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 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.CategoryTheory.Monoidal.Category | {
"line": 387,
"column": 2
} | {
"line": 387,
"column": 61
} | {
"line": 389,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : MonoidalCategory C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsIso f\nZ : C\n⊢ f ▷ Z ≫ inv f ▷ Z = 𝟙 (X ⊗ Z)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | [] | rw [← comp_whiskerRight, IsIso.hom_inv_id, id_whiskerRight] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Monoidal.Category | {
"line": 387,
"column": 2
} | {
"line": 387,
"column": 61
} | {
"line": 389,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : MonoidalCategory C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsIso f\nZ : C\n⊢ f ▷ Z ≫ inv f ▷ Z = 𝟙 (X ⊗ Z)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | [] | rw [← comp_whiskerRight, IsIso.hom_inv_id, id_whiskerRight] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Category | {
"line": 387,
"column": 2
} | {
"line": 387,
"column": 61
} | {
"line": 389,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : MonoidalCategory C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsIso f\nZ : C\n⊢ f ▷ Z ≫ inv f ▷ Z = 𝟙 (X ⊗ Z)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | [] | rw [← comp_whiskerRight, IsIso.hom_inv_id, id_whiskerRight] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Category | {
"line": 621,
"column": 2
} | {
"line": 622,
"column": 6
} | {
"line": 624,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nW X Y Z : C\n⊢ (α_ W X (Y ⊗ Z)).hom ≫ W ◁ (α_ X Y Z).inv ≫ (α_ W (X ⊗ Y) Z).inv = (α_ (W ⊗ X) Y Z).inv ≫ (α_ W X Y).hom ▷ Z",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Cat... | [] | rw [← cancel_epi (α_ W X (Y ⊗ Z)).inv, ← cancel_mono ((α_ W X Y).inv ▷ Z)]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Category | {
"line": 621,
"column": 2
} | {
"line": 622,
"column": 6
} | {
"line": 624,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nW X Y Z : C\n⊢ (α_ W X (Y ⊗ Z)).hom ≫ W ◁ (α_ X Y Z).inv ≫ (α_ W (X ⊗ Y) Z).inv = (α_ (W ⊗ X) Y Z).inv ≫ (α_ W X Y).hom ▷ Z",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Cat... | [] | rw [← cancel_epi (α_ W X (Y ⊗ Z)).inv, ← cancel_mono ((α_ W X Y).inv ▷ Z)]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Functor | {
"line": 927,
"column": 4
} | {
"line": 927,
"column": 79
} | {
"line": 928,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C\ninst✝⁶ : MonoidalCategory C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : MonoidalCategory D\nE : Type u₃\ninst✝³ : Category.{v₃, u₃} E\ninst✝² : MonoidalCategory E\nC' : Type u₁'\ninst✝¹ : Category.{v₁', u₁'} C'\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\ninst... | [
"C : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C\ninst✝⁶ : MonoidalCategory C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : MonoidalCategory D\nE : Type u₃\ninst✝³ : Category.{v₃, u₃} E\ninst✝² : MonoidalCategory E\nC' : Type u₁'\ninst✝¹ : Category.{v₁', u₁'} C'\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\ninst✝ : F.OplaxM... | rw [← δ_natural_left_assoc, ← δ_natural_left_assoc, ← δ_natural_left_assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Adjunction.Mates | {
"line": 132,
"column": 14
} | {
"line": 132,
"column": 54
} | {
"line": 133,
"column": 2
} | [
{
"pp": "C : Type u₁\nD : Type u₂\nE : Type u₃\nF : Type u₄\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ninst✝ : Category.{v₄, u₄} F\nG : C ⥤ E\nH : D ⥤ F\nL₁ : C ⥤ D\nR₁ : D ⥤ C\nL₂ : E ⥤ F\nR₂ : F ⥤ E\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nα : TwoSquare R₁ H G R₂\nd : D... | [
"C : Type u₁\nD : Type u₂\nE : Type u₃\nF : Type u₄\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ninst✝ : Category.{v₄, u₄} F\nG : C ⥤ E\nH : D ⥤ F\nL₁ : C ⥤ D\nR₁ : D ⥤ C\nL₂ : E ⥤ F\nR₂ : F ⥤ E\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nα : TwoSquare R₁ H G R₂\nd : D\n| L₂.map (... | rw [← (mateEquiv adj₁ adj₂).right_inv α] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.CategoryTheory.Adjunction.Mates | {
"line": 132,
"column": 14
} | {
"line": 132,
"column": 54
} | {
"line": 133,
"column": 2
} | [
{
"pp": "C : Type u₁\nD : Type u₂\nE : Type u₃\nF : Type u₄\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ninst✝ : Category.{v₄, u₄} F\nG : C ⥤ E\nH : D ⥤ F\nL₁ : C ⥤ D\nR₁ : D ⥤ C\nL₂ : E ⥤ F\nR₂ : F ⥤ E\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nα : TwoSquare R₁ H G R₂\nd : D... | [
"C : Type u₁\nD : Type u₂\nE : Type u₃\nF : Type u₄\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ninst✝ : Category.{v₄, u₄} F\nG : C ⥤ E\nH : D ⥤ F\nL₁ : C ⥤ D\nR₁ : D ⥤ C\nL₂ : E ⥤ F\nR₂ : F ⥤ E\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nα : TwoSquare R₁ H G R₂\nd : D\n| L₂.map (... | rw [← (mateEquiv adj₁ adj₂).right_inv α] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.CategoryTheory.Adjunction.Mates | {
"line": 132,
"column": 14
} | {
"line": 132,
"column": 54
} | {
"line": 133,
"column": 2
} | [
{
"pp": "C : Type u₁\nD : Type u₂\nE : Type u₃\nF : Type u₄\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ninst✝ : Category.{v₄, u₄} F\nG : C ⥤ E\nH : D ⥤ F\nL₁ : C ⥤ D\nR₁ : D ⥤ C\nL₂ : E ⥤ F\nR₂ : F ⥤ E\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nα : TwoSquare R₁ H G R₂\nd : D... | [
"C : Type u₁\nD : Type u₂\nE : Type u₃\nF : Type u₄\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ninst✝ : Category.{v₄, u₄} F\nG : C ⥤ E\nH : D ⥤ F\nL₁ : C ⥤ D\nR₁ : D ⥤ C\nL₂ : E ⥤ F\nR₂ : F ⥤ E\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nα : TwoSquare R₁ H G R₂\nd : D\n| L₂.map (... | rw [← (mateEquiv adj₁ adj₂).right_inv α] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.CategoryTheory.Adjunction.Mates | {
