module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 401,
"column": 2
} | {
"line": 408,
"column": 81
} | [
{
"pp": "b n : ℕ\n⊢ ⌊logb ↑b ↑n⌋₊ = Nat.log b n",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"CharP.cast_eq_zero",
"Iff.mpr",
"Real.instIsOrderedRing",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NonAssocSemiring.toAddCommMonoidWithOn... | obtain _ | _ | b := b
· simp [Real.logb]
· simp [Real.logb]
obtain rfl | hn := eq_or_ne n 0
· simp
rw [← Nat.cast_inj (R := ℤ), Int.natCast_floor_eq_floor, floor_logb_natCast (by simp),
Int.log_natCast]
exact logb_nonneg (by simp [Nat.cast_add_one_pos]) (Nat.one_le_cast.2 (by lia)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.InverseDeriv | {
"line": 102,
"column": 4
} | {
"line": 102,
"column": 54
} | [
{
"pp": "case neg\nx : ℝ\nh : ¬(x ≠ -1 ∧ x ≠ 1)\n⊢ 0 = 1 / √(1 - x ^ 2)",
"usedConstants": [
"Real",
"_private.Mathlib.Analysis.SpecialFunctions.Trigonometric.InverseDeriv.0.Real.deriv_arcsin._simp_1_2",
"congrArg",
"Eq.mp",
"Ne",
"Real.instOne",
"And",
"_priv... | simp only [not_and_or, Ne, Classical.not_not] at h | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Stirling | {
"line": 61,
"column": 30
} | {
"line": 61,
"column": 39
} | [
{
"pp": "⊢ ↑0! / (√(2 * 0) * (0 / rexp 1) ^ 0) = 0",
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"MulZeroClass.toMul",
"Real.instZero",
"congrArg",
"Real.instDivInvMonoid",
"Stirlin... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Stirling | {
"line": 178,
"column": 90
} | {
"line": 183,
"column": 58
} | [
{
"pp": "⊢ ∃ a, 0 < a ∧ Tendsto stirlingSeq atTop (𝓝 a)",
"usedConstants": [
"Eq.mpr",
"ConditionallyCompleteLattice.toInfSet",
"Real.instLE",
"Real",
"Real.instZero",
"lowerBounds",
"congrArg",
"PartialOrder.toPreorder",
"le_csInf",
"setOf",
... | by
obtain ⟨x, x_pos, hx⟩ := stirlingSeq'_bounded_by_pos_constant
have hx' : x ∈ lowerBounds (Set.range (stirlingSeq ∘ succ)) := by simpa [lowerBounds] using hx
refine ⟨_, lt_of_lt_of_le x_pos (le_csInf (Set.range_nonempty _) hx'), ?_⟩
rw [← Filter.tendsto_add_atTop_iff_nat 1]
exact tendsto_atTop_ciInf stirlin... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Lagrange | {
"line": 171,
"column": 25
} | {
"line": 171,
"column": 36
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\nx : F\n⊢ degree 0 = ⊥",
"usedConstants": [
"Eq.mpr",
"WithBot",
"congrArg",
"id",
"Bot.bot",
"Polynomial.degree",
"Polynomial.degree_zero",
"Field.toSemifield",
"Polynomial",
"Semifield.toDivisionSemiring",
... | degree_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Lagrange | {
"line": 237,
"column": 38
} | {
"line": 237,
"column": 54
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns : Finset ι\nv : ι → F\ni j : ι\nhij : i ≠ j\nhj : j ∈ s\n⊢ ∏ j_1 ∈ s.erase i, eval (v j) (basisDivisor (v i) (v j_1)) = 0",
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"IsDomain.to_noZeroDivisors",
"N... | prod_eq_zero_iff | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Lagrange | {
"line": 586,
"column": 4
} | {
"line": 586,
"column": 40
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝¹ : CommRing R\nι : Type u_2\ns : Finset ι\nv : ι → R\ninst✝ : DecidableEq ι\ni : ι\nt : Finset ι\nhit : i ∉ t\nIH : derivative (nodal t v) = ∑ i ∈ t, nodal (t.erase i) v\n⊢ ∑ i_1 ∈ t, (X - C (v i)) * nodal (t.erase i_1) v = ∑ x ∈ t, nodal ((insert i t).erase x) v",
... | refine sum_congr rfl fun j hjt => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Algebra.InfiniteSum.TsumUniformlyOn | {
"line": 41,
"column": 51
} | {
"line": 43,
"column": 58
} | [
{
"pp": "β : Type u_2\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : CompleteSpace F\nι : Type u_4\nf : ι → β → F\nu : ι → ℝ\nhu : Summable u\ns : Set β\nhfu : ∀ᶠ (n : ι) in cofinite, ∀ x ∈ s, ‖f n x‖ ≤ u n\n⊢ HasSumUniformlyOn f (fun x ↦ ∑' (n : ι), f n x) s",
"usedConstants": [
"HasSumUniform... | by
simp [hasSumUniformlyOn_iff_tendstoUniformlyOn,
tendstoUniformlyOn_tsum_of_cofinite_eventually hu hfu] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 271,
"column": 2
} | {
"line": 271,
"column": 29
} | [
{
"pp": "z : ℍ\na b : ℤ\n⊢ (fun m ↦ ((↑(m 0) + ↑a) * ↑z + ↑(m 1) + ↑b)⁻¹) =O[cofinite] fun m ↦ ‖![m 0 + a, m 1 + b]‖⁻¹",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Int.cast",
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"UpperHalfPlane.co... | rw [Asymptotics.isBigO_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 289,
"column": 3
} | {
"line": 289,
"column": 30
} | [
{
"pp": "a b : ℤ\nthis : ∀ (x : Fin 2 → ℤ), ![x 0 + a, x 1 + b] = x + ![a, b]\n⊢ (fun x ↦ x) =Θ[cofinite] fun x ↦ ‖x‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"congrArg",
"id",
"Asymptotics.IsTheta",
"instOfNat... | by rw [← isTheta_norm_left] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecificLimits.ArithmeticGeometric | {
"line": 55,
"column": 28
} | {
"line": 55,
"column": 36
} | [
{
"pp": "case succ\nR : Type u_1\na b u₀ : R\ninst✝ : CommSemiring R\nn : ℕ\nhn : arithGeom a b u₀ n = a ^ n * u₀ + b * ∑ k ∈ Finset.range n, a ^ k\n⊢ a * (a ^ n * u₀ + b * ∑ k ∈ Finset.range n, a ^ k) + b = a ^ (n + 1) * u₀ + b * ∑ k ∈ Finset.range (n + 1), a ^ k",
"usedConstants": [
"Distrib.leftDis... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecificLimits.ArithmeticGeometric | {
"line": 57,
"column": 72
} | {
"line": 57,
"column": 80
} | [
{
"pp": "case succ.e_a\nR : Type u_1\na b u₀ : R\ninst✝ : CommSemiring R\nn : ℕ\nhn : arithGeom a b u₀ n = a ^ n * u₀ + b * ∑ k ∈ Finset.range n, a ^ k\n⊢ a * ∑ i ∈ Finset.range n, b * a ^ i + b = b * (∑ k ∈ Finset.range n, a ^ (k + 1) + 1)",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mp... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecificLimits.ArithmeticGeometric | {
"line": 64,
"column": 51
} | {
"line": 64,
"column": 59
} | [
{
"pp": "R : Type u_1\na b : R\ninst✝ : CommSemiring R\nn : ℕ\n⊢ a ^ n * b + b * ∑ k ∈ Finset.range n, a ^ k = b * (∑ x ∈ Finset.range n, a ^ x + a ^ n)",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"AddMono... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 42,
