarith)
· simpa [cos_sub_pi_div_two] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 675,
"column": 4
} | {
"line": 675,
"column": 78
} | {
"line": 677,
"column": 2
} | [
{
"pp": "L : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\nκ : ℝ\nhκ : κ > 0\nhκ' : Metric.ball 1 κ ⊆ (fun x_1 ↦ x_1 * ‖z - x‖) ⁻¹' (↑(upperClosure (dist x '' (↑L.lattice \\ {l₀}))))ᶜ\nl : ↥L.lattice\nhl : ↑l ≠ l₀\nthis : ∀ l ∈ L.lattice, l ≠ l₀ → (κ / 2 + 1) * ‖z - x‖ < dist x ... | [] | simpa only [Complex.dist_eq, norm_sub_rev x, mul_comm] using this _ l.2 hl | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 9
} | {
"line": 132,
"column": 4
} | [
{
"pp": "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, Even k ∧ x = cos (↑k * π / ↑n)",
"ppTerm": "?mp",
"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⊢ k ≤ n ∧ Even k ∧ x = cos (↑k * π / ↑n)"
] | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema | {
"line": 145,
"column": 4
} | {
"line": 145,
"column": 9
} | {
"line": 146,
"column": 4
} | [
{
"pp": "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)",
"ppTerm": "?mp",
"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⊢ k ≤ n ∧ Odd k ∧ x = cos (↑k * π / ↑n)"
] | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 696,
"column": 4
} | {
"line": 698,
"column": 52
} | {
"line": 699,
"column": 4
} | [
{
"pp": "case neg\nL : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\nκ : ℝ\nhκ : 1 < κ\nhκ' : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ * κ < ‖↑l - x‖\ne : ℕ × ↥L.lattice ≃ ↥L.lattice ⊕ ℕ × ↥L.lattice :=\n (Equiv.prodCongrLeft fun x ↦ (Denumerable.eqv (Option ℕ)).symm).trans option... | [
"case neg\nL : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\nκ : ℝ\nhκ : 1 < κ\nhκ' : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ * κ < ‖↑l - x‖\ne : ℕ × ↥L.lattice ≃ ↥L.lattice ⊕ ℕ × ↥L.lattice :=\n (Equiv.prodCongrLeft fun x ↦ (Denumerable.eqv (Option ℕ)).symm).trans optionProdEquiv\nH... | have hpx : ‖p.2 - x‖ ≠ 0 := fun h ↦ by
obtain rfl : p.2 = x := by simpa [sub_eq_zero] using h
simpa [(norm_nonneg _).not_gt] using hx p.2 hp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema | {
"line": 171,
"column": 4
} | {
"line": 171,
"column": 9
} | {
"line": 172,
"column": 4
} | [
{
"pp": "case refine_1\nn : ℕ\nhn : n ≠ 0\nx : ℝ\nhx✝ : x ∈ Finset.image (fun k ↦ cos ((2 * ↑k + 1) * π / (2 * ↑n))) (Finset.range n)\nk : ℕ\nhk : k ∈ Finset.range n\nhx : cos ((2 * ↑k + 1) * π / (2 * ↑n)) = x\n⊢ ∃ k_1, ↑↑n * ((2 * ↑k + 1) * π / (2 * ↑n)) = (2 * ↑k_1 + 1) * π / 2",
"ppTerm": "?refine_1",
... | [
"case h\nn : ℕ\nhn : n ≠ 0\nx : ℝ\nhx✝ : x ∈ Finset.image (fun k ↦ cos ((2 * ↑k + 1) * π / (2 * ↑n))) (Finset.range n)\nk : ℕ\nhk : k ∈ Finset.range n\nhx : cos ((2 * ↑k + 1) * π / (2 * ↑n)) = x\n⊢ ↑↑n * ((2 * ↑k + 1) * π / (2 * ↑n)) = (2 * ↑↑k + 1) * π / 2"
] | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 222,
"column": 58
} | {
"line": 222,
"column": 71
} | {
"line": 222,
"column": 72
} | [
{
"pp": "k : ℝ\nhk : 2 < k\nn : ℕ\nhn : 0 < n\n⊢ ↑(8 * n) * ↑n ^ (-k) ≤ 8 * (↑n * ↑n ^ (-k))",
"ppTerm": "?m.189",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.instPow",
"Real.instLE",
"Real",
"NonUnitalCommRing.t... | [
"k : ℝ\nhk : 2 < k\nn : ℕ\nhn : 0 < n\n⊢ ↑8 * ↑n * ↑n ^ (-k) ≤ 8 * (↑n * ↑n ^ (-k))"
] | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema | {
"line": 304,
"column": 6
} | {
"line": 304,
"column": 52
} | {
"line": 305,
"column": 6
} | [
{
"pp": "case mp.refine_1\nn : ℕ\nhn : n ≠ 0\nx : ℝ\nhx : x ∈ Set.Icc (-1) 1\nh✝ : IsExtrOn (fun x ↦ eval x (T ℝ ↑n)) (Set.Icc (-1) 1) x\nh : IsMinOn (fun x ↦ eval x (T ℝ ↑n)) (Set.Icc (-1) 1) x\n⊢ 1 ≤ |eval x (T ℝ ↑n)|",
"ppTerm": "?mp.refine_1",
"assigned": true,
"usedConstants": [
"Iff.mpr"... | [
"case mp.refine_1\nn : ℕ\nhn : n ≠ 0\nx : ℝ\nhx : x ∈ Set.Icc (-1) 1\nh✝ : IsExtrOn (fun x ↦ eval x (T ℝ ↑n)) (Set.Icc (-1) 1) x\nh : IsMinOn (fun x ↦ eval x (T ℝ ↑n)) (Set.Icc (-1) 1) x\n⊢ eval x (T ℝ ↑n) ≤ -1"
] | refine le_abs.mpr (.inr (le_neg_of_le_neg ?_)) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 807,
"column": 4
} | {
"line": 807,
"column": 84
} | {
"line": 808,
"column": 4
} | [
{
"pp": "case refine_1\nL : PeriodPair\nl₀ x : ℂ\nr : NNReal\nhr0 : 0 < r\nhr : Metric.closedBall x ↑r ⊆ (↑L.lattice \\ {l₀})ᶜ\nthis : HasFPowerSeriesOnBall (fderiv ℂ ℘[L - l₀]) (L.weierstrassPExceptSeries l₀ x).derivSeries x ↑r\n⊢ HasFPowerSeriesOnBall (deriv ℘[L - l₀]) (L.derivWeierstrassPExceptSeries l₀ x) x... | [
"case e'_10\nL : PeriodPair\nl₀ x : ℂ\nr : NNReal\nhr0 : 0 < r\nhr : Metric.closedBall x ↑r ⊆ (↑L.lattice \\ {l₀})ᶜ\nthis : HasFPowerSeriesOnBall (fderiv ℂ ℘[L - l₀]) (L.weierstrassPExceptSeries l₀ x).derivSeries x ↑r\n⊢ L.derivWeierstrassPExceptSeries l₀ x =\n ((ContinuousLinearMap.apply ℂ ℂ) 1).compFormalMulti... | convert! (ContinuousLinearMap.apply ℂ ℂ (1 : ℂ)).comp_hasFPowerSeriesOnBall this | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Analysis.SpecificLimits.ArithmeticGeometric | {
"line": 127,
"column": 2
} | {
"line": 127,
"column": 44
} | {
"line": 129,
"column": 0
} | [
{
"pp": "R : Type u_1\na b u₀ : R\ninst✝³ : Field R\ninst✝² : LinearOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : Archimedean R\nha : 1 < a\nh0 : b / (1 - a) < u₀\n⊢ Tendsto (HPow.hPow a) atTop atTop",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"AddGroupWithOne.toAddGroup",
... | [] | exact tendsto_pow_atTop_atTop_of_one_lt ha | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 922,
"column": 2
} | {
"line": 935,
"column": 12
} | {
"line": 937,
"column": 0
} | [
{
"pp": "L : PeriodPair\nl₀ : ℂ\nh : l₀ ∈ L.lattice\n⊢ meromorphicOrderAt ℘[L] l₀ = -2",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"NormedCommRing.toNormedRing",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"InnerProductSpace.toNorme... | [] | trans ↑(-2 : ℤ)
· rw [meromorphicOrderAt_eq_int_iff (L.meromorphic_weierstrassP l₀)]
refine ⟨fun z ↦ (z - l₀) ^ 2 * ℘[L - l₀] z + 1 - (z - l₀) ^ 2 / l₀ ^ 2, ?_, ?_, ?_⟩
