module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 336,
"column": 6
} | {
"line": 336,
"column": 21
} | [
{
"pp": "b : ℝ\nb_pos : 0 < b\nb_lt_one : b < 1\nx : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\nhxy : x < y\n⊢ logb b x < logb b y",
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"Real.lattice",
"abs",
"congrArg",
"id",
"Real.instAddGroup",
"Real.logb",
... | ← logb_abs b y, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 338,
"column": 37
} | {
"line": 338,
"column": 51
} | [
{
"pp": "b : ℝ\nb_pos : 0 < b\nb_lt_one : b < 1\nx : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\nhxy : x < y\n⊢ -y < -x",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"neg_lt_neg_iff",
"Rea... | neg_lt_neg_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 514,
"column": 2
} | {
"line": 514,
"column": 68
} | [
{
"pp": "b : ℝ\nn : ℕ\n⊢ Tendsto (fun x ↦ logb b x ^ n / id x) atTop (𝓝 0)",
"usedConstants": [
"Real",
"instHDiv",
"HMul.hMul",
"Real.instZero",
"Real.instAddMonoid",
"congrArg",
"Real.instDivInvMonoid",
"AddMonoid.toAddZeroClass",
"NormedDivisionRing.... | · simpa using tendsto_pow_logb_div_mul_add_atTop 1 0 n one_ne_zero | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Log.InvLog | {
"line": 36,
"column": 4
} | {
"line": 36,
"column": 43
} | [
{
"pp": "x : ℝ\nx✝ : x ∈ Set.Ioo (-1) 0\nhx₁ : -1 < x\nhx₂ : x < 0\n⊢ log x < 0",
"usedConstants": [
"Real.partialOrder",
"Real",
"Preorder.toLT",
"Real.log_neg_eq_log",
"PartialOrder.toPreorder",
"Eq.rec",
"Real.semiring",
"Real.log",
"LT.lt",
"Re... | refine log_neg_eq_log x ▸ log_neg ?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecialFunctions.Log.InvLog | {
"line": 60,
"column": 2
} | {
"line": 60,
"column": 10
} | [
{
"pp": "case inr.inr.inr\nx : ℝ\nh0 : x ≠ 0\nh1 : x ≠ 1\nh2 : x ≠ -1\n⊢ deriv (fun x ↦ (log x)⁻¹) x = -x⁻¹ / log x ^ 2",
"usedConstants": [
"NormedCommRing.toNormedRing",
"NegZeroClass.toNeg",
"NormedCommRing.toSeminormedCommRing",
"False",
"Real",
"instHDiv",
"Sem... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 397,
"column": 2
} | {
"line": 397,
"column": 91
} | [
{
"pp": "p t : ℝ\nhp : p ∈ Ioo 1 2\nht : 0 < t\n⊢ MonotoneOn (p.rpowIntegrand₁₂ t) (Ici 0)",
"usedConstants": [
"Real",
"HMul.hMul",
"Set.Ici",
"Real.instZero",
"Real.instSub",
"HSub.hSub",
"MonotoneOn.congr",
"Membership.mem",
"Real.rpowIntegrand₀₁",
... | refine MonotoneOn.congr ?_ fun x hx ↦ (rpowIntegrand₁₂_eq_mul_rpowIntegrand₀₁ hx ht).symm | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas | {
"line": 104,
"column": 6
} | {
"line": 104,
"column": 57
} | [
{
"pp": "case inr\nn a fuel : ℕ\nih : ∀ {guess : ℕ}, a < (guess + 1) ^ (n + 2) → a < (go n a fuel guess + 1) ^ (n + 2)\nguess : ℕ\nhlt : a < (guess + 1) ^ (n + 2)\nh : (a / guess ^ (n + 1) + (n + 1) * guess) / (n + 2) < guess\nhguess : guess ≠ 0\n⊢ a < (go n a fuel ((a / guess ^ (n + 1) + (n + 1) * guess) / (n ... | · exact ih <| Nat.nthRoot.lt_pow_go_succ_aux hguess | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Constructions.Polish.EmbeddingReal | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 37
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : StandardBorelSpace α\nhα : ¬Countable α\n⊢ ¬univ.Countable",
"usedConstants": [
"Cardinal.not_countable_real"
]
}
] | exact Cardinal.not_countable_real | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Constructions.HaarToSphere | {
"line": 279,
"column": 6
} | {
"line": 279,
"column": 56
} | [
{
"pp": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : Nontrivial E\nμ : Measure E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : BorelSpace E\ninst✝ : μ.IsAddHaarMeasure\nf : ℝ → F\nr... | rw [← integrable_indicator_iff measurableSet_ball] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.MulExpNegMulSqIntegral | {
"line": 165,
"column": 2
} | {
"line": 252,
"column": 27
} | [
{
"pp": "ε : ℝ\nE : Type u_2\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : PseudoEMetricSpace E\ninst✝⁴ : BorelSpace E\ninst✝³ : CompleteSpace E\ninst✝² : SecondCountableTopology E\nP P' : Measure E\ninst✝¹ : IsFiniteMeasure P\ninst✝ : IsFiniteMeasure P'\nf : E →ᵇ ℝ\nA : Subalgebra ℝ (E →ᵇ ℝ)\nhA : (Subalgebra.map (toC... | by_cases hPP' : P = 0 ∧ P' = 0
· simp only [hPP', integral_zero_measure, sub_self, abs_zero, Nat.ofNat_pos,
mul_nonneg_iff_of_pos_left, (le_of_lt (sqrt_pos_of_pos hε))]
let const : ℝ := (max (P.real Set.univ) (P'.real Set.univ))
have pos_of_measure : 0 < const := by
rw [not_and_or] at hPP'
rcases hPP'... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.MulExpNegMulSqIntegral | {
"line": 165,
"column": 2
} | {
"line": 252,
"column": 27
} | [
{
"pp": "ε : ℝ\nE : Type u_2\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : PseudoEMetricSpace E\ninst✝⁴ : BorelSpace E\ninst✝³ : CompleteSpace E\ninst✝² : SecondCountableTopology E\nP P' : Measure E\ninst✝¹ : IsFiniteMeasure P\ninst✝ : IsFiniteMeasure P'\nf : E →ᵇ ℝ\nA : Subalgebra ℝ (E →ᵇ ℝ)\nhA : (Subalgebra.map (toC... | by_cases hPP' : P = 0 ∧ P' = 0
· simp only [hPP', integral_zero_measure, sub_self, abs_zero, Nat.ofNat_pos,
mul_nonneg_iff_of_pos_left, (le_of_lt (sqrt_pos_of_pos hε))]
let const : ℝ := (max (P.real Set.univ) (P'.real Set.univ))
have pos_of_measure : 0 < const := by
rw [not_and_or] at hPP'
rcases hPP'... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 346,
"column": 33
} | {
"line": 346,
"column": 41
} | [
{
"pp": "L : PeriodPair\nl₀ : ℂ\nr : ℝ\nhr : r > 0\ns : ℂ\nhs : ‖s‖ < r\nh : ‖↑0‖ ≥ 2 * r\nh✝ : ¬↑0 = l₀\nthis : s ≠ ↑0\n⊢ False",
"usedConstants": [
"Norm.norm",
"NormedCommRing.toSeminormedCommRing",