"column": 8
} | {
"line": 42,
"column": 16
} | [
{
"pp": "z : ℂ\n⊢ cexp (z * I) + cexp (-z * I) = cexp (-(z * I)) * (cexp (2 * I * z) + 1)",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"Complex.ins... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 246,
"column": 53
} | {
"line": 246,
"column": 81
} | [
{
"pp": "L : PeriodPair\nl₀ : ℂ\ns : Finset ↥L.lattice\ni : ↥L.lattice\nhi : i ∈ s\nh✝ : ¬↑i = l₀\n⊢ ∀ x ∈ (↑L.lattice \\ {l₀})ᶜ, (x - ↑i) ^ 2 ≠ 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Submodule",
"SetLike.mem_coe.... | aesop (add simp sub_eq_zero) | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 246,
"column": 53
} | {
"line": 246,
"column": 81
} | [
{
"pp": "L : PeriodPair\nl₀ : ℂ\ns : Finset ↥L.lattice\ni : ↥L.lattice\nhi : i ∈ s\nh✝ : ¬↑i = l₀\n⊢ ∀ x ∈ (↑L.lattice \\ {l₀})ᶜ, (x - ↑i) ^ 2 ≠ 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Submodule",
"SetLike.mem_coe.... | aesop (add simp sub_eq_zero) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 246,
"column": 53
} | {
"line": 246,
"column": 81
} | [
{
"pp": "L : PeriodPair\nl₀ : ℂ\ns : Finset ↥L.lattice\ni : ↥L.lattice\nhi : i ∈ s\nh✝ : ¬↑i = l₀\n⊢ ∀ x ∈ (↑L.lattice \\ {l₀})ᶜ, (x - ↑i) ^ 2 ≠ 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Submodule",
"SetLike.mem_coe.... | aesop (add simp sub_eq_zero) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Real.GoldenRatio | {
"line": 229,
"column": 57
} | {
"line": 231,
"column": 59
} | [
{
"pp": "n : ℕ\nih : φ * ↑(Nat.fib (n + 1)) + ↑(Nat.fib n) = φ ^ (n + 1)\n⊢ φ * ↑(Nat.fib (n + 1 + 1)) + ↑(Nat.fib (n + 1)) = φ * ↑(Nat.fib n) + φ ^ 2 * ↑(Nat.fib (n + 1))",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"Ma... | by
simp only [Nat.fib_add_one (Nat.succ_ne_zero n), Nat.succ_sub_succ_eq_sub,
Nat.cast_add, goldenRatio_sq, Nat.sub_zero]; ring | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 134,
"column": 54
} | {
"line": 136,
"column": 74
} | [
{
"pp": "⊢ HasProdLocallyUniformlyOn (fun n z ↦ 1 + sineTerm z n) (fun x ↦ Complex.sin (↑π * x) / (↑π * x)) ℂ_ℤ",
"usedConstants": [
"Int.cast",
"NormedCommRing.toSeminormedCommRing",
"locallyCompact_of_proper",
"instHDiv",
"HasProdUniformlyOn_sineTerm_prod_on_compact",
"... | by
apply hasProdLocallyUniformlyOn_of_forall_compact isOpen_compl_range_intCast
exact fun _ hZ hZC => HasProdUniformlyOn_sineTerm_prod_on_compact hZ hZC | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 380,
"column": 4
} | {
"line": 385,
"column": 23
} | [
{
"pp": "case refine_1\nL : PeriodPair\nl₀ : ℂ\ns : Finset ↥L.lattice\n⊢ DifferentiableOn ℂ (fun x2 ↦ ∑ i ∈ s, (fun l z ↦ if ↑l = l₀ then 0 else 1 / (z - ↑l) ^ 2 - 1 / ↑l ^ 2) i x2)\n (↑L.lattice \\ {l₀})ᶜ",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"InnerProductSpace.... | refine .fun_sum fun i hi ↦ ?_
split_ifs
· simp
refine .sub (.div (by fun_prop) (by fun_prop) fun x hx ↦ ?_) (by fun_prop)
have : x ≠ i := by rintro rfl; simp_all
simpa [sub_eq_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 380,
"column": 4
} | {
"line": 385,
"column": 23
} | [
{
"pp": "case refine_1\nL : PeriodPair\nl₀ : ℂ\ns : Finset ↥L.lattice\n⊢ DifferentiableOn ℂ (fun x2 ↦ ∑ i ∈ s, (fun l z ↦ if ↑l = l₀ then 0 else 1 / (z - ↑l) ^ 2 - 1 / ↑l ^ 2) i x2)\n (↑L.lattice \\ {l₀})ᶜ",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"InnerProductSpace.... | refine .fun_sum fun i hi ↦ ?_
split_ifs
· simp
refine .sub (.div (by fun_prop) (by fun_prop) fun x hx ↦ ?_) (by fun_prop)
have : x ≠ i := by rintro rfl; simp_all
simpa [sub_eq_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 744,
"column": 6
} | {
"line": 745,
"column": 51
} | [
{
"pp": "case r_le.h\nL : PeriodPair\nl₀ x : ℂ\nr : NNReal\nhr0 : 0 < r\nhr : Metric.closedBall x ↑r ⊆ (↑L.lattice \\ {l₀})ᶜ\nl : ↥L.lattice\nhl : ↑l ≠ l₀\n⊢ ‖x + ↑↑r - x‖ < ‖↑l - x‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Submodule",
... | simpa [-Metric.mem_closedBall, mem_closedBall_iff_norm]
using Set.subset_compl_comm.mp hr ⟨l.2, hl⟩ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 887,
"column": 63
} | {
"line": 891,
"column": 59
} | [
{
"pp": "L : PeriodPair\nx : ℂ\nr : NNReal\nhr0 : 0 < r\nhr : Metric.closedBall x ↑r ⊆ (↑L.lattice)ᶜ\n⊢ HasFPowerSeriesOnBall ℘[L] (L.weierstrassPSeries x) x ↑r",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"... | by
simp_rw [← L.weierstrassPExceptSeries_of_notMem _ L.ω₁_div_two_notMem_lattice,
← L.weierstrassPExcept_of_notMem _ L.ω₁_div_two_notMem_lattice]
exact L.hasFPowerSeriesOnBall_weierstrassPExcept _ x r hr0
(hr.trans (Set.compl_subset_compl.mpr Set.diff_subset)) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 1023,
"column": 4
} | {
"line": 1023,
"column": 59
} | [
{
"pp": "case pos\nL : PeriodPair\ni : ℕ\nhi₁ : i < 7\nhi₂ : ¬Odd i\nhi₃ : ¬i = 0\nhi₄ : i = 6\n⊢ iteratedDeriv i (L.relation * id ^ 6) 0 = 0",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"HMul.hMul",
"Complex.instNormedAddCommGroup",
"_private.Mathlib.Analysis.SpecialFunct... | exact hi₄ ▸ L.iteratedDeriv_six_relation_mul_id_pow_six | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 1023,
"column": 4
} | {
"line": 1023,
"column": 59
} | [
{
"pp": "case pos\nL : PeriodPair\ni : ℕ\nhi₁ : i < 7\nhi₂ : ¬Odd i\nhi₃ : ¬i = 0\nhi₄ : i = 6\n⊢ iteratedDeriv i (L.relation * id ^ 6) 0 = 0",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"HMul.hMul",
"Complex.instNormedAddCommGroup",
"_private.Mathlib.Analysis.SpecialFunct... | exact hi₄ ▸ L.iteratedDeriv_six_relation_mul_id_pow_six | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 1023,
"column": 4
} | {
"line": 1023,
"column": 59
} | [
{
"pp": "case pos\nL : PeriodPair\ni : ℕ\nhi₁ : i < 7\nhi₂ : ¬Odd i\nhi₃ : ¬i = 0\nhi₄ : i = 6\n⊢ iteratedDeriv i (L.relation * id ^ 6) 0 = 0",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"HMul.hMul",
"Complex.instNormedAddCommGroup",
"_private.Mathlib.Analysis.SpecialFunct... | exact hi₄ ▸ L.iteratedDeriv_six_relation_mul_id_pow_six | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecificLimits.FloorPow | {
"line": 152,
"column": 8
} | {
"line": 152,