· have : AnalyticAt ℂ ℘[L - l₀] l₀ := L.analyticOnNhd_weierstrassPExcept l₀ l₀ (by simp)
suffices AnalyticAt ℂ (fun z ↦ (z - l₀) ^ 2 / ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 922,
"column": 2
} | {
"line": 935,
"column": 12
} | {
"line": 937,
"column": 0
} | [
{
"pp": "L : PeriodPair\nl₀ : ℂ\nh : l₀ ∈ L.lattice\n⊢ meromorphicOrderAt ℘[L] l₀ = -2",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"NormedCommRing.toNormedRing",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"InnerProductSpace.toNorme... | [] | trans ↑(-2 : ℤ)
· rw [meromorphicOrderAt_eq_int_iff (L.meromorphic_weierstrassP l₀)]
refine ⟨fun z ↦ (z - l₀) ^ 2 * ℘[L - l₀] z + 1 - (z - l₀) ^ 2 / l₀ ^ 2, ?_, ?_, ?_⟩
· have : AnalyticAt ℂ ℘[L - l₀] l₀ := L.analyticOnNhd_weierstrassPExcept l₀ l₀ (by simp)
suffices AnalyticAt ℂ (fun z ↦ (z - l₀) ^ 2 / ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 213,
"column": 2
} | {
"line": 213,
"column": 25
} | {
"line": 214,
"column": 2
} | [
{
"pp": "x : ℂ\nhz : x ∈ ℂ_ℤ\n⊢ Summable fun n ↦ (fun n ↦ 2 * x * (1 / ((x + (↑n + 1)) * (x - (↑n + 1))))) n",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMu... | [
"x : ℂ\nhz : x ∈ ℂ_ℤ\n⊢ Summable fun i ↦ 1 / ((x + (↑i + 1)) * (x - (↑i + 1)))"
] | apply Summable.mul_left | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 217,
"column": 2
} | {
"line": 218,
"column": 53
} | {
"line": 219,
"column": 2
} | [
{
"pp": "x : ℂ\nhz : x ∈ ℂ_ℤ\n⊢ Summable fun i ↦ (x - ↑i)⁻¹ * (x + ↑i)⁻¹",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Int.cast",
"NormedCommRing.toSeminormedCommRing",
"HMul.hMul",
"Complex.instNormedField",
"AddGroupWithOne.toAddMonoidWithOne",
"HSu... | [
"x : ℂ\nhz : x ∈ ℂ_ℤ\n⊢ Summable fun i ↦ (x - ↑i)⁻¹ * (x + ↑i)⁻¹"
] | suffices Summable fun i : ℤ ↦ (x - (↑i : ℂ))⁻¹ * (x + (↑i : ℂ))⁻¹ by
apply this.comp_injective CharZero.cast_injective | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 295,
"column": 2
} | {
"line": 295,
"column": 25
} | {
"line": 296,
"column": 2
} | [
{
"pp": "A B : ℝ\nhB : 0 < B\nk : ℕ\nhk : 1 ≤ k\n⊢ Summable fun a ↦\n ↑k ! * 2 * EisensteinSeries.r { coe := { re := A, im := B }, coe_im_pos := hB } ^ (-1 - ↑k) * ‖(↑a + 1) ^ (-1 - ↑k)‖",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Semigroup.toMul",
"R... | [
"A B : ℝ\nhB : 0 < B\nk : ℕ\nhk : 1 ≤ k\n⊢ Summable fun i ↦ ‖(↑i + 1) ^ (-1 - ↑k)‖"
] | apply Summable.mul_left | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Subadditive | {
"line": 70,
"column": 2
} | {
"line": 72,
"column": 85
} | {
"line": 73,
"column": 2
} | [
{
"pp": "u : ℕ → ℝ\nh : Subadditive u\nL : ℝ\nn : ℕ\nhn : n ≠ 0\nhL : u n / ↑n < L\nr : ℕ\nx✝ : r < n\n⊢ ∀ᶠ (a : ℕ) in atTop, u (n * a + r) / ↑(n * a + r) < L",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Filter.Tendsto.div_atTop",
"Eq.mpr",
"GroupWithZero.toMonoidWith... | [
"u : ℕ → ℝ\nh : Subadditive u\nL : ℝ\nn : ℕ\nhn : n ≠ 0\nhL : u n / ↑n < L\nr : ℕ\nx✝ : r < n\nA : Tendsto (fun x ↦ (u n + u r / x) / (↑n + ↑r / x)) atTop (𝓝 ((u n + 0) / (↑n + 0)))\n⊢ ∀ᶠ (a : ℕ) in atTop, u (n * a + r) / ↑(n * a + r) < L"
] | have A : Tendsto (fun x : ℝ => (u n + u r / x) / (n + r / x)) atTop (𝓝 ((u n + 0) / (n + 0))) :=
(tendsto_const_nhds.add <| tendsto_const_nhds.div_atTop tendsto_id).div
(tendsto_const_nhds.add <| tendsto_const_nhds.div_atTop tendsto_id) <| by simpa | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SumIntegralExpDecay | {
"line": 35,
"column": 8
} | {
"line": 35,
"column": 68
} | {
"line": 36,
"column": 8
} | [
{
"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)\nhint : IntegrableOn (fun x ↦ x ^ (↑k + 1 - 1) * rexp (-(c * x))) (Ioi 0) volume\n⊢ 0 ≤ᵐ[volume.restrict (Ioi 0)] fun x ↦ x ^ (↑k + 1 - 1) * rexp (... | [
"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)\nhint : IntegrableOn (fun x ↦ x ^ (↑k + 1 - 1) * rexp (-(c * x))) (Ioi 0) volume\nx : ℝ\nhx : x ∈ Ioi 0\n⊢ 0 x ≤ x ^ (↑k + 1 - 1) * rexp (-(c * x))"
] | filter_upwards [ae_restrict_mem measurableSet_Ioi] with x hx | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.CategoryTheory.Generator.Preadditive | {
"line": 39,
"column": 33
} | {
"line": 39,
"column": 71
} | {
"line": 39,
"column": 71
} | [
{
"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... | [] | simpa only [Limits.zero_comp] using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Generator.Preadditive | {
"line": 39,
"column": 33
} | {
"line": 39,
"column": 71
} | {
"line": 39,
"column": 71
} | [
{
"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... | [] | simpa only [Limits.zero_comp] using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Generator.Preadditive | {
"line": 39,
"column": 33
} | {
"line": 39,
"column": 71
} | {
"line": 39,
"column": 71
} | [
{
"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... | [] | simpa only [Limits.zero_comp] using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Generator.Preadditive | {
"line": 50,
"column": 37
} | {
"line": 50,
"column": 75
} | {
"line": 50,
"column": 75
} | [
{
"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... | [] | simpa only [Limits.zero_comp] using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Generator.Preadditive | {
"line": 50,
"column": 37
} | {
"line": 50,
"column": 75
} | {
"line": 50,
"column": 75
} | [
{
"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... | [] | simpa only [Limits.zero_comp] using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Generator.Preadditive | {
"line": 50,
"column": 37
} | {
"line": 50,
"column": 75
} | {
"line": 50,
"column": 75
} | [
{
"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... | [] | simpa only [Limits.zero_comp] using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Generator.Abelian | {
"line": 39,
"column": 2
} | {
"line": 54,
"column": 37
} | {