"Submodule",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 371,
"column": 33
} | {
"line": 371,
"column": 41
} | [
{
"pp": "L : PeriodPair\nl₀ : ℂ\ns : Finset ↥L.lattice\ni : ↥L.lattice\nhi : i ∈ s\nh✝ : ¬↑i = l₀\nhx : ↑i ∈ (↑L.lattice \\ {l₀})ᶜ\n⊢ False",
"usedConstants": [
"Submodule",
"False",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"eq_false",
"congrArg",
"and_self",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 382,
"column": 35
} | {
"line": 382,
"column": 43
} | [
{
"pp": "L : PeriodPair\nl₀ : ℂ\ns : Finset ↥L.lattice\ni : ↥L.lattice\nhi : i ∈ s\nh✝ : ¬↑i = l₀\nhx : ↑i ∈ (↑L.lattice \\ {l₀})ᶜ\n⊢ False",
"usedConstants": [
"Submodule",
"False",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"eq_false",
"congrArg",
"and_self",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.NumberTheory.Niven | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 16
} | [
{
"pp": "r : ℚ\nhr : 3 < (Int.fract r).den\n⊢ Irrational (cos (↑r * π))",
"usedConstants": [
"Real",
"Real.pi",
"HMul.hMul",
"Real.cos",
"Classical.byContradiction",
"Real.instRatCast",
"Rat.cast",
"Irrational",
"Real.instMul",
"Not",
"instHM... | by_contra! hnz | Mathlib.Tactic.ByContra._aux_Mathlib_Tactic_ByContra___macroRules_Mathlib_Tactic_ByContra_byContra!_1 | Mathlib.Tactic.ByContra.byContra! |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Extremal | {
"line": 207,
"column": 4
} | {
"line": 211,
"column": 12
} | [
{
"pp": "n : ℕ\nP : ℝ[X]\nhPdeg : P.degree ≤ ↑n\nhPbnd : ∀ x ∈ Set.Icc (-1) 1, |eval x P| ≤ 1\nhP : ¬P = 0\n⊢ P.leadingCoeff ≤ 2 ^ (n - 1)",
"usedConstants": [
"Iff.mpr",
"Real.instIsOrderedRing",
"WithBot.instPreorder",
"Eq.mpr",
"Polynomial.leadingCoeff.eq_1",
"MulOne.t... | lift P.degree to ℕ using degree_ne_bot.mpr hP with d hd
replace hPdeg : d ≤ n := (WithBot.coe_le rfl).mp hPdeg
rw [leadingCoeff, natDegree_eq_of_degree_eq_some hd.symm]
grw [coeff_le_of_forall_abs_le_one (le_of_eq hd.symm) hPbnd, hPdeg]
norm_num | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Extremal | {
"line": 207,
"column": 4
} | {
"line": 211,
"column": 12
} | [
{
"pp": "n : ℕ\nP : ℝ[X]\nhPdeg : P.degree ≤ ↑n\nhPbnd : ∀ x ∈ Set.Icc (-1) 1, |eval x P| ≤ 1\nhP : ¬P = 0\n⊢ P.leadingCoeff ≤ 2 ^ (n - 1)",
"usedConstants": [
"Iff.mpr",
"Real.instIsOrderedRing",
"WithBot.instPreorder",
"Eq.mpr",
"Polynomial.leadingCoeff.eq_1",
"MulOne.t... | lift P.degree to ℕ using degree_ne_bot.mpr hP with d hd
replace hPdeg : d ≤ n := (WithBot.coe_le rfl).mp hPdeg
rw [leadingCoeff, natDegree_eq_of_degree_eq_some hd.symm]
grw [coeff_le_of_forall_abs_le_one (le_of_eq hd.symm) hPbnd, hPdeg]
norm_num | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 177,
"column": 4
} | {
"line": 177,
"column": 22
} | [
{
"pp": "case hΘ\nd : ℤ\nz : ℂ\nhz : z ≠ 0\n⊢ (fun c ↦ ↑c * z) =Θ[cofinite] fun n ↦ ↑n",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"NonUnitalCommRing.toNonUnitalNonA... | simp_rw [mul_comm] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 244,
"column": 48
} | {
"line": 248,
"column": 51
} | [
{
"pp": "z : ℂ\nhz : z ≠ 0\nd k : ℤ\nhk : 2 ≤ k\n⊢ Summable fun c ↦ ((↑c * z + ↑d) ^ k)⁻¹",
"usedConstants": [
"zpow_natCast",
"NormedCommRing.toNormedRing",
"EisensteinSeries.linear_inv_isBigO_left",
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Int... | by
apply summable_inv_of_isBigO_rpow_inv (a := k) (by norm_cast)
lift k to ℕ using (by lia)
simp only [zpow_natCast, Int.cast_natCast, Real.rpow_natCast, ← inv_pow, ← abs_inv]
apply (linear_inv_isBigO_left d hz).abs_right.pow | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema | {
"line": 94,
"column": 8
} | {
"line": 94,
"column": 66
} | [
{
"pp": "case mp\nn : ℕ\nhn : n ≠ 0\nx : ℝ\nhTx : |eval x (T ℝ ↑n)| = 1\nhx : |x| ≤ 1\n⊢ ∃ k ≤ n, x = cos (↑k * π / ↑n)",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Polynomial.eval",
"NegZeroClass.toNeg",
"Real",
"Real.lattice",
"Polynomial.Chebyshev.T",
"R... | ← cos_arccos (neg_le_of_abs_le hx) (le_of_max_le_left hx), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Int.Fib.Basic | {
"line": 52,
"column": 71
} | {
"line": 55,
"column": 25
} | [
{
"pp": "n : ℕ\n⊢ fib (-↑n) = (-1) ^ (n + 1) * ↑(Nat.fib n)",
"usedConstants": [
"Int.instAddCommGroup",
"one_pow",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"Nat.fib_eq_zero._simp_1",
"NonUnitalCommRing.toNonUnitalNonAssocComm... | by
rcases n.even_or_odd with (hn | hn)
· simp [fib, hn, pow_add]
· simp [fib_of_odd, hn] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Int.Fib.Basic | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 78
} | [
{
"pp": "case inr\nn : ℤ\nm : ℕ\nh : -↑m ∣ n\n⊢ fib (-↑m) ∣ fib n",
"usedConstants": [
"_private.Mathlib.Data.Int.Fib.Basic.0.Int.fib_natCast_dvd",
"Int.instCommMonoid",
"False",
"Nat.instMulZeroClass",
"Dvd.dvd",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"is... | · simp [fib_neg_natCast, ← fib_natCast, fib_natCast_dvd <| Int.neg_dvd.mp h] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 968,
"column": 27
} | {
"line": 968,
"column": 35
} | [
{
"pp": "L : PeriodPair\nthis : Meromorphic fun z ↦ ℘'[L] z ^ 2 - 4 * ℘[L] z ^ 3 + L.g₂ * ℘[L] z + L.g₃\nz w : ℂ\nhw : w ∈ {z}ᶜ\nhw' : w ∈ (↑L.lattice \\ {z})ᶜ\n⊢ w ∉ L.lattice",
"usedConstants": [
"Submodule",
"SetLike.mem_coe._simp_1",
"False",
"NonUnitalCommRing.toNonUnitalNonAsso... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 968,
"column": 27
} | {
"line": 968,
"column": 35
} | [
{
"pp": "L : PeriodPair\nthis : Meromorphic fun z ↦ ℘'[L] z ^ 2 - 4 * ℘[L] z ^ 3 + L.g₂ * ℘[L] z + L.g₃\nz w : ℂ\nhw : w ∈ {z}ᶜ\nhw' : w ∈ (↑L.lattice \\ {z})ᶜ\n⊢ w ∉ L.lattice",