"column": 52
} | [
{
"pp": "u : ℕ → ℝ\nl : ℝ\nhmono : Monotone u\nhlim :\n ∀ (a : ℝ),\n 1 < a →\n ∃ c,\n (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧\n Tendsto c atTop atTop ∧ Tendsto (fun n ↦ u (c n) / ↑(c n)) atTop (𝓝 l)\nlnonneg : 0 ≤ l\nA : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤... | apply Tendsto.mono_left _ nhdsWithin_le_nhds | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.SpecificLimits.FloorPow | {
"line": 167,
"column": 8
} | {
"line": 167,
"column": 52
} | [
{
"pp": "u : ℕ → ℝ\nl : ℝ\nhmono : Monotone u\nhlim :\n ∀ (a : ℝ),\n 1 < a →\n ∃ c,\n (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧\n Tendsto c atTop atTop ∧ Tendsto (fun n ↦ u (c n) / ↑(c n)) atTop (𝓝 l)\nlnonneg : 0 ≤ l\nA : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤... | apply Tendsto.mono_left _ nhdsWithin_le_nhds | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.SpecificLimits.FloorPow | {
"line": 280,
"column": 6
} | {
"line": 280,
"column": 58
} | [
{
"pp": "N : ℕ\nj : ℝ\nhj : 0 < j\nc : ℝ\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c⁻¹\n⊢ ∑ i ∈ range N with j < ↑⌊c ^ i⌋₊, 1 / ↑⌊c ^ i⌋₊ ^ 2 ≤ ∑ i ∈ range N with j < c ^ i, 1 / ↑⌊c ^ i⌋₊ ^ 2",
"usedConstants": [
"Real.instIsOrderedRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
"le_refl",
... | gcongr with k hk; exact Nat.floor_le (by positivity) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecificLimits.FloorPow | {
"line": 280,
"column": 6
} | {
"line": 280,
"column": 58
} | [
{
"pp": "N : ℕ\nj : ℝ\nhj : 0 < j\nc : ℝ\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c⁻¹\n⊢ ∑ i ∈ range N with j < ↑⌊c ^ i⌋₊, 1 / ↑⌊c ^ i⌋₊ ^ 2 ≤ ∑ i ∈ range N with j < c ^ i, 1 / ↑⌊c ^ i⌋₊ ^ 2",
"usedConstants": [
"Real.instIsOrderedRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
"le_refl",
... | gcongr with k hk; exact Nat.floor_le (by positivity) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Preserves.Presheaf | {
"line": 141,
"column": 2
} | {
"line": 141,
"column": 52
} | [
{
"pp": "C : Type u\ninst✝⁴ : SmallCategory C\ninst✝³ : HasFiniteColimits C\nA : Cᵒᵖ ⥤ Type u\nJ : Type\ninst✝² : SmallCategory J\ninst✝¹ : FinCategory J\nK : J ⥤ Cᵒᵖ\ninst✝ : IsFiltered (CostructuredArrow yoneda A)\n⊢ (((Presheaf.isColimitTautologicalCocone A).coconePointUniqueUpToIso\n (colimit.i... | rw [Eq.comm, ← Iso.inv_comp_eq, ← Iso.inv_comp_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Limits.FilteredColimitCommutesProduct | {
"line": 184,
"column": 2
} | {
"line": 187,
"column": 80
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasProducts C\ninst✝² : HasFilteredColimitsOfSize.{w, w, v, u} C\ninst✝¹ : IsIPC C\nD : Type u₁\ninst✝ : Category.{v₁, u₁} D\n⊢ IsIPC (D ⥤ C)",
"usedConstants": [
"CategoryTheory.Limits.IsIPC.mk",
"CategoryTheory.Functor.flip",
"Cat... | refine ⟨fun β I _ _ F => ?_⟩
suffices ∀ d, IsIso ((colimitPointwiseProductToProductColimit F).app d) from
NatIso.isIso_of_isIso_app _
exact fun d => colimitPointwiseProductToProductColimit_app F d ▸ inferInstance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.FilteredColimitCommutesProduct | {
"line": 184,
"column": 2
} | {
"line": 187,
"column": 80
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasProducts C\ninst✝² : HasFilteredColimitsOfSize.{w, w, v, u} C\ninst✝¹ : IsIPC C\nD : Type u₁\ninst✝ : Category.{v₁, u₁} D\n⊢ IsIPC (D ⥤ C)",
"usedConstants": [
"CategoryTheory.Limits.IsIPC.mk",
"CategoryTheory.Functor.flip",
"Cat... | refine ⟨fun β I _ _ F => ?_⟩
suffices ∀ d, IsIso ((colimitPointwiseProductToProductColimit F).app d) from
NatIso.isIso_of_isIso_app _
exact fun d => colimitPointwiseProductToProductColimit_app F d ▸ inferInstance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Connected | {
"line": 72,
"column": 32
} | {
"line": 72,
"column": 46
} | [
{
"pp": "J : Type w\ninst✝⁶ : Category.{w', w} J\ninst✝⁵ : IsConnected J\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasPullbacks C\ninst✝¹ : HasColimitsOfShape J C\ninst✝ : HasExactColimitsOfShape J C\nF : J ⥤ C\nc : Cocone F\nhc : IsColimit c\nX Y : C\nf : X ⟶ c.pt\ng : c.pt... | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Connected | {
"line": 72,
"column": 32
} | {
"line": 72,
"column": 46
} | [
{
"pp": "J : Type w\ninst✝⁶ : Category.{w', w} J\ninst✝⁵ : IsConnected J\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasPullbacks C\ninst✝¹ : HasColimitsOfShape J C\ninst✝ : HasExactColimitsOfShape J C\nF : J ⥤ C\nc : Cocone F\nhc : IsColimit c\nX Y : C\nf : X ⟶ c.pt\ng : c.pt... | simpa using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Connected | {
"line": 72,
"column": 32
} | {
"line": 72,
"column": 46
} | [
{
"pp": "J : Type w\ninst✝⁶ : Category.{w', w} J\ninst✝⁵ : IsConnected J\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasPullbacks C\ninst✝¹ : HasColimitsOfShape J C\ninst✝ : HasExactColimitsOfShape J C\nF : J ⥤ C\nc : Cocone F\nhc : IsColimit c\nX Y : C\nf : X ⟶ c.pt\ng : c.pt... | simpa using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Connected | {
"line": 83,
"column": 2
} | {
"line": 84,
"column": 32
} | [
{
"pp": "J : Type w\ninst✝⁵ : Category.{w', w} J\ninst✝⁴ : IsConnected J\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasPushouts C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : HasExactLimitsOfShape J C\nF : J ⥤ C\nc : Cone F\nhc : IsLimit c\nX : C\nf : c.pt ⟶ X\n⊢ IsLimit { pt := X, π := pushout.inr c.π ((Func... | suffices IsIso (limMap (pushout.inr c.π ((Functor.const J).map f))) from
Cone.isLimitOfIsIsoLimMapπ _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Connected | {
"line": 109,
"column": 31
} | {
"line": 109,
"column": 45
} | [
{
"pp": "J : Type w\ninst✝⁶ : Category.{w', w} J\ninst✝⁵ : IsConnected J\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasPushouts C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : HasExactLimitsOfShape J C\nF : J ⥤ C\nc : Cone F\nhc : IsLimit c\nX Y : C\ng : Y ⟶ c.pt\nf : c.pt ⟶ X\nhf ... | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Connected | {
"line": 109,
"column": 31
} | {
"line": 109,
"column": 45
