module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Integral.BoundedContinuousFunction | {
"line": 119,
"column": 27
} | {
"line": 121,
"column": 6
} | {
"line": 123,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝⁸ : MeasurableSpace X\ninst✝⁷ : TopologicalSpace X\nμ : Measure X\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : OpensMeasurableSpace X\ninst✝⁴ : SecondCountableTopology E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : IsProbabilityMeasu... | [] | by
convert! f.norm_integral_le_mul_norm μ
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Pow.Integral | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 51
} | {
"line": 115,
"column": 4
} | [
{
"pp": "E : Type u_2\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : FiniteDimensional ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : NormedAddCommGroup F\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nhd : 1 ≤ Module.finrank ℝ E\nf : E → F\nC α r : ℝ\nhα : α < ↑... | [
"E : Type u_2\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : FiniteDimensional ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : NormedAddCommGroup F\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nhd : 1 ≤ Module.finrank ℝ E\nf : E → F\nC α r : ℝ\nhα : α < ↑(Module.finr... | apply Module.nontrivial_of_finrank_pos (R := ℝ) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas | {
"line": 148,
"column": 2
} | {
"line": 148,
"column": 46
} | {
"line": 150,
"column": 0
} | [
{
"pp": "n : ℕ\nhn : n ≠ 0\na b : ℕ\n⊢ b ^ n ≤ a ^ n ↔ b ≤ a",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Nat.instMonoid",
"NPow.toPow",
"HPow.hPow",
"Nat.instPreorder",
"Nat.pow_left_strictMono",
"Nat",
"instHPow",
"StrictMono.le_iff_le"... | [] | exact (Nat.pow_left_strictMono hn).le_iff_le | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Constructions.Polish.EmbeddingReal | {
"line": 39,
"column": 4
} | {
"line": 39,
"column": 30
} | {
"line": 40,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : StandardBorelSpace α\nhα : Countable α\n⊢ ∃ s, MeasurableSet s ∧ Nonempty (α ≃ᵐ ↑s)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Real",
"MeasurableSet",
"Finite",
"finite_or_infinite",
"M... | [
"case pos.inl\nα : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : StandardBorelSpace α\nhα : Countable α\nh✝ : Finite α\n⊢ ∃ s, MeasurableSet s ∧ Nonempty (α ≃ᵐ ↑s)",
"case pos.inr\nα : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : StandardBorelSpace α\nhα : Countable α\nh✝ : Infinite α\n⊢ ∃ s, MeasurableSet s ∧ N... | cases finite_or_infinite α | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.Analysis.SpecialFunctions.Stirling | {
"line": 207,
"column": 2
} | {
"line": 208,
"column": 94
} | {
"line": 209,
"column": 2
} | [
{
"pp": "⊢ Tendsto (fun n ↦ ↑n / (2 * ↑n + 1)) atTop (𝓝 (2 + 0)⁻¹)",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"NormedCommRing.toSeminormedCommRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
"tendsto_const_div_atTop_nhds_zero_nat",
"Real.part... | [
"n : ℕ\nhn : 1 ≤ n\n⊢ (2 + 1 / ↑n)⁻¹ = (fun n ↦ ↑n / (2 * ↑n + 1)) n"
] | refine (((tendsto_const_div_atTop_nhds_zero_nat 1).const_add (2 : ℝ)).inv₀
((add_zero (2 : ℝ)).symm ▸ two_ne_zero)).congr' (eventually_atTop.mpr ⟨1, fun n hn => ?_⟩) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Basic | {
"line": 98,
"column": 96
} | {
"line": 99,
"column": 31
} | {
"line": 101,
"column": 0
} | [
{
"pp": "θ : ℂ\nn : ℤ\n⊢ eval (2 * cos θ) (S ℂ n) * sin θ = sin ((↑n + 1) * θ)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Int.cast",
"Polynomial.C",
"Polynomial.eval",
"GroupWithZero.toMonoidWithZero",
"False",
"GroupWithZero.toDivisionMonoid",
... | [] | by
simp [S_eq_U_comp_half_mul_X] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Basic | {
"line": 134,
"column": 26
} | {
"line": 135,
"column": 31
} | {
"line": 137,
"column": 0
} | [
{
"pp": "θ : ℂ\nn : ℤ\n⊢ eval (2 * cosh θ) (S ℂ n) * sinh θ = sinh ((↑n + 1) * θ)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Int.cast",
"Polynomial.C",
"Polynomial.eval",
"GroupWithZero.toMonoidWithZero",
"False",
"Complex.sinh",
"GroupWithZe... | [] | by
simp [S_eq_U_comp_half_mul_X] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 405,
"column": 4
} | {
"line": 411,
"column": 76
} | {
"line": 413,
"column": 0
} | [
{
"pp": "case refine_2\nL : PeriodPair\nl₀ : ℂ\ns : Finset ↥L.lattice\nl : ℂ\nhl : l ∈ (↑L.lattice \\ {l₀})ᶜ\n⊢ ∀ i ∈ s, DifferentiableAt ℂ (fun x2 ↦ if ↑i = l₀ then 0 else 1 / (x2 - ↑i) ^ 2 - 1 / ↑i ^ 2) l",
"ppTerm": "?refine_2✝",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormed... | [] | · intros x hxs
split_ifs with hl₁
· simp
have hl₁ : l - x ≠ 0 := fun e ↦ hl₁ (by
obtain rfl := sub_eq_zero.mp e
simpa using hl)
exact .sub (.div (by fun_prop) (by fun_prop) (by simpa)) (by fun_prop) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 648,
"column": 6
} | {
"line": 650,
"column": 57
} | {
"line": 652,
"column": 0
} | [
{
"pp": "case neg\nL : PeriodPair\nl₀ x : ℂ\ni : ℕ\nhl₀ : l₀ ∉ L.lattice\n⊢ (FormalMultilinearSeries.ofScalars ℂ fun i ↦\n if i = 0 then ℘[L - l₀] x else (↑i + 1) * (L.sumInvPow x (i + 2) - 0)).coeff\n (i + 1) =\n ∑' (l : ↥L.lattice),\n if ↑l = l₀ then 0 else (↑(i + 1) + 1) * (↑l - x) ^ (-... | [] | have h₁ (l : L.lattice) : l.1 ≠ l₀ := fun e ↦ hl₀ (e ▸ l.2)
simp [h₁, tsum_mul_left, sumInvPow, add_assoc,
one_add_one_eq_two, ← zpow_natCast, -neg_add_rev] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 648,
"column": 6
} | {
"line": 650,
"column": 57
} | {
"line": 652,
"column": 0
} | [
{
"pp": "case neg\nL : PeriodPair\nl₀ x : ℂ\ni : ℕ\nhl₀ : l₀ ∉ L.lattice\n⊢ (FormalMultilinearSeries.ofScalars ℂ fun i ↦\n if i = 0 then ℘[L - l₀] x else (↑i + 1) * (L.sumInvPow x (i + 2) - 0)).coeff\n (i + 1) =\n ∑' (l : ↥L.lattice),\n if ↑l = l₀ then 0 else (↑(i + 1) + 1) * (↑l - x) ^ (-... | [] | have h₁ (l : L.lattice) : l.1 ≠ l₀ := fun e ↦ hl₀ (e ▸ l.2)
simp [h₁, tsum_mul_left, sumInvPow, add_assoc,
one_add_one_eq_two, ← zpow_natCast, -neg_add_rev] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema | {
"line": 226,
"column": 4
} | {
"line": 229,
"column": 61
} | {
"line": 230,
"column": 2
} | [
{
"pp": "case refine_1\nn k : ℕ\nhn : n ≠ 0\nhk₀ : 0 < k\nhk₁ : k < n\nhk₂ : Even k\nzero_lt : 0 < ↑k * π / ↑n\nlt_pi : ↑k * π / ↑n < π\n⊢ ∀ y ∈ Set.Ioo (-1) 1, (fun x ↦ eval x (T ℝ ↑n)) y ≤ (fun x ↦ eval x (T ℝ ↑n)) (cos (↑k * π / ↑n))",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
... | [] | intro x hx
dsimp
rw [(eval_T_real_eq_one_iff hn _).mpr ⟨k, le_of_lt hk₁, hk₂, rfl⟩]
exact (abs_le.mp (abs_eval_T_real_le_one n (by grind))).2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema | {
"line": 226,
"column": 4
} | {
"line": 229,
"column": 61
} | {
"line": 230,
"column": 2
} | [
{
"pp": "case refine_1\nn k : ℕ\nhn : n ≠ 0\nhk₀ : 0 < k\nhk₁ : k < n\nhk₂ : Even k\nzero_lt : 0 < ↑k * π / ↑n\nlt_pi : ↑k * π / ↑n < π\n⊢ ∀ y ∈ Set.Ioo (-1) 1, (fun x ↦ eval x (T ℝ ↑n)) y ≤ (fun x ↦ eval x (T ℝ ↑n)) (cos (↑k * π / ↑n))",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
... | [] | intro x hx
dsimp
rw [(eval_T_real_eq_one_iff hn _).mpr ⟨k, le_of_lt hk₁, hk₂, rfl⟩]
exact (abs_le.mp (abs_eval_T_real_le_one n (by grind))).2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 785,
"column": 2
} | {
"line": 785,
"column": 69
} | {
"line": 786,
"column": 2
} | [
{
"pp": "L : PeriodPair\nl : ℂ\nn : ℕ\n⊢ iteratedDeriv n ℘[L - l] l / ↑n ! = (if n = 0 then ℘[L - l] l else ↑(n + 1)! * L.sumInvPow l (n + 2)) / ↑n !",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"instHDiv",
"HMul.hMul",
"GroupWi... | [
"L : PeriodPair\nl : ℂ\nn : ℕ\n⊢ iteratedDeriv n ℘[L - l] l / ↑n ! = if n = 0 then ℘[L - l] l else (↑n + 1) * L.sumInvPow l (n + 2)",
"L : PeriodPair\nl : ℂ\nn : ℕ\n⊢ (if n = 0 then ℘[L - l] l else (↑n + 1) * L.sumInvPow l (n + 2)) =\n (if n = 0 then ℘[L - l] l else ↑(n + 1)! * L.sumInvPow l (n + 2)) / ↑n !"