"usedConstants": [
"Submodule",
"SetLike.mem_coe._simp_1",
"False",
"NonUnitalCommRing.toNonUnitalNonAsso... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 968,
"column": 27
} | {
"line": 968,
"column": 35
} | [
{
"pp": "L : PeriodPair\nthis : Meromorphic fun z ↦ ℘'[L] z ^ 2 - 4 * ℘[L] z ^ 3 + L.g₂ * ℘[L] z + L.g₃\nz w : ℂ\nhw : w ∈ {z}ᶜ\nhw' : w ∈ (↑L.lattice \\ {z})ᶜ\n⊢ w ∉ L.lattice",
"usedConstants": [
"Submodule",
"SetLike.mem_coe._simp_1",
"False",
"NonUnitalCommRing.toNonUnitalNonAsso... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Tactic.NormNum.Prime | {
"line": 59,
"column": 4
} | {
"line": 59,
"column": 12
} | [
{
"pp": "case inr.inl\nk : ℕ\nh✝ : MinFacHelper (succ 0) k\nthis : 2 < (succ 0).minFac\nh : succ 0 > 0\n⊢ 1 < succ 0",
"usedConstants": [
"False",
"congrArg",
"False.elim",
"AddMonoid.toAddZeroClass",
"Nat.instAtLeastTwoHAddOfNat",
"Nat.instAddMonoid",
"Nat.instChar... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Tactic.NormNum.Prime | {
"line": 59,
"column": 4
} | {
"line": 59,
"column": 12
} | [
{
"pp": "case inr.inl\nk : ℕ\nh✝ : MinFacHelper (succ 0) k\nthis : 2 < (succ 0).minFac\nh : succ 0 > 0\n⊢ 1 < succ 0",
"usedConstants": [
"False",
"congrArg",
"False.elim",
"AddMonoid.toAddZeroClass",
"Nat.instAtLeastTwoHAddOfNat",
"Nat.instAddMonoid",
"Nat.instChar... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Tactic.NormNum.Prime | {
"line": 59,
"column": 4
} | {
"line": 59,
"column": 12
} | [
{
"pp": "case inr.inl\nk : ℕ\nh✝ : MinFacHelper (succ 0) k\nthis : 2 < (succ 0).minFac\nh : succ 0 > 0\n⊢ 1 < succ 0",
"usedConstants": [
"False",
"congrArg",
"False.elim",
"AddMonoid.toAddZeroClass",
"Nat.instAtLeastTwoHAddOfNat",
"Nat.instAddMonoid",
"Nat.instChar... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 978,
"column": 35
} | {
"line": 978,
"column": 43
} | [
{
"pp": "L : PeriodPair\nz : ℂ\nhz : z ∈ (↑L.lattice \\ {0})ᶜ\nhz0 : ¬z = 0\n⊢ z ∉ L.lattice",
"usedConstants": [
"Submodule",
"SetLike.mem_coe._simp_1",
"False",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"eq_false",
"and_true",
"congrArg",
"Compl.compl... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.CategoryTheory.Filtered.Small | {
"line": 127,
"column": 2
} | {
"line": 127,
"column": 31
} | [
{
"pp": "case coeq\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : IsFilteredOrEmpty C\nα : Type w\nf : α → C\nj j✝ j'✝ : C\nhj₁ : filteredClosure f j✝\nhj₂ : filteredClosure f j'✝\ng g' : j✝ ⟶ j'✝\nih : ∃ a, FilteredClosureSmall.abstractFilteredClosureRealization f a = { obj := j✝, property := hj₁ }.obj\nih' ... | | coeq hj₁ hj₂ g g' ih ih' => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.Coseparator | {
"line": 26,
"column": 2
} | {
"line": 27,
"column": 29
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : IsGrothendieckAbelian.{w, v, u} C\n⊢ HasCoseparator (ShrinkHoms.{u} C)",
"usedConstants": [
"CategoryTheory.IsGrothendieckAbelian.locallySmall",
"CategoryTheory.ShrinkHoms.isGrothendieckAbelian",
"CategoryTheory.I... | obtain ⟨G, -, hG⟩ := Abelian.has_injective_coseparator (separator (ShrinkHoms C))
(isSeparator_separator _) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Abelian.SerreClass.MorphismProperty | {
"line": 107,
"column": 6
} | {
"line": 107,
"column": 21
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\nP : ObjectProperty C\ninst✝¹ : P.IsSerreClass\nX Y : C\nf : X ⟶ Y\ninst✝ : Mono f\nthis : P.monoModSerre f\n⊢ P.isoModSerre f ↔ P.epiModSerre f",
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"And",
"Iff",
... | isoModSerre_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.SerreClass.MorphismProperty | {
"line": 113,
"column": 6
} | {
"line": 113,
"column": 21
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\nP : ObjectProperty C\ninst✝¹ : P.IsSerreClass\nX Y : C\nf : X ⟶ Y\ninst✝ : Epi f\nthis : P.epiModSerre f\n⊢ P.isoModSerre f ↔ P.monoModSerre f",
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"And",
"Iff",
... | isoModSerre_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.SerreClass.MorphismProperty | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 31
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\nP : ObjectProperty C\ninst✝¹ : P.IsSerreClass\nX Y : C\nf : X ⟶ Y\ninst✝ : Mono f\nhf : P.epiModSerre f\n⊢ P.isoModSerre f",
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"CategoryTheory.ObjectProperty.isoModSerre_... | rwa [isoModSerre_iff_of_mono] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.CategoryTheory.Abelian.SerreClass.MorphismProperty | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 31
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\nP : ObjectProperty C\ninst✝¹ : P.IsSerreClass\nX Y : C\nf : X ⟶ Y\ninst✝ : Mono f\nhf : P.epiModSerre f\n⊢ P.isoModSerre f",
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"CategoryTheory.ObjectProperty.isoModSerre_... | rwa [isoModSerre_iff_of_mono] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.SerreClass.MorphismProperty | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 31
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\nP : ObjectProperty C\ninst✝¹ : P.IsSerreClass\nX Y : C\nf : X ⟶ Y\ninst✝ : Mono f\nhf : P.epiModSerre f\n⊢ P.isoModSerre f",
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"CategoryTheory.ObjectProperty.isoModSerre_... | rwa [isoModSerre_iff_of_mono] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.Pseudoelements | {
"line": 446,
"column": 4
} | {
"line": 451,
"column": 58