"line": 56,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasLimits C\ninst✝ : EnoughInjectives C\nG : C\nhG : IsSeparator G\n⊢ ∃ G, Injective G ∧ IsCoseparator G",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"CategoryTheory.Abelian.toPreadditiv... | [] | haveI : WellPowered.{v} C := wellPowered_of_isDetector G hG.isDetector
haveI : HasProductsOfShape (Subobject (op G)) C := hasProductsOfShape_of_small.{v} _ _
let T : C := Injective.under (piObj fun P : Subobject (op G) => unop P)
refine ⟨T, inferInstance, (Preadditive.isCoseparator_iff _).2 fun X Y f hf => ?_⟩
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Generator.Abelian | {
"line": 39,
"column": 2
} | {
"line": 54,
"column": 37
} | {
"line": 56,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasLimits C\ninst✝ : EnoughInjectives C\nG : C\nhG : IsSeparator G\n⊢ ∃ G, Injective G ∧ IsCoseparator G",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"CategoryTheory.Abelian.toPreadditiv... | [] | haveI : WellPowered.{v} C := wellPowered_of_isDetector G hG.isDetector
haveI : HasProductsOfShape (Subobject (op G)) C := hasProductsOfShape_of_small.{v} _ _
let T : C := Injective.under (piObj fun P : Subobject (op G) => unop P)
refine ⟨T, inferInstance, (Preadditive.isCoseparator_iff _).2 fun X Y f hf => ?_⟩
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.Yoneda | {
"line": 65,
"column": 4
} | {
"line": 65,
"column": 41
} | {
"line": 66,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\nD : Type u'\ninst✝² : Category.{v', u'} D\nF : D ⥤ C\ninst✝¹ : F.Full\nG : C\ninst✝ : Projective G\nhG : IsSeparator G\nhG₂ : ∀ (X : D), ∃ p, Epi p\nX Y : D\np : G ⟶ F.obj X\nh✝ : Epi p\nf : F.obj X ⟶ F.obj Y\n⊢ ∃ a, (F ⋙ preadditiveCoyonedaOb... | [
"C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\nD : Type u'\ninst✝² : Category.{v', u'} D\nF : D ⥤ C\ninst✝¹ : F.Full\nG : C\ninst✝ : Projective G\nhG : IsSeparator G\nhG₂ : ∀ (X : D), ∃ p, Epi p\nX Y : D\np : G ⟶ F.obj X\nh✝ : Epi p\nf : X ⟶ Y\n⊢ ∃ a, (F ⋙ preadditiveCoyonedaObj G).map a = (preadditiv... | obtain ⟨f, rfl⟩ := F.map_surjective f | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.SpecificLimits.FloorPow | {
"line": 199,
"column": 4
} | {
"line": 199,
"column": 73
} | {
"line": 200,
"column": 4
} | [
{
"pp": "u : ℕ → ℝ\nl : ℝ\nhmono : Monotone u\nc : ℕ → ℝ\ncone : ∀ (k : ℕ), 1 < c k\nclim : Tendsto c atTop (𝓝 1)\nhc : ∀ (k : ℕ), Tendsto (fun n ↦ u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)\na : ℝ\nha : 1 < a\nk : ℕ\nhk : c k < a\nH : ∀ (n : ℕ), 0 < ↑⌊c k ^ n⌋₊\n⊢ Tendsto (fun n ↦ ↑⌊c k ^ (n + 1)⌋₊ / c k ^ (n +... | [
"case refine_1\nu : ℕ → ℝ\nl : ℝ\nhmono : Monotone u\nc : ℕ → ℝ\ncone : ∀ (k : ℕ), 1 < c k\nclim : Tendsto c atTop (𝓝 1)\nhc : ∀ (k : ℕ), Tendsto (fun n ↦ u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)\na : ℝ\nha : 1 < a\nk : ℕ\nhk : c k < a\nH : ∀ (n : ℕ), 0 < ↑⌊c k ^ n⌋₊\n⊢ Tendsto (fun n ↦ ↑⌊c k ^ (n + 1)⌋₊ / c k ^ (... | refine Tendsto.div (Tendsto.mul ?_ tendsto_const_nhds) ?_ one_ne_zero | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Limits.FunctorCategory.Shapes.Products | {
"line": 41,
"column": 77
} | {
"line": 42,
"column": 17
} | {
"line": 44,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nα : Type w\ninst✝ : HasLimitsOfShape (Discrete α) C\nf : α → D ⥤ C\nd : D\ns : α\n⊢ (piObjIso f d).hom ≫ Pi.π (fun s ↦ (f s).obj d) s = (Pi.π f s).app d",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants":... | [] | by
simp [piObjIso] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.FunctorCategory.Shapes.Products | {
"line": 48,
"column": 77
} | {
"line": 49,
"column": 17
} | {
"line": 51,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nα : Type w\ninst✝ : HasLimitsOfShape (Discrete α) C\nf : α → D ⥤ C\nd : D\ns : α\n⊢ (piObjIso f d).inv ≫ (Pi.π f s).app d = Pi.π (fun s ↦ (f s).obj d) s",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants":... | [] | by
simp [piObjIso] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Indization.Category | {
"line": 199,
"column": 2
} | {
"line": 204,
"column": 67
} | {
"line": 206,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\n⊢ RepresentablyCoflat Ind.yoneda",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor",
"Opposite",
"CategoryTheory.RepresentablyCoflat.mk",
"CategoryTheory.Ind.yonedaCompInclusion",
"CategoryTh... | [] | refine ⟨fun X => ?_⟩
suffices IsFiltered (CostructuredArrow yoneda ((Ind.inclusion C).obj X)) from
IsFiltered.of_equivalence
((CostructuredArrow.post Ind.yoneda (Ind.inclusion C) X).asEquivalence.trans
(CostructuredArrow.mapNatIso Ind.yonedaCompInclusion)).symm
exact ((isIndObject_iff _).1 (Ind.isIn... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Indization.Category | {
"line": 199,
"column": 2
} | {
"line": 204,
"column": 67
} | {
"line": 206,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\n⊢ RepresentablyCoflat Ind.yoneda",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor",
"Opposite",
"CategoryTheory.RepresentablyCoflat.mk",
"CategoryTheory.Ind.yonedaCompInclusion",
"CategoryTh... | [] | refine ⟨fun X => ?_⟩
suffices IsFiltered (CostructuredArrow yoneda ((Ind.inclusion C).obj X)) from
IsFiltered.of_equivalence
((CostructuredArrow.post Ind.yoneda (Ind.inclusion C) X).asEquivalence.trans
(CostructuredArrow.mapNatIso Ind.yonedaCompInclusion)).symm
exact ((isIndObject_iff _).1 (Ind.isIn... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.Injective.Dimension | {
"line": 301,
"column": 19
} | {
"line": 301,
"column": 40
} | {
"line": 301,
"column": 41
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\n⊢ injectiveDimension X = ⊥ ↔ injectiveDimension X < 0",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"WithBot.addMonoidWithOne",
"WithBot.instPreorder",
"Eq.mpr",
"WithBot.some",
"With... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\n⊢ injectiveDimension X < ↑⊥ ↔ injectiveDimension X < 0"
] | ← WithBot.lt_coe_bot, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.Injective.Dimension | {
"line": 316,
"column": 60
} | {
"line": 316,
"column": 70
} | {
"line": 316,
