module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 613,
"column": 6
} | {
"line": 613,
"column": 21
} | {
"line": 613,
"column": 22
} | [
{
"pp": "L : PeriodPair\nx : ℂ\nr : ℕ\nhr : 2 < r\nl : ↥L.lattice\n⊢ ‖((↑l - x) ^ r)⁻¹‖ ≤ ‖↑l - x‖ ^ (-↑r)",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Submodule",
"Real.in... | [
"L : PeriodPair\nx : ℂ\nr : ℕ\nhr : 2 < r\nl : ↥L.lattice\n⊢ ‖((↑l - x) ^ ↑r)⁻¹‖ ≤ ‖↑l - x‖ ^ (-↑r)"
] | ← zpow_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.ContDiff | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 37
} | {
"line": 58,
"column": 2
} | [
{
"pp": "E : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\ninst✝ : CompleteSpace E\nh : ContDiffOn ℝ 1 f (Icc a b)\nhab : a ≤ b\n⊢ derivWithin f (Icc a b) =ᵐ[volume.restrict (Ioc a b)] deriv f",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"M... | [
"E : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\ninst✝ : CompleteSpace E\nh : ContDiffOn ℝ 1 f (Icc a b)\nhab : a ≤ b\n⊢ derivWithin f (Icc a b) =ᵐ[volume.restrict (Ioo a b)] deriv f"
] | rw [← restrict_Ioo_eq_restrict_Ioc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Stirling | {
"line": 112,
"column": 17
} | {
"line": 112,
"column": 24
} | {
"line": 112,
"column": 24
} | [
{
"pp": "case refine_1\nn : ℕ\nh_nonneg : 0 ≤ (1 / (2 * ↑(n + 1) + 1)) ^ 2\n⊢ (2 * ↑(n + 1) + 1)⁻¹ ^ 2 < 1",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"DivInvMonoid.toInv",
"HMul.hMul",
"DivisionCommMonoid.toDivisionMonoid",
"Di... | [
"case refine_1\nn : ℕ\nh_nonneg : 0 ≤ (1 / (2 * ↑(n + 1) + 1)) ^ 2\n⊢ ((2 * ↑(n + 1) + 1) ^ 2)⁻¹ < 1"
] | inv_pow | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 912,
"column": 4
} | {
"line": 912,
"column": 76
} | {
"line": 913,
"column": 4
} | [
{
"pp": "case pos\nL : PeriodPair\nx : ℂ\nhx : x ∈ L.lattice\n⊢ MeromorphicAt (fun x_1 ↦ ℘[L - x] x_1 + (1 / (x_1 - x) ^ 2 - 1 / x ^ 2)) x",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"Submodule",
"SetLike.mem_coe._simp_1",
"Fal... | [
"case pos\nL : PeriodPair\nx : ℂ\nhx : x ∈ L.lattice\nthis : MeromorphicAt ℘[L - x] x\n⊢ MeromorphicAt (fun x_1 ↦ ℘[L - x] x_1 + (1 / (x_1 - x) ^ 2 - 1 / x ^ 2)) x"
] | have := (analyticOnNhd_weierstrassPExcept L x x (by simp)).meromorphicAt | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 164,
"column": 2
} | {
"line": 167,
"column": 96
} | {
"line": 169,
"column": 0
} | [
{
"pp": "c e : ℤ\nz : ℂ\n⊢ (fun d ↦ ↑c * z + ↑d + ↑e) =Θ[cofinite] fun n ↦ ↑n",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Int.cast",
"Eq.mpr",
"Int.cast_complex_isTheta_cast_real",
"NormedCommRing.toSemino... | [] | apply IsTheta.add_isLittleO <;>
[refine Asymptotics.IsLittleO.add_isTheta ?_ (Int.cast_complex_isTheta_cast_real); skip] <;>
simpa [-Int.cofinite_eq] using
.inr <| tendsto_norm_comp_cofinite_atTop_of_isClosedEmbedding Int.isClosedEmbedding_coe_real | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 164,
"column": 2
} | {
"line": 167,
"column": 96
} | {
"line": 169,
"column": 0
} | [
{
"pp": "c e : ℤ\nz : ℂ\n⊢ (fun d ↦ ↑c * z + ↑d + ↑e) =Θ[cofinite] fun n ↦ ↑n",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Int.cast",
"Eq.mpr",
"Int.cast_complex_isTheta_cast_real",
"NormedCommRing.toSemino... | [] | apply IsTheta.add_isLittleO <;>
[refine Asymptotics.IsLittleO.add_isTheta ?_ (Int.cast_complex_isTheta_cast_real); skip] <;>
simpa [-Int.cofinite_eq] using
.inr <| tendsto_norm_comp_cofinite_atTop_of_isClosedEmbedding Int.isClosedEmbedding_coe_real | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 164,
"column": 2
} | {
"line": 167,
"column": 96
} | {
"line": 169,
"column": 0
} | [
{
"pp": "c e : ℤ\nz : ℂ\n⊢ (fun d ↦ ↑c * z + ↑d + ↑e) =Θ[cofinite] fun n ↦ ↑n",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Int.cast",
"Eq.mpr",
"Int.cast_complex_isTheta_cast_real",
"NormedCommRing.toSemino... | [] | apply IsTheta.add_isLittleO <;>
[refine Asymptotics.IsLittleO.add_isTheta ?_ (Int.cast_complex_isTheta_cast_real); skip] <;>
simpa [-Int.cofinite_eq] using
.inr <| tendsto_norm_comp_cofinite_atTop_of_isClosedEmbedding Int.isClosedEmbedding_coe_real | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema | {
"line": 165,
"column": 85
} | {
"line": 175,
"column": 72
} | {
"line": 177,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ (T ℝ ↑n).roots = (Finset.image (fun k ↦ cos ((2 * ↑k + 1) * π / (2 * ↑n))) (Finset.range n)).val",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"CharP.cast_eq_zero",
"Nat.cast_mul._simp_1",
"Iff.mpr",
"zero_le",
"Mathlib.Tactic.FieldSimp.zpo... | [] | by
wlog! hn : n ≠ 0
· simp [hn]
refine roots_eq_of_degree_eq_card (fun x hx ↦ ?_) ?_
· obtain ⟨k, hk, hx⟩ := Finset.mem_image.mp hx
rw [← hx, T_real_cos, cos_eq_zero_iff]
use k
field_simp
norm_cast
· rw [Finset.card_image_of_injOn, Finset.card_range, degree_T, Int.natAbs_natCast]
exact (Fi... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema | {
"line": 233,
"column": 4
} | {
"line": 233,
"column": 76
} | {
"line": 235,
"column": 0
} | [
{
"pp": "case refine_3\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⊢ cos (↑k * π / ↑n) < cos 0",
"ppTerm": "?refine_3",
"assigned": true,
"usedConstants": [
"le_refl",
"Real",
"instHDiv",
"Real.pi",
"... | [] | exact cos_lt_cos_of_nonneg_of_le_pi (le_refl 0) (le_of_lt lt_pi) zero_lt | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema | {
"line": 250,
"column": 4
} | {
"line": 250,
"column": 76
} | {
"line": 252,
"column": 0
} | [
{
"pp": "case refine_3\nn k : ℕ\nhn : n ≠ 0\nhk₁ : k < n\nhk₂ : Odd k\nk_pos : 0 < k\nzero_lt : 0 < ↑k * π / ↑n\nlt_pi : ↑k * π / ↑n < π\n⊢ cos (↑k * π / ↑n) < cos 0",
"ppTerm": "?refine_3",
"assigned": true,
"usedConstants": [
"le_refl",
"Real",
"instHDiv",
"Real.pi",
... | [] | exact cos_lt_cos_of_nonneg_of_le_pi (le_refl 0) (le_of_lt lt_pi) zero_lt | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 119,
"column": 80
} | {
"line": 123,
"column": 12
} | {
"line": 125,
"column": 0
} | [
{
"pp": "Z : Set ℂ\nhZC : IsCompact Z\n⊢ MultipliableUniformlyOn (fun n z ↦ 1 + sineTerm z n) Z",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"NormedCommRing.toSeminormedCommRing",
"Real.instLE",
"Real",
"ins... | [] | by
obtain ⟨u, hu, hu2⟩ := sineTerm_bound_aux hZC
refine Summable.multipliableUniformlyOn_nat_one_add hZC hu ?_ ?_
· filter_upwards with n z hz using hu2 n z hz
· fun_prop | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Int.Fib.Basic | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 47
} | {
"line": 140,
"column": 0
} | [
{
"pp": "case inr.inr\nm n : ℕ\n⊢ fib (-↑m + -↑n) = fib (-↑m - 1) * fib (-↑n) + fib (-↑m) * fib (-↑n + 1)",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Data.Int.Fib.Basic.0.Int.fib_neg_natCast_add_neg_natCast"
],
"usedFVars": [
"m",
"n"
... | [] | · exact fib_neg_natCast_add_neg_natCast _ _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 192,
"column": 2
} | {
"line": 192,
"column": 20
} | {
"line": 193,
"column": 2
} | [
{
"pp": "x : ℂ\nhx : x ∈ ℂ_ℤ\nn : ℕ\n⊢ logDeriv (fun z ↦ ∏ j ∈ Finset.range n, (1 + sineTerm z j)) x = ∑ j ∈ Finset.range n, cotTerm x j",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"logDeriv",
"Eq.mpr",
"Complex.commRing",
"congrArg",
"NormedAlgebra.id",
... | [
"x : ℂ\nhx : x ∈ ℂ_ℤ\nn : ℕ\n⊢ ∑ i ∈ Finset.range n, logDeriv (fun z ↦ 1 + sineTerm z i) x = ∑ j ∈ Finset.range n, cotTerm x j",