"line": 156,
"column": 14
} | {
"line": 156,
"column": 54
} | {
"line": 157,
"column": 2
} | [
{
"pp": "C : Type u₁\nD : Type u₂\nE : Type u₃\nF : Type u₄\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ninst✝ : Category.{v₄, u₄} F\nG : C ⥤ E\nH : D ⥤ F\nL₁ : C ⥤ D\nR₁ : D ⥤ C\nL₂ : E ⥤ F\nR₂ : F ⥤ E\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nα : TwoSquare R₁ H G R₂\nc : C... | [
"C : Type u₁\nD : Type u₂\nE : Type u₃\nF : Type u₄\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ninst✝ : Category.{v₄, u₄} F\nG : C ⥤ E\nH : D ⥤ F\nL₁ : C ⥤ D\nR₁ : D ⥤ C\nL₂ : E ⥤ F\nR₂ : F ⥤ E\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nα : TwoSquare R₁ H G R₂\nc : C\n| G.map (a... | rw [← (mateEquiv adj₁ adj₂).right_inv α] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.CategoryTheory.Adjunction.Mates | {
"line": 156,
"column": 14
} | {
"line": 156,
"column": 54
} | {
"line": 157,
"column": 2
} | [
{
"pp": "C : Type u₁\nD : Type u₂\nE : Type u₃\nF : Type u₄\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ninst✝ : Category.{v₄, u₄} F\nG : C ⥤ E\nH : D ⥤ F\nL₁ : C ⥤ D\nR₁ : D ⥤ C\nL₂ : E ⥤ F\nR₂ : F ⥤ E\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nα : TwoSquare R₁ H G R₂\nc : C... | [
"C : Type u₁\nD : Type u₂\nE : Type u₃\nF : Type u₄\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ninst✝ : Category.{v₄, u₄} F\nG : C ⥤ E\nH : D ⥤ F\nL₁ : C ⥤ D\nR₁ : D ⥤ C\nL₂ : E ⥤ F\nR₂ : F ⥤ E\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nα : TwoSquare R₁ H G R₂\nc : C\n| G.map (a... | rw [← (mateEquiv adj₁ adj₂).right_inv α] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.CategoryTheory.Adjunction.Mates | {
"line": 156,
"column": 14
} | {
"line": 156,
"column": 54
} | {
"line": 157,
"column": 2
} | [
{
"pp": "C : Type u₁\nD : Type u₂\nE : Type u₃\nF : Type u₄\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ninst✝ : Category.{v₄, u₄} F\nG : C ⥤ E\nH : D ⥤ F\nL₁ : C ⥤ D\nR₁ : D ⥤ C\nL₂ : E ⥤ F\nR₂ : F ⥤ E\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nα : TwoSquare R₁ H G R₂\nc : C... | [
"C : Type u₁\nD : Type u₂\nE : Type u₃\nF : Type u₄\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ninst✝ : Category.{v₄, u₄} F\nG : C ⥤ E\nH : D ⥤ F\nL₁ : C ⥤ D\nR₁ : D ⥤ C\nL₂ : E ⥤ F\nR₂ : F ⥤ E\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nα : TwoSquare R₁ H G R₂\nc : C\n| G.map (a... | rw [← (mateEquiv adj₁ adj₂).right_inv α] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.CategoryTheory.Adjunction.Mates | {
"line": 371,
"column": 2
} | {
"line": 371,
"column": 36
} | {
"line": 373,
"column": 0
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : Category.{v₂, u₂} D\nL₁ L₂ L₃ : C ⥤ D\nR₁ R₂ R₃ : D ⥤ C\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nadj₃ : L₃ ⊣ R₃\nα : R₁ ⟶ R₂\nβ : R₂ ⟶ R₃\n⊢ (conjugateEquiv adj₁ adj₂) ((conjugateEquiv adj₁ adj₂).symm α) ≫\n (conjugateEquiv adj₂ adj₃) ((conju... | [] | simp only [Equiv.apply_symm_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Monoidal.Comon_ | {
"line": 68,
"column": 18
} | {
"line": 68,
"column": 39
} | {
"line": 69,
"column": 2
} | [
{
"pp": "C✝ : Type u₁\ninst✝³ : Category.{v₁, u₁} C✝\ninst✝² : MonoidalCategory C✝\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : MonoidalCategory C\n⊢ (λ_ (𝟙_ C)).inv ≫ 𝟙_ C ◁ 𝟙 (𝟙_ C) = (ρ_ (𝟙_ C)).inv",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"CategoryTheory.Monoid... | [] | by monoidal_coherence | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Comon_ | {
"line": 69,
"column": 17
} | {
"line": 69,
"column": 38
} | {
"line": 71,
"column": 0
} | [
{
"pp": "C✝ : Type u₁\ninst✝³ : Category.{v₁, u₁} C✝\ninst✝² : MonoidalCategory C✝\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : MonoidalCategory C\n⊢ (λ_ (𝟙_ C)).inv ≫ 𝟙_ C ◁ (λ_ (𝟙_ C)).inv = (λ_ (𝟙_ C)).inv ≫ (λ_ (𝟙_ C)).inv ▷ 𝟙_ C ≫ (α_ (𝟙_ C) (𝟙_ C) (𝟙_ C)).hom",
"ppTerm": "?m.28",
"... | [] | by monoidal_coherence | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity | {
"line": 34,
"column": 2
} | {
"line": 35,
"column": 64
} | {
"line": 36,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : WfDvdMonoid α\na₀ x : α\nh : a₀ ≠ 0\nhx : ¬IsUnit x\n⊢ ∃ n a, ¬x ∣ a ∧ a₀ = x ^ n * a",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Dvd.dvd",
"HMul.hMul",
"MulZeroClass.... | [
"α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : WfDvdMonoid α\nx : α\nhx : ¬IsUnit x\na : α\nn : ℕ\nh : x ^ n * a ≠ 0\nhm : ∀ x_1 ∈ {a_1 | ∃ n_1, x ^ n_1 * a_1 = x ^ n * a}, ¬DvdNotUnit x_1 a\n⊢ ∃ n_1 a_1, ¬x ∣ a_1 ∧ x ^ n * a = x ^ n_1 * a_1"
] | obtain ⟨a, ⟨n, rfl⟩, hm⟩ := wellFounded_dvdNotUnit.has_min
{a | ∃ n, x ^ n * a = a₀} ⟨a₀, 0, by rw [pow_zero, one_mul]⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.TwoSidedIdeal.Operations | {
"line": 339,
"column": 76
} | {
"line": 341,
"column": 16
} | {
"line": 341,
"column": 16
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\nI : Ideal R\nJ : TwoSidedIdeal R\nh : I ≤ asIdeal J\nx : R\nhx : x ∈ fromIdeal I\n⊢ x ∈ J",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"Ring.toNonAssocRing",
"TwoSidedIdeal",
"PartialOrder.toPreorder",
... | [] | by
simp only [fromIdeal, OrderHom.coe_mk, mem_span_iff] at hx
exact hx _ h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.LocalRing.Basic | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 51
} | {
"line": 74,
"column": 52
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : IsLocalRing R\na b : R\nu : Rˣ\nhu : ↑u⁻¹ * a + ↑u⁻¹ * b = 1\n⊢ IsUnit a ∨ IsUnit b",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Units.val",
"HMul.hMul",
"Monoid.toMulOneClass",
"CommSemiring.toSemiring",
... | [