"column": 70
} | [
{
"pp": "case coe.top\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nhd : injectiveDimension X = ↑⊤\nn : ℕ\nhn : ⊤ ≤ ↑n\n⊢ False",
"ppTerm": "?coe.top",
"assigned": true,
"usedConstants": [
"WithBot",
"instCompleteLinearOrderENat",
"ChainCompletePartialOrder.ins... | [
"case coe.top\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nhd : injectiveDimension X = ↑⊤\nn : ℕ\nhn : ↑n = ⊤\n⊢ False"
] | top_le_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.Pseudoelements | {
"line": 250,
"column": 2
} | {
"line": 251,
"column": 6
} | {
"line": 253,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\n⊢ pseudoApply f 0 = 0",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"Eq.mpr",
"CategoryTheory.Over",
"CategoryTheory.Abelian.Pseudoeleme... | [] | rw [pseudoZero_def, pseudoApply_mk']
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.Pseudoelements | {
"line": 250,
"column": 2
} | {
"line": 251,
"column": 6
} | {
"line": 253,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\n⊢ pseudoApply f 0 = 0",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"Eq.mpr",
"CategoryTheory.Over",
"CategoryTheory.Abelian.Pseudoeleme... | [] | rw [pseudoZero_def, pseudoApply_mk']
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.Pseudoelements | {
"line": 256,
"column": 35
} | {
"line": 258,
"column": 8
} | {
"line": 260,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nP Q : C\na : Pseudoelement P\na' : Over P\n⊢ pseudoApply 0 ⟦a'⟧ = 0",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"Eq.mpr",
"CategoryTheory.Over",
"Category... | [] | by
rw [pseudoZero_def, pseudoApply_mk']
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Abelian.Pseudoelements | {
"line": 305,
"column": 14
} | {
"line": 305,
"column": 31
} | {
"line": 305,
"column": 32
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\nP Q : C\nf : P ⟶ Q\ninst✝ : Epi f\nqbar : Pseudoelement Q\nq : Over Q\n⊢ 𝟙 (pullback f q.hom) ≫ ((fun g ↦ app f g) (Over.mk (pullback.fst f q.hom))).hom = pullback.snd f q.hom ≫ q.hom",
"ppTerm": "?m.77",
"assigned": true,
"usedCo... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\nP Q : C\nf : P ⟶ Q\ninst✝ : Epi f\nqbar : Pseudoelement Q\nq : Over Q\n⊢ ((fun g ↦ app f g) (Over.mk (pullback.fst f q.hom))).hom = pullback.snd f q.hom ≫ q.hom"
] | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.SerreClass.MorphismProperty | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 21
} | {
"line": 204,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\nD : Type u'\ninst✝² : Category.{v', u'} D\ninst✝¹ : Abelian D\nP : ObjectProperty C\ninst✝ : P.IsSerreClass\n⊢ P.isoModSerre.IsStableUnderBaseChange",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"id",
"Categ... | [
"C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\nD : Type u'\ninst✝² : Category.{v', u'} D\ninst✝¹ : Abelian D\nP : ObjectProperty C\ninst✝ : P.IsSerreClass\n⊢ (P.monoModSerre ⊓ P.epiModSerre).IsStableUnderBaseChange"
] | dsimp [isoModSerre] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Abelian.SerreClass.MorphismProperty | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 21
} | {
"line": 218,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\nD : Type u'\ninst✝² : Category.{v', u'} D\ninst✝¹ : Abelian D\nP : ObjectProperty C\ninst✝ : P.IsSerreClass\n⊢ P.isoModSerre.IsStableUnderCobaseChange",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"CategoryTheory.... | [
"C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\nD : Type u'\ninst✝² : Category.{v', u'} D\ninst✝¹ : Abelian D\nP : ObjectProperty C\ninst✝ : P.IsSerreClass\n⊢ (P.monoModSerre ⊓ P.epiModSerre).IsStableUnderCobaseChange"
] | dsimp [isoModSerre] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Monoidal.Rigid.FunctorCategory | {
"line": 48,
"column": 33
} | {
"line": 48,
"column": 50
} | {
"line": 48,
"column": 51
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝³ : Groupoid C\ninst✝² : Category.{v_1, u_2} D\ninst✝¹ : MonoidalCategory D\ninst✝ : RightRigidCategory D\nF : C ⥤ D\nX Y : C\nf : X ⟶ Y\n⊢ 𝟙 (𝟙_ D) ≫ η_ (F.obj Y) (F.obj Y)ᘁ = η_ (F.obj X) (F.obj X)ᘁ ≫ (F.map f ⊗ₘ inv (F.map f)ᘁ)",
"ppTerm": "?m.107",
"assign... | [
"C : Type u_1\nD : Type u_2\ninst✝³ : Groupoid C\ninst✝² : Category.{v_1, u_2} D\ninst✝¹ : MonoidalCategory D\ninst✝ : RightRigidCategory D\nF : C ⥤ D\nX Y : C\nf : X ⟶ Y\n⊢ η_ (F.obj Y) (F.obj Y)ᘁ = η_ (F.obj X) (F.obj X)ᘁ ≫ (F.map f ⊗ₘ inv (F.map f)ᘁ)"
] | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Adjunction.Lifting.Right | {
"line": 161,
"column": 2
} | {
"line": 162,
"column": 73
} | {
"line": 163,
"column": 2
} | [
{
"pp": "A : Type u₁\nB : Type u₂\nC : Type u₃\ninst✝³ : Category.{v₁, u₁} A\ninst✝² : Category.{v₂, u₂} B\ninst✝¹ : Category.{v₃, u₃} C\nU : A ⥤ B\nF : B ⥤ A\nL : C ⥤ B\nU' : A ⥤ C\nadj₁ : F ⊣ U\nadj₂ : L ⋙ F ⊣ U'\ninst✝ : HasCoreflexiveEqualizers C\nh : (X : B) → RegularMono (adj₁.unit.app X)\n⊢ B ⥤ C",
"... | [
"A : Type u₁\nB : Type u₂\nC : Type u₃\ninst✝³ : Category.{v₁, u₁} A\ninst✝² : Category.{v₂, u₂} B\ninst✝¹ : Category.{v₃, u₃} C\nU : A ⥤ B\nF : B ⥤ A\nL : C ⥤ B\nU' : A ⥤ C\nadj₁ : F ⊣ U\nadj₂ : L ⋙ F ⊣ U'\ninst✝ : HasCoreflexiveEqualizers C\nh : (X : B) → RegularMono (adj₁.unit.app X)\n⊢ ∀ (X' X : C) (Y : B) (f :... | refine Adjunction.rightAdjointOfEquiv
(fun X Y => (constructRightAdjointEquiv L _ adj₁ adj₂ h X Y).symm) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Action.Monoidal | {
"line": 309,
"column": 12
} | {
"line": 309,
"column": 29
} | {
"line": 309,
"column": 30
} | [
{
"pp": "V : Type u_1\ninst✝⁵ : Category.{v_1, u_1} V\nG : Type u_2\ninst✝⁴ : Monoid G\nW : Type u_3\ninst✝³ : Category.{v_2, u_3} W\ninst✝² : MonoidalCategory V\ninst✝¹ : MonoidalCategory W\nF : V ⥤ W\ninst✝ : F.LaxMonoidal\ng : G\n⊢ 𝟙 (𝟙_ W) ≫ ε F = ε F ≫ F.map (𝟙 (𝟙_ V))",
"ppTerm": "?m.112",
"as... | [