"case hf\nx : ℂ\nhx : x ∈ ℂ_ℤ\nn : ℕ\n⊢ ∀ i ∈ Finset.range n, 1 + sineTerm x i ≠ 0",
"case hd\nx : ℂ\nhx : x ∈ ℂ_ℤ\nn : ℕ\n⊢ ∀ i ∈ Finset.range n, DifferentiableAt ℂ (fun z ↦ 1 + sine... | rw [logDeriv_prod] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SumIntegralExpDecay | {
"line": 56,
"column": 4
} | {
"line": 56,
"column": 22
} | {
"line": 57,
"column": 2
} | [
{
"pp": "case gpos\nk M : ℕ\nc : ℝ\nhc : 0 < c\n⊢ 0 ≤ rexp (-(c * (↑M - 1)))",
"ppTerm": "?gpos",
"assigned": true,
"usedConstants": [
"Real",
"HMul.hMul",
"Real.instSub",
"HSub.hSub",
"Nat.cast",
"Real.instOne",
"instHSub",
"Real.instMul",
"Real... | [] | · apply exp_nonneg | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecificLimits.FloorPow | {
"line": 187,
"column": 10
} | {
"line": 187,
"column": 12
} | {
"line": 188,
"column": 2
} | [
{
"pp": "u : ℕ → ℝ\nl : ℝ\nhmono : Monotone u\nc : ℕ → ℝ\ncone : ∀ (k : ℕ), 1 < c k\nclim : Tendsto c atTop (𝓝 1)\nhc : ∀ (k : ℕ), Tendsto (fun n ↦ u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)\na : ℝ\n⊢ 1 < a →\n ∃ c,\n (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧\n Tendsto c atTop atTop ∧ Ten... | [
"u : ℕ → ℝ\nl : ℝ\nhmono : Monotone u\nc : ℕ → ℝ\ncone : ∀ (k : ℕ), 1 < c k\nclim : Tendsto c atTop (𝓝 1)\nhc : ∀ (k : ℕ), Tendsto (fun n ↦ u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)\na : ℝ\nha : 1 < a\n⊢ ∃ c,\n (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧\n Tendsto c atTop atTop ∧ Tendsto (fun n ↦ u (... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.ModuleEmbedding.Opposite | {
"line": 61,
"column": 72
} | {
"line": 64,
"column": 39
} | {
"line": 66,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nD : Type v\ninst✝³ : SmallCategory D\nF : D ⥤ Cᵒᵖ\ninst✝² : Abelian C\ninst✝¹ : IsGrothendieckAbelian.{v, v, u} C\ninst✝ : Nonempty D\n⊢ IsSeparator (generator F)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.... | [] | by
apply isSeparator_sigma_of_isSeparator _ Classical.ofNonempty
apply isSeparator_sigma_of_isSeparator _ 0
exact isSeparator_projectiveSeparator | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.FilteredColimitCommutesProduct | {
"line": 215,
"column": 37
} | {
"line": 222,
"column": 73
} | {
"line": 224,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\nι : Type u_2\ninst✝² : HasProductsOfShape ι C\nι' : Type u_3\ninst✝¹ : HasProductsOfShape ι' C\ninst✝ : IsIPCOfShape.{w, u_2, v_1, u_1} ι C\ne : ι ≃ ι'\nJ : ι' → Type w\nx✝¹ : (i : ι') → SmallCategory (J i)\nx✝ : ∀ (i : ι'), IsFiltered (J i)\nF : (i : ι') →... | [] | by
obtain ⟨h⟩ := nonempty_isColimit fun i : ι ↦ hc (e i)
constructor
apply IsColimit.equivOfNatIsoOfIso _ _ _ _ <|
h.whiskerEquivalence (Pi.equivalenceOfEquiv J e).symm
· exact (Pi.equivalenceOfEquivCompPointwiseProduct F e)
· -- Without the double `symm`, one runs into DTT hell
exact ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.ModuleEmbedding.GabrielPopescu | {
"line": 125,
"column": 4
} | {
"line": 125,
"column": 66
} | {
"line": 127,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : IsGrothendieckAbelian.{v, v, u} C\nG : C\nhG : IsSeparator G\nA B : C\nf : (preadditiveCoyonedaObj G).obj A ⟶ (preadditiveCoyonedaObj G).obj B\nthis : Epi (Sigma.desc fun f ↦ f)\nh : (kernel.ι (Sigma.desc fun m ↦ m) ≫ Sigma.desc fun m ... | [] | simpa [-comp_epiDesc] using! Sigma.ι _ q ≫= comp_epiDesc _ _ h | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.CategoryTheory.Abelian.Injective.Resolution | {
"line": 259,
"column": 59
} | {
"line": 261,
"column": 75
} | {
"line": 263,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasInjectiveResolutions C\nX Y : C\nf : X ⟶ Y\nI : InjectiveResolution X\nJ : InjectiveResolution Y\nφ : I.cocomplex ⟶ J.cocomplex\ncomm : I.ι.f 0 ≫ φ.f 0 = f ≫ J.ι.f 0\n⊢ I.iso.inv ≫ (injectiveResolutions C).map f = (HomotopyCategory.... | [] | by
rw [← cancel_mono (J.iso).hom, Category.assoc, iso_hom_naturality f I J φ comm,
Iso.inv_hom_id_assoc, Category.assoc, Iso.inv_hom_id, Category.comp_id] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Abelian.Injective.Resolution | {
"line": 342,
"column": 10
} | {
"line": 342,
"column": 32
} | {
"line": 342,
"column": 32
} | [
{
"pp": "case succ\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nZ : C\nn✝ : ℕ\n⊢ QuasiIsoAt (((ofCocomplex Z).fromSingle₀Equiv Z).symm ⟨Injective.ι Z, ⋯⟩) (n✝ + 1)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPr... | [
"case succ\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nZ : C\nn✝ : ℕ\n⊢ HomologicalComplex.ExactAt (ofCocomplex Z) (n✝ + 1)",
"case succ.hK\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nZ : C\nn✝ : ℕ\n⊢ HomologicalComplex.Exac... | quasiIsoAt_iff_exactAt | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.Injective.Resolution | {
"line": 342,
"column": 6
} | {
"line": 342,
"column": 33
} | {
"line": 343,
"column": 6
} | [
{
"pp": "case succ\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nZ : C\nn✝ : ℕ\n⊢ QuasiIsoAt (((ofCocomplex Z).fromSingle₀Equiv Z).symm ⟨Injective.ι Z, ⋯⟩) (n✝ + 1)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPr... | [
"case succ\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nZ : C\nn✝ : ℕ\n⊢ HomologicalComplex.ExactAt (ofCocomplex Z) (n✝ + 1)",
"case succ.hK\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nZ : C\nn✝ : ℕ\n⊢ HomologicalComplex.Exac... | rw [quasiIsoAt_iff_exactAt] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.MorphismProperty.OfObjectProperty | {
"line": 60,
"column": 4
} | {
"line": 60,
"column": 46
} | {
"line": 62,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP Q : ObjectProperty C\ninst✝ : P.IsClosedUnderIsomorphisms\nX Y Z : C\ni : X ⟶ Y\nhi : IsIso i\nf : Y ⟶ Z\nhY : P Y\nhZ : Q Z\n⊢ ofObjectProperty P Q (i ≫ f)",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"CategoryTheory.Object... | [] | exact ⟨(P.prop_iff_of_isIso i).mpr hY, hZ⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Abelian.RightDerived | {
"line": 331,
"column": 2
} | {
"line": 331,
"column": 56
} | {
"line": 332,
"column": 2
} | [
{
"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\n⊢ F.toRightDerivedZero.app X =\n I.toRightDerivedZero' F ≫\n (Cochai... | [
"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\n⊢ (injectiveResolution X).toRightDerivedZero' F ≫\n ((F.mapHomologicalComplex (Comp... | dsimp [Functor.toRightDerivedZero, isoRightDerivedObj] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Abelian.SerreClass.Localization | {
"line": 160,
"column": 10
} | {
"line": 160,
"column": 34
} | {
"line": 160,
"column": 35
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : Abelian C\nD : Type u'\ninst✝⁴ : Category.{v', u'} D\nL : C ⥤ D\nP : ObjectProperty C\ninst✝³ : P.IsSerreClass\ninst✝² : L.IsLocalization P.isoModSerre\ninst✝¹ : Preadditive D\ninst✝ : L.Additive\nX Y : C\nf : X ⟶ Y\nthis : L.EssSurj\nx✝ : Mono (L.map f)... | [
"C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : Abelian C\nD : Type u'\ninst✝⁴ : Category.{v', u'} D\nL : C ⥤ D\nP : ObjectProperty C\ninst✝³ : P.IsSerreClass\ninst✝² : L.IsLocalization P.isoModSerre\ninst✝¹ : Preadditive D\ninst✝ : L.Additive\nX Y : C\nf : X ⟶ Y\nthis : L.EssSurj\nx✝ : Mono (L.map f)\n⊢ L.map (k... | ← cancel_mono (L.map f), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Action.Concrete | {
"line": 38,
"column": 14
} | {
"line": 38,
"column": 27
} | {
"line": 39,