] | trans if n = 0 then ℘[L - l] l else (n + 1) * L.sumInvPow l (n + 2) | Batteries.Tactic._aux_Batteries_Tactic_Trans___elabRules_Batteries_Tactic_tacticTrans____1 | Batteries.Tactic.tacticTrans___ |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 862,
"column": 60
} | {
"line": 866,
"column": 14
} | {
"line": 868,
"column": 0
} | [
{
"pp": "L : PeriodPair\nl₀ : ℂ\nhl₀ : l₀ ∉ L.lattice\n⊢ L.weierstrassPExceptSeries l₀ = L.weierstrassPSeries",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Submodule",
"PeriodPair.weierstrassPExceptSeries._proof_1",... | [] | by
delta weierstrassPSeries weierstrassPExceptSeries
congr! with z i f
· rw [L.weierstrassPExcept_of_notMem _ hl₀]
· simp [hl₀] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 82,
"column": 6
} | {
"line": 82,
"column": 29
} | {
"line": 82,
"column": 30
} | [
{
"pp": "x : ℂ\nhx : x ∈ ℂ_ℤ\nn : ℕ\n⊢ ¬1 + -x ^ 2 / (↑n + 1) ^ 2 = 0",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"instHDiv",
"AddGroupWithOne.toAddGroup",
"congrArg",
"AddMonoid.t... | [
"x : ℂ\nhx : x ∈ ℂ_ℤ\nn : ℕ\n⊢ ¬1 = -(-x ^ 2 / (↑n + 1) ^ 2)"
] | add_eq_zero_iff_eq_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 928,
"column": 6
} | {
"line": 928,
"column": 42
} | {
"line": 929,
"column": 6
} | [
{
"pp": "case pos\nL : PeriodPair\nl₀ : ℂ\nh : l₀ ∈ L.lattice\nthis : AnalyticAt ℂ ℘[L - l₀] l₀\nhl₀ : l₀ = 0\n⊢ AnalyticAt ℂ (fun z ↦ (z - l₀) ^ 2 / l₀ ^ 2) l₀",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"InnerProductSpace.... | [
"case neg\nL : PeriodPair\nl₀ : ℂ\nh : l₀ ∈ L.lattice\nthis : AnalyticAt ℂ ℘[L - l₀] l₀\nhl₀ : ¬l₀ = 0\n⊢ AnalyticAt ℂ (fun z ↦ (z - l₀) ^ 2 / l₀ ^ 2) l₀"
] | · simpa [hl₀] using analyticAt_const | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Int.Fib.Basic | {
"line": 73,
"column": 4
} | {
"line": 73,
"column": 8
} | {
"line": 74,
"column": 4
} | [
{
"pp": "n : ℕ\nhn0 : ¬n = 0\n⊢ fib (-↑n + 1) = (-1) ^ (n + 1) * ↑(Nat.fib n) + (-1) ^ (1 + (n + 1)) * ↑(Nat.fib (n + 1))",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Int.instAddSemigroup",
"HMul.hMul",
"Int.fib",
"AddGroupWithOne.t... | [
"n : ℕ\nhn0 : ¬n = 0\n⊢ (-1) ^ (n + 1) * ↑(Nat.fib n) + (-1) ^ (1 + (n + 1)) * ↑(Nat.fib (n + 1)) = fib (-↑n + 1)"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 1007,
"column": 56
} | {
"line": 1007,
"column": 62
} | {
"line": 1007,
"column": 62
} | [
{
"pp": "L : PeriodPair\n⊢ Odd 3",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"Odd",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
"Nat.instSemiring",
"Eq.refl",
"OfNat.ofNat",
"Decidable.decide",... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 1007,
"column": 56
} | {
"line": 1007,
"column": 62
} | {
"line": 1007,
"column": 62
} | [
{
"pp": "L : PeriodPair\n⊢ Odd 3",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"Odd",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
"Nat.instSemiring",
"Eq.refl",
"OfNat.ofNat",
"Decidable.decide",... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 1007,
"column": 56
} | {
"line": 1007,
"column": 62
} | {
"line": 1007,
"column": 62
} | [
{
"pp": "L : PeriodPair\n⊢ Odd 3",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"Odd",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
"Nat.instSemiring",
"Eq.refl",
"OfNat.ofNat",
"Decidable.decide",... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Int.Fib.Basic | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 71
} | {
"line": 95,
"column": 0
} | [
{
"pp": "n : ℤ\n⊢ fib n = 0 ↔ n = 0",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"NegZeroClass.toNeg",
"False",
"Nat.instMulZeroClass",
"IsDomain.to_noZeroDivisors",
"Nat.fib_eq_zero._simp_1",
"HMul.hMul",
"Nat.instO... | [] | obtain ⟨n, (rfl | rfl)⟩ := n.eq_nat_or_neg <;> simp [fib_neg_natCast] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Data.Int.Fib.Basic | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 71
} | {
"line": 95,
"column": 0
} | [
{
"pp": "n : ℤ\n⊢ fib n = 0 ↔ n = 0",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"NegZeroClass.toNeg",
"False",
"Nat.instMulZeroClass",
"IsDomain.to_noZeroDivisors",
"Nat.fib_eq_zero._simp_1",
"HMul.hMul",
"Nat.instO... | [] | obtain ⟨n, (rfl | rfl)⟩ := n.eq_nat_or_neg <;> simp [fib_neg_natCast] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Int.Fib.Basic | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 71
} | {
"line": 95,
"column": 0
} | [
{
"pp": "n : ℤ\n⊢ fib n = 0 ↔ n = 0",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"NegZeroClass.toNeg",
"False",
"Nat.instMulZeroClass",
"IsDomain.to_noZeroDivisors",
"Nat.fib_eq_zero._simp_1",
"HMul.hMul",
"Nat.instO... | [] | obtain ⟨n, (rfl | rfl)⟩ := n.eq_nat_or_neg <;> simp [fib_neg_natCast] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 1023,
"column": 60
} | {
"line": 1023,
"column": 66
} | {
"line": 1023,
"column": 66
} | [
{
"pp": "L : PeriodPair\ni : ℕ\nhi₁ : i < 7\nhi₂ : Odd i\n⊢ Even 6",
"ppTerm": "?m.208",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"Nat.instDecidablePredEven",
"Bool.true",
"Nat",
"Even",
"Bool",
"instAddNat"... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 1023,
"column": 60
} | {
"line": 1023,
"column": 66
} | {
"line": 1023,
"column": 66
} | [
{
"pp": "L : PeriodPair\ni : ℕ\nhi₁ : i < 7\nhi₂ : Odd i\n⊢ Even 6",
"ppTerm": "?m.208",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"Nat.instDecidablePredEven",
"Bool.true",
"Nat",
"Even",
"Bool",
"instAddNat"... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 1023,
"column": 60
} | {
"line": 1023,
"column": 66
} | {
"line": 1023,
"column": 66
} | [
{
"pp": "L : PeriodPair\ni : ℕ\nhi₁ : i < 7\nhi₂ : Odd i\n⊢ Even 6",
"ppTerm": "?m.208",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"Nat.instDecidablePredEven",
"Bool.true",
"Nat",
"Even",
"Bool",
"instAddNat"... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 194,
"column": 2
} | {
"line": 194,
"column": 41
} | {
"line": 195,
"column": 2
} | [
{
"pp": "case hf\nx : ℂ\nhx : x ∈ ℂ_ℤ\nn : ℕ\n⊢ ∀ i ∈ Finset.range n, 1 + sineTerm x i ≠ 0",
"ppTerm": "?hf",
"assigned": true,
"usedConstants": [
"Finset",
"Membership.mem",
"sineTerm_ne_zero",
"Finset.range",
"Finset.instSetLike",
"Nat",
"SetLike.instMembe... | [
"case hd\nx : ℂ\nhx : x ∈ ℂ_ℤ\nn : ℕ\n⊢ ∀ i ∈ Finset.range n, DifferentiableAt ℂ (fun z ↦ 1 + sineTerm z i) x"
] | · exact fun i _ ↦ sineTerm_ne_zero hx i | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 315,
"column": 4
} | {
"line": 316,
"column": 29
} | {
"line": 317,
"column": 2
} | [
{
"pp": "case hbc.h₁\nk : ℕ\nK : Set ℂ\nA B : ℝ\nhB : 0 < B\nhKAB : K ⊆ UpperHalfPlane.coe '' verticalStrip A B\nn : ℕ\na : ℍ\nhaAB : a ∈ verticalStrip A B\nha : ↑a ∈ K\nh1 :\n ‖↑(![1, ↑n + 1] 0) * ↑a + ↑(![1, ↑n + 1] 1)‖ ^ (-(↑k + 1)) ≤\n r { coe := { re := A, im := B }, coe_im_pos := hB } ^ (-(↑k + 1)) * ... | [] | simpa (disch := positivity) [sub_eq_add_neg, ← Real.rpow_intCast, abs_norm_eq_max_natAbs,