} | [
{
"pp": "J : Type w\ninst✝⁶ : Category.{w', w} J\ninst✝⁵ : IsConnected J\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasPushouts C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : HasExactLimitsOfShape J C\nF : J ⥤ C\nc : Cone F\nhc : IsLimit c\nX Y : C\ng : Y ⟶ c.pt\nf : c.pt ⟶ X\nhf ... | simpa using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Connected | {
"line": 109,
"column": 31
} | {
"line": 109,
"column": 45
} | [
{
"pp": "J : Type w\ninst✝⁶ : Category.{w', w} J\ninst✝⁵ : IsConnected J\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasPushouts C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : HasExactLimitsOfShape J C\nF : J ⥤ C\nc : Cone F\nhc : IsLimit c\nX Y : C\ng : Y ⟶ c.pt\nf : c.pt ⟶ X\nhf ... | simpa using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.ModuleEmbedding.GabrielPopescu | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 35
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : IsGrothendieckAbelian.{v, v, u} C\nG : C\nhG : IsSeparator G\nA B : C\nM : ModuleCat (End G)ᵐᵒᵖ\ng : M ⟶ ModuleCat.of (End G)ᵐᵒᵖ (G ⟶ A)\nhg : Mono g\nf : M ⟶ ModuleCat.of (End G)ᵐᵒᵖ (G ⟶ B)\nF : Finset (Discrete ↑M)\nh : G ⟶ pullback ... | simp only [← map_smul, ← map_sum] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Abelian.Injective.Ext | {
"line": 147,
"column": 28
} | {
"line": 147,
"column": 44
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX Y : C\nR : InjectiveResolution Y\nn✝ n : ℕ\nf g : X ⟶ R.cocomplex.X n\nm : ℕ\nhm : n + 1 = m\nhf : f ≫ R.cocomplex.d n m = 0\nhg : g ≫ R.cocomplex.d n m = 0\n⊢ (f + g) ≫ R.cocomplex.d n m = 0",
"usedConstants": [
... | by simp [hf, hg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Abelian.Injective.Ext | {
"line": 157,
"column": 28
} | {
"line": 157,
"column": 44
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX Y : C\nR : InjectiveResolution Y\nn✝ n : ℕ\nf g : X ⟶ R.cocomplex.X n\nm : ℕ\nhm : n + 1 = m\nhf : f ≫ R.cocomplex.d n m = 0\nhg : g ≫ R.cocomplex.d n m = 0\n⊢ (f - g) ≫ R.cocomplex.d n m = 0",
"usedConstants": [
... | by simp [hf, hg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Abelian.SerreClass.MorphismProperty | {
"line": 126,
"column": 45
} | {
"line": 127,
"column": 24
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\nP : ObjectProperty C\ninst✝ : P.IsSerreClass\nX Y : C\n⊢ P.isoModSerre 0 ↔ P X ∧ P Y",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
... | by
simp [isoModSerre_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Abelian.SerreClass.MorphismProperty | {
"line": 176,
"column": 2
} | {
"line": 177,
"column": 53
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Abelian C\nD : Type u'\ninst✝³ : Category.{v', u'} D\ninst✝² : Abelian D\nP : ObjectProperty C\ninst✝¹ : P.IsSerreClass\nF : C ⥤ D\ninst✝ : F.PreservesZeroMorphisms\nhF : P.isoModSerre.IsInvertedBy F\nX : C\nhX : P X\nf : 0 ⟶ X := 0\n⊢ F.kernel X",
"... | have := hF _ ((P.isoModSerre_iff_of_mono f).2
((P.prop_iff_of_iso cokernelZeroIsoTarget).2 hX)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Abelian.Projective.Ext | {
"line": 155,
"column": 28
} | {
"line": 155,
"column": 44
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX Y : C\nR : ProjectiveResolution X\nn✝ n : ℕ\nf g : R.complex.X n ⟶ Y\nm : ℕ\nhm : n + 1 = m\nhf : R.complex.d m n ≫ f = 0\nhg : R.complex.d m n ≫ g = 0\n⊢ R.complex.d m n ≫ (f + g) = 0",
"usedConstants": [
"Catego... | by simp [hf, hg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Abelian.Projective.Ext | {
"line": 165,
"column": 28
} | {
"line": 165,
"column": 44
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX Y : C\nR : ProjectiveResolution X\nn✝ n : ℕ\nf g : R.complex.X n ⟶ Y\nm : ℕ\nhm : n + 1 = m\nhf : R.complex.d m n ≫ f = 0\nhg : R.complex.d m n ≫ g = 0\n⊢ R.complex.d m n ≫ (f - g) = 0",
"usedConstants": [
"Catego... | by simp [hf, hg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.MorphismProperty.OfObjectProperty | {
"line": 58,
"column": 23
} | {
"line": 58,
"column": 31
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP Q : ObjectProperty C\ninst✝ : P.IsClosedUnderIsomorphisms\nX Y Z : C\ni : X ⟶ Y\nhi : isomorphisms C i\nf : Y ⟶ Z\n⊢ ofObjectProperty P Q f → ofObjectProperty P Q (i ≫ f)",
"usedConstants": [
"CategoryTheory.MorphismProperty.ofObjectProperty"
... | ⟨hY, hZ⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Term.anonymousCtor |
Mathlib.CategoryTheory.MorphismProperty.OfObjectProperty | {
"line": 64,
"column": 23
} | {
"line": 64,
"column": 31
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP Q : ObjectProperty C\ninst✝ : Q.IsClosedUnderIsomorphisms\nX Y Z : C\ni : Y ⟶ Z\nhi : isomorphisms C i\nf : X ⟶ Y\n⊢ ofObjectProperty P Q f → ofObjectProperty P Q (f ≫ i)",
"usedConstants": [
"CategoryTheory.MorphismProperty.ofObjectProperty"
... | ⟨hY, hZ⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Term.anonymousCtor |
Mathlib.CategoryTheory.Abelian.Projective.Ext | {
"line": 253,
"column": 2
} | {
"line": 269,
"column": 53
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX Y : C\nR : ProjectiveResolution X\nn : ℕ\nf : R.complex.X n ⟶ Y\nm : ℕ\nhm : n + 1 = m\nhf : R.complex.d m n ≫ f = 0\nX' : C\nR' : ProjectiveResolution X'\ng : X' ⟶ X\nφ : R'.Hom R g\n⊢ (Ext.mk₀ g).comp (R.extMk f m hm hf) ... | have := HasDerivedCategory.standard C
ext
have : (R'.cochainComplexXIso (-n) n (by lia)).hom ≫ φ.hom.f n =
φ.hom'.f (-n) ≫ (R.cochainComplexXIso (-n) n (by lia)).hom := by
simp [φ.hom'_f _ _ rfl]
simp only [Ext.comp_hom, extMk_hom, Ext.mk₀_hom, reassoc_of% this]
rw [Cocycle.toSingleMk_precomp _ _ _ (b... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.Projective.Ext | {
"line": 253,
"column": 2
} | {
"line": 269,
"column": 53
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX Y : C\nR : ProjectiveResolution X\nn : ℕ\nf : R.complex.X n ⟶ Y\nm : ℕ\nhm : n + 1 = m\nhf : R.complex.d m n ≫ f = 0\nX' : C\nR' : ProjectiveResolution X'\ng : X' ⟶ X\nφ : R'.Hom R g\n⊢ (Ext.mk₀ g).comp (R.extMk f m hm hf) ... | have := HasDerivedCategory.standard C
ext