"V : Type u_1\ninst✝⁵ : Category.{v_1, u_1} V\nG : Type u_2\ninst✝⁴ : Monoid G\nW : Type u_3\ninst✝³ : Category.{v_2, u_3} W\ninst✝² : MonoidalCategory V\ninst✝¹ : MonoidalCategory W\nF : V ⥤ W\ninst✝ : F.LaxMonoidal\ng : G\n⊢ ε F = ε F ≫ F.map (𝟙 (𝟙_ V))"
] | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Action.Monoidal | {
"line": 335,
"column": 20
} | {
"line": 335,
"column": 37
} | {
"line": 335,
"column": 38
} | [
{
"pp": "V : Type u_1\ninst✝⁵ : Category.{v_1, u_1} V\nG : Type u_2\ninst✝⁴ : Monoid G\nW : Type u_3\ninst✝³ : Category.{v_2, u_3} W\ninst✝² : MonoidalCategory V\ninst✝¹ : MonoidalCategory W\nF : V ⥤ W\ninst✝ : F.OplaxMonoidal\ng : G\n⊢ 𝟙 (F.obj (𝟙_ V)) ≫ η F = η F ≫ 𝟙 (𝟙_ W)",
"ppTerm": "?m.122",
"... | [
"V : Type u_1\ninst✝⁵ : Category.{v_1, u_1} V\nG : Type u_2\ninst✝⁴ : Monoid G\nW : Type u_3\ninst✝³ : Category.{v_2, u_3} W\ninst✝² : MonoidalCategory V\ninst✝¹ : MonoidalCategory W\nF : V ⥤ W\ninst✝ : F.OplaxMonoidal\ng : G\n⊢ η F = η F ≫ 𝟙 (𝟙_ W)"
] | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Pseudo | {
"line": 227,
"column": 80
} | {
"line": 228,
"column": 38
} | {
"line": 230,
"column": 0
} | [
{
"pp": "B : Type u₁\ninst✝¹ : Bicategory B\nC : Type u₂\ninst✝ : Bicategory C\nF G : B ⥤ᵖ C\nα : F ⟶ G\na b : B\nf g : a ⟶ b\nη : f ≅ g\n⊢ (α.naturality g).inv = α.app a ◁ G.map₂ η.inv ≫ (α.naturality f).inv ≫ F.map₂ η.hom ▷ α.app b",
"ppTerm": "?m.108",
"assigned": true,
"usedConstants": [
"... | [] | by
simp [naturality_naturality_iso α η] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Pseudo | {
"line": 251,
"column": 39
} | {
"line": 252,
"column": 34
} | {
"line": 254,
"column": 0
} | [
{
"pp": "B : Type u₁\ninst✝¹ : Bicategory B\nC : Type u₂\ninst✝ : Bicategory C\nF G : B ⥤ᵖ C\nα : F ⟶ G\na b c : B\nf : a ⟶ b\ng : b ⟶ c\n⊢ (α.naturality (f ≫ g)).inv =\n α.app a ◁ (G.mapComp f g).hom ≫\n (α_ (α.app a) (G.map f) (G.map g)).inv ≫\n (α.naturality f).inv ▷ G.map g ≫\n (α_ (... | [] | by
simp [naturality_comp_iso α f g] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Oplax | {
"line": 137,
"column": 6
} | {
"line": 137,
"column": 26
} | {
"line": 138,
"column": 6
} | [
{
"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\n⊢ 𝟙 ((η.app a ≫ θ.app a) ≫ H.map (𝟙 a)) ⊗≫\n η.app a ◁ θ.naturality (𝟙 a) ⊗≫\n (η.naturality (𝟙 a) ≫ F.mapId a ▷ η.app a) ▷ θ.app a ⊗≫ 𝟙 (𝟙 (F.obj a) ≫ η... | [
"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\n⊢ 𝟙 ((η.app a ≫ θ.app a) ≫ H.map (𝟙 a)) ⊗≫\n η.app a ◁ θ.naturality (𝟙 a) ⊗≫\n (η.app a ◁ G.mapId a ≫ (ρ_ (η.app a)).hom ≫ (λ_ (η.app a)).inv) ▷ θ.app a ⊗≫ 𝟙 (𝟙 (F.ob... | rw [η.naturality_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Oplax | {
"line": 150,
"column": 2
} | {
"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⊢ η.vCompNaturality θ (f ≫ g) ≫ F.mapComp f g ▷ η.vCompApp θ c =\n η.vCompApp θ a ◁ H.mapComp f g ≫\n (α_ (η.vCompApp θ a) (H.map f) (H.m... | [] | calc
_ = 𝟙 _ ⊗≫ η.app a ◁ θ.naturality (f ≫ g) ⊗≫
(η.naturality (f ≫ g) ≫ F.mapComp f g ▷ η.app c) ▷ θ.app c ⊗≫ 𝟙 _ := by
bicategory
_ = 𝟙 _ ⊗≫ η.app a ◁ (θ.naturality (f ≫ g) ≫ G.mapComp f g ▷ θ.app c) ⊗≫
(η.naturality f ▷ G.map g ⊗≫ F.map f ◁ η.naturality g) ▷ θ.app c ⊗≫ 𝟙 _ := b... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Oplax | {
"line": 150,
"column": 2
} | {
"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⊢ η.vCompNaturality θ (f ≫ g) ≫ F.mapComp f g ▷ η.vCompApp θ c =\n η.vCompApp θ a ◁ H.mapComp f g ≫\n (α_ (η.vCompApp θ a) (H.map f) (H.m... | [] | calc
_ = 𝟙 _ ⊗≫ η.app a ◁ θ.naturality (f ≫ g) ⊗≫
(η.naturality (f ≫ g) ≫ F.mapComp f g ▷ η.app c) ▷ θ.app c ⊗≫ 𝟙 _ := by
bicategory
_ = 𝟙 _ ⊗≫ η.app a ◁ (θ.naturality (f ≫ g) ≫ G.mapComp f g ▷ θ.app c) ⊗≫
(η.naturality f ▷ G.map g ⊗≫ F.map f ◁ η.naturality g) ▷ θ.app c ⊗≫ 𝟙 _ := b... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Oplax | {
"line": 150,
"column": 2
} | {
"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⊢ η.vCompNaturality θ (f ≫ g) ≫ F.mapComp f g ▷ η.vCompApp θ c =\n η.vCompApp θ a ◁ H.mapComp f g ≫\n (α_ (η.vCompApp θ a) (H.map f) (H.m... | [] | calc
_ = 𝟙 _ ⊗≫ η.app a ◁ θ.naturality (f ≫ g) ⊗≫
(η.naturality (f ≫ g) ≫ F.mapComp f g ▷ η.app c) ▷ θ.app c ⊗≫ 𝟙 _ := by
bicategory
_ = 𝟙 _ ⊗≫ η.app a ◁ (θ.naturality (f ≫ g) ≫ G.mapComp f g ▷ θ.app c) ⊗≫
(η.naturality f ▷ G.map g ⊗≫ F.map f ◁ η.naturality g) ▷ θ.app c ⊗≫ 𝟙 _ := b... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Lax | {
"line": 266,
"column": 6
} | {
"line": 266,
"column": 26
} | {
"line": 267,
"column": 6
} | [
{
"pp": "B : Type u₁\ninst✝¹ : Bicategory B\nC : Type u₂\ninst✝ : Bicategory C\nF G H : B ⥤ᴸ C\nη : OplaxTrans F G\nθ : OplaxTrans G H\na : B\n⊢ 𝟙 (𝟙 (F.obj a) ≫ η.vCompApp θ a) ⊗≫\n (F.mapId a ▷ η.app a ≫ η.naturality (𝟙 a)) ▷ θ.app a ⊗≫\n η.app a ◁ θ.naturality (𝟙 a) ⊗≫ 𝟙 ((η.app a ≫ θ.app a)... | [
"B : Type u₁\ninst✝¹ : Bicategory B\nC : Type u₂\ninst✝ : Bicategory C\nF G H : B ⥤ᴸ C\nη : OplaxTrans F G\nθ : OplaxTrans G H\na : B\n⊢ 𝟙 (𝟙 (F.obj a) ≫ η.vCompApp θ a) ⊗≫\n ((λ_ (η.app a)).hom ≫ (ρ_ (η.app a)).inv ≫ η.app a ◁ G.mapId a) ▷ θ.app a ⊗≫\n η.app a ◁ θ.naturality (𝟙 a) ⊗≫ 𝟙 ((η.app a ≫ ... | rw [η.naturality_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Category.Cat.Limit | {
"line": 72,
"column": 4
} | {
"line": 73,
"column": 43
} | {
"line": 74,
"column": 2
} | [
{
"pp": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ Cat\nX Y Z : limit (F ⋙ objects)\nf : limit (homDiagram X Y)\ng : limit (homDiagram Y Z)\nj j' : J\nh : j ⟶ j'\n⊢ (hom ((homDiagram X Z).map h)) ((hom (limit.π (homDiagram X Y) j)) f ≫ (hom (limit.π (homDiagram Y Z) j)) g) =\n (hom (limit.π (homDiagram X ... | [] | simp [-homDiagram_obj, ← limit.w_apply (homDiagram X Y) h f,
← limit.w_apply (homDiagram Y Z) h g] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Category.Cat.Limit | {
"line": 72,
"column": 4
} | {
"line": 73,
"column": 43
} | {
"line": 74,
"column": 2
} | [
{