abs_of_nonneg] using h1 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 315,
"column": 4
} | {
"line": 316,
"column": 29
} | {
"line": 317,
"column": 2
} | [
{
"pp": "case hbc.h₁\nk : ℕ\nK : Set ℂ\nA B : ℝ\nhB : 0 < B\nhKAB : K ⊆ UpperHalfPlane.coe '' verticalStrip A B\nn : ℕ\na : ℍ\nhaAB : a ∈ verticalStrip A B\nha : ↑a ∈ K\nh1 :\n ‖↑(![1, ↑n + 1] 0) * ↑a + ↑(![1, ↑n + 1] 1)‖ ^ (-(↑k + 1)) ≤\n r { coe := { re := A, im := B }, coe_im_pos := hB } ^ (-(↑k + 1)) * ... | [] | simpa (disch := positivity) [sub_eq_add_neg, ← Real.rpow_intCast, abs_norm_eq_max_natAbs,
abs_of_nonneg] using h1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 315,
"column": 4
} | {
"line": 316,
"column": 29
} | {
"line": 317,
"column": 2
} | [
{
"pp": "case hbc.h₁\nk : ℕ\nK : Set ℂ\nA B : ℝ\nhB : 0 < B\nhKAB : K ⊆ UpperHalfPlane.coe '' verticalStrip A B\nn : ℕ\na : ℍ\nhaAB : a ∈ verticalStrip A B\nha : ↑a ∈ K\nh1 :\n ‖↑(![1, ↑n + 1] 0) * ↑a + ↑(![1, ↑n + 1] 1)‖ ^ (-(↑k + 1)) ≤\n r { coe := { re := A, im := B }, coe_im_pos := hB } ^ (-(↑k + 1)) * ... | [] | simpa (disch := positivity) [sub_eq_add_neg, ← Real.rpow_intCast, abs_norm_eq_max_natAbs,
abs_of_nonneg] using h1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Subadditive | {
"line": 90,
"column": 4
} | {
"line": 91,
"column": 84
} | {
"line": 92,
"column": 2
} | [
{
"pp": "case refine_1\nu : ℕ → ℝ\nh : Subadditive u\nhbdd : BddBelow (range fun n ↦ u n / ↑n)\nl : ℝ\nhl : l < h.lim\n⊢ ∀ᶠ (b : ℕ) in atTop, l < u b / ↑b",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Nat.instMulZeroClass",
"Real",
"Preorder.toLT",... | [] | refine eventually_atTop.2
⟨1, fun n hn => hl.trans_le (h.lim_le_div hbdd (zero_lt_one.trans_le hn).ne')⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Subadditive | {
"line": 90,
"column": 4
} | {
"line": 91,
"column": 84
} | {
"line": 92,
"column": 2
} | [
{
"pp": "case refine_1\nu : ℕ → ℝ\nh : Subadditive u\nhbdd : BddBelow (range fun n ↦ u n / ↑n)\nl : ℝ\nhl : l < h.lim\n⊢ ∀ᶠ (b : ℕ) in atTop, l < u b / ↑b",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Nat.instMulZeroClass",
"Real",
"Preorder.toLT",... | [] | refine eventually_atTop.2
⟨1, fun n hn => hl.trans_le (h.lim_le_div hbdd (zero_lt_one.trans_le hn).ne')⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Subadditive | {
"line": 90,
"column": 4
} | {
"line": 91,
"column": 84
} | {
"line": 92,
"column": 2
} | [
{
"pp": "case refine_1\nu : ℕ → ℝ\nh : Subadditive u\nhbdd : BddBelow (range fun n ↦ u n / ↑n)\nl : ℝ\nhl : l < h.lim\n⊢ ∀ᶠ (b : ℕ) in atTop, l < u b / ↑b",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Nat.instMulZeroClass",
"Real",
"Preorder.toLT",... | [] | refine eventually_atTop.2
⟨1, fun n hn => hl.trans_le (h.lim_le_div hbdd (zero_lt_one.trans_le hn).ne')⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Preadditive.Yoneda.Projective | {
"line": 50,
"column": 4
} | {
"line": 51,
"column": 87
} | {
"line": 53,
"column": 0
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nP : C\n⊢ (preadditiveCoyonedaObj P).PreservesEpimorphisms → (coyoneda.obj (op P)).PreservesEpimorphisms",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"ModuleCat.forget_preservesEpimorphisms",... | [] | intro
exact (inferInstance : (preadditiveCoyonedaObj P ⋙ forget _).PreservesEpimorphisms) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Preadditive.Yoneda.Projective | {
"line": 50,
"column": 4
} | {
"line": 51,
"column": 87
} | {
"line": 53,
"column": 0
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nP : C\n⊢ (preadditiveCoyonedaObj P).PreservesEpimorphisms → (coyoneda.obj (op P)).PreservesEpimorphisms",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"ModuleCat.forget_preservesEpimorphisms",... | [] | intro
exact (inferInstance : (preadditiveCoyonedaObj P ⋙ forget _).PreservesEpimorphisms) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Generator.Preadditive | {
"line": 33,
"column": 33
} | {
"line": 33,
"column": 71
} | {
"line": 33,
"column": 71
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nP : ObjectProperty C\nh𝒢 : P.IsSeparating\nX Y : C\nf : X ⟶ Y\nhf : ∀ (G : C), P G → ∀ (h : G ⟶ X), h ≫ f = 0\n⊢ ∀ (G : C), P G → ∀ (h : G ⟶ X), h ≫ f = h ≫ 0",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.... | [] | simpa only [Limits.comp_zero] using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Generator.Preadditive | {
"line": 33,
"column": 33
} | {
"line": 33,
"column": 71
} | {
"line": 33,
"column": 71
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nP : ObjectProperty C\nh𝒢 : P.IsSeparating\nX Y : C\nf : X ⟶ Y\nhf : ∀ (G : C), P G → ∀ (h : G ⟶ X), h ≫ f = 0\n⊢ ∀ (G : C), P G → ∀ (h : G ⟶ X), h ≫ f = h ≫ 0",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.... | [] | simpa only [Limits.comp_zero] using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Generator.Preadditive | {
"line": 33,
"column": 33
} | {
"line": 33,
"column": 71
} | {
"line": 33,
"column": 71
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nP : ObjectProperty C\nh𝒢 : P.IsSeparating\nX Y : C\nf : X ⟶ Y\nhf : ∀ (G : C), P G → ∀ (h : G ⟶ X), h ≫ f = 0\n⊢ ∀ (G : C), P G → ∀ (h : G ⟶ X), h ≫ f = h ≫ 0",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.... | [] | simpa only [Limits.comp_zero] using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Generator.Preadditive | {
"line": 44,
"column": 37
} | {
"line": 44,
"column": 75
} | {
"line": 44,
"column": 75
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nG : C\nhG : IsSeparator G\nX Y : C\nf : X ⟶ Y\nhf : ∀ (h : G ⟶ X), h ≫ f = 0\n⊢ ∀ (h : G ⟶ X), h ≫ f = h ≫ 0",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver... | [] | simpa only [Limits.comp_zero] using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Generator.Preadditive | {
"line": 44,
"column": 37
} | {
"line": 44,
"column": 75
} | {
"line": 44,
"column": 75
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nG : C\nhG : IsSeparator G\nX Y : C\nf : X ⟶ Y\nhf : ∀ (h : G ⟶ X), h ≫ f = 0\n⊢ ∀ (h : G ⟶ X), h ≫ f = h ≫ 0",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver... | [] | simpa only [Limits.comp_zero] using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Generator.Preadditive | {
"line": 44,
"column": 37
} | {
"line": 44,
"column": 75
} | {
"line": 44,
"column": 75
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nG : C\nhG : IsSeparator G\nX Y : C\nf : X ⟶ Y\nhf : ∀ (h : G ⟶ X), h ≫ f = 0\n⊢ ∀ (h : G ⟶ X), h ≫ f = h ≫ 0",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver... | [] | simpa only [Limits.comp_zero] using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp | {
"line": 102,
"column": 2
} | {
"line": 102,
"column": 14
} | {
"line": 103,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ φ f g ≫ π f g = 0",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ biprod.desc (f ≫ biprod.inl) (biprod.lift (-𝟙 Y) g) ≫ biprod.desc (g ≫ cokernel.π (f ≫ g)) (cokernel.π (f ≫ g)) = 0"