have : (R'.cochainComplexXIso (-n) n (by lia)).hom ≫ φ.hom.f n =
φ.hom'.f (-n) ≫ (R.cochainComplexXIso (-n) n (by lia)).hom := by
simp [φ.hom'_f _ _ rfl]
simp only [Ext.comp_hom, extMk_hom, Ext.mk₀_hom, reassoc_of% this]
rw [Cocycle.toSingleMk_precomp _ _ _ (b... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.RightDerived | {
"line": 314,
"column": 6
} | {
"line": 314,
"column": 53
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\nD : Type u_1\ninst✝⁴ : Category.{v_1, u_1} D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : F.Additive\nX Y : C\nf : X ⟶ Y\n⊢ (injectiveResolution X).toRightDerivedZero' F ≫\n HomologicalComplex.cyclesMap\n ... | ← HomologicalComplex.homologyπ_naturality_assoc | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.RightDerived | {
"line": 334,
"column": 24
} | {
"line": 334,
"column": 71
} | [
{
"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 : C\nI : InjectiveResolution X\nF : C ⥤ D\ninst✝ : F.Additive\nh₁ :\n I.toRightDerivedZero' F =\n (injectiveResolution X).toRightDerive... | ← HomologicalComplex.homologyπ_naturality_assoc | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.SerreClass.Localization | {
"line": 445,
"column": 57
} | {
"line": 449,
"column": 29
} | [
{
"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\n⊢ PreservesFiniteLimits L",
"usedConstants": [
... | by
letI := abelian L P
rw [((Functor.preservesFiniteLimits_tfae L).out 3 2:)]
intro _ _ f
exact preservesKernel L P f | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Abelian.SerreClass.Localization | {
"line": 503,
"column": 2
} | {
"line": 509,
"column": 74
} | [
{
"pp": "C : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : Abelian C\nD : Type u'\ninst✝⁶ : Category.{v', u'} D\nL : C ⥤ D\nP : ObjectProperty C\ninst✝⁵ : P.IsSerreClass\nE : Type u''\ninst✝⁴ : Category.{v'', u''} E\ninst✝³ : Abelian E\ninst✝² : L.IsLocalization P.isoModSerre\ninst✝¹ : Preadditive D\ninst✝ : L.A... | have : PreservesColimit (parallelPair (L.map f'.hom) 0) G :=
preservesColimit_of_preserves_colimit_cocone
(CokernelCofork.isColimitMapCoconeEquiv _ _
(isColimitOfPreserves L (cokernelIsCokernel f'.hom)))
((CokernelCofork.isColimitMapCoconeEquiv _ G).symm
(CokernelCofork.isColimit... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Adhesive.Subobject | {
"line": 45,
"column": 19
} | {
"line": 48,
"column": 46
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Adhesive C\nX✝ : C\ninst✝ : Adhesive C\nX : C\nF : Discrete WalkingPair ⥤ Subobject X\n⊢ HasColimit F",
"usedConstants": [
"CategoryTheory.Limits.pullback",
"CategoryTheory.Subobject.arrow",
"CategoryTheory.Limits.pushout.inr_desc",... | by
have : HasColimit (pair (F.obj ⟨WalkingPair.left⟩) (F.obj ⟨WalkingPair.right⟩)) :=
⟨⟨⟨_, isColimitBinaryCofan (F.obj ⟨WalkingPair.left⟩) (F.obj ⟨WalkingPair.right⟩)⟩⟩⟩
apply hasColimit_of_iso (diagramIsoPair F) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Adjunction.Lifting.Left | {
"line": 96,
"column": 6
} | {
"line": 96,
"column": 47
} | [
{
"pp": "case refine_3\nA : Type u₁\nB : Type u₂\nC : Type u₃\ninst✝² : Category.{v₁, u₁} A\ninst✝¹ : Category.{v₂, u₂} B\ninst✝ : Category.{v₃, u₃} C\nU : B ⥤ C\nF : C ⥤ B\nR : A ⥤ B\nF' : C ⥤ A\nadj₁ : F ⊣ U\nadj₂ : F' ⊣ R ⋙ U\nh : (X : B) → RegularEpi (adj₁.counit.app X)\nX : B\ns : Cofork (F.map (U.map (adj... | apply hm.trans ((h _).desc' s.π _).2.symm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Adjunction.Triple | {
"line": 173,
"column": 61
} | {
"line": 173,
"column": 90
} | [
{
"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 : G.Full\ninst✝ : G.Faithful\nX : C\n⊢ Epi (t.rightToLeft.app X ≫ t.adj₂.unit.app (F.obj X)) ↔ Epi (H.map (t.adj₁.unit.app X))",
"usedConstants": [
... | rightToLeft_app_adj₂_unit_app | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Adjunction.Lifting.Right | {
"line": 203,
"column": 4
} | {
"line": 205,
"column": 56
} | [
{
"pp": "A : Type u₁\nB : Type u₂\nC : Type u₃\ninst✝⁵ : Category.{v₁, u₁} A\ninst✝⁴ : Category.{v₂, u₂} B\ninst✝³ : Category.{v₃, u₃} C\nF : B ⥤ A\ninst✝² : ComonadicLeftAdjoint F\nL : C ⥤ B\ninst✝¹ : HasCoreflexiveEqualizers C\ninst✝ : (L ⋙ F).IsLeftAdjoint\nL' : C ⥤ (comonadicAdjunction F).toComonad.Coalgebr... | intro X
simp only [Comonad.adj_unit]
exact ⟨_, _, _, _, Comonad.beckCoalgebraEqualizer X⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Adjunction.Lifting.Right | {
"line": 203,
"column": 4
} | {
"line": 205,
"column": 56
} | [
{
"pp": "A : Type u₁\nB : Type u₂\nC : Type u₃\ninst✝⁵ : Category.{v₁, u₁} A\ninst✝⁴ : Category.{v₂, u₂} B\ninst✝³ : Category.{v₃, u₃} C\nF : B ⥤ A\ninst✝² : ComonadicLeftAdjoint F\nL : C ⥤ B\ninst✝¹ : HasCoreflexiveEqualizers C\ninst✝ : (L ⋙ F).IsLeftAdjoint\nL' : C ⥤ (comonadicAdjunction F).toComonad.Coalgebr... | intro X
simp only [Comonad.adj_unit]
exact ⟨_, _, _, _, Comonad.beckCoalgebraEqualizer X⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Bicategory.Coherence | {
"line": 146,
"column": 27
} | {
"line": 146,
"column": 53
} | [
{
"pp": "case mk.h.whisker_left\nB : Type u\ninst✝ : Quiver B\na b c : B\nf g : Hom b c\na✝ b✝ c✝ : FreeBicategory B\nf✝ : a✝ ⟶ b✝\ng✝ h✝ : b✝ ⟶ c✝\nη✝ : Hom₂ g✝ h✝\nih : (fun p ↦ normalizeAux p g✝) = fun p ↦ normalizeAux p h✝\n⊢ (fun p ↦ normalizeAux p (f✝ ≫ g✝)) = fun p ↦ normalizeAux p (f✝ ≫ h✝)",
"usedC... | funext; apply congr_fun ih | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Bicategory.Coherence | {
"line": 146,
"column": 27
} | {
"line": 146,
"column": 53
} | [
{
"pp": "case mk.h.whisker_left\nB : Type u\ninst✝ : Quiver B\na b c : B\nf g : Hom b c\na✝ b✝ c✝ : FreeBicategory B\nf✝ : a✝ ⟶ b✝\ng✝ h✝ : b✝ ⟶ c✝\nη✝ : Hom₂ g✝ h✝\nih : (fun p ↦ normalizeAux p g✝) = fun p ↦ normalizeAux p h✝\n⊢ (fun p ↦ normalizeAux p (f✝ ≫ g✝)) = fun p ↦ normalizeAux p (f✝ ≫ h✝)",
"usedC... | funext; apply congr_fun ih | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Bicategory.Functor.StrictlyUnitary | {
"line": 167,
"column": 24
} | {
"line": 169,
"column": 33
} | [
{