"pp": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ Cat\nX Y Z : limit (F ⋙ objects)\nf : limit (homDiagram X Y)\ng : limit (homDiagram Y Z)\nj j' : J\nh : j ⟶ j'\n⊢ (hom ((homDiagram X Z).map h)) ((hom (limit.π (homDiagram X Y) j)) f ≫ (hom (limit.π (homDiagram Y Z) j)) g) =\n (hom (limit.π (homDiagram X ... | [] | simp [-homDiagram_obj, ← limit.w_apply (homDiagram X Y) h f,
← limit.w_apply (homDiagram Y Z) h g] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Category.Cat.Limit | {
"line": 72,
"column": 4
} | {
"line": 73,
"column": 43
} | {
"line": 74,
"column": 2
} | [
{
"pp": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ Cat\nX Y Z : limit (F ⋙ objects)\nf : limit (homDiagram X Y)\ng : limit (homDiagram Y Z)\nj j' : J\nh : j ⟶ j'\n⊢ (hom ((homDiagram X Z).map h)) ((hom (limit.π (homDiagram X Y) j)) f ≫ (hom (limit.π (homDiagram Y Z) j)) g) =\n (hom (limit.π (homDiagram X ... | [] | simp [-homDiagram_obj, ← limit.w_apply (homDiagram X Y) h f,
← limit.w_apply (homDiagram Y Z) h g] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Category.PartialFun | {
"line": 149,
"column": 31
} | {
"line": 149,
"column": 79
} | {
"line": 149,
"column": 79
} | [
{
"pp": "X : Pointed\n⊢ ((pointedToPartialFun ⋙ partialFunToPointed).obj X).X ≃ ((𝟭 Pointed).obj X).X",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Classical.propDecidable",
"Pointed.point",
"Equiv.optionSubtypeNe",
"Pointed.X",
"Eq"
],
"usedFVars"... | [] | by classical exact Equiv.optionSubtypeNe X.point | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Category.Cat.Limit | {
"line": 141,
"column": 17
} | {
"line": 143,
"column": 43
} | {
"line": 144,
"column": 2
} | [
{
"pp": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ Cat\ns : Cone F\nj : J\nX Y : ↑s.pt\nf : X ⟶ Y\n⊢ (limitConeLift F s ≫ (limitCone F).π.app j).toFunctor.map f = eqToHom ⋯ ≫ (s.π.app j).toFunctor.map f ≫ eqToHom ⋯",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"CategoryTheory.Li... | [] | by
dsimp [limitConeLift]
exact Types.Limit.π_mk.{v, v} _ _ _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Comma.StructuredArrow.Final | {
"line": 44,
"column": 2
} | {
"line": 45,
"column": 66
} | {
"line": 46,
"column": 2
} | [
{
"pp": "A : Type u₁\ninst✝⁴ : SmallCategory A\nB : Type u₁\ninst✝³ : SmallCategory B\nT : Type u₁\ninst✝² : SmallCategory T\nL : A ⥤ T\nR : B ⥤ T\ninst✝¹ : R.Final\ninst✝ : ∀ (b : B), (CostructuredArrow.toOver L (R.obj b)).Final\nG : T ⥤ Type u₁\n⊢ IsIso (colimit.pre G L)",
"ppTerm": "?m.32",
"assigned... | [
"A : Type u₁\ninst✝⁴ : SmallCategory A\nB : Type u₁\ninst✝³ : SmallCategory B\nT : Type u₁\ninst✝² : SmallCategory T\nL : A ⥤ T\nR : B ⥤ T\ninst✝¹ : R.Final\ninst✝ : ∀ (b : B), (CostructuredArrow.toOver L (R.obj b)).Final\nG : T ⥤ Type u₁\nthis : ∀ (b : B), ((R.whiskerLeft (preFunctor L (𝟭 T))).app b).toFunctor.Fi... | have : ∀ (b : B), Final ((whiskerLeft R (preFunctor L (𝟭 T))).app b).toFunctor := fun b =>
inferInstanceAs (Final (CostructuredArrow.toOver L (R.obj b))) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Localization.Prod | {
"line": 82,
"column": 12
} | {
"line": 82,
"column": 14
} | {
"line": 82,
"column": 15
} | [
{
"pp": "C₁ : Type u₁\nC₂ : Type u₂\nD₁ : Type u₃\nD₂ : Type u₄\ninst✝⁶ : Category.{v₁, u₁} C₁\ninst✝⁵ : Category.{v₂, u₂} C₂\ninst✝⁴ : Category.{v₃, u₃} D₁\ninst✝³ : Category.{v₄, u₄} D₂\nL₁ : C₁ ⥤ D₁\nW₁ : MorphismProperty C₁\nL₂ : C₂ ⥤ D₂\nW₂ : MorphismProperty C₂\nE : Type u₅\ninst✝² : Category.{v₅, u₅} E\n... | [
"C₁ : Type u₁\nC₂ : Type u₂\nD₁ : Type u₃\nD₂ : Type u₄\ninst✝⁶ : Category.{v₁, u₁} C₁\ninst✝⁵ : Category.{v₂, u₂} C₂\ninst✝⁴ : Category.{v₃, u₃} D₁\ninst✝³ : Category.{v₄, u₄} D₂\nL₁ : C₁ ⥤ D₁\nW₁ : MorphismProperty C₁\nL₂ : C₂ ⥤ D₂\nW₂ : MorphismProperty C₂\nE : Type u₅\ninst✝² : Category.{v₅, u₅} E\nF : C₁ × C₂ ... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Localization.Monoidal.Basic | {
"line": 243,
"column": 2
} | {
"line": 243,
"column": 63
} | {
"line": 244,
"column": 2
} | [
{
"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\nX₁ X₂ X₃ : C\n⊢ (((Localization.associator L' L' L' L' L' L'... | [
"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\nX₁ X₂ X₃ : C\n⊢ (((Localization.associator L' L' L' L' L' L' W W W W W (... | simp only [Functor.map_id, comp_id, NatTrans.id_app, id_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Discrete.SumsProducts | {
"line": 42,
"column": 69
} | {
"line": 42,
"column": 71
} | {
"line": 42,
"column": 71
} | [
{
"pp": "J : Type u_1\nK : Type u_2\nX✝ Y✝ : Discrete J × Discrete K\nx✝ : X✝ ⟶ Y✝\nf₁ : X✝.1.as = Y✝.1.as\nf₂ : X✝.2.as = Y✝.2.as\n⊢ { as := (Y✝.1.as, X✝.2.as) } = { as := (Y✝.1.as, Y✝.2.as) }",
"ppTerm": "?m.132",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
... | [
"J : Type u_1\nK : Type u_2\nX✝ Y✝ : Discrete J × Discrete K\nx✝ : X✝ ⟶ Y✝\nf₁ : X✝.1.as = Y✝.1.as\nf₂ : X✝.2.as = Y✝.2.as\n⊢ { as := (Y✝.1.as, Y✝.2.as) } = { as := (Y✝.1.as, Y✝.2.as) }"
] | f₂ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Distributive.Monoidal | {
"line": 213,
"column": 2
} | {
"line": 213,
"column": 25
} | {
"line": 215,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v, u_1} C\ninst✝² : MonoidalCategory C\ninst✝¹ : HasBinaryCoproducts C\ninst✝ : BraidedCategory C\nX Y Z : C\n⊢ coprodComparison (tensorLeft X) Y Z ≫ (β_ X (Y ⨿ Z)).hom =\n coprod.map (β_ X Y).hom (β_ X Z).hom ≫ coprodComparison (tensorRight X) Y Z",
"ppTerm": "?... | [] | simp [coprodComparison] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Distributive.Monoidal | {
"line": 213,
"column": 2
} | {
"line": 213,
"column": 25
} | {
"line": 215,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v, u_1} C\ninst✝² : MonoidalCategory C\ninst✝¹ : HasBinaryCoproducts C\ninst✝ : BraidedCategory C\nX Y Z : C\n⊢ coprodComparison (tensorLeft X) Y Z ≫ (β_ X (Y ⨿ Z)).hom =\n coprod.map (β_ X Y).hom (β_ X Z).hom ≫ coprodComparison (tensorRight X) Y Z",
"ppTerm": "?... | [] | simp [coprodComparison] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Distributive.Monoidal | {
"line": 213,
"column": 2
} | {
"line": 213,
"column": 25