] | dsimp [φ, π] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Generator.Abelian | {
"line": 46,
"column": 2
} | {
"line": 46,
"column": 52
} | {
"line": 47,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasLimits C\ninst✝ : EnoughInjectives C\nG : C\nhG : IsSeparator G\nthis✝ : WellPowered.{v, v, u} C\nthis : HasProductsOfShape (Subobject (op G)) C\nT : C := Injective.under (∏ᶜ fun P ↦ unop (Subobject.underlying.obj P))\nX Y : C\nf :... | [
"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasLimits C\ninst✝ : EnoughInjectives C\nG : C\nhG : IsSeparator G\nthis✝ : WellPowered.{v, v, u} C\nthis : HasProductsOfShape (Subobject (op G)) C\nT : C := Injective.under (∏ᶜ fun P ↦ unop (Subobject.underlying.obj P))\nX Y : C\nf : X ⟶ Y\nhf :... | let R := Subobject.mk (factorThruImage (h ≫ f)).op | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.CategoryTheory.Limits.Preserves.Yoneda | {
"line": 82,
"column": 2
} | {
"line": 83,
"column": 38
} | {
"line": 84,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nJ : Type u₂\ninst✝² : Category.{v₂, u₂} J\ninst✝¹ : HasColimitsOfShape J (Type v₁)\ninst✝ : HasColimitsOfShape J (Type (max u₁ v₁))\nF : J ⥤ Cᵒᵖ ⥤ Type v₁\nX : C\n⊢ PreservesColimit F (coyoneda.obj (op (yoneda.obj X)))",
"ppTerm": "?m.28",
"assigned": ... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nJ : Type u₂\ninst✝² : Category.{v₂, u₂} J\ninst✝¹ : HasColimitsOfShape J (Type v₁)\ninst✝ : HasColimitsOfShape J (Type (max u₁ v₁))\nF : J ⥤ Cᵒᵖ ⥤ Type v₁\nX : C\n⊢ IsIso (colimit.post F (coyoneda.obj (op (yoneda.obj X))))"
] | suffices IsIso (colimit.post F (coyoneda.obj (op (yoneda.obj X)))) from
preservesColimit_of_isIso_post _ _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.CategoryTheory.Abelian.Injective.Dimension | {
"line": 286,
"column": 76
} | {
"line": 286,
"column": 82
} | {
"line": 286,
"column": 82
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nh : HasInjectiveDimensionLT X 0\n⊢ ⊥ < ↑0",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"WithBot",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"instCompleteLinearOrderENat",
"of_deci... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.CategoryTheory.Abelian.Injective.Dimension | {
"line": 286,
"column": 76
} | {
"line": 286,
"column": 82
} | {
"line": 286,
"column": 82
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nh : HasInjectiveDimensionLT X 0\n⊢ ⊥ < ↑0",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"WithBot",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"instCompleteLinearOrderENat",
"of_deci... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.Injective.Dimension | {
"line": 286,
"column": 76
} | {
"line": 286,
"column": 82
} | {
"line": 286,
"column": 82
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nh : HasInjectiveDimensionLT X 0\n⊢ ⊥ < ↑0",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"WithBot",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"instCompleteLinearOrderENat",
"of_deci... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.Injective.Dimension | {
"line": 308,
"column": 4
} | {
"line": 308,
"column": 62
} | {
"line": 309,
"column": 4
} | [
{
"pp": "case bot\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nhd : injectiveDimension X = ⊥\n⊢ ⊥ ≠ ⊤ ↔ ∃ n, HasInjectiveDimensionLE X n",
"ppTerm": "?bot",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"WithBot.instBoundedOrder",
"False",
"WithBot",
... | [
"case bot\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nhd : injectiveDimension X = ⊥\n⊢ ∃ n, HasInjectiveDimensionLE X n"
] | simp only [ne_eq, bot_ne_top, not_false_eq_true, true_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Abelian.Projective.Extend | {
"line": 62,
"column": 4
} | {
"line": 62,
"column": 76
} | {
"line": 63,
"column": 2
} | [
{
"pp": "case pos\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroObject C\ninst✝ : Preadditive C\nX : C\nR : ProjectiveResolution X\nk : ℕ\nhn : -↑k ≤ 0\n⊢ Projective (R.cochainComplex.X (-↑k))",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Nat.instOne",
"AddGroupWi... | [] | exact Projective.of_iso (R.cochainComplexXIso (-k) k).symm inferInstance | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Abelian.Injective.Ext | {
"line": 254,
"column": 24
} | {
"line": 254,
"column": 48
} | {
"line": 254,
"column": 49
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX Y : C\nR : InjectiveResolution Y\nn : ℕ\nf : X ⟶ R.cocomplex.X n\nm : ℕ\nhm : n + 1 = m\nhf : f ≫ R.cocomplex.d n m = 0\nY' : C\nR' : InjectiveResolution Y'\ng : Y ⟶ Y'\nφ : R.Hom R' g\nthis✝ : HasDerivedCategory C\nthis : ... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX Y : C\nR : InjectiveResolution Y\nn : ℕ\nf : X ⟶ R.cocomplex.X n\nm : ℕ\nhm : n + 1 = m\nhf : f ≫ R.cocomplex.d n m = 0\nY' : C\nR' : InjectiveResolution Y'\ng : Y ⟶ Y'\nφ : R.Hom R' g\nthis✝ : HasDerivedCategory C\nthis : (f ≫ φ.hom.f... | ShiftedHom.mk₀_comp_mk₀, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.Projective.Ext | {
"line": 243,
"column": 4
} | {
"line": 243,
"column": 28
} | {
"line": 243,
"column": 29
} | [
{
"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\nY' : C\ng : Y ⟶ Y'\nthis : HasDerivedCategory C\n⊢ (ShiftedHom.mk₀ 0 ⋯ ((DerivedCategory.singleFunctorIso... | [
"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\nY' : C\ng : Y ⟶ Y'\nthis : HasDerivedCategory C\n⊢ (ShiftedHom.mk₀ 0 ⋯ ((DerivedCategory.singleFunctorIsoCompQ C 0).h... | ShiftedHom.mk₀_comp_mk₀, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.Projective.Ext | {
"line": 243,
"column": 29
} | {
"line": 243,
"column": 53
} | {
"line": 243,
"column": 54
} | [
{
"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\nY' : C\ng : Y ⟶ Y'\nthis : HasDerivedCategory C\n⊢ (ShiftedHom.mk₀ 0 ⋯ ((DerivedCategory.singleFunctorIso... | [
"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\nY' : C\ng : Y ⟶ Y'\nthis : HasDerivedCategory C\n⊢ (ShiftedHom.mk₀ 0 ⋯ ((DerivedCategory.singleFunctorIsoCompQ C 0).h... | ShiftedHom.mk₀_comp_mk₀, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.Projective.Ext | {
"line": 267,
"column": 24
} | {
"line": 267,
"column": 48
} | {
"line": 267,
"column": 49
} | [
{
"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\nthis✝ : HasDerivedCategory C\nthis : (R... | [
"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\nthis✝ : HasDerivedCategory C\nthis : (R'.cochainCom... | ShiftedHom.mk₀_comp_mk₀, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monoidal.Rigid.FunctorCategory | {
"line": 34,
"column": 32
} | {
"line": 34,
"column": 60
} | {
"line": 34,
"column": 61
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝³ : Groupoid C\ninst✝² : Category.{v_1, u_2} D\ninst✝¹ : MonoidalCategory D\ninst✝ : RightRigidCategory D\nF : C ⥤ D\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\n⊢ F.map (inv (f ≫ g))ᘁ = F.map (inv f)ᘁ ≫ F.map (inv g)ᘁ",