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝ : Ring R\nG : ModuleCat R\nx y : Over G\nP : ModuleCat R\np : P ⟶ x.left\nq : P ⟶ y.left\nhp : Epi p\nhq : Epi q\nH : p ≫ x.hom = q ≫ y.hom\na : ↑G\nha : a ∈ (ModuleCat.Hom.hom x.hom).range\n⊢ a ∈ (ModuleCat.Hom.hom y.hom).range",
"usedConstants": [
"Eq.mpr"... | obtain ⟨a', ha'⟩ := ha
obtain ⟨a'', ha''⟩ := (ModuleCat.epi_iff_surjective p).1 hp a'
refine ⟨q a'', ?_⟩
dsimp at ha' ⊢
rw [← LinearMap.comp_apply, ← ModuleCat.hom_comp, ← H,
ModuleCat.hom_comp, LinearMap.comp_apply, ha'', ha'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.Pseudoelements | {
"line": 446,
"column": 4
} | {
"line": 451,
"column": 58
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝ : Ring R\nG : ModuleCat R\nx y : Over G\nP : ModuleCat R\np : P ⟶ x.left\nq : P ⟶ y.left\nhp : Epi p\nhq : Epi q\nH : p ≫ x.hom = q ≫ y.hom\na : ↑G\nha : a ∈ (ModuleCat.Hom.hom x.hom).range\n⊢ a ∈ (ModuleCat.Hom.hom y.hom).range",
"usedConstants": [
"Eq.mpr"... | obtain ⟨a', ha'⟩ := ha
obtain ⟨a'', ha''⟩ := (ModuleCat.epi_iff_surjective p).1 hp a'
refine ⟨q a'', ?_⟩
dsimp at ha' ⊢
rw [← LinearMap.comp_apply, ← ModuleCat.hom_comp, ← H,
ModuleCat.hom_comp, LinearMap.comp_apply, ha'', ha'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.RightDerived | {
"line": 334,
"column": 20
} | {
"line": 334,
"column": 23
} | [
{
"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... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.MorphismProperty.OfObjectProperty | {
"line": 43,
"column": 2
} | {
"line": 44,
"column": 20
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nP Q Q' : ObjectProperty C\nh : Q ≤ Q'\n⊢ ofObjectProperty P Q ≤ ofObjectProperty P Q'",
"usedConstants": [
"CategoryTheory.MorphismProperty.ofObjectProperty",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"_private.Mathl... | intro _ _ _ ⟨hX, hY⟩
exact ⟨hX, h _ hY⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.MorphismProperty.OfObjectProperty | {
"line": 43,
"column": 2
} | {
"line": 44,
"column": 20
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nP Q Q' : ObjectProperty C\nh : Q ≤ Q'\n⊢ ofObjectProperty P Q ≤ ofObjectProperty P Q'",
"usedConstants": [
"CategoryTheory.MorphismProperty.ofObjectProperty",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"_private.Mathl... | intro _ _ _ ⟨hX, hY⟩
exact ⟨hX, h _ hY⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Action.Concrete | {
"line": 213,
"column": 4
} | {
"line": 214,
"column": 8
} | [
{
"pp": "V : Type (u + 1)\ninst✝³ : LargeCategory V\nFV : V → V → Type u_1\nCV : V → Type u_2\ninst✝² : (X Y : V) → FunLike (FV X Y) (CV X) (CV Y)\ninst✝¹ : ConcreteCategory V FV\nG : Type u_3\ninst✝ : Monoid G\nX : Action V G\nx : ToType X\n⊢ 1 • x = x",
"usedConstants": [
"CategoryTheory.End.one",
... | change ConcreteCategory.hom (X.ρ 1) x = x
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Action.Concrete | {
"line": 213,
"column": 4
} | {
"line": 214,
"column": 8
} | [
{
"pp": "V : Type (u + 1)\ninst✝³ : LargeCategory V\nFV : V → V → Type u_1\nCV : V → Type u_2\ninst✝² : (X Y : V) → FunLike (FV X Y) (CV X) (CV Y)\ninst✝¹ : ConcreteCategory V FV\nG : Type u_3\ninst✝ : Monoid G\nX : Action V G\nx : ToType X\n⊢ 1 • x = x",
"usedConstants": [
"CategoryTheory.End.one",
... | change ConcreteCategory.hom (X.ρ 1) x = x
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SemidirectProduct | {
"line": 258,
"column": 12
} | {
"line": 258,
"column": 20
} | [
{
"pp": "case left\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst✝⁶ : Group N\ninst✝⁵ : Group G\ninst✝⁴ : Group H\nφ : G →* MulAut N\nN₁ : Type u_4\nG₁ : Type u_5\nN₂ : Type u_6\nG₂ : Type u_7\ninst✝³ : Group N₁\ninst✝² : Group G₁\ninst✝¹ : Group N₂\ninst✝ : Group G₂\nφ₁ : G₁ →* MulAut N₁\nφ₂ : G₂ →* MulAut N₂... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.SemidirectProduct | {
"line": 258,
"column": 12
} | {
"line": 258,
"column": 20
} | [
{
"pp": "case right\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst✝⁶ : Group N\ninst✝⁵ : Group G\ninst✝⁴ : Group H\nφ : G →* MulAut N\nN₁ : Type u_4\nG₁ : Type u_5\nN₂ : Type u_6\nG₂ : Type u_7\ninst✝³ : Group N₁\ninst✝² : Group G₁\ninst✝¹ : Group N₂\ninst✝ : Group G₂\nφ₁ : G₁ →* MulAut N₁\nφ₂ : G₂ →* MulAut N... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.SemidirectProduct | {
"line": 300,
"column": 12
} | {
"line": 300,
"column": 20
} | [
{
"pp": "case left\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst✝⁶ : Group N\ninst✝⁵ : Group G\ninst✝⁴ : Group H\nφ : G →* MulAut N\nN₁ : Type u_4\nG₁ : Type u_5\nN₂ : Type u_6\nG₂ : Type u_7\ninst✝³ : Group N₁\ninst✝² : Group G₁\ninst✝¹ : Group N₂\ninst✝ : Group G₂\nφ₁ : G₁ →* MulAut N₁\nφ₂ : G₂ →* MulAut N₂... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.SemidirectProduct | {
"line": 300,
"column": 12
} | {
"line": 300,
"column": 20
} | [
{
"pp": "case right\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst✝⁶ : Group N\ninst✝⁵ : Group G\ninst✝⁴ : Group H\nφ : G →* MulAut N\nN₁ : Type u_4\nG₁ : Type u_5\nN₂ : Type u_6\nG₂ : Type u_7\ninst✝³ : Group N₁\ninst✝² : Group G₁\ninst✝¹ : Group N₂\ninst✝ : Group G₂\nφ₁ : G₁ →* MulAut N₁\nφ₂ : G₂ →* MulAut N... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.CategoryTheory.Abelian.SerreClass.Localization | {
"line": 456,
"column": 65
} | {
"line": 456,
"column": 80
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : Abelian C\nD : Type u'\ninst✝⁴ : Category.{v', u'} D\nL : C ⥤ D\nP : ObjectProperty C\ninst✝³ : P.IsSerreClass\ninst✝² : L.IsLocalization P.isoModSerre\ninst✝¹ : Preadditive D\ninst✝ : L.Additive\nX Y : C\nf : X ⟶ Y\nthis : Abelian D := abelian L P\n⊢ P.... | isoModSerre_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Adhesive.Subobject | {