"R : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : IsLocalRing R\na b : R\nu : Rˣ\nhu : ↑u⁻¹ * a + ↑u⁻¹ * b = 1\n⊢ IsUnit (↑u⁻¹ * a) → IsUnit a",
"R : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : IsLocalRing R\na b : R\nu : Rˣ\nhu : ↑u⁻¹ * a + ↑u⁻¹ * b = 1\n⊢ IsUnit (↑u⁻¹ * b) → IsUnit b"
] | apply Or.imp _ _ (isUnit_or_isUnit_of_add_one hu) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Monoidal.Mon | {
"line": 133,
"column": 13
} | {
"line": 133,
"column": 34
} | {
"line": 135,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\nM X Y : C\ninst✝ : MonObj M\n⊢ 𝟙_ C ◁ 𝟙 (𝟙_ C) ≫ (λ_ (𝟙_ C)).hom = (ρ_ (𝟙_ C)).hom",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"CategoryTheory.MonoidalCategoryStruct.whiskerLeft",
"Categor... | [] | by monoidal_coherence | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Mon | {
"line": 132,
"column": 15
} | {
"line": 132,
"column": 36
} | {
"line": 133,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\nM X Y : C\ninst✝ : MonObj M\n⊢ (λ_ (𝟙_ C)).hom ▷ 𝟙_ C ≫ (λ_ (𝟙_ C)).hom = (α_ (𝟙_ C) (𝟙_ C) (𝟙_ C)).hom ≫ 𝟙_ C ◁ (λ_ (𝟙_ C)).hom ≫ (λ_ (𝟙_ C)).hom",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
... | [] | by monoidal_coherence | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.StrictInitial | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 24
} | {
"line": 76,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasStrictInitialObjects C\nI : C\nhI : IsInitial I\nA : C\nf g : A ⟶ I\nthis : IsIso f\n⊢ f = g",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.IsInitial.isIso_to"
],
"usedFVars": [
"C",
... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasStrictInitialObjects C\nI : C\nhI : IsInitial I\nA : C\nf g : A ⟶ I\nthis✝ : IsIso f\nthis : IsIso g\n⊢ f = g"
] | haveI := hI.isIso_to g | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.CategoryTheory.Limits.Shapes.StrictInitial | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 44
} | {
"line": 106,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasStrictInitialObjects C\nI X : C\ninst✝ : HasBinaryProduct I X\nhI : IsInitial I\n⊢ I ⨯ X ≅ I",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"CategoryTheory.IsIso",
"CategoryTheory.Limits.IsInitial.isIso_to",
"C... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasStrictInitialObjects C\nI X : C\ninst✝ : HasBinaryProduct I X\nhI : IsInitial I\nthis : IsIso prod.fst\n⊢ I ⨯ X ≅ I"
] | have := hI.isIso_to (prod.fst : I ⨯ X ⟶ I) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Localization.Module | {
"line": 330,
"column": 2
} | {
"line": 330,
"column": 47
} | {
"line": 331,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹⁰ : CommSemiring R\nS : Submonoid R\nM : Type u_2\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\nN : Type u_5\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : Module R N\nM' : Type u_3\nN' : Type u_4\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : AddCommMonoid N'\ninst✝³ : Module R M'\ninst✝² : Module R ... | [
"R : Type u_1\ninst✝¹⁰ : CommSemiring R\nS : Submonoid R\nM : Type u_2\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\nN : Type u_5\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : Module R N\nM' : Type u_3\nN' : Type u_4\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : AddCommMonoid N'\ninst✝³ : Module R M'\ninst✝² : Module R N'\ng₁ : M →... | apply IsLocalizedModule.linearMap_ext S g₁ g₂ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Limits.Shapes.Opposites.Pullbacks | {
"line": 322,
"column": 46
} | {
"line": 324,
"column": 31
} | {
"line": 326,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y Z : C\nf : X ⟶ Y\ng : X ⟶ Z\n⊢ HasPullback f.op g.op ↔ HasPushout f g",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor.op",
"Eq.mpr",
"CategoryTheory.Limits.WalkingSpan",
"CategoryTheory.L... | [] | by
rw [HasPullback, hasLimit_iff_of_iso (cospanOp f g), hasLimit_inverse_equivalence_comp_iff,
hasLimit_op_iff_hasColimit] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Localization.BaseChange | {
"line": 223,
"column": 4
} | {
"line": 223,
"column": 35
} | {
"line": 225,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝¹⁸ : CommSemiring R\nS : Submonoid R\nA : Type u_2\ninst✝¹⁷ : CommSemiring A\ninst✝¹⁶ : Algebra R A\ninst✝¹⁵ : IsLocalization S A\nM : Type u_3\ninst✝¹⁴ : AddCommMonoid M\ninst✝¹³ : Module R M\nM' : Type u_4\ninst✝¹² : AddCommMonoid M'\ninst✝¹¹ : Module R M'\ninst✝¹⁰ : Modu... | [] | simp only [tmul_zero, map_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Localization.BaseChange | {
"line": 223,
"column": 4
} | {
"line": 223,
"column": 35
} | {
"line": 225,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝¹⁸ : CommSemiring R\nS : Submonoid R\nA : Type u_2\ninst✝¹⁷ : CommSemiring A\ninst✝¹⁶ : Algebra R A\ninst✝¹⁵ : IsLocalization S A\nM : Type u_3\ninst✝¹⁴ : AddCommMonoid M\ninst✝¹³ : Module R M\nM' : Type u_4\ninst✝¹² : AddCommMonoid M'\ninst✝¹¹ : Module R M'\ninst✝¹⁰ : Modu... | [] | simp only [tmul_zero, map_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Localization.BaseChange | {
"line": 223,
"column": 4
} | {
"line": 223,
"column": 35
} | {
"line": 225,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝¹⁸ : CommSemiring R\nS : Submonoid R\nA : Type u_2\ninst✝¹⁷ : CommSemiring A\ninst✝¹⁶ : Algebra R A\ninst✝¹⁵ : IsLocalization S A\nM : Type u_3\ninst✝¹⁴ : AddCommMonoid M\ninst✝¹³ : Module R M\nM' : Type u_4\ninst✝¹² : AddCommMonoid M'\ninst✝¹¹ : Module R M'\ninst✝¹⁰ : Modu... | [] | simp only [tmul_zero, map_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.LocalizedModule.Basic | {
"line": 234,
"column": 26
} | {
"line": 234,
"column": 60
} | {
"line": 234,
"column": 60
} | [
{