"ppTerm": "?m.95",
"assigned": true,
"use... | [] | simp [comp_rightAdjointMate] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Monoidal.Rigid.FunctorCategory | {
"line": 34,
"column": 32
} | {
"line": 34,
"column": 60
} | {
"line": 34,
"column": 61
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝³ : Groupoid C\ninst✝² : Category.{v_1, u_2} D\ninst✝¹ : MonoidalCategory D\ninst✝ : RightRigidCategory D\nF : C ⥤ D\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\n⊢ F.map (inv (f ≫ g))ᘁ = F.map (inv f)ᘁ ≫ F.map (inv g)ᘁ",
"ppTerm": "?m.95",
"assigned": true,
"use... | [] | simp [comp_rightAdjointMate] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Rigid.FunctorCategory | {
"line": 34,
"column": 32
} | {
"line": 34,
"column": 60
} | {
"line": 34,
"column": 61
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝³ : Groupoid C\ninst✝² : Category.{v_1, u_2} D\ninst✝¹ : MonoidalCategory D\ninst✝ : RightRigidCategory D\nF : C ⥤ D\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\n⊢ F.map (inv (f ≫ g))ᘁ = F.map (inv f)ᘁ ≫ F.map (inv g)ᘁ",
"ppTerm": "?m.95",
"assigned": true,
"use... | [] | simp [comp_rightAdjointMate] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Action.Limits | {
"line": 188,
"column": 2
} | {
"line": 194,
"column": 34
} | {
"line": 196,
"column": 0
} | [
{
"pp": "V : Type u_1\ninst✝² : Category.{v_1, u_1} V\nG : Type u_2\ninst✝¹ : Monoid G\ninst✝ : HasFiniteLimits V\n⊢ PreservesFiniteLimits (forget V G)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor",
"CategoryTheory.Limits.PreservesFiniteLimits",
... | [] | change PreservesFiniteLimits ((Action.functorCategoryEquivalence V G).functor ⋙
(evaluation (SingleObj G) V).obj (SingleObj.star G))
have : PreservesFiniteLimits ((evaluation (SingleObj G) V).obj (SingleObj.star G)) := by
constructor
intro _ _ _
infer_instance
apply comp_preservesFiniteLimits | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Action.Limits | {
"line": 188,
"column": 2
} | {
"line": 194,
"column": 34
} | {
"line": 196,
"column": 0
} | [
{
"pp": "V : Type u_1\ninst✝² : Category.{v_1, u_1} V\nG : Type u_2\ninst✝¹ : Monoid G\ninst✝ : HasFiniteLimits V\n⊢ PreservesFiniteLimits (forget V G)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor",
"CategoryTheory.Limits.PreservesFiniteLimits",
... | [] | change PreservesFiniteLimits ((Action.functorCategoryEquivalence V G).functor ⋙
(evaluation (SingleObj G) V).obj (SingleObj.star G))
have : PreservesFiniteLimits ((evaluation (SingleObj G) V).obj (SingleObj.star G)) := by
constructor
intro _ _ _
infer_instance
apply comp_preservesFiniteLimits | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Adjunction.FullyFaithfulLimits | {
"line": 60,
"column": 2
} | {
"line": 60,
"column": 39
} | {
"line": 61,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nE : Type u₃\ninst✝³ : Category.{v₃, u₃} E\nH : D ⥤ E\ninst✝² : HasColimitsOfSize.{v, u, v₁, u₁} C\ninst✝¹ : G.Full\ninst✝ : G.Faithful\nthis : F.IsLeftAdjoint\nx✝¹ : PreservesColimit... | [
"C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nE : Type u₃\ninst✝³ : Category.{v₃, u₃} E\nH : D ⥤ E\ninst✝² : HasColimitsOfSize.{v, u, v₁, u₁} C\ninst✝¹ : G.Full\ninst✝ : G.Faithful\nthis : F.IsLeftAdjoint\nx✝¹ : PreservesColimitsOfSize.{v, ... | rw [adj.preservesColimitsOfShape_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Monad.Coequalizer | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 16
} | {
"line": 104,
"column": 6
} | [
{
"pp": "case refine_3\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⊢ ∀ {m : (beckAlgebraCofork X).pt ⟶ s.pt},\n (beckA... | [
"case refine_3\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\nm : (beckAlgebraCofork X).pt ⟶ s.pt\nhm : (beckAlgebraCofork X).π ... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Adjunction.Lifting.Left | {
"line": 93,
"column": 6
} | {
"line": 93,
"column": 16
} | {
"line": 94,
"column": 6
} | [
{
"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... | [
"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 (adj₁.counit.app... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Adjunction.Lifting.Right | {
"line": 96,
"column": 6
} | {
"line": 96,
"column": 16
} | {
"line": 97,
"column": 6
} | [
{
"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 : A ⥤ B\nF : B ⥤ A\nL : C ⥤ B\nU' : A ⥤ C\nadj₁ : F ⊣ U\nadj₂ : L ⋙ F ⊣ U'\nh : (X : B) → RegularMono (adj₁.unit.app X)\nX : B\ns : Fork (U.map (F.map (adj₁.u... | [
"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 : A ⥤ B\nF : B ⥤ A\nL : C ⥤ B\nU' : A ⥤ C\nadj₁ : F ⊣ U\nadj₂ : L ⋙ F ⊣ U'\nh : (X : B) → RegularMono (adj₁.unit.app X)\nX : B\ns : Fork (U.map (F.map (adj₁.unit.app X)))... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Bicategory.Free | {
"line": 41,
"column": 2
} | {
"line": 42,
"column": 12
} | {
"line": 44,
"column": 0
} | [
{
"pp": "B : Type u\n⊢ [Inhabited B] → Inhabited (FreeBicategory B)",
"ppTerm": "?m.1",
"assigned": true,
"usedConstants": [
"id",
"CategoryTheory.FreeBicategory",
"Inhabited"
],
"usedFVars": [
"B"
],
"usedGoals": []
}
] | [] | intro h
exact id h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Bicategory.Free | {
"line": 41,
"column": 2
} | {
"line": 42,
"column": 12
} | {
"line": 44,
"column": 0
} | [
{
"pp": "B : Type u\n⊢ [Inhabited B] → Inhabited (FreeBicategory B)",
"ppTerm": "?m.1",
"assigned": true,
"usedConstants": [
"id",
"CategoryTheory.FreeBicategory",
"Inhabited"
],
"usedFVars": [
"B"
],
"usedGoals": []
}
] | [] | intro h
exact id h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Lax | {
"line": 159,
"column": 6
} | {
"line": 160,
"column": 16
} | {
"line": 161,
"column": 4
} | [
{
"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.app a ◁ H.mapComp f g ≫ θ.naturality (f ≫ g)) ⊗≫\n η.naturality (f ≫ g) ▷ θ... | [] | rw [naturality_comp θ]
bicategory | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Lax | {
"line": 159,
"column": 6
} | {
"line": 160,
"column": 16
} | {
"line": 161,
"column": 4
} | [
{
"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.app a ◁ H.mapComp f g ≫ θ.naturality (f ≫ g)) ⊗≫\n η.naturality (f ≫ g) ▷ θ... | [] | rw [naturality_comp θ]
bicategory | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Bicategory.Kan.Adjunction | {
"line": 75,
"column": 15
} | {
"line": 76,
"column": 57
} | {
"line": 78,
"column": 0
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b c : B\nf : a ⟶ b\nu : b ⟶ a\nadj : f ⊣ u\nx : B\nh : a ⟶ x\ns : LeftExtension f (𝟙 a ≫ h)\nτ₀ : (LeftExtension.mk u adj.unit).whisker h ⟶ s\nτ : u ≫ h ⟶ s.extension := StructuredArrow.Hom.right τ₀\nhτ : adj.unit ▷ h ⊗≫ f ◁ τ = s.unit\n⊢ 𝟙 (u ≫ h) ⊗≫ u ◁ (adj.unit... | [] | by
rw [hτ]; dsimp only [StructuredArrow.homMk_right] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Bicategory.LocallyGroupoid | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 27
} | {
"line": 107,