"line": 47,
"column": 4
} | {
"line": 47,
"column": 46
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Adhesive C\nX : C\nF : Discrete WalkingPair ⥤ Subobject X\nthis : HasColimit (pair (F.obj { as := WalkingPair.left }) (F.obj { as := WalkingPair.right }))\n⊢ HasColimit F",
"usedConstants": [
"PartialOrder.toPreorder",
"CategoryTheory.Limi... | apply hasColimit_of_iso (diagramIsoPair F) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Adjunction.FullyFaithfulLimits | {
"line": 43,
"column": 2
} | {
"line": 44,
"column": 92
} | [
{
"pp": "C : Type u₁\ninst✝⁶ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nE : Type u₃\ninst✝⁴ : Category.{v₃, u₃} E\nH : D ⥤ E\nJ : Type u\ninst✝³ : Category.{v, u} J\ninst✝² : HasColimitsOfShape J C\ninst✝¹ : G.Full\ninst✝ : G.Faithful\nthis : F.IsLeftAdj... | let iso : (K ⋙ G) ⋙ F ≅ K :=
Functor.associator _ _ _ ≪≫ Functor.isoWhiskerLeft _ (asIso adj.counit) ≪≫ K.rightUnitor | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.CategoryTheory.Monad.Coequalizer | {
"line": 97,
"column": 44
} | {
"line": 97,
"column": 47
} | [
{
"pp": "case refine_1\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nT : Monad C\nX : T.Algebra\ns : Cofork (FreeCoequalizer.topMap X) (FreeCoequalizer.bottomMap X)\nh₁ : T.map X.a ≫ s.π.f = T.μ.app X.A ≫ s.π.f\nh₂ : T.map s.π.f ≫ s.pt.a = T.μ.app X.A ≫ s.π.f\n⊢ T.map (T.η.app X.1) ≫ T.map s.π.f ≫ s.pt.a = X.a ≫ T... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Adjunction.Triple | {
"line": 151,
"column": 22
} | {
"line": 151,
"column": 61
} | [
{
"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⊢ G.map (t.adj₁.counit.app (H.obj X) ≫ t.rightToLeft.app X) = G.map (F.map (t.adj₂.counit.app X))",
"usedConstan... | simp [← cancel_epi (t.adj₁.unit.app _)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Adjunction.Triple | {
"line": 151,
"column": 22
} | {
"line": 151,
"column": 61
} | [
{
"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⊢ G.map (t.adj₁.counit.app (H.obj X) ≫ t.rightToLeft.app X) = G.map (F.map (t.adj₂.counit.app X))",
"usedConstan... | simp [← cancel_epi (t.adj₁.unit.app _)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Adjunction.Triple | {
"line": 151,
"column": 22
} | {
"line": 151,
"column": 61
} | [
{
"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⊢ G.map (t.adj₁.counit.app (H.obj X) ≫ t.rightToLeft.app X) = G.map (F.map (t.adj₂.counit.app X))",
"usedConstan... | simp [← cancel_epi (t.adj₁.unit.app _)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Oplax | {
"line": 151,
"column": 4
} | {
"line": 165,
"column": 16
} | [] | _ = 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 _ := by
r... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.CategoryTheory.Bicategory.Grothendieck | {
"line": 191,
"column": 6
} | {
"line": 191,
"column": 89
} | [
{
"pp": "case hfg₂\n𝒮 : Type u₁\ninst✝ : Category.{v₁, u₁} 𝒮\nF G H : LocallyDiscrete 𝒮 ⥤ᵖ Cat\nα : F ⟶ G\na b c : ∫ F\nf : a ⟶ b\ng : b ⟶ c\n⊢ 𝟙 ((G.map (f.base.toLoc ≫ g.base.toLoc)).toFunctor.obj ((α.app { as := a.base }).toFunctor.obj a.fiber)) ≫\n (α.naturality (f.base.toLoc ≫ g.base.toLoc)).inv.t... | simp [naturality_comp_inv_app, ← Functor.map_comp, ← reassoc_of% Cat.Hom₂.comp_app] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Bicategory.Grothendieck | {
"line": 219,
"column": 4
} | {
"line": 221,
"column": 16
} | [
{
"pp": "𝒮 : Type u₁\ninst✝ : Category.{v₁, u₁} 𝒮\nF G H : LocallyDiscrete 𝒮 ⥤ᵖ Cat\nα : F ⟶ G\nβ : G ⟶ H\nX✝ Y✝ : ∫ F\nf : X✝ ⟶ Y✝\n⊢ (map (α ≫ β)).map f ≫ ((fun x ↦ eqToIso ⋯) Y✝).hom = ((fun x ↦ eqToIso ⋯) X✝).hom ≫ (map α ⋙ map β).map f",
"usedConstants": [
"CategoryTheory.Pseudofunctor.StrongT... | dsimp
simp only [comp_id, id_comp]
ext <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Bicategory.Grothendieck | {
"line": 219,
"column": 4
} | {
"line": 221,
"column": 16
} | [
{
"pp": "𝒮 : Type u₁\ninst✝ : Category.{v₁, u₁} 𝒮\nF G H : LocallyDiscrete 𝒮 ⥤ᵖ Cat\nα : F ⟶ G\nβ : G ⟶ H\nX✝ Y✝ : ∫ F\nf : X✝ ⟶ Y✝\n⊢ (map (α ≫ β)).map f ≫ ((fun x ↦ eqToIso ⋯) Y✝).hom = ((fun x ↦ eqToIso ⋯) X✝).hom ≫ (map α ⋙ map β).map f",
"usedConstants": [
"CategoryTheory.Pseudofunctor.StrongT... | dsimp
simp only [comp_id, id_comp]
ext <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Bicategory.Grothendieck | {
"line": 381,
"column": 4
} | {
"line": 383,
"column": 16
} | [
{
"pp": "𝒮 : Type u₁\ninst✝ : Category.{v₁, u₁} 𝒮\nF G H : LocallyDiscrete 𝒮ᵒᵖ ⥤ᵖ Cat\nα : F ⟶ G\nβ : G ⟶ H\nX✝ Y✝ : ∫ᶜ F\nf : X✝ ⟶ Y✝\n⊢ (map (α ≫ β)).map f ≫ ((fun x ↦ eqToIso ⋯) Y✝).hom = ((fun x ↦ eqToIso ⋯) X✝).hom ≫ (map α ⋙ map β).map f",
"usedConstants": [
"CategoryTheory.Pseudofunctor.Stro... | dsimp
simp only [comp_id, id_comp]
ext <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Bicategory.Grothendieck | {
"line": 381,
"column": 4
} | {
"line": 383,
"column": 16
} | [
{
"pp": "𝒮 : Type u₁\ninst✝ : Category.{v₁, u₁} 𝒮\nF G H : LocallyDiscrete 𝒮ᵒᵖ ⥤ᵖ Cat\nα : F ⟶ G\nβ : G ⟶ H\nX✝ Y✝ : ∫ᶜ F\nf : X✝ ⟶ Y✝\n⊢ (map (α ≫ β)).map f ≫ ((fun x ↦ eqToIso ⋯) Y✝).hom = ((fun x ↦ eqToIso ⋯) X✝).hom ≫ (map α ⋙ map β).map f",
"usedConstants": [
"CategoryTheory.Pseudofunctor.Stro... | dsimp
simp only [comp_id, id_comp]
ext <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sigma.Basic | {
"line": 243,
"column": 4
} | {
"line": 245,
"column": 25
} | [
{
"pp": "I : Type w₁\nC : I → Type u₁\ninst✝¹ : (i : I) → Category.{v₁, u₁} (C i)\nD : I → Type u₁\ninst✝ : (i : I) → Category.{v₁, u₁} (D i)\nF G : (i : I) → C i ⥤ D i\nα : (i : I) → F i ⟶ G i\n⊢ ∀ ⦃X Y : (i : I) × C i⦄ (f : X ⟶ Y),\n (Functor.sigma F).map f ≫ SigmaHom.mk ((α Y.fst).app Y.snd) =\n Sigm... | rintro ⟨i, X⟩ ⟨_, _⟩ ⟨f⟩