"pp": "R : Type u\ninst✝¹⁰ : CommSemiring R\nS✝ : Submonoid R\nM : Type v\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\nT : Type u_1\ninst✝⁷ : CommSemiring T\ninst✝⁶ : Algebra R T\ninst✝⁵ : IsLocalization S✝ T\nT' : Type u_2\ninst✝⁴ : CommSemiring T'\ninst✝³ : Algebra R T'\ninst✝² : IsLocalization S✝ T'\nA ... | [] | by simp only [zero_mul, smul_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monad.Algebra | {
"line": 169,
"column": 16
} | {
"line": 169,
"column": 39
} | {
"line": 169,
"column": 40
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nT : Monad C\nX : C\nY : T.Algebra\nf : X ⟶ T.forget.obj Y\n⊢ T.η.app X ≫ T.map f ≫ Y.a = f",
"ppTerm": "?m.91",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryT... | [
"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nT : Monad C\nX : C\nY : T.Algebra\nf : X ⟶ T.forget.obj Y\n⊢ (𝟭 C).map f ≫ T.η.app Y.A ≫ Y.a = f"
] | ← T.η.naturality_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Category.Ring.Constructions | {
"line": 360,
"column": 51
} | {
"line": 368,
"column": 35
} | {
"line": 370,
"column": 0
} | [
{
"pp": "A B : CommRingCat\nf g : A ⟶ B\n⊢ IsLocalHom (Hom.hom (equalizerFork f g).ι)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"CommRingCat.Hom.hom",
"HMul.hMul",
... | [] | by
constructor
rintro ⟨a, h₁ : _ = _⟩ (⟨⟨x, y, h₃, h₄⟩, rfl : x = _⟩ : IsUnit a)
have : y ∈ RingHom.eqLocus f.hom g.hom := by
apply (f.hom.isUnit_map ⟨⟨x, y, h₃, h₄⟩, rfl⟩ : IsUnit (f x)).mul_left_inj.mp
conv_rhs => rw [h₁]
rw [← f.hom.map_mul, ← g.hom.map_mul, h₄, f.hom.map_one, g.hom.map_one]
rw [... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Adjunction.Comma | {
"line": 98,
"column": 17
} | {
"line": 105,
"column": 7
} | {
"line": 107,
"column": 0
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Category.{v₂, u₂} D\nG : D ⥤ C\ninst✝ : ∀ (A : C), HasTerminal (CostructuredArrow G A)\nB : D\nA : C\ng : B ⟶ (⊤_ CostructuredArrow G A).left\n⊢ (fun g ↦ (terminal.from (CostructuredArrow.mk g)).left) ((fun g ↦ G.map g ≫ (⊤_ CostructuredA... | [] | by
let B' : CostructuredArrow G A :=
CostructuredArrow.mk (G.map g ≫ (⊤_ CostructuredArrow G A).hom)
let g' : B' ⟶ ⊤_ CostructuredArrow G A := CostructuredArrow.homMk g rfl
have : terminal.from _ = g' := by cat_disch
change CommaMorphism.left (terminal.from B') = _
rw [this]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Bicategory.Functor.Prelax | {
"line": 201,
"column": 36
} | {
"line": 201,
"column": 49
} | {
"line": 201,
"column": 49
} | [
{
"pp": "B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nF : PrelaxFunctor B C\nx y : B\nf g : x ⟶ y\nhfg : f = g\n⊢ F.map f = F.map g",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategorySt... | [] | by rw [← hfg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Bicategory.Functor.Prelax | {
"line": 207,
"column": 39
} | {
"line": 207,
"column": 52
} | {
"line": 207,
"column": 52
} | [
{
"pp": "B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nF : PrelaxFunctor B C\nx y : B\nf g : x ⟶ y\nhfg : f = g\n⊢ F.map f = F.map g",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategorySt... | [] | by rw [← hfg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.LocalizedModule.Basic | {
"line": 1202,
"column": 2
} | {
"line": 1202,
"column": 70
} | {
"line": 1203,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : Module R M\ninst✝³ : Module R M'\nf : M →ₗ[R] M'\ninst✝² : IsLocalizedModule S f\nN : Type u_6\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\ng : M' →ₗ[R] N... | [
"R : Type u_1\ninst✝⁷ : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : Module R M\ninst✝³ : Module R M'\nf : M →ₗ[R] M'\ninst✝² : IsLocalizedModule S f\nN : Type u_6\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\ng : M' →ₗ[R] N\nH : ∀ {x y... | obtain ⟨⟨y, n⟩, (hym : n • b = f y)⟩ := IsLocalizedModule.surj S f b | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.MorphismProperty.Composition | {
"line": 208,
"column": 4
} | {
"line": 209,
"column": 18
} | {
"line": 211,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\nhf : isomorphisms C f\nhg : isomorphisms C g\n⊢ isomorphisms C (f ≫ g)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Is... | [] | rw [isomorphisms.iff] at hf hg ⊢
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.MorphismProperty.Composition | {
"line": 208,
"column": 4
} | {
"line": 209,
"column": 18
} | {
"line": 211,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\nhf : isomorphisms C f\nhg : isomorphisms C g\n⊢ isomorphisms C (f ≫ g)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Is... | [] | rw [isomorphisms.iff] at hf hg ⊢
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.MorphismProperty.Composition | {
"line": 221,
"column": 4
} | {
"line": 221,
"column": 18
} | {
"line": 223,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\nhf : Epi f\nhg : Epi g\n⊢ Epi (f ≫ g)",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"CategoryTheory.epi_comp"
],
"usedFVars": [
"C",
... | [] | apply epi_comp | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.FinitePresentation | {
"line": 78,
"column": 2
} | {
"line": 93,
"column": 77
} | {
"line": 95,
"column": 0
} | [
{
"pp": "R : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : FinitePresentation R A\ne : A ≃ₐ[R] B\n⊢ FinitePresentation R B",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [] | obtain ⟨n, f, hf⟩ := FinitePresentation.out (R := R) (A := A)
use n, AlgHom.comp (↑e) f
constructor
· rw [AlgHom.coe_comp]
exact Function.Surjective.comp e.surjective hf.1
suffices (RingHom.ker (AlgHom.comp (e : A →ₐ[R] B) f).toRingHom) = RingHom.ker f.toRingHom by
rw [this]
exact hf.2
have hco : ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.FinitePresentation | {
"line": 78,
"column": 2
} | {