"column": 0
} | [
{
"pp": "B : Type u₁\ninst✝ : Bicategory B\na✝ b✝ c✝ : Pith B\nf✝ g✝ : a✝ ⟶ b✝\nh✝ i✝ : b✝ ⟶ c✝\nη : f✝ ⟶ g✝\nθ : h✝ ⟶ i✝\n⊢ ({ iso := f✝.of ◁ᵢ θ.iso } ≫ { iso := η.iso ▷ᵢ i✝.of }).iso.hom =\n ({ iso := η.iso ▷ᵢ h✝.of } ≫ { iso := g✝.of ◁ᵢ θ.iso }).iso.hom",
"ppTerm": "?m.130",
"assigned": true,
... | [] | simp [whisker_exchange] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Bicategory.Kan.Adjunction | {
"line": 238,
"column": 14
} | {
"line": 238,
"column": 29
} | {
"line": 238,
"column": 29
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b c : B\nf : a ⟶ b\ng : a ⟶ c\nt : LeftExtension f g\nH : t.IsKan\nx : B\nh : c ⟶ x\nu : x ⟶ c\nadj : h ⊣ u\nη' : 𝟙 c ⟶ h ≫ u := adj.unit\nH' : (LeftLift.mk h η').IsAbsKan := fun {x_1} ↦ adj.isAbsoluteLeftKanLift\ns : LeftExtension f (g ≫ h)\nk : b ⟶ x := s.extensio... | [
"B : Type u\ninst✝ : Bicategory B\na b c : B\nf : a ⟶ b\ng : a ⟶ c\nt : LeftExtension f g\nH : t.IsKan\nx : B\nh : c ⟶ x\nu : x ⟶ c\nadj : h ⊣ u\nη' : 𝟙 c ⟶ h ≫ u := adj.unit\nH' : (LeftLift.mk h η').IsAbsKan := fun {x_1} ↦ adj.isAbsoluteLeftKanLift\ns : LeftExtension f (g ≫ h)\nk : b ⟶ x := s.extension\nθ : g ≫ h... | IsKan.fac_assoc | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Category.PartialFun | {
"line": 114,
"column": 8
} | {
"line": 114,
"column": 33
} | {
"line": 115,
"column": 8
} | [
{
"pp": "X : PartialFun\no : { X := Option X, point := none }.X\na : X\n⊢ { toFun := Option.elim' none fun a ↦ (𝟙 X a).toOption, map_point := ⋯ }.toFun (some a) =\n (𝟙 { X := Option X, point := none }).toFun (some a)",
"ppTerm": "?m.71",
"assigned": true,
"usedConstants": [
"Option.elim'"... | [
"X : PartialFun\no : { X := Option X, point := none }.X\na : X\n⊢ (Part.some a).toOption = some a"
] | dsimp [CategoryStruct.id] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Category.Cat.Limit | {
"line": 75,
"column": 4
} | {
"line": 76,
"column": 26
} | {
"line": 77,
"column": 2
} | [
{
"pp": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ Cat\nX✝ Y✝ : limit (F ⋙ objects)\nx✝ : limit (homDiagram X✝ Y✝)\n⊢ Types.Limit.mk (homDiagram X✝ Y✝)\n (fun j ↦\n (hom (limit.π (homDiagram X✝ X✝) j))\n (Types.Limit.mk (homDiagram X✝ X✝) (fun x ↦ 𝟙 ((hom (limit.π (F ⋙ objects) x)) X✝... | [] | apply Types.limit_ext.{v, v}
simp [-homDiagram_obj] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Category.Cat.Limit | {
"line": 75,
"column": 4
} | {
"line": 76,
"column": 26
} | {
"line": 77,
"column": 2
} | [
{
"pp": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ Cat\nX✝ Y✝ : limit (F ⋙ objects)\nx✝ : limit (homDiagram X✝ Y✝)\n⊢ Types.Limit.mk (homDiagram X✝ Y✝)\n (fun j ↦\n (hom (limit.π (homDiagram X✝ X✝) j))\n (Types.Limit.mk (homDiagram X✝ X✝) (fun x ↦ 𝟙 ((hom (limit.π (F ⋙ objects) x)) X✝... | [] | apply Types.limit_ext.{v, v}
simp [-homDiagram_obj] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Category.Cat.Limit | {
"line": 78,
"column": 4
} | {
"line": 79,
"column": 26
} | {
"line": 80,
"column": 2
} | [
{
"pp": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ Cat\nX✝ Y✝ : limit (F ⋙ objects)\nx✝ : limit (homDiagram X✝ Y✝)\n⊢ Types.Limit.mk (homDiagram X✝ Y✝)\n (fun j ↦\n (hom (limit.π (homDiagram X✝ Y✝) j)) x✝ ≫\n (hom (limit.π (homDiagram Y✝ Y✝) j))\n (Types.Limit.mk (homDiagram Y... | [] | apply Types.limit_ext.{v, v}
simp [-homDiagram_obj] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Category.Cat.Limit | {
"line": 78,
"column": 4
} | {
"line": 79,
"column": 26
} | {
"line": 80,
"column": 2
} | [
{
"pp": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ Cat\nX✝ Y✝ : limit (F ⋙ objects)\nx✝ : limit (homDiagram X✝ Y✝)\n⊢ Types.Limit.mk (homDiagram X✝ Y✝)\n (fun j ↦\n (hom (limit.π (homDiagram X✝ Y✝) j)) x✝ ≫\n (hom (limit.π (homDiagram Y✝ Y✝) j))\n (Types.Limit.mk (homDiagram Y... | [] | apply Types.limit_ext.{v, v}
simp [-homDiagram_obj] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Category.Cat.Limit | {
"line": 81,
"column": 4
} | {
"line": 82,
"column": 26
} | {
"line": 84,
"column": 0
} | [
{
"pp": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ Cat\nW✝ X✝ Y✝ Z✝ : limit (F ⋙ objects)\nx✝² : limit (homDiagram W✝ X✝)\nx✝¹ : limit (homDiagram X✝ Y✝)\nx✝ : limit (homDiagram Y✝ Z✝)\n⊢ Types.Limit.mk (homDiagram W✝ Z✝)\n (fun j ↦\n (hom (limit.π (homDiagram W✝ Y✝) j))\n (Types.Limit... | [] | apply Types.limit_ext.{v, v}
simp [-homDiagram_obj] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Category.Cat.Limit | {
"line": 81,
"column": 4
} | {
"line": 82,
"column": 26
} | {
"line": 84,
"column": 0
} | [
{
"pp": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ Cat\nW✝ X✝ Y✝ Z✝ : limit (F ⋙ objects)\nx✝² : limit (homDiagram W✝ X✝)\nx✝¹ : limit (homDiagram X✝ Y✝)\nx✝ : limit (homDiagram Y✝ Z✝)\n⊢ Types.Limit.mk (homDiagram W✝ Z✝)\n (fun j ↦\n (hom (limit.π (homDiagram W✝ Y✝) j))\n (Types.Limit... | [] | apply Types.limit_ext.{v, v}
simp [-homDiagram_obj] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Category.TwoP | {
"line": 113,
"column": 18
} | {
"line": 115,
"column": 44
} | {
"line": 117,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nX✝ Y✝ Z✝ : Pointed\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\n⊢ { hom := { toFun := Option.map (f ≫ g).toFun, map_fst := ⋯, map_snd := ⋯ } } =\n { hom := { toFun := Option.map f.toFun, map_fst := ⋯, map_snd := ⋯ } } ≫\n { hom := { toFun := Option.map g.toFun, map_fst := ⋯, map_snd :... | [] | by
ext : 3
exact (Option.map_comp_map f.1 g.1).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Category.TwoP | {
"line": 122,
"column": 18
} | {
"line": 124,
"column": 44
} | {
"line": 126,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nX✝ Y✝ Z✝ : Pointed\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\n⊢ { hom := { toFun := Option.map (f ≫ g).toFun, map_fst := ⋯, map_snd := ⋯ } } =\n { hom := { toFun := Option.map f.toFun, map_fst := ⋯, map_snd := ⋯ } } ≫\n { hom := { toFun := Option.map g.toFun, map_fst := ⋯, map_snd :... | [] | by
ext : 3
exact (Option.map_comp_map f.1 g.1).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Category.Cat.Limit | {
"line": 145,
"column": 4
} | {
"line": 145,
"column": 8
} | {
"line": 146,
"column": 4
} | [
{
"pp": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ Cat\ns : Cone F\nm : s.pt ⟶ (limitCone F).pt\nw : ∀ (j : J), m ≫ (limitCone F).π.app j = s.π.app j\n⊢ m = limitConeLift F s",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"CategoryTheory.Cat.category",
"CategoryTheory.Cat.H... | [
"J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ Cat\ns : Cone F\nm : s.pt ⟶ (limitCone F).pt\nw : ∀ (j : J), m ≫ (limitCone F).π.app j = s.π.app j\n⊢ limitConeLift F s = m"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Comma.StructuredArrow.Final | {
"line": 61,
"column": 2
} | {
"line": 61,
"column": 26
} | {
"line": 62,
"column": 2
} | [
{