} | {
"line": 215,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v, u_1} C\ninst✝² : MonoidalCategory C\ninst✝¹ : HasBinaryCoproducts C\ninst✝ : BraidedCategory C\nX Y Z : C\n⊢ coprodComparison (tensorLeft X) Y Z ≫ (β_ X (Y ⨿ Z)).hom =\n coprod.map (β_ X Y).hom (β_ X Z).hom ≫ coprodComparison (tensorRight X) Y Z",
"ppTerm": "?... | [] | simp [coprodComparison] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Distributive.Monoidal | {
"line": 222,
"column": 2
} | {
"line": 222,
"column": 25
} | {
"line": 224,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v, u_1} C\ninst✝² : MonoidalCategory C\ninst✝¹ : HasBinaryCoproducts C\ninst✝ : SymmetricCategory C\nX Y Z : C\n⊢ coprodComparison (tensorRight X) Y Z ≫ (β_ (Y ⨿ Z) X).hom =\n coprod.map (β_ Y X).hom (β_ Z X).hom ≫ coprodComparison (tensorLeft X) Y Z",
"ppTerm": ... | [] | simp [coprodComparison] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Distributive.Monoidal | {
"line": 222,
"column": 2
} | {
"line": 222,
"column": 25
} | {
"line": 224,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v, u_1} C\ninst✝² : MonoidalCategory C\ninst✝¹ : HasBinaryCoproducts C\ninst✝ : SymmetricCategory C\nX Y Z : C\n⊢ coprodComparison (tensorRight X) Y Z ≫ (β_ (Y ⨿ Z) X).hom =\n coprod.map (β_ Y X).hom (β_ Z X).hom ≫ coprodComparison (tensorLeft X) Y Z",
"ppTerm": ... | [] | simp [coprodComparison] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Distributive.Monoidal | {
"line": 222,
"column": 2
} | {
"line": 222,
"column": 25
} | {
"line": 224,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v, u_1} C\ninst✝² : MonoidalCategory C\ninst✝¹ : HasBinaryCoproducts C\ninst✝ : SymmetricCategory C\nX Y Z : C\n⊢ coprodComparison (tensorRight X) Y Z ≫ (β_ (Y ⨿ Z) X).hom =\n coprod.map (β_ Y X).hom (β_ Z X).hom ≫ coprodComparison (tensorLeft X) Y Z",
"ppTerm": ... | [] | simp [coprodComparison] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.FiberedCategory.HasFibers | {
"line": 188,
"column": 19
} | {
"line": 188,
"column": 30
} | {
"line": 188,
"column": 30
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝³ : Category.{v₁, u₁} 𝒮\ninst✝² : Category.{v₂, u₂} 𝒳\np : 𝒳 ⥤ 𝒮\ninst✝¹ : HasFibers p\ninst✝ : p.IsPreFibered\nR S : 𝒮\na : 𝒳\nf : R ⟶ S\nha : p.obj a = S\n| f",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStr... | [
"𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝³ : Category.{v₁, u₁} 𝒮\ninst✝² : Category.{v₂, u₂} 𝒳\np : 𝒳 ⥤ 𝒮\ninst✝¹ : HasFibers p\ninst✝ : p.IsPreFibered\nR S : 𝒮\na : 𝒳\nf : R ⟶ S\nha : p.obj a = S\n| 𝟙 R ≫ f"
] | ← id_comp f | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.Limits.Preserves.Grothendieck | {
"line": 87,
"column": 2
} | {
"line": 105,
"column": 48
} | {
"line": 106,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C\nH : Type u₂\ninst✝⁶ : Category.{v₂, u₂} H\nJ : Type u₃\ninst✝⁵ : Category.{v₃, u₃} J\nF : C ⥤ Cat\ninst✝⁴ : HasColimitsOfShape C H\ninst✝³ : HasLimitsOfShape J H\ninst✝² : ∀ (c : C), HasColimitsOfShape (↑(F.obj c)) H\ninst✝¹ : PreservesLimitsOfShape J colim\ni... | [
"C : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C\nH : Type u₂\ninst✝⁶ : Category.{v₂, u₂} H\nJ : Type u₃\ninst✝⁵ : Category.{v₃, u₃} J\nF : C ⥤ Cat\ninst✝⁴ : HasColimitsOfShape C H\ninst✝³ : HasLimitsOfShape J H\ninst✝² : ∀ (c : C), HasColimitsOfShape (↑(F.obj c)) H\ninst✝¹ : PreservesLimitsOfShape J colim\ninst✝ : ∀ (c ... | haveI : IsIso (limit.post K colim) := by
convert! Iso.isIso_hom i₂
ext
simp only [colim_obj, Functor.comp_obj, limit.post_π, colim_map, Iso.trans_def,
Iso.trans_assoc, Iso.trans_hom, Category.assoc, HasLimit.isoOfNatIso_hom_π,
fiberwiseColim_obj, isoWhiskerLeft_hom, NatTrans.comp_app, Functor.as... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.CategoryTheory.Functor.Derived.Adjunction | {
"line": 73,
"column": 6
} | {
"line": 73,
"column": 23
} | {
"line": 73,
"column": 24
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nD₁ : Type u_3\nD₂ : Type u_4\ninst✝⁷ : Category.{v_1, u_1} C₁\ninst✝⁶ : Category.{v_2, u_2} C₂\ninst✝⁵ : Category.{v_3, u_3} D₁\ninst✝⁴ : Category.{v_4, u_4} D₂\nG : C₁ ⥤ C₂\nF : C₂ ⥤ C₁\nadj : G ⊣ F\nL₁ : C₁ ⥤ D₁\nL₂ : C₂ ⥤ D₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismPrope... | [
"C₁ : Type u_1\nC₂ : Type u_2\nD₁ : Type u_3\nD₂ : Type u_4\ninst✝⁷ : Category.{v_1, u_1} C₁\ninst✝⁶ : Category.{v_2, u_2} C₂\ninst✝⁵ : Category.{v_3, u_3} D₁\ninst✝⁴ : Category.{v_4, u_4} D₂\nG : C₁ ⥤ C₂\nF : C₂ ⥤ C₁\nadj : G ⊣ F\nL₁ : C₁ ⥤ D₁\nL₂ : C₂ ⥤ D₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\ninst... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Functor.Derived.Adjunction | {
"line": 73,
"column": 24
} | {
"line": 73,
"column": 41
} | {
"line": 73,
"column": 42
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nD₁ : Type u_3\nD₂ : Type u_4\ninst✝⁷ : Category.{v_1, u_1} C₁\ninst✝⁶ : Category.{v_2, u_2} C₂\ninst✝⁵ : Category.{v_3, u_3} D₁\ninst✝⁴ : Category.{v_4, u_4} D₂\nG : C₁ ⥤ C₂\nF : C₂ ⥤ C₁\nadj : G ⊣ F\nL₁ : C₁ ⥤ D₁\nL₂ : C₂ ⥤ D₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismPrope... | [
"C₁ : Type u_1\nC₂ : Type u_2\nD₁ : Type u_3\nD₂ : Type u_4\ninst✝⁷ : Category.{v_1, u_1} C₁\ninst✝⁶ : Category.{v_2, u_2} C₂\ninst✝⁵ : Category.{v_3, u_3} D₁\ninst✝⁴ : Category.{v_4, u_4} D₂\nG : C₁ ⥤ C₂\nF : C₂ ⥤ C₁\nadj : G ⊣ F\nL₁ : C₁ ⥤ D₁\nL₂ : C₂ ⥤ D₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\ninst... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Functor.Derived.Adjunction | {
"line": 73,
"column": 42
} | {
"line": 73,
"column": 59
} | {
"line": 73,
"column": 60
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nD₁ : Type u_3\nD₂ : Type u_4\ninst✝⁷ : Category.{v_1, u_1} C₁\ninst✝⁶ : Category.{v_2, u_2} C₂\ninst✝⁵ : Category.{v_3, u_3} D₁\ninst✝⁴ : Category.{v_4, u_4} D₂\nG : C₁ ⥤ C₂\nF : C₂ ⥤ C₁\nadj : G ⊣ F\nL₁ : C₁ ⥤ D₁\nL₂ : C₂ ⥤ D₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismPrope... | [