"line": 93,
"column": 77
} | {
"line": 95,
"column": 0
} | [
{
"pp": "R : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : FinitePresentation R A\ne : A ≃ₐ[R] B\n⊢ FinitePresentation R B",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [] | obtain ⟨n, f, hf⟩ := FinitePresentation.out (R := R) (A := A)
use n, AlgHom.comp (↑e) f
constructor
· rw [AlgHom.coe_comp]
exact Function.Surjective.comp e.surjective hf.1
suffices (RingHom.ker (AlgHom.comp (e : A →ₐ[R] B) f).toRingHom) = RingHom.ker f.toRingHom by
rw [this]
exact hf.2
have hco : ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Types.Products | {
"line": 175,
"column": 4
} | {
"line": 175,
"column": 58
} | {
"line": 176,
"column": 4
} | [
{
"pp": "case refine_1\nX : Type u\n⊢ binaryProductFunctor.obj X ≅ prod.functor.obj X",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.Limits.prod.functor",
"C... | [
"case refine_1.refine_1\nX Y : Type u\n⊢ (binaryProductFunctor.obj X).obj Y ≅ (prod.functor.obj X).obj Y",
"case refine_1.refine_2\nX X✝ Y✝ : Type u\nx✝ : X✝ ⟶ Y✝\n⊢ (binaryProductFunctor.obj X).map x✝ ≫ Iso.hom ?refine_1.refine_1 =\n Iso.hom ?refine_1.refine_1 ≫ (prod.functor.obj X).map x✝"
] | refine NatIso.ofComponents (fun Y => ?_) (fun _ => ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.FinitePresentation | {
"line": 164,
"column": 2
} | {
"line": 176,
"column": 35
} | {
"line": 178,
"column": 0
} | [
{
"pp": "R : Type w₁\nA : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : FinitePresentation R A\nι : Type v\ninst✝ : Finite ι\nhfp : ∃ ι x f, Surjective ⇑f ∧ (RingHom.ker f.toRingHom).FG\n⊢ ∃ ι_1 x f, Surjective ⇑f ∧ (RingHom.ker f.toRingHom).FG",
"ppTerm": "?m.34",
"a... | [] | classical
-- Make universe level `v` explicit so it matches that of `ι`
obtain ⟨(ι' : Type v), _, f, hf_surj, hf_ker⟩ := hfp
let g := (MvPolynomial.mapAlgHom f).comp (MvPolynomial.sumAlgEquiv R ι ι').toAlgHom
cases nonempty_fintype (ι ⊕ ι')
refine
⟨ι ⊕ ι', by infer_instance, g,
(MvPolynomial.map_sur... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.CategoryTheory.WithTerminal.Basic | {
"line": 227,
"column": 4
} | {
"line": 227,
"column": 27
} | {
"line": 228,
"column": 4
} | [
{
"pp": "case of\nC : Type u\ninst✝ : Category.{v, u} C\na✝¹ b✝ : Cat\nf✝ : a✝¹ ⟶ b✝\na✝ : ↑a✝¹\n⊢ (prelaxfunctor.map₂ (Bicategory.rightUnitor f✝).hom).toNatTrans.app (of a✝) =\n ((Cat.Hom.isoMk (mapComp f✝.toFunctor (𝟙 b✝).toFunctor)).hom ≫\n Bicategory.whiskerLeft (prelaxfunctor.map f✝) (Cat.Ho... | [
"case star\nC : Type u\ninst✝ : Category.{v, u} C\na✝ b✝ : Cat\nf✝ : a✝ ⟶ b✝\n⊢ (prelaxfunctor.map₂ (Bicategory.rightUnitor f✝).hom).toNatTrans.app star =\n ((Cat.Hom.isoMk (mapComp f✝.toFunctor (𝟙 b✝).toFunctor)).hom ≫\n Bicategory.whiskerLeft (prelaxfunctor.map f✝) (Cat.Hom.isoMk (mapId ↑b✝)).hom ≫... | · simpa using! (refl _) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Grothendieck | {
"line": 515,
"column": 4
} | {
"line": 515,
"column": 62
} | {
"line": 517,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u₁\ninst✝ : Category.{v₁, u₁} D\nF : C ⥤ Cat\nG✝ : C ⥤ Type w\nG : D ≌ C\nX : Grothendieck (G.functor ⋙ F)\n⊢ (pre F G.functor).map\n (eqToHom ⋯ ≫\n ((map (G.functor.whiskerLeft (whiskerRight G.counitInv F))).whiskerLeft\n ... | [] | fapply Grothendieck.ext <;> simp [preNatIso, transportIso] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.WithTerminal.Basic | {
"line": 639,
"column": 4
} | {
"line": 639,
"column": 27
} | {
"line": 640,
"column": 4
} | [
{
"pp": "case of\nC : Type u\ninst✝ : Category.{v, u} C\na✝¹ b✝ : Cat\nf✝ : a✝¹ ⟶ b✝\na✝ : ↑a✝¹\n⊢ (prelaxfunctor.map₂ (Bicategory.rightUnitor f✝).hom).toNatTrans.app (of a✝) =\n ((Cat.Hom.isoMk (mapComp f✝.toFunctor (𝟙 b✝).toFunctor)).hom ≫\n Bicategory.whiskerLeft (prelaxfunctor.map f✝) (Cat.Ho... | [
"case star\nC : Type u\ninst✝ : Category.{v, u} C\na✝ b✝ : Cat\nf✝ : a✝ ⟶ b✝\n⊢ (prelaxfunctor.map₂ (Bicategory.rightUnitor f✝).hom).toNatTrans.app star =\n ((Cat.Hom.isoMk (mapComp f✝.toFunctor (𝟙 b✝).toFunctor)).hom ≫\n Bicategory.whiskerLeft (prelaxfunctor.map f✝) (Cat.Hom.isoMk (mapId ↑b✝)).hom ≫... | · simpa using! (refl _) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Limits.Shapes.Opposites.Equalizers | {
"line": 205,
"column": 30
} | {
"line": 205,
"column": 46
} | {
"line": 205,
"column": 46
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nJ : Type u₂\ninst✝ : Category.{v₂, u₂} J\nX Y : C\nf g : X ⟶ Y\nc : Cofork f g\n⊢ c.op.unop.π ≫ (Iso.refl c.op.unop.pt).hom = c.π",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quive... | [] | simp [op_unop_π] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Limits.Shapes.Opposites.Equalizers | {
"line": 205,
"column": 30
} | {
"line": 205,
"column": 46
} | {
"line": 205,
"column": 46
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nJ : Type u₂\ninst✝ : Category.{v₂, u₂} J\nX Y : C\nf g : X ⟶ Y\nc : Cofork f g\n⊢ c.op.unop.π ≫ (Iso.refl c.op.unop.pt).hom = c.π",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quive... | [] | simp [op_unop_π] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Opposites.Equalizers | {
"line": 205,
"column": 30
} | {
"line": 205,
"column": 46
} | {
"line": 205,
"column": 46
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nJ : Type u₂\ninst✝ : Category.{v₂, u₂} J\nX Y : C\nf g : X ⟶ Y\nc : Cofork f g\n⊢ c.op.unop.π ≫ (Iso.refl c.op.unop.pt).hom = c.π",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quive... | [] | simp [op_unop_π] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Final | {
"line": 854,
"column": 69
} | {
"line": 869,
"column": 16
} | {
"line": 871,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nE : Type u₃\ninst✝ : Category.{v₃, u₃} E\nF : C ⥤ D\nG : D ⥤ E\nhF : F.Final\nhG : G.Final\n⊢ (F ⋙ G).Final",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.E... | [] | by