"column": 2
} | [
{
"pp": "X : Type u\n⊢ Function.LeftInverse (fun f ↦ ⇑(ConcreteCategory.hom f)) fun f ↦ ↾f",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Function.End",
"CategoryTheory.ConcreteCategory.hom",
"TypeCat.instFunLikeFun",
"TypeCat.ofHom",
"CategoryTheory.End",
... | [] | by intro; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Action.Concrete | {
"line": 39,
"column": 15
} | {
"line": 39,
"column": 28
} | {
"line": 40,
"column": 2
} | [
{
"pp": "X : Type u\n⊢ Function.RightInverse (fun f ↦ ⇑(ConcreteCategory.hom f)) fun f ↦ ↾f",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Function.End",
"CategoryTheory.ConcreteCategory.hom",
"TypeCat.instFunLikeFun",
"TypeCat.ofHom",
"CategoryTheory.End",
... | [] | by intro; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Action.Basic | {
"line": 444,
"column": 33
} | {
"line": 444,
"column": 70
} | {
"line": 444,
"column": 71
} | [
{
"pp": "V : Type u_1\ninst✝² : Category.{v_1, u_1} V\nW : Type u_2\ninst✝¹ : Category.{v_2, u_2} W\nF : V ⥤ W\nG : Type u_3\ninst✝ : Monoid G\nX✝ Y✝ : Action V G\nf : X✝ ⟶ Y✝\ng : G\n⊢ F.map (X✝.ρ g) ≫ F.map f.hom = F.map f.hom ≫ F.map (Y✝.ρ g)",
"ppTerm": "?m.112",
"assigned": true,
"usedConstants... | [] | rw [← F.map_comp, f.comm, F.map_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Adjunction.Lifting.Left | {
"line": 92,
"column": 6
} | {
"line": 92,
"column": 33
} | {
"line": 93,
"column": 4
} | [
{
"pp": "case refine_2\nA : Type u₁\nB : Type u₂\nC : Type u₃\ninst✝² : Category.{v₁, u₁} A\ninst✝¹ : Category.{v₂, u₂} B\ninst✝ : Category.{v₃, u₃} C\nU : B ⥤ C\nF : C ⥤ B\nR : A ⥤ B\nF' : C ⥤ A\nadj₁ : F ⊣ U\nadj₂ : F' ⊣ R ⋙ U\nh : (X : B) → RegularEpi (adj₁.counit.app X)\nX : B\ns : Cofork (F.map (U.map (adj... | [] | apply ((h X).desc' s.π _).2 | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Adjunction.Lifting.Left | {
"line": 92,
"column": 6
} | {
"line": 92,
"column": 33
} | {
"line": 93,
"column": 4
} | [
{
"pp": "case refine_2\nA : Type u₁\nB : Type u₂\nC : Type u₃\ninst✝² : Category.{v₁, u₁} A\ninst✝¹ : Category.{v₂, u₂} B\ninst✝ : Category.{v₃, u₃} C\nU : B ⥤ C\nF : C ⥤ B\nR : A ⥤ B\nF' : C ⥤ A\nadj₁ : F ⊣ U\nadj₂ : F' ⊣ R ⋙ U\nh : (X : B) → RegularEpi (adj₁.counit.app X)\nX : B\ns : Cofork (F.map (U.map (adj... | [] | apply ((h X).desc' s.π _).2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Adjunction.Lifting.Left | {
"line": 92,
"column": 6
} | {
"line": 92,
"column": 33
} | {
"line": 93,
"column": 4
} | [
{
"pp": "case refine_2\nA : Type u₁\nB : Type u₂\nC : Type u₃\ninst✝² : Category.{v₁, u₁} A\ninst✝¹ : Category.{v₂, u₂} B\ninst✝ : Category.{v₃, u₃} C\nU : B ⥤ C\nF : C ⥤ B\nR : A ⥤ B\nF' : C ⥤ A\nadj₁ : F ⊣ U\nadj₂ : F' ⊣ R ⋙ U\nh : (X : B) → RegularEpi (adj₁.counit.app X)\nX : B\ns : Cofork (F.map (U.map (adj... | [] | apply ((h X).desc' s.π _).2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Action.Monoidal | {
"line": 348,
"column": 8
} | {
"line": 349,
"column": 55
} | {
"line": 349,
"column": 56
} | [
{
"pp": "V : Type u_1\ninst✝⁵ : Category.{v_1, u_1} V\nG : Type u_2\ninst✝⁴ : Monoid G\nW : Type u_3\ninst✝³ : Category.{v_2, u_3} W\ninst✝² : MonoidalCategory V\ninst✝¹ : MonoidalCategory W\nF : V ⥤ W\ninst✝ : F.OplaxMonoidal\ng : G\n⊢ ((F.mapAction G).obj (𝟙_ (Action V G))).ρ g ≫ η F = η F ≫ (𝟙_ (Action W G... | [] | dsimp [FunctorCategoryEquivalence.inverse, Functor.mapAction]
rw [map_id, Category.id_comp, Category.comp_id] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Action.Monoidal | {
"line": 348,
"column": 8
} | {
"line": 349,
"column": 55
} | {
"line": 349,
"column": 56
} | [
{
"pp": "V : Type u_1\ninst✝⁵ : Category.{v_1, u_1} V\nG : Type u_2\ninst✝⁴ : Monoid G\nW : Type u_3\ninst✝³ : Category.{v_2, u_3} W\ninst✝² : MonoidalCategory V\ninst✝¹ : MonoidalCategory W\nF : V ⥤ W\ninst✝ : F.OplaxMonoidal\ng : G\n⊢ ((F.mapAction G).obj (𝟙_ (Action V G))).ρ g ≫ η F = η F ≫ (𝟙_ (Action W G... | [] | dsimp [FunctorCategoryEquivalence.inverse, Functor.mapAction]
rw [map_id, Category.id_comp, Category.comp_id] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Adjunction.Triple | {
"line": 296,
"column": 6
} | {
"line": 296,
"column": 49
} | {
"line": 296,
"column": 49
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\nF : C ⥤ D\nG : D ⥤ C\nH : C ⥤ D\nt : Triple F G H\ninst✝¹ : F.Full\ninst✝ : F.Faithful\nh : ∀ (X : D), Mono (t.adj₁.counit.app X ≫ t.adj₂.unit.app X)\nX : C\n⊢ Mono (t.leftToRight.app X)",
"ppTerm": "?m.106"... | [
"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\nF : C ⥤ D\nG : D ⥤ C\nH : C ⥤ D\nt : Triple F G H\ninst✝¹ : F.Full\ninst✝ : F.Faithful\nh : ∀ (X : D), Mono (t.adj₁.counit.app X ≫ t.adj₂.unit.app X)\nX : C\n⊢ Mono (t.adj₂.unit.app (F.obj X))"
] | mono_leftToRight_app_iff_mono_adj₂_unit_app | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Bicategory.Coherence | {
"line": 200,
"column": 79
} | {
"line": 200,
"column": 96
} | {
"line": 200,
"column": 96
} | [
{
"pp": "case mk.whisker_right\nB : Type u\ninst✝ : Quiver B\na b c : B\nf g : Hom b c\na✝ b✝ c✝ : FreeBicategory B\nf✝ g✝ : a✝ ⟶ b✝\nh : b✝ ⟶ c✝\nη' : Hom₂ f✝ g✝\nih :\n ∀ (p : Path a a✝),\n (preinclusion B).map { as := p } ◁ Quot.mk Rel η' ≫ (normalizeIso p g✝).hom =\n (normalizeIso p f✝).hom ≫ (prei... | [
"case mk.whisker_right\nB : Type u\ninst✝ : Quiver B\na b c : B\nf g : Hom b c\na✝ b✝ c✝ : FreeBicategory B\nf✝ g✝ : a✝ ⟶ b✝\nh : b✝ ⟶ c✝\nη' : Hom₂ f✝ g✝\nih :\n ∀ (p : Path a a✝),\n (preinclusion B).map { as := p } ◁ Quot.mk Rel η' ≫ (normalizeIso p g✝).hom =\n (normalizeIso p f✝).hom ≫ (preinclusion B).... | comp_whiskerRight | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Bicategory.Monad.Basic | {
"line": 89,
"column": 4
} | {
"line": 90,
"column": 74
} | {
"line": 91,
"column": 2
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na : B\nF : LocallyDiscrete (Discrete Unit) ⥤ᵒᵖᴸ B\n⊢ F.map₂ (ρ_ (𝟙 { as := { as := () } })).inv ≫\n F.map₂ (𝟙 { as := { as := () } } ◁ (ρ_ (𝟙 { as := { as := () } })).inv) ≫\n F.mapComp (𝟙 { as := { as := () } }) (𝟙 { as := { as := () } } ≫ 𝟙 { as := ... | [] | simp only [whiskerLeft_rightUnitor_inv, PrelaxFunctor.map₂_comp, Category.assoc,
OplaxFunctor.map₂_associator, whiskerRight_id, Iso.hom_inv_id_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Bicategory.Kan.Adjunction | {
"line": 116,
"column": 2
} | {
"line": 117,
"column": 89
} | {
"line": 118,
"column": 2
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b : B\nf : a ⟶ b\n⊢ [IsLeftAdjoint f, HasAbsLeftKanExtension f (𝟙 a), ∃ (x : HasLeftKanExtension f (𝟙 a)), Lan.CommuteWith f (𝟙 a) f].TFAE",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"CategoryTheory.Bicategory.Adjunction.isAbsoluteL... | [
"B : Type u\ninst✝ : Bicategory B\na b : B\nf : a ⟶ b\ntfae_1_to_2 : IsLeftAdjoint f → HasAbsLeftKanExtension f (𝟙 a)\n⊢ [IsLeftAdjoint f, HasAbsLeftKanExtension f (𝟙 a), ∃ (x : HasLeftKanExtension f (𝟙 a)), Lan.CommuteWith f (𝟙 a) f].TFAE"
] | tfae_have 1 → 2
| h => IsAbsKan.hasAbsLeftKanExtension (Adjunction.ofIsLeftAdjoint f).isAbsoluteLeftKan | Mathlib.Tactic.TFAE._aux_Mathlib_Tactic_TFAE___macroRules_Mathlib_Tactic_TFAE_tfaeHave_1 | Mathlib.Tactic.TFAE.tfaeHave |
Mathlib.CategoryTheory.Bicategory.Strict.Pseudofunctor | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 62
} | {
"line": 156,
"column": 2
} | [
{
"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' f₀₁ f₁₃ f ⋯).hom... | [