"pp": "A : Type u₁\ninst✝⁴ : SmallCategory A\nB : Type u₁\ninst✝³ : SmallCategory B\nT : Type u₁\ninst✝² : SmallCategory T\nL : A ⥤ T\nR : B ⥤ T\ninst✝¹ : R.Final\ninst✝ : ∀ (b : B), (CostructuredArrow.toOver L (R.obj b)).Final\nG : T ⥤ Type u₁\nthis : ∀ (b : B), ((R.whiskerLeft (preFunctor L (𝟭 T))).app b).... | [
"A : Type u₁\ninst✝⁴ : SmallCategory A\nB : Type u₁\ninst✝³ : SmallCategory B\nT : Type u₁\ninst✝² : SmallCategory T\nL : A ⥤ T\nR : B ⥤ T\ninst✝¹ : R.Final\ninst✝ : ∀ (b : B), (CostructuredArrow.toOver L (R.obj b)).Final\nG : T ⥤ Type u₁\nthis : ∀ (b : B), ((R.whiskerLeft (preFunctor L (𝟭 T))).app b).toFunctor.Fi... | convert! Iso.isIso_hom i | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.CategoryTheory.Localization.Monoidal.Basic | {
"line": 335,
"column": 28
} | {
"line": 375,
"column": 19
} | {
"line": 377,
"column": 0
} | [
{
"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\nY₁ Y₂ Y₃ Y₄ : LocalizedMonoidal L W ε\n⊢ Pentagon Y₁ Y₂ Y₃ Y... | [] | by
obtain ⟨X₁, ⟨e₁⟩⟩ : ∃ X₁, Nonempty ((L').obj X₁ ≅ Y₁) := ⟨_, ⟨(L').objObjPreimageIso Y₁⟩⟩
obtain ⟨X₂, ⟨e₂⟩⟩ : ∃ X₂, Nonempty ((L').obj X₂ ≅ Y₂) := ⟨_, ⟨(L').objObjPreimageIso Y₂⟩⟩
obtain ⟨X₃, ⟨e₃⟩⟩ : ∃ X₃, Nonempty ((L').obj X₃ ≅ Y₃) := ⟨_, ⟨(L').objObjPreimageIso Y₃⟩⟩
obtain ⟨X₄, ⟨e₄⟩⟩ : ∃ X₄, Nonempty ((L'... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Localization.Monoidal.Basic | {
"line": 409,
"column": 2
} | {
"line": 410,
"column": 28
} | {
"line": 411,
"column": 2
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\nL : C ⥤ D\nW : MorphismProperty C\ninst✝² : MonoidalCategory C\ninst✝¹ : W.IsMonoidal\ninst✝ : L.IsLocalization W\nunit : D\nε : L.obj (𝟙_ C) ≅ unit\nX Y : LocalizedMonoidal L W ε\nX' Y' : C\ne₁ : L'.obj X' ≅ X... | [
"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' Y' : C\ne₁ : L'.obj X' ≅ X\ne₂ : L'.ob... | simp only [← tensorHom_id, ← id_tensorHom, ← tensor_comp, assoc, comp_id,
id_comp, Iso.inv_hom_id] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Limits.Shapes.End | {
"line": 319,
"column": 4
} | {
"line": 319,
"column": 26
} | {
"line": 319,
"column": 26
} | [
{
"pp": "J : Type u\ninst✝³ : Category.{v, u} J\nC : Type u'\ninst✝² : Category.{v', u'} C\nF : Jᵒᵖ ⥤ J ⥤ C\ninst✝¹ : HasCoend F\nX : C\nf✝ : (j : J) → (F.obj (op j)).obj j ⟶ X\nhf : ∀ ⦃i j : J⦄ (g : i ⟶ j), (F.map g.op).app i ≫ f✝ i = (F.obj (op j)).map g ≫ f✝ j\nF' : Jᵒᵖ ⥤ J ⥤ C\ninst✝ : HasCoend F'\nf : F ⟶ ... | [] | simp [coend.condition] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Limits.Shapes.End | {
"line": 319,
"column": 4
} | {
"line": 319,
"column": 26
} | {
"line": 319,
"column": 26
} | [
{
"pp": "J : Type u\ninst✝³ : Category.{v, u} J\nC : Type u'\ninst✝² : Category.{v', u'} C\nF : Jᵒᵖ ⥤ J ⥤ C\ninst✝¹ : HasCoend F\nX : C\nf✝ : (j : J) → (F.obj (op j)).obj j ⟶ X\nhf : ∀ ⦃i j : J⦄ (g : i ⟶ j), (F.map g.op).app i ≫ f✝ i = (F.obj (op j)).map g ≫ f✝ j\nF' : Jᵒᵖ ⥤ J ⥤ C\ninst✝ : HasCoend F'\nf : F ⟶ ... | [] | simp [coend.condition] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.End | {
"line": 319,
"column": 4
} | {
"line": 319,
"column": 26
} | {
"line": 319,
"column": 26
} | [
{
"pp": "J : Type u\ninst✝³ : Category.{v, u} J\nC : Type u'\ninst✝² : Category.{v', u'} C\nF : Jᵒᵖ ⥤ J ⥤ C\ninst✝¹ : HasCoend F\nX : C\nf✝ : (j : J) → (F.obj (op j)).obj j ⟶ X\nhf : ∀ ⦃i j : J⦄ (g : i ⟶ j), (F.map g.op).app i ≫ f✝ i = (F.obj (op j)).map g ≫ f✝ j\nF' : Jᵒᵖ ⥤ J ⥤ C\ninst✝ : HasCoend F'\nf : F ⟶ ... | [] | simp [coend.condition] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Enriched.Opposite | {
"line": 50,
"column": 4
} | {
"line": 52,
"column": 38
} | {
"line": 53,
"column": 2
} | [
{
"pp": "V : Type u₁\ninst✝³ : Category.{v₁, u₁} V\ninst✝² : MonoidalCategory V\ninst✝¹ : BraidedCategory V\nC : Type u\ninst✝ : EnrichedCategory V C\nx✝¹ x✝ : Cᵒᵖ\n⊢ (λ_ (Opposite.unop x✝ ⟶[V] Opposite.unop x✝¹)).inv ≫\n (id (Opposite.unop x✝¹) ▷ Opposite.unop x✝ ⟶[V] Opposite.unop x✝¹) ≫\n (β_ (Op... | [] | simp only [braiding_naturality_left_assoc, braiding_tensorUnit_left,
Category.assoc, Iso.inv_hom_id_assoc]
exact EnrichedCategory.comp_id _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Enriched.Opposite | {
"line": 50,
"column": 4
} | {
"line": 52,
"column": 38
} | {
"line": 53,
"column": 2
} | [
{
"pp": "V : Type u₁\ninst✝³ : Category.{v₁, u₁} V\ninst✝² : MonoidalCategory V\ninst✝¹ : BraidedCategory V\nC : Type u\ninst✝ : EnrichedCategory V C\nx✝¹ x✝ : Cᵒᵖ\n⊢ (λ_ (Opposite.unop x✝ ⟶[V] Opposite.unop x✝¹)).inv ≫\n (id (Opposite.unop x✝¹) ▷ Opposite.unop x✝ ⟶[V] Opposite.unop x✝¹) ≫\n (β_ (Op... | [] | simp only [braiding_naturality_left_assoc, braiding_tensorUnit_left,
Category.assoc, Iso.inv_hom_id_assoc]
exact EnrichedCategory.comp_id _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.FiberedCategory.Cartesian | {
"line": 123,
"column": 24
} | {
"line": 123,
"column": 28
} | {
"line": 124,
"column": 2
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₁} 𝒮\ninst✝¹ : Category.{v₂, u₂} 𝒳\np : 𝒳 ⥤ 𝒮\na✝ b✝ : 𝒳\nφ : a✝ ⟶ b✝\nR S : 𝒮\na b : 𝒳\ninst✝ : p.IsCartesian (p.map φ) φ\n⊢ IsCartesian.map p (p.map φ) φ φ = 𝟙 a✝",
"ppTerm": "?m.138",
"assigned": true,
"usedConstants": [
"... | [
"𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₁} 𝒮\ninst✝¹ : Category.{v₂, u₂} 𝒳\np : 𝒳 ⥤ 𝒮\na✝ b✝ : 𝒳\nφ : a✝ ⟶ b✝\nR S : 𝒮\na b : 𝒳\ninst✝ : p.IsCartesian (p.map φ) φ\n⊢ 𝟙 a✝ = IsCartesian.map p (p.map φ) φ φ"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.FiberedCategory.Cocartesian | {
"line": 118,
"column": 24
} | {
"line": 118,
"column": 28
} | {
"line": 119,
"column": 2
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₁} 𝒮\ninst✝¹ : Category.{v₂, u₂} 𝒳\np : 𝒳 ⥤ 𝒮\na✝ b✝ : 𝒳\nφ : a✝ ⟶ b✝\nR S : 𝒮\na b : 𝒳\ninst✝ : p.IsCocartesian (p.map φ) φ\n⊢ IsCocartesian.map p (p.map φ) φ φ = 𝟙 b✝",
"ppTerm": "?m.138",
"assigned": true,
"usedConstants": [
... | [
"𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₁} 𝒮\ninst✝¹ : Category.{v₂, u₂} 𝒳\np : 𝒳 ⥤ 𝒮\na✝ b✝ : 𝒳\nφ : a✝ ⟶ b✝\nR S : 𝒮\na b : 𝒳\ninst✝ : p.IsCocartesian (p.map φ) φ\n⊢ 𝟙 b✝ = IsCocartesian.map p (p.map φ) φ φ"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.FiberedCategory.Cocartesian | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 39
} | {
"line": 132,
"column": 2
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝³ : Category.{v₁, u₁} 𝒮\ninst✝² : Category.{v₂, u₂} 𝒳\np : 𝒳 ⥤ 𝒮\na✝ b✝ : 𝒳\nφ : a✝ ⟶ b✝\nb' : 𝒳\nφ' : a✝ ⟶ b'\nR S : 𝒮\na b : 𝒳\ninst✝¹ : p.IsCocartesian (p.map φ) φ\ninst✝ : p.IsCocartesian (p.map φ) φ'\n⊢ φ ≫ IsCocartesian.map p (p.map φ) φ φ' ≫ IsCocartesian... | [] | simp only [fac_assoc, fac, comp_id] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.FiberedCategory.Cocartesian | {
"line": 135,
"column": 4
} | {
"line": 135,
"column": 39
} | {
"line": 137,
"column": 0