change SigmaHom.mk _ = SigmaHom.mk _
rw [(α i).naturality] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sigma.Basic | {
"line": 243,
"column": 4
} | {
"line": 245,
"column": 25
} | [
{
"pp": "I : Type w₁\nC : I → Type u₁\ninst✝¹ : (i : I) → Category.{v₁, u₁} (C i)\nD : I → Type u₁\ninst✝ : (i : I) → Category.{v₁, u₁} (D i)\nF G : (i : I) → C i ⥤ D i\nα : (i : I) → F i ⟶ G i\n⊢ ∀ ⦃X Y : (i : I) × C i⦄ (f : X ⟶ Y),\n (Functor.sigma F).map f ≫ SigmaHom.mk ((α Y.fst).app Y.snd) =\n Sigm... | rintro ⟨i, X⟩ ⟨_, _⟩ ⟨f⟩
change SigmaHom.mk _ = SigmaHom.mk _
rw [(α i).naturality] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Bicategory.Strict.Pseudofunctor | {
"line": 98,
"column": 4
} | {
"line": 98,
"column": 80
} | [
{
"pp": "B : Type u₁\nC : Type u₂\ninst✝² : Bicategory B\ninst✝¹ : Strict B\ninst✝ : Bicategory C\nF : B ⥤ᵖ C\nb₀ b₁ b₂ b₃ : B\nf₀₁ : b₀ ⟶ b₁\nf₁₂ : b₁ ⟶ b₂\nf₂₃ : b₂ ⟶ b₃\nf₀₂ : b₀ ⟶ b₂\nf₁₃ : b₁ ⟶ b₃\nf : b₀ ⟶ b₃\nh₀₂ : f₀₁ ≫ f₁₂ = f₀₂\nh₁₃ : f₁₂ ≫ f₂₃ = f₁₃\nhf : f₀₁ ≫ f₁₃ = f\n⊢ (F.mapComp' f₀₂ f₂₃ f ⋯).hom... | F.mapComp'₀₁₃_hom_comp_whiskerLeft_mapComp'_hom_assoc _ _ _ _ _ _ h₀₂ h₁₃ hf | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.PFun | {
"line": 319,
"column": 2
} | {
"line": 321,
"column": 51
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nC : α → Sort u_7\nf : α →. β ⊕ α\nb : β\na : α\nh : b ∈ f.fix a\nfa : Sum.inl b ∈ f a\nhbase : (a_final : α) → Sum.inl b ∈ f a_final → C a_final\nhind : (a₀ a₁ : α) → b ∈ f.fix a₁ → Sum.inr a₁ ∈ f a₀ → C a₁ → C a₀\n⊢ Sum.casesOn (motive := fun x ↦ (f a).get ⋯ = x → C a) ((f ... | refine Eq.rec (motive := fun x e ↦
Sum.casesOn x ?_ ?_ (Eq.trans (Part.get_eq_of_mem fa (dom_of_mem_fix h)) e) = hbase a fa) ?_
(Part.get_eq_of_mem fa (dom_of_mem_fix h)).symm | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Category.RelCat | {
"line": 96,
"column": 44
} | {
"line": 99,
"column": 67
} | [
{
"pp": "X Y : RelCat\nr : X ⟶ Y\nh : IsIso r\nh1 : ∀ (a b : X), (∃ y, (a, y) ∈ r.rel ∧ (y, b) ∈ (inv r).rel) ↔ a = b\nh2 : ∀ (a b : Y), (∃ y, (a, y) ∈ (inv r).rel ∧ (y, b) ∈ r.rel) ↔ a = b\nf : X → Y\nhf : ∀ (x : X), (x, f x) ∈ r.rel ∧ (f x, x) ∈ (inv r).rel\ng : Y → X\nhg : ∀ (x : Y), (x, g x) ∈ (inv r).rel ∧... | by
use asIso (↾f)
ext ⟨x, y⟩
exact ⟨by aesop, fun hxy ↦ (h2 (f x) y).1 ⟨x, (hf x).2, hxy⟩⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Distributive.Cartesian | {
"line": 79,
"column": 6
} | {
"line": 79,
"column": 49
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : CartesianMonoidalCategory C\ninst✝¹ : HasBinaryCoproducts C\ninst✝ : IsCartesianDistributive C\nA B Z : C\nf g : Z ⟶ (pair A B).obj { as := WalkingPair.left }\nhe : f ≫ (BinaryCofan.mk coprod.inl coprod.inr).inl = g ≫ (BinaryCofan.mk coprod.inl coprod.in... | simpa only [lift_snd] using this =≫ snd _ _ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Localization.Monoidal.Basic | {
"line": 426,
"column": 2
} | {
"line": 428,
"column": 60
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\nL : C ⥤ D\nW : MorphismProperty C\ninst✝² : MonoidalCategory C\ninst✝¹ : W.IsMonoidal\ninst✝ : L.IsLocalization W\nunit : D\nε : L.obj (𝟙_ C) ≅ unit\nX Y : LocalizedMonoidal L W ε\nX' : C\ne₁ : L'.obj X' ≅ X\nY... | rw [← μ_natural_left, tensorHom_id, ← whiskerRight_comp_assoc,
← μ_natural_right, ← Iso.comp_inv_eq, assoc, assoc, assoc,
Iso.hom_inv_id, comp_id, ← whiskerLeft_comp, ← h₂] at h₃ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.FiberedCategory.HomLift | {
"line": 69,
"column": 54
} | {
"line": 70,
"column": 33
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝¹ : Category.{v₁, u₂} 𝒳\ninst✝ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\na : 𝒳\n⊢ p.IsHomLift (𝟙 (p.obj a)) (𝟙 a)",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.IsHomLift.map",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"con... | by
rw [← p.map_id]; infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.FiberedCategory.HomLift | {
"line": 176,
"column": 2
} | {
"line": 176,
"column": 10
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₂} 𝒳\ninst✝¹ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ⟶ b\ninst✝ : p.IsHomLift f φ\n⊢ p.IsHomLift f (𝟙 a ≫ φ)",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrA... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.CategoryTheory.FiberedCategory.HomLift | {
"line": 176,
"column": 2
} | {
"line": 176,
"column": 10
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₂} 𝒳\ninst✝¹ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ⟶ b\ninst✝ : p.IsHomLift f φ\n⊢ p.IsHomLift f (𝟙 a ≫ φ)",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrA... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.FiberedCategory.HomLift | {
"line": 176,
"column": 2
} | {
"line": 176,
"column": 10
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₂} 𝒳\ninst✝¹ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ⟶ b\ninst✝ : p.IsHomLift f φ\n⊢ p.IsHomLift f (𝟙 a ≫ φ)",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrA... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.FiberedCategory.HomLift | {
"line": 175,
"column": 51
} | {
"line": 176,
"column": 10
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₂} 𝒳\ninst✝¹ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ⟶ b\ninst✝ : p.IsHomLift f φ\n⊢ p.IsHomLift f (𝟙 a ≫ φ)",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrA... | by