"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⊢ 𝟙 (F.map f₀₁ ≫ F.map f₁₃) =\n (F.map... | rw [← cancel_epi (F.mapComp' f₀₁ f₁₃ f).inv, Iso.inv_hom_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Bicategory.Strict.Pseudofunctor | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 44
} | {
"line": 204,
"column": 0
} | [
{
"pp": "B : Type u₁\nC : Type u₂\ninst✝² : Bicategory B\ninst✝¹ : Strict B\ninst✝ : Bicategory C\nF : B ⥤ᵖ C\nX₁ X₂ Y₁ Y₂ : B\nt : X₁ ⟶ Y₁\nl : X₁ ⟶ X₂\nr : Y₁ ⟶ Y₂\nb : X₂ ⟶ Y₂\nsq : CommSq t l r b\n⊢ F.isoMapOfCommSq sq = (F.mapComp' t r (t ≫ r) ⋯).symm ≪≫ F.mapComp' l b (t ≫ r) ⋯",
"ppTerm": "?m.86",
... | [] | simp [isoMapOfCommSq, mapComp'_eq_mapComp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Category.PartialFun | {
"line": 83,
"column": 21
} | {
"line": 85,
"column": 44
} | {
"line": 88,
"column": 0
} | [
{
"pp": "X✝ Y✝ : Type u_1\na₁✝ a₂✝ : X✝ ⟶ Y✝\nh : typeToPartialFun.map a₁✝ = typeToPartialFun.map a₂✝\n⊢ a₁✝ = a₂✝",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"CategoryTheory.ConcreteCategory.hom",
"typeToPartialFun",
"TypeCat.instFunLikeFun",
"id",
"TypeC... | [] | by
ext x
exact congrFun (PFun.lift_injective h) x | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Category.RelCat | {
"line": 102,
"column": 4
} | {
"line": 102,
"column": 89
} | {
"line": 103,
"column": 4
} | [
{
"pp": "case mp\nX Y : RelCat\nr : X ⟶ Y\nh : IsIso r\nh1 : ∀ (a b : X), ((a, b) ∈ (r ≫ inv r).rel) = ((a, b) ∈ (𝟙 X).rel)\nh2 : ∀ (a b : Y), ((a, b) ∈ (inv r ≫ r).rel) = ((a, b) ∈ (𝟙 Y).rel)\n⊢ ∃ f, graphFunctor.map f.hom = r",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"SetRel"... | [
"case mp\nX 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\n⊢ ∃ f, graphFunctor.map f.hom = r"
] | simp only [RelCat.Hom.rel_comp_apply₂, RelCat.Hom.rel_id_apply₂, eq_iff_iff] at h1 h2 | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.EffectiveEpi.Coproduct | {
"line": 88,
"column": 2
} | {
"line": 89,
"column": 66
} | {
"line": 90,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nB : C\nα : Type u_2\nX : α → C\nπ : (a : α) → X a ⟶ B\ninst✝³ : HasCoproduct X\ninst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)\ninst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a ↦ pullback g (Sigma.ι X a)\ninst✝ : ∀ {Z : C} (g : Z ... | [
"C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nB : C\nα : Type u_2\nX : α → C\nπ : (a : α) → X a ⟶ B\ninst✝³ : HasCoproduct X\ninst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)\ninst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a ↦ pullback g (Sigma.ι X a)\ninst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi ... | apply_fun ((Sigma.ι (fun a ↦ pullback _ _) b) ≫ (Sigma.desc fun a ↦
pullback.fst (pullback.fst _ _ ≫ g₂) (Sigma.ι X a)) ≫ ·) at hg | Mathlib.Tactic._aux_Mathlib_Tactic_ApplyFun___elabRules_Mathlib_Tactic_applyFun_1 | Mathlib.Tactic.applyFun |
Mathlib.CategoryTheory.FiberedCategory.HomLift | {
"line": 264,
"column": 2
} | {
"line": 265,
"column": 70
} | {
"line": 267,
"column": 0
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₂} 𝒳\ninst✝¹ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ≅ b\ninst✝ : p.IsHomLift f φ.hom\n⊢ p.IsHomLift (isoOfIsoLift p f φ).inv φ.inv",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Categor... | [] | apply of_commSq
apply CommSq.horiz_inv (f := p.mapIso φ) (by apply commSq p f φ.hom) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.FiberedCategory.HomLift | {
"line": 264,
"column": 2
} | {
"line": 265,
"column": 70
} | {
"line": 267,
"column": 0
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₂} 𝒳\ninst✝¹ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ≅ b\ninst✝ : p.IsHomLift f φ.hom\n⊢ p.IsHomLift (isoOfIsoLift p f φ).inv φ.inv",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Categor... | [] | apply of_commSq
apply CommSq.horiz_inv (f := p.mapIso φ) (by apply commSq p f φ.hom) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.FiberedCategory.BasedCategory | {
"line": 83,
"column": 14
} | {
"line": 83,
"column": 28
} | {
"line": 83,
"column": 29
} | [
{
"pp": "𝒮 : Type u₁\ninst✝ : Category.{v₁, u₁} 𝒮\n𝒳 : BasedCategory 𝒮\n𝒴 : BasedCategory 𝒮\n𝒵 : BasedCategory 𝒮\nF : 𝒳 ⥤ᵇ 𝒴\nG : 𝒴 ⥤ᵇ 𝒵\n⊢ (F.toFunctor ⋙ G.toFunctor) ⋙ 𝒵.p = 𝒳.p",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor",
... | [
"𝒮 : Type u₁\ninst✝ : Category.{v₁, u₁} 𝒮\n𝒳 : BasedCategory 𝒮\n𝒴 : BasedCategory 𝒮\n𝒵 : BasedCategory 𝒮\nF : 𝒳 ⥤ᵇ 𝒴\nG : 𝒴 ⥤ᵇ 𝒵\n⊢ F.toFunctor ⋙ G.toFunctor ⋙ 𝒵.p = 𝒳.p"
] | Functor.assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.FiberedCategory.Cartesian | {
"line": 393,
"column": 2
} | {
"line": 394,
"column": 39
} | {
"line": 395,
"column": 2
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝³ : Category.{v₁, u₁} 𝒮\ninst✝² : Category.{v₂, u₂} 𝒳\np : 𝒳 ⥤ 𝒮\nR R' S : 𝒮\na a' b : 𝒳\nf : R ⟶ S\nf' : R' ⟶ S\ng : R' ≅ R\nh : f' = g.hom ≫ f\nφ : a ⟶ b\nφ' : a' ⟶ b\ninst✝¹ : p.IsStronglyCartesian f φ\ninst✝ : p.IsStronglyCartesian f' φ'\n⊢ p.IsHomLift g.inv (... | [
"𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝³ : Category.{v₁, u₁} 𝒮\ninst✝² : Category.{v₂, u₂} 𝒳\np : 𝒳 ⥤ 𝒮\nR R' S : 𝒮\na a' b : 𝒳\nf : R ⟶ S\nf' : R' ⟶ S\ng : R' ≅ R\nh : f' = g.hom ≫ f\nφ : a ⟶ b\nφ' : a' ⟶ b\ninst✝¹ : p.IsStronglyCartesian f φ\ninst✝ : p.IsStronglyCartesian f' φ'\nthis : p.IsHomLift ((fun x ↦ g.in... | have : p.IsHomLift ((fun x ↦ g.inv ≫ x) (g.hom ≫ f)) φ := by
simpa using IsCartesian.toIsHomLift | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Galois.Examples | {
"line": 71,
"column": 2
} | {
"line": 73,
"column": 29
} | {
"line": 75,
"column": 0
} | [
{
"pp": "G : Type u\ninst✝ : Group G\nX Y : Action FintypeCat G\nf : X ⟶ Y\n⊢ Mono (Action.imageComplementIncl G f)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"CategoryTheory.instFaithfulForget",
"Action.instFunLikeHomSubtypeV",
"CategoryTheory.ConcreteCategory.hom",... | [] | apply Functor.mono_of_mono_map (forget _)
apply ConcreteCategory.mono_of_injective
exact Subtype.val_injective | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Galois.Examples | {
"line": 71,
"column": 2
} | {
"line": 73,
"column": 29
} | {
"line": 75,
"column": 0
} | [
{
"pp": "G : Type u\ninst✝ : Group G\nX Y : Action FintypeCat G\nf : X ⟶ Y\n⊢ Mono (Action.imageComplementIncl G f)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"CategoryTheory.instFaithfulForget",
"Action.instFunLikeHomSubtypeV",
"CategoryTheory.ConcreteCategory.hom",... | [] | apply Functor.mono_of_mono_map (forget _)
apply ConcreteCategory.mono_of_injective
exact Subtype.val_injective | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Galois.Examples | {
"line": 137,
"column": 35
} | {
"line": 151,
"column": 53
} | {
"line": 153,
"column": 0
} | [
{
"pp": "G : Type u\ninst✝² : Group G\nX : FintypeCat\ninst✝¹ : MulAction G X.obj\ninst✝ : MulAction.IsPretransitive G X.obj\nh : Nonempty X.obj\nY : Action FintypeCat G\ni : Y ⟶ Action.FintypeCat.ofMulAction G X\nhm : Mono i\nhni : ∀ (a : IsInitial Y), False\n⊢ IsIso i",
"ppTerm": "?m.24",
"assigned": ... | [] | by
/- We show that the induced inclusion `i.hom` of finite sets is surjective, using the
transitivity of the `G`-action. -/
obtain ⟨(y : Y.V)⟩ := (not_initial_iff_fiber_nonempty (Action.forget _ _) Y).mp hni
have : IsIso i.hom := by