let s₁ : C ≌ AsSmall.{max u₁ v₁ u₂ v₂ u₃ v₃} C := AsSmall.equiv
let s₂ : D ≌ AsSmall.{max u₁ v₁ u₂ v₂ u₃ v₃} D := AsSmall.equiv
let s₃ : E ≌ AsSmall.{max u₁ v₁ u₂ v₂ u₃ v₃} E := AsSmall.equiv
let i : s₁.inverse ⋙ (F ⋙ G) ⋙ s₃.functor ≅
(s₁.inverse ⋙ F ⋙ s₂.functor) ⋙ (s₂.inverse ⋙ G ⋙ s₃.functor) :=
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Final | {
"line": 931,
"column": 4
} | {
"line": 934,
"column": 18
} | {
"line": 936,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nc : C\nhc : IsTerminal c\nc' : C\n⊢ IsConnected (StructuredArrow c' (fromPUnit c))",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.IsTerminal.from",
"Inhabited.default",
"CategoryTheory.eqToHom",
... | [] | letI : Inhabited (StructuredArrow c' (fromPUnit c)) := ⟨.mk (Y := default) (hc.from c')⟩
letI : Subsingleton (StructuredArrow c' (fromPUnit c)) :=
⟨fun i j ↦ StructuredArrow.obj_ext _ _ (by cat_disch) (hc.hom_ext _ _)⟩
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Final | {
"line": 931,
"column": 4
} | {
"line": 934,
"column": 18
} | {
"line": 936,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nc : C\nhc : IsTerminal c\nc' : C\n⊢ IsConnected (StructuredArrow c' (fromPUnit c))",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.IsTerminal.from",
"Inhabited.default",
"CategoryTheory.eqToHom",
... | [] | letI : Inhabited (StructuredArrow c' (fromPUnit c)) := ⟨.mk (Y := default) (hc.from c')⟩
letI : Subsingleton (StructuredArrow c' (fromPUnit c)) :=
⟨fun i j ↦ StructuredArrow.obj_ext _ _ (by cat_disch) (hc.hom_ext _ _)⟩
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.FinitePresentation | {
"line": 440,
"column": 2
} | {
"line": 440,
"column": 29
} | {
"line": 441,
"column": 2
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : CommRing C\ng : B →+* C\nf : A →+* B\nhg : g.FinitePresentation\nhf : f.FinitePresentation\n⊢ (g.comp f).FinitePresentation",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Algebra.al... | [
"A : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : CommRing C\ng : B →+* C\nf : A →+* B\nhg : g.FinitePresentation\nhf : f.FinitePresentation\nalgInst✝² : Algebra A B := f.toAlgebra\nalgInst✝¹ : Algebra B C := g.toAlgebra\nalgInst✝ : Algebra A C := (g.comp f).toAlgebra\nsca... | algebraize [f, g, g.comp f] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.RingTheory.FinitePresentation | {
"line": 445,
"column": 2
} | {
"line": 445,
"column": 29
} | {
"line": 446,
"column": 2
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : CommRing C\nf : A →+* B\ng : B →+* C\nhg : (g.comp f).FinitePresentation\nhf : f.FiniteType\n⊢ g.FinitePresentation",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Algebra.algebraMap... | [
"A : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : CommRing C\nf : A →+* B\ng : B →+* C\nhg : (g.comp f).FinitePresentation\nhf : f.FiniteType\nalgInst✝² : Algebra A B := f.toAlgebra\nalgInst✝¹ : Algebra B C := g.toAlgebra\nalgInst✝ : Algebra A C := (g.comp f).toAlgebra\nsc... | algebraize [f, g, g.comp f] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.CategoryTheory.Limits.Shapes.KernelPair | {
"line": 109,
"column": 43
} | {
"line": 118,
"column": 72
} | {
"line": 118,
"column": 72
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nR X Y Z : C\na b : R ⟶ X\nf₁ : X ⟶ Y\nf₂ : Y ⟶ Z\ncomm : a ≫ f₁ = b ≫ f₁\nbig_k : IsKernelPair (f₁ ≫ f₂) a b\ns : PullbackCone f₁ f₁\n⊢ { l //\n l ≫ (PullbackCone.mk a b comm).fst = s.fst ∧\n l ≫ (PullbackCone.mk a b comm).snd = s.snd ∧\n ∀ {m : s.p... | [] | by
let s' : PullbackCone (f₁ ≫ f₂) (f₁ ≫ f₂) :=
PullbackCone.mk s.fst s.snd (s.condition_assoc _)
refine ⟨big_k.isLimit.lift s', big_k.isLimit.fac _ WalkingCospan.left,
big_k.isLimit.fac _ WalkingCospan.right, fun m₁ m₂ => ?_⟩
apply big_k.isLimit.hom_ext
refine (Pullb... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Assoc | {
"line": 118,
"column": 10
} | {
"line": 118,
"column": 90
} | {
"line": 118,
"column": 90
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nX₁ X₂ X₃ Y₁ Y₂ : C\nf₁ : X₁ ⟶ Y₁\nf₂ : X₂ ⟶ Y₁\nf₃ : X₂ ⟶ Y₂\nf₄ : X₃ ⟶ Y₂\ninst✝² : HasPullback f₁ f₂\ninst✝¹ : HasPullback f₃ f₄\ninst✝ : HasPullback (pullback.snd f₁ f₂ ≫ f₃) f₄\n⊢ (pullback.fst (pullback.snd f₁ f₂ ≫ f₃) f₄ ≫ pullback.fst f₁ f₂) ≫ f₁ =\n pu... | [] | rw [pullback.lift_fst_assoc, Category.assoc, Category.assoc, pullback.condition] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Assoc | {
"line": 118,
"column": 10
} | {
"line": 118,
"column": 90
} | {
"line": 118,
"column": 90
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nX₁ X₂ X₃ Y₁ Y₂ : C\nf₁ : X₁ ⟶ Y₁\nf₂ : X₂ ⟶ Y₁\nf₃ : X₂ ⟶ Y₂\nf₄ : X₃ ⟶ Y₂\ninst✝² : HasPullback f₁ f₂\ninst✝¹ : HasPullback f₃ f₄\ninst✝ : HasPullback (pullback.snd f₁ f₂ ≫ f₃) f₄\n⊢ (pullback.fst (pullback.snd f₁ f₂ ≫ f₃) f₄ ≫ pullback.fst f₁ f₂) ≫ f₁ =\n pu... | [] | rw [pullback.lift_fst_assoc, Category.assoc, Category.assoc, pullback.condition] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Assoc | {
"line": 118,
"column": 10
} | {
"line": 118,
"column": 90
} | {
"line": 118,
"column": 90
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nX₁ X₂ X₃ Y₁ Y₂ : C\nf₁ : X₁ ⟶ Y₁\nf₂ : X₂ ⟶ Y₁\nf₃ : X₂ ⟶ Y₂\nf₄ : X₃ ⟶ Y₂\ninst✝² : HasPullback f₁ f₂\ninst✝¹ : HasPullback f₃ f₄\ninst✝ : HasPullback (pullback.snd f₁ f₂ ≫ f₃) f₄\n⊢ (pullback.fst (pullback.snd f₁ f₂ ≫ f₃) f₄ ≫ pullback.fst f₁ f₂) ≫ f₁ =\n pu... | [] | rw [pullback.lift_fst_assoc, Category.assoc, Category.assoc, pullback.condition] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Diagonal | {
"line": 67,
"column": 24
} | {
"line": 68,
"column": 41
} | {
"line": 68,
"column": 41