simp_all | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.FiberedCategory.HomLift | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 10
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₂} 𝒳\ninst✝¹ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ⟶ b\ninst✝ : p.IsHomLift f φ\n⊢ p.IsHomLift f (φ ≫ 𝟙 b)",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrA... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.CategoryTheory.FiberedCategory.HomLift | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 10
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₂} 𝒳\ninst✝¹ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ⟶ b\ninst✝ : p.IsHomLift f φ\n⊢ p.IsHomLift f (φ ≫ 𝟙 b)",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrA... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.FiberedCategory.HomLift | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 10
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₂} 𝒳\ninst✝¹ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ⟶ b\ninst✝ : p.IsHomLift f φ\n⊢ p.IsHomLift f (φ ≫ 𝟙 b)",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrA... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.FiberedCategory.HomLift | {
"line": 178,
"column": 51
} | {
"line": 179,
"column": 10
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₂} 𝒳\ninst✝¹ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ⟶ b\ninst✝ : p.IsHomLift f φ\n⊢ p.IsHomLift f (φ ≫ 𝟙 b)",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrA... | by
simp_all | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.FiberedCategory.HomLift | {
"line": 182,
"column": 2
} | {
"line": 182,
"column": 10
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₂} 𝒳\ninst✝¹ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ⟶ b\ninst✝ : p.IsHomLift f φ\n⊢ p.IsHomLift (𝟙 R ≫ f) φ",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrA... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.CategoryTheory.FiberedCategory.HomLift | {
"line": 182,
"column": 2
} | {
"line": 182,
"column": 10
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₂} 𝒳\ninst✝¹ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ⟶ b\ninst✝ : p.IsHomLift f φ\n⊢ p.IsHomLift (𝟙 R ≫ f) φ",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrA... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.FiberedCategory.HomLift | {
"line": 182,
"column": 2
} | {
"line": 182,
"column": 10
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₂} 𝒳\ninst✝¹ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ⟶ b\ninst✝ : p.IsHomLift f φ\n⊢ p.IsHomLift (𝟙 R ≫ f) φ",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrA... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.FiberedCategory.HomLift | {
"line": 181,
"column": 51
} | {
"line": 182,
"column": 10
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₂} 𝒳\ninst✝¹ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ⟶ b\ninst✝ : p.IsHomLift f φ\n⊢ p.IsHomLift (𝟙 R ≫ f) φ",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrA... | by
simp_all | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.FiberedCategory.HomLift | {
"line": 185,
"column": 2
} | {
"line": 185,
"column": 10
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₂} 𝒳\ninst✝¹ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ⟶ b\ninst✝ : p.IsHomLift f φ\n⊢ p.IsHomLift (f ≫ 𝟙 S) φ",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrA... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.CategoryTheory.FiberedCategory.HomLift | {
"line": 185,
"column": 2
} | {
"line": 185,
"column": 10
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₂} 𝒳\ninst✝¹ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ⟶ b\ninst✝ : p.IsHomLift f φ\n⊢ p.IsHomLift (f ≫ 𝟙 S) φ",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrA... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.FiberedCategory.HomLift | {
"line": 185,
"column": 2
} | {
"line": 185,
"column": 10
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₂} 𝒳\ninst✝¹ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ⟶ b\ninst✝ : p.IsHomLift f φ\n⊢ p.IsHomLift (f ≫ 𝟙 S) φ",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrA... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.FiberedCategory.HomLift | {
"line": 184,
"column": 51
} | {
"line": 185,
"column": 10
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₂} 𝒳\ninst✝¹ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ⟶ b\ninst✝ : p.IsHomLift f φ\n⊢ p.IsHomLift (f ≫ 𝟙 S) φ",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrA... | by
simp_all | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.FiberedCategory.HomLift | {
"line": 188,
"column": 11
} | {
"line": 188,
"column": 19
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₂} 𝒳\ninst✝¹ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\nb : 𝒳\nf : R ⟶ S\na' : 𝒳\nφ : a' ⟶ b\ninst✝ : p.IsHomLift f φ\n⊢ p.IsHomLift f (eqToHom ⋯ ≫ φ)",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.CategoryTheory.FiberedCategory.HomLift | {
"line": 191,
"column": 11
} | {
"line": 191,
"column": 19
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₂} 𝒳\ninst✝¹ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ⟶ b\ninst✝ : p.IsHomLift f φ\n⊢ p.IsHomLift f (φ ≫ eqToHom ⋯)",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"c... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.CategoryTheory.FiberedCategory.HomLift | {
"line": 194,
"column": 11
} | {
"line": 194,
"column": 19
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₂} 𝒳\ninst✝¹ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nS : 𝒮\na b : 𝒳\nφ : a ⟶ b\nR' : 𝒮\nf : R' ⟶ S\ninst✝ : p.IsHomLift f φ\n⊢ p.IsHomLift (eqToHom ⋯ ≫ f) φ",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.CategoryTheory.FiberedCategory.HomLift | {