"C₁ : Type u_1\nC₂ : Type u_2\nD₁ : Type u_3\nD₂ : Type u_4\ninst✝⁷ : Category.{v_1, u_1} C₁\ninst✝⁶ : Category.{v_2, u_2} C₂\ninst✝⁵ : Category.{v_3, u_3} D₁\ninst✝⁴ : Category.{v_4, u_4} D₂\nG : C₁ ⥤ C₂\nF : C₂ ⥤ C₁\nadj : G ⊣ F\nL₁ : C₁ ⥤ D₁\nL₂ : C₂ ⥤ D₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\ninst... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Functor.Derived.Adjunction | {
"line": 87,
"column": 44
} | {
"line": 87,
"column": 61
} | {
"line": 87,
"column": 62
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nD₁ : Type u_3\nD₂ : Type u_4\ninst✝⁷ : Category.{v_1, u_1} C₁\ninst✝⁶ : Category.{v_2, u_2} C₂\ninst✝⁵ : Category.{v_3, u_3} D₁\ninst✝⁴ : Category.{v_4, u_4} D₂\nG : C₁ ⥤ C₂\nF : C₂ ⥤ C₁\nadj : G ⊣ F\nL₁ : C₁ ⥤ D₁\nL₂ : C₂ ⥤ D₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismPrope... | [
"C₁ : Type u_1\nC₂ : Type u_2\nD₁ : Type u_3\nD₂ : Type u_4\ninst✝⁷ : Category.{v_1, u_1} C₁\ninst✝⁶ : Category.{v_2, u_2} C₂\ninst✝⁵ : Category.{v_3, u_3} D₁\ninst✝⁴ : Category.{v_4, u_4} D₂\nG : C₁ ⥤ C₂\nF : C₂ ⥤ C₁\nadj : G ⊣ F\nL₁ : C₁ ⥤ D₁\nL₂ : C₂ ⥤ D₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\ninst... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Functor.Derived.Adjunction | {
"line": 87,
"column": 62
} | {
"line": 87,
"column": 79
} | {
"line": 88,
"column": 6
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nD₁ : Type u_3\nD₂ : Type u_4\ninst✝⁷ : Category.{v_1, u_1} C₁\ninst✝⁶ : Category.{v_2, u_2} C₂\ninst✝⁵ : Category.{v_3, u_3} D₁\ninst✝⁴ : Category.{v_4, u_4} D₂\nG : C₁ ⥤ C₂\nF : C₂ ⥤ C₁\nadj : G ⊣ F\nL₁ : C₁ ⥤ D₁\nL₂ : C₂ ⥤ D₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismPrope... | [
"C₁ : Type u_1\nC₂ : Type u_2\nD₁ : Type u_3\nD₂ : Type u_4\ninst✝⁷ : Category.{v_1, u_1} C₁\ninst✝⁶ : Category.{v_2, u_2} C₂\ninst✝⁵ : Category.{v_3, u_3} D₁\ninst✝⁴ : Category.{v_4, u_4} D₂\nG : C₁ ⥤ C₂\nF : C₂ ⥤ C₁\nadj : G ⊣ F\nL₁ : C₁ ⥤ D₁\nL₂ : C₂ ⥤ D₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\ninst... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Galois.Topology | {
"line": 51,
"column": 2
} | {
"line": 51,
"column": 13
} | {
"line": 52,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{u₂, u₁} C\nF : C ⥤ FintypeCat\n⊢ Function.Injective ⇑(autEmbedding F)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor",
"MonoidHom.instFunLike",
"MonoidHom",
"Monoid.toMulOneClass",
"Finite",
... | [
"C : Type u₁\ninst✝ : Category.{u₂, u₁} C\nF : C ⥤ FintypeCat\nσ τ : Aut F\nh : (autEmbedding F) σ = (autEmbedding F) τ\n⊢ σ = τ"
] | intro σ τ h | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Galois.Prorepresentability | {
"line": 318,
"column": 39
} | {
"line": 318,
"column": 74
} | {
"line": 318,
"column": 74
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{u₂, u₁} C\ninst✝¹ : GaloisCategory C\nF : C ⥤ FintypeCat\ninst✝ : FiberFunctor F\nf : End F\nA : PointedGaloisObject F\n⊢ ((ConcreteCategory.hom (limit.π (incl F ⋙ (F ⋙ FintypeCat.incl) ⋙ uliftFunctor.{u₁, u₂}) A))\n ((colimitCoyonedaHomIsoLimit' (incl F) (F ⋙ Fin... | [
"C : Type u₁\ninst✝² : Category.{u₂, u₁} C\ninst✝¹ : GaloisCategory C\nF : C ⥤ FintypeCat\ninst✝ : FiberFunctor F\nf : End F\nA : PointedGaloisObject F\n⊢ {\n down :=\n (ConcreteCategory.hom\n ((((yoneda.obj (F ⋙ FintypeCat.incl)).mapIso\n (colimit.isoColimitCocon... | colimitCoyonedaHomIsoLimit'_π_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Galois.IsFundamentalgroup | {
"line": 181,
"column": 2
} | {
"line": 209,
"column": 35
} | {
"line": 211,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝⁹ : Category.{u₂, u₁} C\nF : C ⥤ FintypeCat\nG : Type u_1\ninst✝⁸ : Group G\ninst✝⁷ : (X : C) → MulAction G (F.obj X).obj\ninst✝⁶ : IsNaturalSMul F G\ninst✝⁵ : GaloisCategory C\ninst✝⁴ : FiberFunctor F\ninst✝³ : TopologicalSpace G\ninst✝² : IsTopologicalGroup G\ninst✝¹ : CompactSpace ... | [] | intro t
choose gi hgi using (fun X : PointedGaloisObject F ↦ toAut_surjective_isGalois F G t X)
let cl (X : PointedGaloisObject F) : Set G := gi X • MulAction.stabilizer G X.pt
let c : Set G := ⋂ i, cl i
have hne : c.Nonempty := by
rw [← Set.univ_inter c]
apply CompactSpace.isCompact_univ.inter_iInter_n... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Galois.IsFundamentalgroup | {
"line": 181,
"column": 2
} | {
"line": 209,
"column": 35
} | {
"line": 211,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝⁹ : Category.{u₂, u₁} C\nF : C ⥤ FintypeCat\nG : Type u_1\ninst✝⁸ : Group G\ninst✝⁷ : (X : C) → MulAction G (F.obj X).obj\ninst✝⁶ : IsNaturalSMul F G\ninst✝⁵ : GaloisCategory C\ninst✝⁴ : FiberFunctor F\ninst✝³ : TopologicalSpace G\ninst✝² : IsTopologicalGroup G\ninst✝¹ : CompactSpace ... | [] | intro t
choose gi hgi using (fun X : PointedGaloisObject F ↦ toAut_surjective_isGalois F G t X)
let cl (X : PointedGaloisObject F) : Set G := gi X • MulAction.stabilizer G X.pt
let c : Set G := ⋂ i, cl i
have hne : c.Nonempty := by
rw [← Set.univ_inter c]
apply CompactSpace.isCompact_univ.inter_iInter_n... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Galois.EssSurj | {
"line": 164,
"column": 4
} | {
"line": 164,
"column": 34
} | {
"line": 165,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝⁶ : Category.{u₂, u₁} C\nF : C ⥤ FintypeCat\ninst✝⁵ : GaloisCategory C\ninst✝⁴ : FiberFunctor F\nG : Type u_1\ninst✝³ : Group G\ninst✝² : TopologicalSpace G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : CompactSpace G\nV U : OpenSubgroup (Aut F)\nh : (↑U).Normal\nA : C\nu : (functorToAction... | [
"C : Type u₁\ninst✝⁶ : Category.{u₂, u₁} C\nF : C ⥤ FintypeCat\ninst✝⁵ : GaloisCategory C\ninst✝⁴ : FiberFunctor F\nG : Type u_1\ninst✝³ : Group G\ninst✝² : TopologicalSpace G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : CompactSpace G\nV U : OpenSubgroup (Aut F)\nh : (↑U).Normal\nA : C\nu : (functorToAction F).obj A ≅ ... | ext (x : Aut F ⧸ U.toSubgroup) | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
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