refine (ConcreteCategory.isIso_iff_bijective i.hom).mpr ⟨?_, fun ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Groupoid.FreeGroupoidOfCategory | {
"line": 203,
"column": 51
} | {
"line": 204,
"column": 30
} | {
"line": 206,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\n⊢ map (𝟭 C) = 𝟭 (FreeGroupoid C)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"CategoryTheory.FreeGroupoid",
"CategoryTheory.Functor",
"CategoryTheory.FreeGroupoid.map",
"CategoryTheory.Functor.comp",
"Catego... | [] | by
symm; apply lift_unique; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Groupoid.FreeGroupoidOfCategory | {
"line": 218,
"column": 77
} | {
"line": 219,
"column": 30
} | {
"line": 221,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nE : Type u₂\ninst✝ : Category.{v₂, u₂} E\nφ : C ⥤ D\nφ' : D ⥤ E\n⊢ map (φ ⋙ φ') = map φ ⋙ map φ'",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"CategoryTheory.FreeGroupoid",
"CategoryT... | [] | by
symm; apply lift_unique; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Groupoid.FreeGroupoidOfCategory | {
"line": 232,
"column": 36
} | {
"line": 232,
"column": 50
} | {
"line": 232,
"column": 51
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nE : Type u₂\ninst✝ : Groupoid E\nF : C ⥤ D\nG : D ⥤ E\n⊢ (F ⋙ of D) ⋙ lift G = F ⋙ G",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"CategoryTheory.FreeGroupoid",
"Eq.mpr",
"Categ... | [
"C : Type u\ninst✝² : Category.{v, u} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nE : Type u₂\ninst✝ : Groupoid E\nF : C ⥤ D\nG : D ⥤ E\n⊢ F ⋙ of D ⋙ lift G = F ⋙ G"
] | Functor.assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Groupoid.FreeGroupoidOfCategory | {
"line": 267,
"column": 17
} | {
"line": 267,
"column": 40
} | {
"line": 268,
"column": 2
} | [
{
"pp": "C✝ : Type u\ninst✝ : Category.{v, u} C✝\nC : Cat\n⊢ map (𝟙 C).toFunctor = 𝟙 (of (FreeGroupoid ↑C))",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"CategoryTheory.FreeGroupoid",
"CategoryTheory.Cat.category",
"CategoryTheory.Functor",
"CategoryTheory.Cate... | [] | simp [map_id, id_eq_id] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Groupoid.FreeGroupoidOfCategory | {
"line": 267,
"column": 17
} | {
"line": 267,
"column": 40
} | {
"line": 268,
"column": 2
} | [
{
"pp": "C✝ : Type u\ninst✝ : Category.{v, u} C✝\nC : Cat\n⊢ map (𝟙 C).toFunctor = 𝟙 (of (FreeGroupoid ↑C))",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"CategoryTheory.FreeGroupoid",
"CategoryTheory.Cat.category",
"CategoryTheory.Functor",
"CategoryTheory.Cate... | [] | simp [map_id, id_eq_id] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Groupoid.FreeGroupoidOfCategory | {
"line": 267,
"column": 17
} | {
"line": 267,
"column": 40
} | {
"line": 268,
"column": 2
} | [
{
"pp": "C✝ : Type u\ninst✝ : Category.{v, u} C✝\nC : Cat\n⊢ map (𝟙 C).toFunctor = 𝟙 (of (FreeGroupoid ↑C))",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"CategoryTheory.FreeGroupoid",
"CategoryTheory.Cat.category",
"CategoryTheory.Functor",
"CategoryTheory.Cate... | [] | simp [map_id, id_eq_id] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.GuitartExact.Over | {
"line": 65,
"column": 4
} | {
"line": 65,
"column": 28
} | {
"line": 66,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nF : C ⥤ D\nX : C\ninst✝¹ : ∀ (Y : C), HasBinaryProduct X Y\ninst✝ : ∀ (Y : C), PreservesLimit (pair X Y) F\nW : Over (F.obj X)\nZ : C\ng : (Over.forget (F.obj X)).obj W ⟶ F.obj Z\nP : (TwoSquare.overPost F X).Structur... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nF : C ⥤ D\nX : C\ninst✝¹ : ∀ (Y : C), HasBinaryProduct X Y\ninst✝ : ∀ (Y : C), PreservesLimit (pair X Y) F\nW : Over (F.obj X)\nZ : C\ng : (Over.forget (F.obj X)).obj W ⟶ F.obj Z\nP : (TwoSquare.overPost F X).StructuredArrowRight... | have := Nonempty.intro P | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.GradedObject.Braiding | {
"line": 119,
"column": 14
} | {
"line": 125,
"column": 53
} | {
"line": 126,
"column": 2
} | [
{
"pp": "I : Type u_1\ninst✝²⁰ : AddCommMonoid I\nC : Type u_2\ninst✝¹⁹ : Category.{v_1, u_2} C\ninst✝¹⁸ : MonoidalCategory C\nX Y Z : GradedObject I C\ninst✝¹⁷ : BraidedCategory C\ninst✝¹⁶ : X.HasTensor Y\ninst✝¹⁵ : Y.HasTensor Z\ninst✝¹⁴ : Z.HasTensor X\ninst✝¹³ : Z.HasTensor Y\ninst✝¹² : X.HasTensor Z\ninst✝... | [
"I : Type u_1\ninst✝²⁰ : AddCommMonoid I\nC : Type u_2\ninst✝¹⁹ : Category.{v_1, u_2} C\ninst✝¹⁸ : MonoidalCategory C\nX Y Z : GradedObject I C\ninst✝¹⁷ : BraidedCategory C\ninst✝¹⁶ : X.HasTensor Y\ninst✝¹⁵ : Y.HasTensor Z\ninst✝¹⁴ : Z.HasTensor X\ninst✝¹³ : Z.HasTensor Y\ninst✝¹² : X.HasTensor Z\ninst✝¹¹ : (tensor... | rw [ιTensorObj₃_associator_inv_assoc, ιTensorObj₃'_eq X Y Z i₁ i₂ i₃ k h _ rfl, assoc,
ι_tensorObjDesc_assoc, assoc, ← MonoidalCategory.tensorHom_id,
BraidedCategory.braiding_naturality_assoc,
BraidedCategory.braiding_tensor_left_hom, assoc, assoc, assoc, assoc, Iso.inv_hom_id_assoc,
MonoidalCategory.id... | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.CategoryTheory.GradedObject.Braiding | {
"line": 119,
"column": 14
} | {
"line": 125,
"column": 53
} | {
"line": 126,
"column": 2
} | [
{
"pp": "I : Type u_1\ninst✝²⁰ : AddCommMonoid I\nC : Type u_2\ninst✝¹⁹ : Category.{v_1, u_2} C\ninst✝¹⁸ : MonoidalCategory C\nX Y Z : GradedObject I C\ninst✝¹⁷ : BraidedCategory C\ninst✝¹⁶ : X.HasTensor Y\ninst✝¹⁵ : Y.HasTensor Z\ninst✝¹⁴ : Z.HasTensor X\ninst✝¹³ : Z.HasTensor Y\ninst✝¹² : X.HasTensor Z\ninst✝... | [
"I : Type u_1\ninst✝²⁰ : AddCommMonoid I\nC : Type u_2\ninst✝¹⁹ : Category.{v_1, u_2} C\ninst✝¹⁸ : MonoidalCategory C\nX Y Z : GradedObject I C\ninst✝¹⁷ : BraidedCategory C\ninst✝¹⁶ : X.HasTensor Y\ninst✝¹⁵ : Y.HasTensor Z\ninst✝¹⁴ : Z.HasTensor X\ninst✝¹³ : Z.HasTensor Y\ninst✝¹² : X.HasTensor Z\ninst✝¹¹ : (tensor... | rw [ιTensorObj₃_associator_inv_assoc, ιTensorObj₃'_eq X Y Z i₁ i₂ i₃ k h _ rfl, assoc,
ι_tensorObjDesc_assoc, assoc, ← MonoidalCategory.tensorHom_id,
BraidedCategory.braiding_naturality_assoc,
BraidedCategory.braiding_tensor_left_hom, assoc, assoc, assoc, assoc, Iso.inv_hom_id_assoc,
MonoidalCategory.id... | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.CategoryTheory.GradedObject.Braiding | {
"line": 119,
"column": 14
} | {
"line": 125,
"column": 53
} | {
"line": 126,
"column": 2
} | [
{
"pp": "I : Type u_1\ninst✝²⁰ : AddCommMonoid I\nC : Type u_2\ninst✝¹⁹ : Category.{v_1, u_2} C\ninst✝¹⁸ : MonoidalCategory C\nX Y Z : GradedObject I C\ninst✝¹⁷ : BraidedCategory C\ninst✝¹⁶ : X.HasTensor Y\ninst✝¹⁵ : Y.HasTensor Z\ninst✝¹⁴ : Z.HasTensor X\ninst✝¹³ : Z.HasTensor Y\ninst✝¹² : X.HasTensor Z\ninst✝... | [
"I : Type u_1\ninst✝²⁰ : AddCommMonoid I\nC : Type u_2\ninst✝¹⁹ : Category.{v_1, u_2} C\ninst✝¹⁸ : MonoidalCategory C\nX Y Z : GradedObject I C\ninst✝¹⁷ : BraidedCategory C\ninst✝¹⁶ : X.HasTensor Y\ninst✝¹⁵ : Y.HasTensor Z\ninst✝¹⁴ : Z.HasTensor X\ninst✝¹³ : Z.HasTensor Y\ninst✝¹² : X.HasTensor Z\ninst✝¹¹ : (tensor... | rw [ιTensorObj₃_associator_inv_assoc, ιTensorObj₃'_eq X Y Z i₁ i₂ i₃ k h _ rfl, assoc,
ι_tensorObjDesc_assoc, assoc, ← MonoidalCategory.tensorHom_id,
BraidedCategory.braiding_naturality_assoc,
BraidedCategory.braiding_tensor_left_hom, assoc, assoc, assoc, assoc, Iso.inv_hom_id_assoc,
MonoidalCategory.id... | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.CategoryTheory.IsoCat | {
"line": 75,
"column": 8
} | {
"line": 75,
"column": 22
} | {
"line": 75,
"column": 23