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝³ : Category.{v₁, u₁} 𝒮\ninst✝² : Category.{v₂, u₂} 𝒳\np : 𝒳 ⥤ 𝒮\na✝ b✝ : 𝒳\nφ : a✝ ⟶ b✝\nb' : 𝒳\nφ' : a✝ ⟶ b'\nR S : 𝒮\na b : 𝒳\ninst✝¹ : p.IsCocartesian (p.map φ') φ\ninst✝ : p.IsCocartesian (p.map φ') φ'\n⊢ φ' ≫ IsCocartesian.map p (p.map φ') φ' φ ≫ IsCocarte... | [] | simp only [fac_assoc, fac, comp_id] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.FiberedCategory.Cartesian | {
"line": 247,
"column": 24
} | {
"line": 247,
"column": 28
} | {
"line": 248,
"column": 2
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₁} 𝒮\ninst✝¹ : Category.{v₂, u₂} 𝒳\np : 𝒳 ⥤ 𝒮\na✝ b✝ : 𝒳\nφ : a✝ ⟶ b✝\nR S : 𝒮\na b : 𝒳\ninst✝ : p.IsStronglyCartesian (p.map φ) φ\n⊢ map p (p.map φ) φ ⋯ φ = 𝟙 a✝",
"ppTerm": "?m.148",
"assigned": true,
"usedConstants": [
"Ca... | [
"𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₁} 𝒮\ninst✝¹ : Category.{v₂, u₂} 𝒳\np : 𝒳 ⥤ 𝒮\na✝ b✝ : 𝒳\nφ : a✝ ⟶ b✝\nR S : 𝒮\na b : 𝒳\ninst✝ : p.IsStronglyCartesian (p.map φ) φ\n⊢ 𝟙 a✝ = map p (p.map φ) φ ⋯ φ"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.FiberedCategory.Cocartesian | {
"line": 237,
"column": 24
} | {
"line": 237,
"column": 28
} | {
"line": 238,
"column": 2
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₁} 𝒮\ninst✝¹ : Category.{v₂, u₂} 𝒳\np : 𝒳 ⥤ 𝒮\na✝ b✝ : 𝒳\nφ : a✝ ⟶ b✝\nR S : 𝒮\na b : 𝒳\ninst✝ : p.IsStronglyCocartesian (p.map φ) φ\n⊢ map p (p.map φ) φ ⋯ φ = 𝟙 b✝",
"ppTerm": "?m.148",
"assigned": true,
"usedConstants": [
"... | [
"𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₁} 𝒮\ninst✝¹ : Category.{v₂, u₂} 𝒳\np : 𝒳 ⥤ 𝒮\na✝ b✝ : 𝒳\nφ : a✝ ⟶ b✝\nR S : 𝒮\na b : 𝒳\ninst✝ : p.IsStronglyCocartesian (p.map φ) φ\n⊢ 𝟙 b✝ = map p (p.map φ) φ ⋯ φ"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.FiberedCategory.Fibered | {
"line": 128,
"column": 15
} | {
"line": 128,
"column": 18
} | {
"line": 129,
"column": 2
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₁} 𝒮\ninst✝¹ : Category.{v₂, u₂} 𝒳\np : 𝒳 ⥤ 𝒮\nh : ∀ (a : 𝒳) (R : 𝒮) (f : R ⟶ p.obj a), ∃ b φ, p.IsStronglyCartesian f φ\nR S : 𝒮\nf : R ⟶ S\na b : 𝒳\nφ : a ⟶ b\ninst✝ : p.IsCartesian f φ\nc : 𝒳\ng : p.obj c ⟶ R\nφ' : c ⟶ b\n⊢ ∀ [p.IsHomLift ... | [
"𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₁} 𝒮\ninst✝¹ : Category.{v₂, u₂} 𝒳\np : 𝒳 ⥤ 𝒮\nh : ∀ (a : 𝒳) (R : 𝒮) (f : R ⟶ p.obj a), ∃ b φ, p.IsStronglyCartesian f φ\nR S : 𝒮\nf : R ⟶ S\na b : 𝒳\nφ : a ⟶ b\ninst✝ : p.IsCartesian f φ\nc : 𝒳\ng : p.obj c ⟶ R\nφ' : c ⟶ b\nhφ' : p.IsHomLift (g ≫ f) φ'\... | hφ' | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Functor.Derived.PointwiseLeftDerived | {
"line": 58,
"column": 55
} | {
"line": 62,
"column": 42
} | {
"line": 64,
"column": 0
} | [
{
"pp": "C : Type u₁\nD : Type u₂\nH : Type u₃\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} H\nF : C ⥤ H\nL : C ⥤ D\nW : MorphismProperty C\ninst✝ : L.IsLocalization W\nX : C\n⊢ F.HasPointwiseLeftDerivedFunctorAt W X ↔ L.HasPointwiseRightKanExtensionAt F (L.obj X)",
... | [] | by
rw [← hasPointwiseRightKanExtensionAt_iff_of_equivalence W.Q L F
(Localization.uniq W.Q L W) (Localization.compUniqFunctor W.Q L W) (W.Q.obj X) (L.obj X)
((Localization.compUniqFunctor W.Q L W).app X)]
exact ⟨fun h ↦ h.hasLimit', fun h ↦ ⟨h⟩⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Galois.Examples | {
"line": 111,
"column": 24
} | {
"line": 130,
"column": 22
} | {
"line": 132,
"column": 0
} | [
{
"pp": "G : Type u\ninst✝¹ : Group G\nX : Action FintypeCat G\ninst✝ : IsConnected X\nx y : X.V.obj\n⊢ ∃ g, g • x = y",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"MulAction.nonempty_orbit",
"CategoryTheory.instFaithfulForget",
"False",
"Action.... | [] | by
/- We show that the `G`-orbit of `x` is a non-initial subobject of `X` and hence by
connectedness, the orbit equals `X.V`. -/
let T : Set X.V := MulAction.orbit G x
have : Fintype T := Fintype.ofFinite T
letI : MulAction G (FintypeCat.of T) := inferInstanceAs <| MulAction G
↑(MulAction.orbi... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Galois.GaloisObjects | {
"line": 110,
"column": 4
} | {
"line": 110,
"column": 46
} | {
"line": 111,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{u₂, u₁} C\ninst✝² : GaloisCategory C\nF : C ⥤ FintypeCat\ninst✝¹ : FiberFunctor F\nX : C\ninst✝ : IsGalois X\nx : (F.obj X).obj\nn : Aut F\nninstab : n ∈ MulAction.stabilizer (Aut F) x\ng : Aut F\nφ : Aut X\nh : (ConcreteCategory.hom (F.map φ.hom)) x = g⁻¹ • x\n⊢ g • n •... | [
"C : Type u₁\ninst✝³ : Category.{u₂, u₁} C\ninst✝² : GaloisCategory C\nF : C ⥤ FintypeCat\ninst✝¹ : FiberFunctor F\nX : C\ninst✝ : IsGalois X\nx : (F.obj X).obj\nn : Aut F\nninstab : n ∈ MulAction.stabilizer (Aut F) x\ng : Aut F\nφ : Aut X\nh : (ConcreteCategory.hom (F.map φ.hom)) x = g⁻¹ • x\n⊢ g • g⁻¹ • x = x"
] | rw [← h, mulAction_naturality, ninstab, h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Galois.Full | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 48
} | {
"line": 65,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nF : C ⥤ FintypeCat\ninst✝³ : GaloisCategory C\ninst✝² : FiberFunctor F\nX : C\nY : Action FintypeCat (Aut F)\ni : Y ⟶ (functorToAction F).obj X\ninst✝¹ : Mono i\ninst✝ : IsConnected Y\ny : ((forget₂ (Action FintypeCat (Aut F)) FintypeCat).obj Y).obj\nZ : C\... | [
"C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nF : C ⥤ FintypeCat\ninst✝³ : GaloisCategory C\ninst✝² : FiberFunctor F\nX : C\nY : Action FintypeCat (Aut F)\ni : Y ⟶ (functorToAction F).obj X\ninst✝¹ : Mono i\ninst✝ : IsConnected Y\ny : ((forget₂ (Action FintypeCat (Aut F)) FintypeCat).obj Y).obj\nZ : C\nf : Z ⟶ X\n... | suffices h : i.hom y = F.map f z by simpa [hu] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.CategoryTheory.Galois.Decomposition | {
"line": 207,
"column": 6
} | {
"line": 207,
"column": 27
} | {
"line": 207,
"column": 27
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{u₂, u₁} C\ninst✝¹ : GaloisCategory C\nF : C ⥤ FintypeCat\ninst✝ : FiberFunctor F\nX : C\nt : (F.obj X).obj\n⊢ (ConcreteCategory.hom (F.map (Pi.π (fun x ↦ X) t))) (mkSelfProdFib F X) = t",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [
"C : Type u₁\ninst✝² : Category.{u₂, u₁} C\ninst✝¹ : GaloisCategory C\nF : C ⥤ FintypeCat\ninst✝ : FiberFunctor F\nX : C\nt : (F.obj X).obj\n⊢ (ConcreteCategory.hom ((piComparison F fun x ↦ X) ≫ Pi.π (fun b ↦ F.obj X) t)) (mkSelfProdFib F X) = t"
] | ← piComparison_comp_π | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Galois.Full | {
"line": 119,
"column": 4
} | {
"line": 119,
"column": 66
} | {
"line": 121,
"column": 0
} | [
{
"pp": "case h\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\nF : C ⥤ FintypeCat\ninst✝¹ : GaloisCategory C\ninst✝ : FiberFunctor F\nX Y : C\nf : (functorToAction F).obj X ⟶ (functorToAction F).obj Y\nu : (functorToAction F).obj X ⟶ (functorToAction F).obj X ⨯ (functorToAction F).obj Y :=\n prod.lift (𝟙 ((fu... | [] | simp [-FintypeCat.comp_apply, -Action.comp_hom, i, u, ψ, hgvi] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
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