"pp": "B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nF : StrictlyUnitaryLaxFunctor B C\nG : StrictlyUnitaryLaxFunctor C D\nx✝ : B\n⊢ (F.comp G.toLaxFunctor).mapId x✝ = eqToHom ⋯",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver"... | by
simp [StrictlyUnitaryLaxFunctor.mapId_eq_eqToHom,
PrelaxFunctor.map₂_eqToHom] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Oplax | {
"line": 231,
"column": 27
} | {
"line": 231,
"column": 49
} | [
{
"pp": "B : Type u₁\ninst✝¹ : Bicategory B\nC : Type u₂\ninst✝ : Bicategory C\nF G : B ⥤ᵒᵖᴸ C\nη : OplaxTrans F G\na b : B\na' : C\nf g : a ⟶ b\nβ : f ⟶ g\nh : G.obj b ⟶ a'\n⊢ (F.map₂ β ▷ η.app b ≫ η.naturality g) ▷ h =\n η.naturality f ▷ h ≫ (α_ (η.app a) (G.map f) h).hom ≫ η.app a ◁ G.map₂ β ▷ h ≫ (α_ (η.... | naturality_naturality, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Lax | {
"line": 165,
"column": 10
} | {
"line": 165,
"column": 27
} | [
{
"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 ⊗≫ G.map f ◁ θ.naturality g) ⊗≫\n (η.app a ◁ G.mapCo... | naturality_comp η | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Localization.Monoidal.Basic | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 57
} | [
{
"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 : C\n⊢ failed to pretty print expression (use 'set_option ... | apply Lifting₂.liftingLift₂ (hF := isInvertedBy₂ L W ε) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Localization.Monoidal.Basic | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 57
} | [
{
"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 : C\n⊢ failed to pretty print expression (use 'set_option ... | apply Lifting₂.liftingLift₂ (hF := isInvertedBy₂ L W ε) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Localization.Monoidal.Basic | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 57
} | [
{
"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 : C\n⊢ failed to pretty print expression (use 'set_option ... | apply Lifting₂.liftingLift₂ (hF := isInvertedBy₂ L W ε) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Dialectica.Monoidal | {
"line": 67,
"column": 65
} | {
"line": 67,
"column": 97
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasFiniteProducts C\ninst✝ : HasPullbacks C\nX : Dial C\n⊢ (tensorUnitImpl.tensorObjImpl X).rel =\n (Subobject.pullback (prod.map (Limits.prod.leftUnitor X.src).hom (Limits.prod.leftUnitor X.tgt).hom)).obj X.rel",
"usedConstants": [
"congrAr... | by simp [Subobject.pullback_top] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Dialectica.Monoidal | {
"line": 72,
"column": 67
} | {
"line": 72,
"column": 99
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasFiniteProducts C\ninst✝ : HasPullbacks C\nX : Dial C\n⊢ (X.tensorObjImpl tensorUnitImpl).rel =\n (Subobject.pullback (prod.map (Limits.prod.rightUnitor X.src).hom (Limits.prod.rightUnitor X.tgt).hom)).obj X.rel",
"usedConstants": [
"congr... | by simp [Subobject.pullback_top] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Localization.Monoidal.Basic | {
"line": 298,
"column": 83
} | {
"line": 299,
"column": 68
} | [
{
"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 Y₁ : LocalizedMonoidal L W ε\nf₁ : X₁ ⟶ Y₁\n⊢ f₁ ▷ ... | by
simp only [← tensorHom_id, associator_naturality, id_tensorHom_id] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.FiberedCategory.HomLift | {
"line": 103,
"column": 74
} | {
"line": 104,
"column": 21
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝¹ : Category.{v₁, u₂} 𝒳\ninst✝ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ⟶ b\nha : p.obj a = R\nhb : p.obj b = S\nh : f = eqToHom ⋯ ≫ p.map φ ≫ eqToHom hb\n⊢ p.IsHomLift f φ",
"usedConstants": [
"CategoryTheory.CategoryStruct.t... | by
subst ha hb h; simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Functor.Derived.Adjunction | {
"line": 76,
"column": 4
} | {
"line": 88,
"column": 70
} | [
{
"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... | suffices F'.leftUnitor.inv ≫ whiskerLeft F' η ≫ (Functor.associator _ _ _).inv ≫
whiskerRight ε F' ≫ F'.rightUnitor.hom = 𝟙 _ from
fun Y₂ ↦ by simpa using congr_app this Y₂
apply F'.rightDerived_ext β W₂
ext X₂
have eq₁ := η.naturality (β.app X₂)
have eq₂ := F'.congr_map (hε X₂)
have ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Functor.Derived.Adjunction | {
"line": 76,
"column": 4
} | {
"line": 88,
"column": 70
} | [
{
"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... | suffices F'.leftUnitor.inv ≫ whiskerLeft F' η ≫ (Functor.associator _ _ _).inv ≫
whiskerRight ε F' ≫ F'.rightUnitor.hom = 𝟙 _ from
fun Y₂ ↦ by simpa using congr_app this Y₂
apply F'.rightDerived_ext β W₂
ext X₂
have eq₁ := η.naturality (β.app X₂)
have eq₂ := F'.congr_map (hε X₂)
have ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.EpiMono | {
"line": 76,
"column": 4
} | {
"line": 77,
"column": 48
} | [
{
"pp": "case mpr\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX Y : C\nf : X ⟶ Y\n⊢ IsPullback (𝟙 X) (𝟙 X) f f → Mono f",
"usedConstants": [
"Iff.mpr",
"CategoryTheory.Mono",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.IsPullback",
"Categor... | intro hf
exact (mono_iff_fst_eq_snd hf.isLimit).2 rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.EpiMono | {
"line": 76,
"column": 4
} | {
"line": 77,
"column": 48
} | [
{
"pp": "case mpr\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX Y : C\nf : X ⟶ Y\n⊢ IsPullback (𝟙 X) (𝟙 X) f f → Mono f",
"usedConstants": [
"Iff.mpr",
"CategoryTheory.Mono",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.IsPullback",
"Categor... | intro hf
exact (mono_iff_fst_eq_snd hf.isLimit).2 rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.EpiMono | {
"line": 107,
"column": 4
} | {
"line": 107,
"column": 34
} | [
{
"pp": "case mpr\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX Y : C\nf : X ⟶ Y\nc : PushoutCocone f f\nhc : IsColimit c\na✝ : IsIso c.inl\nφ : c.pt ⟶ Y\nhφ₁ : c.inl ≫ φ = 𝟙 Y\nhφ₂ : c.inr ≫ φ = 𝟙 Y\nthis : IsSplitMono φ\n⊢ c.inl = c.inr",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits... | rw [← cancel_mono φ, hφ₁, hφ₂] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Functor.TypeValuedFlat | {