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Category.{v_3, u_3} E\nF : C ⥤ D\nG : D ⥤ E\ne : IsoCat C D\nf : IsoCat D E\n⊢ 𝟭 C = (e.functor ⋙ f.functor) ⋙ f.inverse ⋙ e.inverse",
"ppTerm": "?m.45",
"assigned": true,
"use... | [
"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Category.{v_3, u_3} E\nF : C ⥤ D\nG : D ⥤ E\ne : IsoCat C D\nf : IsoCat D E\n⊢ 𝟭 C = e.functor ⋙ f.functor ⋙ f.inverse ⋙ e.inverse"
] | Functor.assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.IsoCat | {
"line": 78,
"column": 8
} | {
"line": 78,
"column": 22
} | {
"line": 78,
"column": 23
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Category.{v_3, u_3} E\nF : C ⥤ D\nG : D ⥤ E\ne : IsoCat C D\nf : IsoCat D E\n⊢ (f.inverse ⋙ e.inverse) ⋙ e.functor ⋙ f.functor = 𝟭 E",
"ppTerm": "?m.88",
"assigned": true,
"use... | [
"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Category.{v_3, u_3} E\nF : C ⥤ D\nG : D ⥤ E\ne : IsoCat C D\nf : IsoCat D E\n⊢ f.inverse ⋙ e.inverse ⋙ e.functor ⋙ f.functor = 𝟭 E"
] | Functor.assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.FormalCoproducts.Basic | {
"line": 98,
"column": 19
} | {
"line": 98,
"column": 32
} | {
"line": 99,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u₁\ninst✝ : Category.{v₁, u₁} A\nX Y : FormalCoproduct C\ne : X.I ≃ Y.I\nh : (i : X.I) → X.obj i ≅ Y.obj (e i)\n⊢ { f := ⇑e, φ := fun i ↦ (h i).hom } ≫ { f := ⇑e.symm, φ := fun i ↦ eqToHom ⋯ ≫ (h (e.symm i)).inv } = 𝟙 X",
"ppTerm": "?m.54",
"ass... | [] | ext <;> aesop | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Limits.FormalCoproducts.Basic | {
"line": 98,
"column": 19
} | {
"line": 98,
"column": 32
} | {
"line": 99,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u₁\ninst✝ : Category.{v₁, u₁} A\nX Y : FormalCoproduct C\ne : X.I ≃ Y.I\nh : (i : X.I) → X.obj i ≅ Y.obj (e i)\n⊢ { f := ⇑e, φ := fun i ↦ (h i).hom } ≫ { f := ⇑e.symm, φ := fun i ↦ eqToHom ⋯ ≫ (h (e.symm i)).inv } = 𝟙 X",
"ppTerm": "?m.54",
"ass... | [] | ext <;> aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.FormalCoproducts.Basic | {
"line": 98,
"column": 19
} | {
"line": 98,
"column": 32
} | {
"line": 99,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u₁\ninst✝ : Category.{v₁, u₁} A\nX Y : FormalCoproduct C\ne : X.I ≃ Y.I\nh : (i : X.I) → X.obj i ≅ Y.obj (e i)\n⊢ { f := ⇑e, φ := fun i ↦ (h i).hom } ≫ { f := ⇑e.symm, φ := fun i ↦ eqToHom ⋯ ≫ (h (e.symm i)).inv } = 𝟙 X",
"ppTerm": "?m.54",
"ass... | [] | ext <;> aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.FormalCoproducts.Basic | {
"line": 99,
"column": 19
} | {
"line": 99,
"column": 32
} | {
"line": 101,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u₁\ninst✝ : Category.{v₁, u₁} A\nX Y : FormalCoproduct C\ne : X.I ≃ Y.I\nh : (i : X.I) → X.obj i ≅ Y.obj (e i)\n⊢ { f := ⇑e.symm, φ := fun i ↦ eqToHom ⋯ ≫ (h (e.symm i)).inv } ≫ { f := ⇑e, φ := fun i ↦ (h i).hom } = 𝟙 Y",
"ppTerm": "?m.72",
"ass... | [] | ext <;> aesop | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Limits.FormalCoproducts.Basic | {
"line": 99,
"column": 19
} | {
"line": 99,
"column": 32
} | {
"line": 101,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u₁\ninst✝ : Category.{v₁, u₁} A\nX Y : FormalCoproduct C\ne : X.I ≃ Y.I\nh : (i : X.I) → X.obj i ≅ Y.obj (e i)\n⊢ { f := ⇑e.symm, φ := fun i ↦ eqToHom ⋯ ≫ (h (e.symm i)).inv } ≫ { f := ⇑e, φ := fun i ↦ (h i).hom } = 𝟙 Y",
"ppTerm": "?m.72",
"ass... | [] | ext <;> aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.FormalCoproducts.Basic | {
"line": 99,
"column": 19
} | {
"line": 99,
"column": 32
} | {
"line": 101,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u₁\ninst✝ : Category.{v₁, u₁} A\nX Y : FormalCoproduct C\ne : X.I ≃ Y.I\nh : (i : X.I) → X.obj i ≅ Y.obj (e i)\n⊢ { f := ⇑e.symm, φ := fun i ↦ eqToHom ⋯ ≫ (h (e.symm i)).inv } ≫ { f := ⇑e, φ := fun i ↦ (h i).hom } = 𝟙 Y",
"ppTerm": "?m.72",
"ass... | [] | ext <;> aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.FormalCoproducts.Basic | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 15
} | {
"line": 127,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : C\nY : FormalCoproduct C\nf : (incl C).obj X ⟶ Y\n⊢ fromIncl (asSigma f).fst (asSigma f).snd = f",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.FormalCoproduct.Hom.asSigma",
"CategoryThe... | [] | ext <;> aesop | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Limits.FormalCoproducts.Basic | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 15
} | {
"line": 127,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : C\nY : FormalCoproduct C\nf : (incl C).obj X ⟶ Y\n⊢ fromIncl (asSigma f).fst (asSigma f).snd = f",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.FormalCoproduct.Hom.asSigma",
"CategoryThe... | [] | ext <;> aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.FormalCoproducts.Basic | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 15
} | {
"line": 127,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : C\nY : FormalCoproduct C\nf : (incl C).obj X ⟶ Y\n⊢ fromIncl (asSigma f).fst (asSigma f).snd = f",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.FormalCoproduct.Hom.asSigma",
"CategoryThe... | [] | ext <;> aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.PiProd | {
"line": 57,
"column": 8
} | {
"line": 57,
"column": 30
} | {
"line": 57,
"column": 30
} | [
{
"pp": "case neg\nC : Type u_1\nI : Type u_2\ninst✝⁷ : Category.{v_1, u_1} C\nX Y : I → C\nf : (i : I) → X i ⟶ Y i\nP : I → Prop\ninst✝⁶ : HasProduct X\ninst✝⁵ : HasProduct Y\ninst✝⁴ : HasProduct fun i ↦ X ↑i\ninst✝³ : HasProduct fun i ↦ X ↑i\ninst✝² : HasProduct fun i ↦ Y ↑i\ninst✝¹ : HasProduct fun i ↦ Y ↑i\... | [] | simp [← h₂, dif_neg h] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Limits.Shapes.PiProd | {
"line": 57,
"column": 8
} | {
"line": 57,
"column": 30
} | {
"line": 57,
"column": 30
} | [
{
"pp": "case neg\nC : Type u_1\nI : Type u_2\ninst✝⁷ : Category.{v_1, u_1} C\nX Y : I → C\nf : (i : I) → X i ⟶ Y i\nP : I → Prop\ninst✝⁶ : HasProduct X\ninst✝⁵ : HasProduct Y\ninst✝⁴ : HasProduct fun i ↦ X ↑i\ninst✝³ : HasProduct fun i ↦ X ↑i\ninst✝² : HasProduct fun i ↦ Y ↑i\ninst✝¹ : HasProduct fun i ↦ Y ↑i\... | [] | simp [← h₂, dif_neg h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.PiProd | {
"line": 57,
"column": 8
} | {
"line": 57,
"column": 30
} | {
"line": 57,
"column": 30
} | [
{
"pp": "case neg\nC : Type u_1\nI : Type u_2\ninst✝⁷ : Category.{v_1, u_1} C\nX Y : I → C\nf : (i : I) → X i ⟶ Y i\nP : I → Prop\ninst✝⁶ : HasProduct X\ninst✝⁵ : HasProduct Y\ninst✝⁴ : HasProduct fun i ↦ X ↑i\ninst✝³ : HasProduct fun i ↦ X ↑i\ninst✝² : HasProduct fun i ↦ Y ↑i\ninst✝¹ : HasProduct fun i ↦ Y ↑i\... | [] | simp [← h₂, dif_neg h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.ChosenPullback | {
"line": 117,
"column": 34
} | {
"line": 117,
"column": 76
} | {
"line": 119,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX S : C\nf : X ⟶ S\nh : ChosenPullback f f\n⊢ Nonempty h.Diagonal",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryTheory.CategoryStruct.id",
... | [] | by apply LiftStruct.nonempty <;> cat_disch | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.WeakLimits.Basic | {
"line": 62,
"column": 65
} | {
"line": 62,
"column": 73
} | {
"line": 62,
"column": 73
} | [
{