"line": 197,
"column": 11
} | {
"line": 197,
"column": 19
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₂} 𝒳\ninst✝¹ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ⟶ b\ninst✝ : p.IsHomLift f φ\n⊢ p.IsHomLift (f ≫ eqToHom ⋯) φ",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"c... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.CategoryTheory.FiberedCategory.HomLift | {
"line": 270,
"column": 55
} | {
"line": 270,
"column": 63
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝⁴ : Category.{v₁, u₂} 𝒳\ninst✝³ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ⟶ b\ninst✝² : IsIso f\ninst✝¹ : IsIso φ\ninst✝ : p.IsHomLift f φ\n⊢ p.IsHomLift (asIso f).hom (asIso φ).hom",
"usedConstants": []
}
] | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.CategoryTheory.FiberedCategory.HomLift | {
"line": 270,
"column": 55
} | {
"line": 270,
"column": 63
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝⁴ : Category.{v₁, u₂} 𝒳\ninst✝³ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ⟶ b\ninst✝² : IsIso f\ninst✝¹ : IsIso φ\ninst✝ : p.IsHomLift f φ\n⊢ p.IsHomLift (asIso f).hom (asIso φ).hom",
"usedConstants": []
}
] | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.FiberedCategory.HomLift | {
"line": 270,
"column": 55
} | {
"line": 270,
"column": 63
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝⁴ : Category.{v₁, u₂} 𝒳\ninst✝³ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ⟶ b\ninst✝² : IsIso f\ninst✝¹ : IsIso φ\ninst✝ : p.IsHomLift f φ\n⊢ p.IsHomLift (asIso f).hom (asIso φ).hom",
"usedConstants": []
}
] | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.FiberedCategory.HomLift | {
"line": 279,
"column": 51
} | {
"line": 279,
"column": 59
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₂} 𝒳\ninst✝¹ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nS : 𝒮\na b : 𝒳\nφ : a ≅ b\ninst✝ : p.IsHomLift (𝟙 S) φ.hom\n⊢ p.IsHomLift (asIso (𝟙 S)).hom φ.hom",
"usedConstants": []
}
] | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.CategoryTheory.FiberedCategory.HomLift | {
"line": 279,
"column": 51
} | {
"line": 279,
"column": 59
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₂} 𝒳\ninst✝¹ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nS : 𝒮\na b : 𝒳\nφ : a ≅ b\ninst✝ : p.IsHomLift (𝟙 S) φ.hom\n⊢ p.IsHomLift (asIso (𝟙 S)).hom φ.hom",
"usedConstants": []
}
] | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.FiberedCategory.HomLift | {
"line": 279,
"column": 51
} | {
"line": 279,
"column": 59
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₂} 𝒳\ninst✝¹ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nS : 𝒮\na b : 𝒳\nφ : a ≅ b\ninst✝ : p.IsHomLift (𝟙 S) φ.hom\n⊢ p.IsHomLift (asIso (𝟙 S)).hom φ.hom",
"usedConstants": []
}
] | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.FiberedCategory.BasedCategory | {
"line": 242,
"column": 45
} | {
"line": 242,
"column": 53
} | [
{
"pp": "𝒮 : Type u₁\ninst✝¹ : Category.{v₁, u₁} 𝒮\n𝒳 : BasedCategory 𝒮\n𝒴 : BasedCategory 𝒮\nF G : 𝒳 ⥤ᵇ 𝒴\nα : F ⟶ G\ninst✝ : IsIso α.toNatTrans\n⊢ IsIso ((forgetful 𝒳 𝒴).map α)",
"usedConstants": []
}
] | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.CategoryTheory.FiberedCategory.BasedCategory | {
"line": 242,
"column": 45
} | {
"line": 242,
"column": 53
} | [
{
"pp": "𝒮 : Type u₁\ninst✝¹ : Category.{v₁, u₁} 𝒮\n𝒳 : BasedCategory 𝒮\n𝒴 : BasedCategory 𝒮\nF G : 𝒳 ⥤ᵇ 𝒴\nα : F ⟶ G\ninst✝ : IsIso α.toNatTrans\n⊢ IsIso ((forgetful 𝒳 𝒴).map α)",
"usedConstants": []
}
] | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.FiberedCategory.BasedCategory | {
"line": 242,
"column": 45
} | {
"line": 242,
"column": 53
} | [
{
"pp": "𝒮 : Type u₁\ninst✝¹ : Category.{v₁, u₁} 𝒮\n𝒳 : BasedCategory 𝒮\n𝒴 : BasedCategory 𝒮\nF G : 𝒳 ⥤ᵇ 𝒴\nα : F ⟶ G\ninst✝ : IsIso α.toNatTrans\n⊢ IsIso ((forgetful 𝒳 𝒴).map α)",
"usedConstants": []
}
] | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.FiberedCategory.HasFibers | {
"line": 86,
"column": 16
} | {
"line": 86,
"column": 76
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝¹ : Category.{v₁, u₁} 𝒮\ninst✝ : Category.{v₂, u₂} 𝒳\np : 𝒳 ⥤ 𝒮\nS : 𝒮\n⊢ (Fiber.inducedFunctor ⋯).IsEquivalence",
"usedConstants": [
"CategoryTheory.Functor",
"CategoryTheory.Functor.Fiber.fiberInclusion_comp_eq_const",
"CategoryTheory.Functo... | exact isEquivalence_of_iso (F := 𝟭 (Fiber p S)) (Iso.refl _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.FiberedCategory.HasFibers | {
"line": 86,
"column": 16
} | {
"line": 86,
"column": 76
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝¹ : Category.{v₁, u₁} 𝒮\ninst✝ : Category.{v₂, u₂} 𝒳\np : 𝒳 ⥤ 𝒮\nS : 𝒮\n⊢ (Fiber.inducedFunctor ⋯).IsEquivalence",
"usedConstants": [
"CategoryTheory.Functor",
"CategoryTheory.Functor.Fiber.fiberInclusion_comp_eq_const",
"CategoryTheory.Functo... | exact isEquivalence_of_iso (F := 𝟭 (Fiber p S)) (Iso.refl _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.FiberedCategory.HasFibers | {
"line": 86,
"column": 16
} | {
"line": 86,
"column": 76
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝¹ : Category.{v₁, u₁} 𝒮\ninst✝ : Category.{v₂, u₂} 𝒳\np : 𝒳 ⥤ 𝒮\nS : 𝒮\n⊢ (Fiber.inducedFunctor ⋯).IsEquivalence",
"usedConstants": [
"CategoryTheory.Functor",
"CategoryTheory.Functor.Fiber.fiberInclusion_comp_eq_const",
"CategoryTheory.Functo... | exact isEquivalence_of_iso (F := 𝟭 (Fiber p S)) (Iso.refl _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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