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nX Y : C\nf : X ⟶ Y\ninst✝ : HasPullback f f\nH : IsIso (diagonal f)\nZ✝ : C\nx✝¹ x✝ : Z✝ ⟶ X\ne : x✝¹ ≫ f = x✝ ≫ f\n⊢ x✝¹ = x✝",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.pullback",
... | [] | by rw [← lift_fst _ _ e, (cancel_epi (g := fst f f) (h := snd f f)
(diagonal f)).mp (by simp), lift_snd] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.CommAlgCat.FiniteType | {
"line": 81,
"column": 4
} | {
"line": 81,
"column": 81
} | {
"line": 82,
"column": 4
} | [
{
"pp": "case refine_1\nR : Type u\ninst✝ : CommRing R\n⊢ Small.{u, max (u + 1) (v + 1)} (Skeleton (FGAlgCat R))",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"CategoryTheory.toSkeleton",
"CommAlgCat.instCommRingObjForgetAlgHomCarrier",
"FGAlgCatSkeleton.eval",
... | [
"case refine_1\nR : Type u\ninst✝ : CommRing R\nf : FGAlgCatSkeleton R → Skeleton (FGAlgCat R) := toSkeleton ∘ (FGAlgCat.uliftFunctor R).obj ∘ FGAlgCatSkeleton.eval R\n⊢ Small.{u, max (u + 1) (v + 1)} (Skeleton (FGAlgCat R))"
] | let f := toSkeleton ∘ (FGAlgCat.uliftFunctor R).obj ∘ FGAlgCatSkeleton.eval R | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.CategoryTheory.Limits.Shapes.Diagonal | {
"line": 120,
"column": 4
} | {
"line": 121,
"column": 40
} | {
"line": 122,
"column": 2
} | [
{
"pp": "case h₀\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\nX Y Z : C\ninst✝¹ : HasPullbacks C\nU V₁ V₂ : C\nf : X ⟶ Y\ni : U ⟶ Y\ni₁ : V₁ ⟶ pullback f i\ni₂ : V₂ ⟶ pullback f i\ninst✝ : HasPullback i₁ i₂\n⊢ ((snd (diagonal f) (map (i₁ ≫ snd f i) (i₂ ≫ snd f i) f f (i₁ ≫ fst f i) (i₂ ≫ fst f i) i ⋯ ⋯) ≫\n ... | [] | simp only [Category.assoc, pullback_diagonal_map_snd_fst_fst,
pullback_diagonal_map_snd_snd_fst] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Limits.Shapes.Diagonal | {
"line": 120,
"column": 4
} | {
"line": 121,
"column": 40
} | {
"line": 122,
"column": 2
} | [
{
"pp": "case h₀\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\nX Y Z : C\ninst✝¹ : HasPullbacks C\nU V₁ V₂ : C\nf : X ⟶ Y\ni : U ⟶ Y\ni₁ : V₁ ⟶ pullback f i\ni₂ : V₂ ⟶ pullback f i\ninst✝ : HasPullback i₁ i₂\n⊢ ((snd (diagonal f) (map (i₁ ≫ snd f i) (i₂ ≫ snd f i) f f (i₁ ≫ fst f i) (i₂ ≫ fst f i) i ⋯ ⋯) ≫\n ... | [] | simp only [Category.assoc, pullback_diagonal_map_snd_fst_fst,
pullback_diagonal_map_snd_snd_fst] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Diagonal | {
"line": 120,
"column": 4
} | {
"line": 121,
"column": 40
} | {
"line": 122,
"column": 2
} | [
{
"pp": "case h₀\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\nX Y Z : C\ninst✝¹ : HasPullbacks C\nU V₁ V₂ : C\nf : X ⟶ Y\ni : U ⟶ Y\ni₁ : V₁ ⟶ pullback f i\ni₂ : V₂ ⟶ pullback f i\ninst✝ : HasPullback i₁ i₂\n⊢ ((snd (diagonal f) (map (i₁ ≫ snd f i) (i₂ ≫ snd f i) f f (i₁ ≫ fst f i) (i₂ ≫ fst f i) i ⋯ ⋯) ≫\n ... | [] | simp only [Category.assoc, pullback_diagonal_map_snd_fst_fst,
pullback_diagonal_map_snd_snd_fst] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.RingHomProperties | {
"line": 150,
"column": 25
} | {
"line": 150,
"column": 83
} | {
"line": 151,
"column": 2
} | [
{
"pp": "P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nh₁ : RespectsIso P\nh₂ :\n ∀ ⦃R S T : Type u⦄ [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S]\n [inst_4 : Algebra R T], P (algebraMap R T) → P (algebraMap S (S ⊗[R] T))\nR S R... | [] | by simp [e, f', IsBaseChange.equiv_tmul, Algebra.smul_def] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.MorphismProperty.Limits | {
"line": 197,
"column": 7
} | {
"line": 197,
"column": 31
} | {
"line": 197,
"column": 32
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nP : MorphismProperty C\ninst✝ : P.RespectsIso\nH : ∀ {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) [HasPullback f g], P g → ∃ T fst snd, IsPullback fst snd f g ∧ P fst\nX Y S : C\nf : X ⟶ S\ng : Y ⟶ S\nx✝ : HasPullback f g\nhg : P g\nT : C\nfst : T ⟶ X\nsnd : T ⟶ Y\nh : Is... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nP : MorphismProperty C\ninst✝ : P.RespectsIso\nH : ∀ {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) [HasPullback f g], P g → ∃ T fst snd, IsPullback fst snd f g ∧ P fst\nX Y S : C\nf : X ⟶ S\ng : Y ⟶ S\nx✝ : HasPullback f g\nhg : P g\nT : C\nfst : T ⟶ X\nsnd : T ⟶ Y\nh : IsPullback fst... | ← h.isoPullback_inv_fst, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts | {
"line": 193,
"column": 4
} | {
"line": 199,
"column": 9
} | {
"line": 200,
"column": 2
} | [
{
"pp": "J : Type v\ninst✝² : SmallCategory J\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nn : ℕ\nf : Fin (n + 1) → C\nc₁ : Cofan fun i ↦ f i.succ\nc₂ : BinaryCofan (f 0) c₁.pt\nt₁ : IsColimit c₁\nt₂ : IsColimit c₂\ns : Cocone (Discrete.functor f)\n⊢ ∀ (j : Discrete (Fin (n... | [] | rintro ⟨j⟩
refine Fin.inductionOn j ?_ ?_
· apply (BinaryCofan.IsColimit.desc' t₂ _ _).2.1
· rintro i -
dsimp only [extendCofan_ι_app]
rw [Fin.cases_succ, assoc, (BinaryCofan.IsColimit.desc' t₂ _ _).2.2, t₁.fac]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts | {
"line": 193,
"column": 4
} | {
"line": 199,
"column": 9
} | {
"line": 200,
"column": 2
} | [
{
"pp": "J : Type v\ninst✝² : SmallCategory J\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nn : ℕ\nf : Fin (n + 1) → C\nc₁ : Cofan fun i ↦ f i.succ\nc₂ : BinaryCofan (f 0) c₁.pt\nt₁ : IsColimit c₁\nt₂ : IsColimit c₂\ns : Cocone (Discrete.functor f)\n⊢ ∀ (j : Discrete (Fin (n... | [] | rintro ⟨j⟩
refine Fin.inductionOn j ?_ ?_
· apply (BinaryCofan.IsColimit.desc' t₂ _ _).2.1
· rintro i -
dsimp only [extendCofan_ι_app]
rw [Fin.cases_succ, assoc, (BinaryCofan.IsColimit.desc' t₂ _ _).2.2, t₁.fac]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.ObjectProperty.Small | {
"line": 178,
"column": 4
} | {
"line": 179,