"pp": "J : Type u_1\ninst✝³ : Category.{v_1, u_1} J\nK : Type u_2\ninst✝² : Category.{v_2, u_2} K\nC : Type u_3\ninst✝¹ : Category.{v_3, u_3} C\nF : J ⥤ C\nD : Type u_4\ninst✝ : Category.{v_4, u_4} D\nG : K ⥤ D\nt t' : Cone F\nl : IsLimit t\nl' : IsWeakLimit t'\nx✝ : J\n⊢ l'.lift t ≫ t'.π.app x✝ = t.π.app x✝"... | [
"J : Type u_1\ninst✝³ : Category.{v_1, u_1} J\nK : Type u_2\ninst✝² : Category.{v_2, u_2} K\nC : Type u_3\ninst✝¹ : Category.{v_3, u_3} C\nF : J ⥤ C\nD : Type u_4\ninst✝ : Category.{v_4, u_4} D\nG : K ⥤ D\nt t' : Cone F\nl : IsLimit t\nl' : IsWeakLimit t'\nx✝ : J\n⊢ t.π.app x✝ = t.π.app x✝"
] | l'.fac t | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.LocallyCartesianClosed.ExponentiableMorphism | {
"line": 216,
"column": 2
} | {
"line": 216,
"column": 62
} | {
"line": 218,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nI J K : C\nf : I ⟶ J\ng : J ⟶ K\ninst✝⁵ : ChosenPullbacksAlong f\ninst✝⁴ : ChosenPullbacksAlong g\ninst✝³ : ChosenPullbacksAlong (f ≫ g)\ninst✝² : ExponentiableMorphism f\ninst✝¹ : ExponentiableMorphism g\ninst✝ : ExponentiableMorphism (f ≫ g)\n⊢ Functor.whiskerR... | [] | rw [pushforwardComp, Adjunction.rightAdjointUniq_hom_counit] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.LocallyCartesianClosed.ExponentiableMorphism | {
"line": 216,
"column": 2
} | {
"line": 216,
"column": 62
} | {
"line": 218,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nI J K : C\nf : I ⟶ J\ng : J ⟶ K\ninst✝⁵ : ChosenPullbacksAlong f\ninst✝⁴ : ChosenPullbacksAlong g\ninst✝³ : ChosenPullbacksAlong (f ≫ g)\ninst✝² : ExponentiableMorphism f\ninst✝¹ : ExponentiableMorphism g\ninst✝ : ExponentiableMorphism (f ≫ g)\n⊢ Functor.whiskerR... | [] | rw [pushforwardComp, Adjunction.rightAdjointUniq_hom_counit] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.LocallyCartesianClosed.ExponentiableMorphism | {
"line": 216,
"column": 2
} | {
"line": 216,
"column": 62
} | {
"line": 218,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nI J K : C\nf : I ⟶ J\ng : J ⟶ K\ninst✝⁵ : ChosenPullbacksAlong f\ninst✝⁴ : ChosenPullbacksAlong g\ninst✝³ : ChosenPullbacksAlong (f ≫ g)\ninst✝² : ExponentiableMorphism f\ninst✝¹ : ExponentiableMorphism g\ninst✝ : ExponentiableMorphism (f ≫ g)\n⊢ Functor.whiskerR... | [] | rw [pushforwardComp, Adjunction.rightAdjointUniq_hom_counit] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Localization.Monoidal.Functor | {
"line": 115,
"column": 4
} | {
"line": 116,
"column": 62
} | {
"line": 117,
"column": 2
} | [
{
"pp": "case refine_1\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝¹⁰ : Category.{v_1, u_1} C\ninst✝⁹ : Category.{v_2, u_2} D\ninst✝⁸ : Category.{v_3, u_3} E\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : MonoidalCategory D\ninst✝⁵ : MonoidalCategory E\nL : C ⥤ D\nW : MorphismProperty C\ninst✝⁴ : L.IsLocalization W... | [] | simp [← whisker_exchange_assoc, tensor_whiskerLeft_symm, -tensor_whiskerLeft,
← LaxMonoidal.associativity_assoc G, ← Functor.map_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Localization.Monoidal.Functor | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 82
} | {
"line": 152,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝¹⁰ : Category.{v_1, u_1} C\ninst✝⁹ : Category.{v_2, u_2} D\ninst✝⁸ : Category.{v_3, u_3} E\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : MonoidalCategory D\ninst✝⁵ : MonoidalCategory E\nL : C ⥤ D\nW : MorphismProperty C\ninst✝⁴ : L.IsLocalization W\ninst✝³ : L.Mo... | [] | simp [Functor.CoreMonoidal.toLaxMonoidal_μ, curriedTensorPreIsoPost_hom_app_app] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Localization.Monoidal.Functor | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 82
} | {
"line": 152,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝¹⁰ : Category.{v_1, u_1} C\ninst✝⁹ : Category.{v_2, u_2} D\ninst✝⁸ : Category.{v_3, u_3} E\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : MonoidalCategory D\ninst✝⁵ : MonoidalCategory E\nL : C ⥤ D\nW : MorphismProperty C\ninst✝⁴ : L.IsLocalization W\ninst✝³ : L.Mo... | [] | simp [Functor.CoreMonoidal.toLaxMonoidal_μ, curriedTensorPreIsoPost_hom_app_app] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Localization.Monoidal.Functor | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 82
} | {
"line": 152,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝¹⁰ : Category.{v_1, u_1} C\ninst✝⁹ : Category.{v_2, u_2} D\ninst✝⁸ : Category.{v_3, u_3} E\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : MonoidalCategory D\ninst✝⁵ : MonoidalCategory E\nL : C ⥤ D\nW : MorphismProperty C\ninst✝⁴ : L.IsLocalization W\ninst✝³ : L.Mo... | [] | simp [Functor.CoreMonoidal.toLaxMonoidal_μ, curriedTensorPreIsoPost_hom_app_app] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Action.Basic | {
"line": 209,
"column": 2
} | {
"line": 210,
"column": 6
} | {
"line": 212,
"column": 0
} | [
{
"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\ninst✝¹ : MonoidalCategory C\ninst✝ : MonoidalLeftAction C D\nx y : D\nf : x ⟶ y\n⊢ 𝟙_ C ⊴ₗ f = (λₗ x).hom ≫ f ≫ (λₗ y).inv",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | rw [← Category.assoc, actionUnitIso_hom_naturality]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Action.Basic | {
"line": 209,
"column": 2
} | {
"line": 210,
"column": 6
} | {
"line": 212,
"column": 0
} | [
{
"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\ninst✝¹ : MonoidalCategory C\ninst✝ : MonoidalLeftAction C D\nx y : D\nf : x ⟶ y\n⊢ 𝟙_ C ⊴ₗ f = (λₗ x).hom ≫ f ≫ (λₗ y).inv",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | rw [← Category.assoc, actionUnitIso_hom_naturality]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Action.Basic | {
"line": 520,
"column": 2
} | {
"line": 521,
"column": 6
} | {
"line": 523,
"column": 0
} | [
{
"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\ninst✝¹ : MonoidalCategory C\ninst✝ : MonoidalRightAction C D\nx y : D\nf : x ⟶ y\n⊢ f ⊵ᵣ 𝟙_ C = (ρᵣ x).hom ≫ f ≫ (ρᵣ y).inv",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | rw [← Category.assoc, actionUnitIso_hom_naturality]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Action.Basic | {
"line": 520,
"column": 2
} | {
"line": 521,
"column": 6
} | {
"line": 523,
"column": 0
} | [
{
"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\ninst✝¹ : MonoidalCategory C\ninst✝ : MonoidalRightAction C D\nx y : D\nf : x ⟶ y\n⊢ f ⊵ᵣ 𝟙_ C = (ρᵣ x).hom ≫ f ≫ (ρᵣ y).inv",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | rw [← Category.assoc, actionUnitIso_hom_naturality]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 211,
"column": 66
} | {
"line": 211,
"column": 83
} | {
"line": 211,
"column": 83
} | [
{
"pp": "case a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\nA B : Mon C\nM : Bimod A B\ninst✝¹ : HasCoequalizers C\nR S T : Mon C\nP : Bimod R S\nQ : Bimod S T\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\n| ((α_ R.X P.X S.X).hom ≫ R.X ◁ P.actR... | [
"case a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\nA B : Mon C\nM : Bimod A B\ninst✝¹ : HasCoequalizers C\nR S T : Mon C\nP : Bimod R S\nQ : Bimod S T\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\n| (α_ R.X P.X S.X).hom ▷ Q.X ≫ (R.X ◁ P.actRight ≫... | comp_whiskerRight | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 248,
"column": 75
} | {
"line": 248,
"column": 92
} | {
"line": 248,
"column": 92
} | [
{