"line": 42,
"column": 15
} | {
"line": 48,
"column": 92
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nF : C ⥤ Type w\ninst✝¹ : HasFiniteLimits C\ninst✝ : PreservesFiniteLimits F\n⊢ ∀ (X Y : F.Elements), ∃ W x x, True",
"usedConstants": [
"CategoryTheory.categoryOfElements",
"CategoryTheory.Limits.Cone.π",
"CategoryTheory.Functor.Elements",
... | by
rintro ⟨X, x⟩ ⟨Y, y⟩
let h := mapIsLimitOfPreservesOfIsLimit F _ _ (prodIsProd X Y)
let h' := Types.binaryProductLimit (F.obj X) (F.obj Y)
exact ⟨⟨X ⨯ Y, (h'.conePointUniqueUpToIso h).hom ⟨x, y⟩⟩,
⟨prod.fst, congr_fun (h'.conePointUniqueUpToIso_hom_comp h (.mk .left)) _⟩,
⟨prod.snd, congr... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Galois.Prorepresentability | {
"line": 258,
"column": 2
} | {
"line": 259,
"column": 43
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{u₂, u₁} C\ninst✝¹ : GaloisCategory C\nF : C ⥤ FintypeCat\ninst✝ : FiberFunctor F\nA : PointedGaloisObject F\nσ : Aut A.obj\n⊢ ∃ a, (π F A) a = σ",
"usedConstants": [
"GrpCat.instConcreteCategoryMonoidHomCarrier",
"GrpCat",
"MonoidHom.instFunLike",
... | have (i : PointedGaloisObject F) : Finite ((autGaloisSystem F ⋙ forget _).obj i) :=
inferInstanceAs <| Finite (Aut (i.obj)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Galois.EssSurj | {
"line": 98,
"column": 16
} | {
"line": 98,
"column": 39
} | [
{
"pp": "G : Type u_1\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalSpace G\ninst✝⁴ : IsTopologicalGroup G\ninst✝³ : CompactSpace G\nX : Action FintypeCat G\ninst✝² : TopologicalSpace X.V.obj\ninst✝¹ : DiscreteTopology X.V.obj\ninst✝ : ContinuousSMul G X.V.obj\nι : Type\nhf : Finite ι\nf : ι → Action FintypeCat G\nu :... | continuous_induced_rng, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Galois.IsFundamentalgroup | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 33
} | [
{
"pp": "case h\nC : 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 : Compa... | simp only [toAut_hom_app_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Galois.EssSurj | {
"line": 162,
"column": 4
} | {
"line": 174,
"column": 8
} | [
{
"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... | intro (m : V ⧸ Subgroup.subgroupOf U V)
simp only [const_obj_obj, Functor.comp_map, const_obj_map, Category.comp_id]
rw [← cancel_epi (u.inv), Iso.inv_hom_id_assoc]
apply Action.hom_ext
ext (x : Aut F ⧸ U.toSubgroup)
induction m, x using Quotient.inductionOn₂ with | _ σ μ
suffices h : ⟦μ * σ⁻¹⟧ ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Galois.EssSurj | {
"line": 162,
"column": 4
} | {
"line": 174,
"column": 8
} | [
{
"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... | intro (m : V ⧸ Subgroup.subgroupOf U V)
simp only [const_obj_obj, Functor.comp_map, const_obj_map, Category.comp_id]
rw [← cancel_epi (u.inv), Iso.inv_hom_id_assoc]
apply Action.hom_ext
ext (x : Aut F ⧸ U.toSubgroup)
induction m, x using Quotient.inductionOn₂ with | _ σ μ
suffices h : ⟦μ * σ⁻¹⟧ ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Groupoid.FreeGroupoid | {
"line": 166,
"column": 4
} | {
"line": 166,
"column": 36
} | [
{
"pp": "V : Type u\ninst✝¹ : Quiver V\nV' : Type u'\ninst✝ : Groupoid V'\nφ : V ⥤q V'\nΦ : Quiver.FreeGroupoid V ⥤ V'\nhΦ : of V ⋙q Φ.toPrefunctor = φ\n⊢ Symmetrify.of ⋙q (Paths.of (Symmetrify V) ⋙q (Quotient.functor redStep ⋙ Φ).toPrefunctor) = φ",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.... | rw [← Functor.toPrefunctor_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Groupoid.Subgroupoid | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 75
} | [
{
"pp": "C : Type u\ninst✝ : Groupoid C\nS T : Subgroupoid C\nh : S ≤ T\ns t : ↑S.objs\nx✝¹ x✝ : s ⟶ t\nf : ↑s ⟶ ↑t\nhf : f ∈ S.arrows ↑s ↑t\ng : ↑s ⟶ ↑t\nhg : g ∈ S.arrows ↑s ↑t\n⊢ (fun f ↦ (inclusion h).map f) ⟨f, hf⟩ = (fun f ↦ (inclusion h).map f) ⟨g, hg⟩ → ⟨f, hf⟩ = ⟨g, hg⟩",
"usedConstants": [
"... | dsimp only [inclusion]; rw [Subtype.mk_eq_mk, Subtype.mk_eq_mk]; exact id | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Groupoid.Subgroupoid | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 75
} | [
{
"pp": "C : Type u\ninst✝ : Groupoid C\nS T : Subgroupoid C\nh : S ≤ T\ns t : ↑S.objs\nx✝¹ x✝ : s ⟶ t\nf : ↑s ⟶ ↑t\nhf : f ∈ S.arrows ↑s ↑t\ng : ↑s ⟶ ↑t\nhg : g ∈ S.arrows ↑s ↑t\n⊢ (fun f ↦ (inclusion h).map f) ⟨f, hf⟩ = (fun f ↦ (inclusion h).map f) ⟨g, hg⟩ → ⟨f, hf⟩ = ⟨g, hg⟩",
"usedConstants": [
"... | dsimp only [inclusion]; rw [Subtype.mk_eq_mk, Subtype.mk_eq_mk]; exact id | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Idempotents.Biproducts | {
"line": 123,
"column": 6
} | {
"line": 128,
"column": 66
} | [
{
"pp": "case h₀\nC : Type u_1\ninst✝¹ : Category.{v, u_1} C\ninst✝ : Preadditive C\nP : Karoubi C\n⊢ biprod.inl ≫ biprod.desc P.decompId_i P.complement.decompId_i ≫ biprod.lift P.decompId_p P.complement.decompId_p =\n biprod.inl ≫ 𝟙 (P ⊞ P.complement)",
"usedConstants": [
"CategoryTheory.Limits.b... | rw [biprod.inl_desc_assoc, comp_id, biprod.lift_eq, comp_add, ← decompId_assoc,
add_eq_left, ← assoc]
refine (?_ =≫ _).trans zero_comp
ext
simp only [comp_f, toKaroubi_obj_X, decompId_i_f, decompId_p_f,
complement_p, comp_sub, comp_id, idem, sub_self, zero_def] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Idempotents.Biproducts | {
"line": 123,
"column": 6
} | {
"line": 128,
"column": 66
} | [
{
"pp": "case h₀\nC : Type u_1\ninst✝¹ : Category.{v, u_1} C\ninst✝ : Preadditive C\nP : Karoubi C\n⊢ biprod.inl ≫ biprod.desc P.decompId_i P.complement.decompId_i ≫ biprod.lift P.decompId_p P.complement.decompId_p =\n biprod.inl ≫ 𝟙 (P ⊞ P.complement)",
"usedConstants": [
"CategoryTheory.Limits.b... | rw [biprod.inl_desc_assoc, comp_id, biprod.lift_eq, comp_add, ← decompId_assoc,
add_eq_left, ← assoc]
refine (?_ =≫ _).trans zero_comp
ext
simp only [comp_f, toKaroubi_obj_X, decompId_i_f, decompId_p_f,
complement_p, comp_sub, comp_id, idem, sub_self, zero_def] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Groupoid.Subgroupoid | {
"line": 472,
"column": 57
} | {
"line": 480,
"column": 28
} | [
{
"pp": "C : Type u\ninst✝¹ : Groupoid C\nS : Subgroupoid C\nD : Type u_1\ninst✝ : Groupoid D\nφ : C ⥤ D\nhφ : Function.Injective φ.obj\nd : D\n⊢ d ∈ (map φ hφ S).objs ↔ ∃ c ∈ S.objs, φ.obj c = d",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
... | by
dsimp [objs, map]
constructor
· rintro ⟨f, hf⟩
change Map.Arrows φ hφ S d d f at hf; rw [Map.arrows_iff] at hf
obtain ⟨c, d, g, ec, ed, eg, gS, eg⟩ := hf
exact ⟨c, ⟨mem_objs_of_src S eg, ec⟩⟩
· rintro ⟨c, ⟨γ, γS⟩, rfl⟩
exact ⟨φ.map γ, ⟨γ, γS⟩⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.FunctorToTypes | {
"line": 106,
"column": 63
} | {
"line": 107,
"column": 49
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF G : C ⥤ Type w\n⊢ (binaryProductIso F G).inv ≫ Limits.prod.fst = prod.fst",
"usedConstants": [
"CategoryTheory.FunctorToTypes.binaryProductIso._proof_1",
"CategoryTheory.Functor.flip",
"CategoryTheory.Functor",
"CategoryTheory.Categor... | by
simp [binaryProductIso, binaryProductLimitCone] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.FunctorToTypes | {
"line": 117,
"column": 63
} | {
"line": 118,
"column": 49
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF G : C ⥤ Type w\n⊢ (binaryProductIso F G).inv ≫ Limits.prod.snd = prod.snd",
"usedConstants": [
"CategoryTheory.FunctorToTypes.binaryProductIso._proof_1",
"CategoryTheory.Functor.flip",
"CategoryTheory.Functor",
"CategoryTheory.Categor... | by
simp [binaryProductIso, binaryProductLimitCone] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.ChosenPullback | {
"line": 228,
"column": 44
} | {
"line": 229,
"column": 43
} | [
{
"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\nY : C\nφ φ' : Y ⟶ h.pullback\nh₁ : φ ≫ h.p₁ = φ' ≫ h.p₁\nh₂ : φ ≫ h.p₂ = φ' ≫ h.p₂\nh₃ : φ ≫... | by
apply h.isPullback₂.hom_ext <;> cat_disch | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.ChosenPullback | {
"line": 237,
"column": 24
} | {
"line": 237,
"column": 48
} | [
{
"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\nx✝ : C\na : x✝ ⟶ h₁₂.pullback\nb : x✝ ⟶ h₁₃.pullback\nw : a ≫ h₁₂.p₁ = b ≫ h₁₃.p₁\n⊢ (b ≫ h₁... | simpa using w.symm =≫ f₁ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.ChosenPullback | {
"line": 237,
"column": 24
} | {
"line": 237,
"column": 48
} | [
{
"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\nx✝ : C\na : x✝ ⟶ h₁₂.pullback\nb : x✝ ⟶ h₁₃.pullback\nw : a ≫ h₁₂.p₁ = b ≫ h₁₃.p₁\n⊢ (b ≫ h₁... | simpa using w.symm =≫ f₁ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.ChosenPullback | {
"line": 237,
"column": 24
} | {
"line": 237,
"column": 48
} | [
{
"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\nx✝ : C\na : x✝ ⟶ h₁₂.pullback\nb : x✝ ⟶ h₁₃.pullback\nw : a ≫ h₁₂.p₁ = b ≫ h₁₃.p₁\n⊢ (b ≫ h₁... | simpa using w.symm =≫ f₁ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Types.ColimitTypeFiltered | {
"line": 59,
"column": 74
} | {
"line": 59,
"column": 77
} | [
{
"pp": "case mp.trans\nJ : Type u\ninst✝¹ : Category.{v, u} J\ninst✝ : IsFiltered J\nF : J ⥤ Type w₀\nx✝ y✝ x y z : (j : J) × F.obj j\na✝¹ : Relation.EqvGen F.ColimitTypeRel x y\na✝ : Relation.EqvGen F.ColimitTypeRel y z\nk : J\nf : x.fst ⟶ k\ng : y.fst ⟶ k\nh : F.map f x.snd = F.map g y.snd\nk' : J\nf' : y.fs... | h'' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Types.ColimitTypeFiltered | {
"line": 44,
"column": 4
} | {
"line": 59,
"column": 78
} | [
{
"pp": "case mp\nJ : Type u\ninst✝¹ : Category.{v, u} J\ninst✝ : IsFiltered J\nF : J ⥤ Type w₀\nx y : (j : J) × F.obj j\n⊢ Relation.EqvGen F.ColimitTypeRel x y → ∃ k f g, F.map f x.snd = F.map g y.snd",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | intro h
induction h with
| rel x y h =>
obtain ⟨f, h⟩ := h
exact ⟨y.1, f, 𝟙 _, by simpa using h.symm⟩
| refl x => exact ⟨x.1, 𝟙 _, 𝟙 _, rfl⟩
| symm _ _ _ h =>
obtain ⟨k, f, g, h⟩ := h
exact ⟨k, g, f, h.symm⟩
| trans x y z _ _ h h' =>
obtain ⟨k, f, g, h⟩ := h
ob... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Types.ColimitTypeFiltered | {
"line": 44,
"column": 4
} | {
"line": 59,
"column": 78
} | [
{
"pp": "case mp\nJ : Type u\ninst✝¹ : Category.{v, u} J\ninst✝ : IsFiltered J\nF : J ⥤ Type w₀\nx y : (j : J) × F.obj j\n⊢ Relation.EqvGen F.ColimitTypeRel x y → ∃ k f g, F.map f x.snd = F.map g y.snd",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | intro h
induction h with
| rel x y h =>
obtain ⟨f, h⟩ := h
exact ⟨y.1, f, 𝟙 _, by simpa using h.symm⟩
| refl x => exact ⟨x.1, 𝟙 _, 𝟙 _, rfl⟩
| symm _ _ _ h =>
obtain ⟨k, f, g, h⟩ := h
exact ⟨k, g, f, h.symm⟩
| trans x y z _ _ h h' =>
obtain ⟨k, f, g, h⟩ := h
ob... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Localization.BousfieldTransfiniteComposition | {
"line": 50,
"column": 10
} | {
"line": 57,
"column": 86
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nP : ObjectProperty C\nJ : Type w\ninst✝³ : LinearOrder J\ninst✝² : SuccOrder J\ninst✝¹ : OrderBot J\ninst✝ : WellFoundedLT J\nX Y : C\nf : X ⟶ Y\nx✝ : P.isLocal.transfiniteCompositionsOfShape J f\nZ : C\nhZ : P Z\nhf : P.isLocal.TransfiniteCompositionOfShape J f\... | let c : Cocone ((Set.principalSegIio j).monotone.functor ⋙ hf.F) :=
{ pt := Z
ι.app k := s.1 (op k)
ι.naturality _ _ g := by
dsimp
simpa only [Category.comp_id] using s.2 g.op }
exact ⟨(hf.F.isColimitOfIsWellOrderContinuous j hj).desc c, ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Localization.BousfieldTransfiniteComposition | {
"line": 50,
"column": 10
} | {
"line": 57,
"column": 86
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nP : ObjectProperty C\nJ : Type w\ninst✝³ : LinearOrder J\ninst✝² : SuccOrder J\ninst✝¹ : OrderBot J\ninst✝ : WellFoundedLT J\nX Y : C\nf : X ⟶ Y\nx✝ : P.isLocal.transfiniteCompositionsOfShape J f\nZ : C\nhZ : P Z\nhf : P.isLocal.TransfiniteCompositionOfShape J f\... | let c : Cocone ((Set.principalSegIio j).monotone.functor ⋙ hf.F) :=
{ pt := Z
ι.app k := s.1 (op k)
ι.naturality _ _ g := by
dsimp
simpa only [Category.comp_id] using s.2 g.op }
exact ⟨(hf.F.isColimitOfIsWellOrderContinuous j hj).desc c, ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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