"pp": "case a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : HasCoequalizers C\nR S T : Mon C\nP : Bimod R S\nQ : Bimod S T\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\n| (α_ R.X R.X P.X).hom ▷ Q.X ≫ (R.X ◁ P.actLeft ≫ P.actLeft) ▷ Q.X... | [
"case a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : HasCoequalizers C\nR S T : Mon C\nP : Bimod R S\nQ : Bimod S T\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\n| (α_ R.X R.X P.X).hom ▷ Q.X ≫ (R.X ◁ P.actLeft) ▷ Q.X ≫ P.actLeft ▷ Q.X",
"... | comp_whiskerRight | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.Monoidal.Arrow | {
"line": 141,
"column": 2
} | {
"line": 142,
"column": 85
} | {
"line": 144,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : HasPushouts C\ninst✝³ : HasInitial C\ninst✝² : CartesianMonoidalCategory C\ninst✝¹ : MonoidalClosed C\ninst✝ : BraidedCategory C\nX Y Z : Arrow C\n⊢ (α_ X Y Z).inv ≫ (braiding (X ⊗ Y) Z).hom ≫ (α_ Z X Y).inv =\n X ◁ (braiding Y Z).hom ≫ (α_ X Z Y).inv... | [] | refine Arrow.hom_ext _ _ (pushout.hom_ext ?_ (by simp)) (by simp)
apply ((tensorLeft _).map_isPushout (IsPushout.of_hasPushout _ _)).hom_ext <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Arrow | {
"line": 141,
"column": 2
} | {
"line": 142,
"column": 85
} | {
"line": 144,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : HasPushouts C\ninst✝³ : HasInitial C\ninst✝² : CartesianMonoidalCategory C\ninst✝¹ : MonoidalClosed C\ninst✝ : BraidedCategory C\nX Y Z : Arrow C\n⊢ (α_ X Y Z).inv ≫ (braiding (X ⊗ Y) Z).hom ≫ (α_ Z X Y).inv =\n X ◁ (braiding Y Z).hom ≫ (α_ X Z Y).inv... | [] | refine Arrow.hom_ext _ _ (pushout.hom_ext ?_ (by simp)) (by simp)
apply ((tensorLeft _).map_isPushout (IsPushout.of_hasPushout _ _)).hom_ext <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 315,
"column": 4
} | {
"line": 315,
"column": 21
} | {
"line": 315,
"column": 21
} | [
{
"pp": "case a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : HasCoequalizers C\nR S T : Mon C\nP : Bimod R S\nQ : Bimod S T\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n| (α_ P.X Q.X T.X).hom ▷ T.X ≫\n (P.X ◁ Q.actRight ≫ coequaliz... | [
"case a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : HasCoequalizers C\nR S T : Mon C\nP : Bimod R S\nQ : Bimod S T\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n| (α_ P.X Q.X T.X).hom ▷ T.X ≫\n (P.X ◁ Q.actRight) ▷ T.X ≫ coequalizer.π ... | comp_whiskerRight | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.Monoidal.Ring | {
"line": 44,
"column": 2
} | {
"line": 47,
"column": 13
} | {
"line": 48,
"column": 2
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : CartesianMonoidalCategory C\nR : C\ninst✝¹ : MonObj R\ninst✝ : AddMonObj R\nh : R ◁ σ ≫ μ = lift (R ◁ fst R R ≫ μ) (R ◁ snd R R ≫ μ) ≫ σ\nx✝ : C\na b c : x✝ ⟶ R\n⊢ a * (b + c) = a * b + a * c",
"ppTerm": "?refine_1",
"assigned": tr... | [
"case refine_2\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : CartesianMonoidalCategory C\nR : C\ninst✝¹ : MonObj R\ninst✝ : AddMonObj R\nh : ∀ ⦃X : C⦄ (a b c : X ⟶ R), a * (b + c) = a * b + a * c\n⊢ R ◁ σ ≫ μ = lift (R ◁ fst R R ≫ μ) (R ◁ snd R R ≫ μ) ≫ σ"
] | · have := lift a (lift b c) ≫= h
simp only [lift_whiskerLeft_assoc] at this
simp only [Hom.add_def, Hom.mul_def, this, ← Category.assoc]
cat_disch | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 334,
"column": 4
} | {
"line": 334,
"column": 21
} | {
"line": 334,
"column": 21
} | [
{
"pp": "case a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : HasCoequalizers C\nR S T : Mon C\nP : Bimod R S\nQ : Bimod S T\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (... | [
"case a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : HasCoequalizers C\nR S T : Mon C\nP : Bimod R S\nQ : Bimod S T\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight ... | comp_whiskerRight | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Basic | {
"line": 74,
"column": 8
} | {
"line": 74,
"column": 85
} | {
"line": 75,
"column": 8
} | [
{
"pp": "case e_a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalClosed C\nJ : Type u₂\ninst✝¹ : Category.{v₂, u₂} J\ninst✝ : ∀ (F₁ F₂ : J ⥤ C), HasFunctorEnrichedHom C F₁ F₂\nF₁ F₂ F₂' F₃ F₃' : J ⥤ C\ng : F₂ ⟶ functorEnrichedHom C F₁ F₃\nj j' : J\nφ : j ⟶ j'\n⊢ enrich... | [
"case e_a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalClosed C\nJ : Type u₂\ninst✝¹ : Category.{v₂, u₂} J\ninst✝ : ∀ (F₁ F₂ : J ⥤ C), HasFunctorEnrichedHom C F₁ F₂\nF₁ F₂ F₂' F₃ F₃' : J ⥤ C\ng : F₂ ⟶ functorEnrichedHom C F₁ F₃\nj j' : J\nφ : j ⟶ j'\nα : Under.mk (𝟙 j) ... | let α : Under.mk (𝟙 j) ⟶ (Under.map φ).obj (Under.mk (𝟙 j')) := Under.homMk φ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 399,
"column": 68
} | {
"line": 399,
"column": 85
} | {
"line": 399,
"column": 85
} | [
{
"pp": "case a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n... | [
"case a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nX Y Z : Mon ... | comp_whiskerRight | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 412,
"column": 56
} | {
"line": 412,
"column": 73
} | {
"line": 412,
"column": 73
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon C\nM : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nX Y Z : M... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon C\nM : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nX Y Z : Mon C\nM₁ M₂ ... | comp_whiskerRight | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 433,
"column": 6
} | {
"line": 433,
"column": 23
} | {
"line": 433,
"column": 23
} | [
{
"pp": "case a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon C\nM : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nX... | [
"case a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon C\nM : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nX Y Z : Mon C... | comp_whiskerRight | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 473,
"column": 27
} | {
"line": 473,
"column": 44
} | {
"line": 473,
"column": 44
} | [
{
"pp": "case a.a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon C\nM : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\... | [
"case a.a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon C\nM : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nR S T U : M... | comp_whiskerRight | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 500,
"column": 25
} | {
"line": 500,
"column": 42
} | {
"line": 500,
"column": 42
} | [
{
"pp": "case a.a.a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nR S T U : Mon C\nP : Bimo... | [
"case a.a.a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nR S T U : Mon C\nP : Bimod R S\nQ : B... | comp_whiskerRight | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 521,
"column": 64
} | {
"line": 521,
"column": 81
} | {
"line": 521,
"column": 81
} | [
{
"pp": "case a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nR S T U : Mon C\nP : Bimod R ... | [
"case a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nR S T U : Mon C\nP : Bimod R S\nQ : Bimod... | comp_whiskerRight | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 530,
"column": 25
} | {
"line": 530,
"column": 42
} | {
"line": 530,
"column": 42
} | [
{
"pp": "case a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nR S T U : Mon C\nP : Bimod R ... | [
"case a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nR S T U : Mon C\nP : Bimod R S\nQ : Bimod... | comp_whiskerRight | Lean.Elab.Tactic.Conv.evalRewrite | null |
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