"column": 89
} | {
"line": 180,
"column": 2
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝ : Category.{v, u} C\nP : ObjectProperty C\nx✝ : ObjectProperty.EssentiallySmall.{w, v, u} P.op\n⊢ ObjectProperty.EssentiallySmall.{w, v, u} P",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"CategoryTheory.ObjectProperty.instSmallUnopOfOp... | [] | obtain ⟨Q, h₁, _, h₂⟩ := EssentiallySmall.exists_small_le P.op
exact ⟨Q.unop, inferInstance, by rwa [← unop_isoClosure, ← op_monotone_iff, op_unop]⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.ObjectProperty.Small | {
"line": 178,
"column": 4
} | {
"line": 179,
"column": 89
} | {
"line": 180,
"column": 2
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝ : Category.{v, u} C\nP : ObjectProperty C\nx✝ : ObjectProperty.EssentiallySmall.{w, v, u} P.op\n⊢ ObjectProperty.EssentiallySmall.{w, v, u} P",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"CategoryTheory.ObjectProperty.instSmallUnopOfOp... | [] | obtain ⟨Q, h₁, _, h₂⟩ := EssentiallySmall.exists_small_le P.op
exact ⟨Q.unop, inferInstance, by rwa [← unop_isoClosure, ← op_monotone_iff, op_unop]⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.ObjectProperty.LimitsOfShape | {
"line": 154,
"column": 4
} | {
"line": 154,
"column": 26
} | {
"line": 155,
"column": 4
} | [
{
"pp": "case refine_1\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP : ObjectProperty C\nJ : Type u'\ninst✝ : Category.{v', u'} J\n⊢ (P.strictLimitsOfShape J).isoClosure ≤ P.limitsOfShape J",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Prop... | [
"case refine_1\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP : ObjectProperty C\nJ : Type u'\ninst✝ : Category.{v', u'} J\n⊢ P.strictLimitsOfShape J ≤ P.limitsOfShape J"
] | rw [isoClosure_le_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.ObjectProperty.LimitsOfShape | {
"line": 183,
"column": 23
} | {
"line": 183,
"column": 36
} | {
"line": 183,
"column": 36
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁸ : Category.{v_1, u_1} C\ninst✝⁷ : Category.{v_2, u_2} D\nP : ObjectProperty C\nJ : Type u'\ninst✝⁶ : Category.{v', u'} J\nJ' : Type u''\ninst✝⁵ : Category.{v'', u''} J'\ninst✝⁴ : ObjectProperty.Small.{w, v_1, u_1} P\ninst✝³ : LocallySmall.{w, v_1, u_1} C\ninst✝² : Sma... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.ObjectProperty.ContainsZero | {
"line": 99,
"column": 4
} | {
"line": 99,
"column": 32
} | {
"line": 101,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nD : Type u'\ninst✝¹ : Category.{v', u'} D\nP✝ Q : ObjectProperty C\nP : ObjectProperty Cᵒᵖ\ninst✝ : P.ContainsZero\nZ : Cᵒᵖ\nhZ : IsZero Z\nmem : P Z\n⊢ ∃ Z, IsZero Z ∧ P.unop Z",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Category... | [] | exact ⟨Z.unop, hZ.unop, mem⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.ObjectProperty.ColimitsOfShape | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 26
} | {
"line": 163,
"column": 4
} | [
{
"pp": "case refine_1\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP : ObjectProperty C\nJ : Type u'\ninst✝ : Category.{v', u'} J\n⊢ (P.strictColimitsOfShape J).isoClosure ≤ P.colimitsOfShape J",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Obj... | [
"case refine_1\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP : ObjectProperty C\nJ : Type u'\ninst✝ : Category.{v', u'} J\n⊢ P.strictColimitsOfShape J ≤ P.colimitsOfShape J"
] | rw [isoClosure_le_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.ObjectProperty.ColimitsOfShape | {
"line": 191,
"column": 23
} | {
"line": 191,
"column": 36
} | {
"line": 191,
"column": 36
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁸ : Category.{v_1, u_1} C\ninst✝⁷ : Category.{v_2, u_2} D\nP : ObjectProperty C\nJ : Type u'\ninst✝⁶ : Category.{v', u'} J\nJ' : Type u''\ninst✝⁵ : Category.{v'', u''} J'\ninst✝⁴ : ObjectProperty.Small.{w, v_1, u_1} P\ninst✝³ : LocallySmall.{w, v_1, u_1} C\ninst✝² : Sma... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.Diagonal | {
"line": 503,
"column": 4
} | {
"line": 503,
"column": 52
} | {
"line": 504,
"column": 4
} | [
{
"pp": "case refine_1.refine_1\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasPullbacks C\nX Y Z S : C\nf : X ⟶ S\ng : Y ⟶ S\nh : Z ⟶ S\nc : PullbackCone (fst g h) (snd f g)\n⊢ (c.snd ≫ fst f g) ≫ f = (c.fst ≫ snd g h) ≫ h",
"ppTerm": "?refine_1.refine_1",
"assigned": true,
"usedConstant... | [
"case refine_1.refine_2\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasPullbacks C\nX Y Z S : C\nf : X ⟶ S\ng : Y ⟶ S\nh : Z ⟶ S\nc : PullbackCone (fst g h) (snd f g)\n⊢ c.snd ≫ fst f g = lift (c.snd ≫ fst f g) (c.fst ≫ snd g h) ⋯ ≫ fst f h"
] | · simp [pullback.condition, ← c.condition_assoc] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Monoidal.Cartesian.Basic | {
"line": 691,
"column": 60
} | {
"line": 691,
"column": 73
} | {
"line": 692,
"column": 2
} | [
{
"pp": "C✝ : Type u\ninst✝⁹ : Category.{v, u} C✝\nC : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : CartesianMonoidalCategory C\nD : Type u₁\ninst✝⁶ : Category.{v₁, u₁} D\ninst✝⁵ : CartesianMonoidalCategory D\nF : C ⥤ D\nE✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} E✝\ninst✝³ : CartesianMonoidalCategory E✝\nG✝ : D ... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Grp | {
"line": 202,
"column": 15
} | {
"line": 202,
"column": 32
} | {
"line": 202,
"column": 33
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : CartesianMonoidalCategory C\nA B : C\ninst✝ : GrpObj B\nf g h : A ⟶ B\n⊢ h = lift (f ≫ ι) g ≫ μ ↔ g = lift f h ≫ μ",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.GrpObj.inv",
"Category... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : CartesianMonoidalCategory C\nA B : C\ninst✝ : GrpObj B\nf g h : A ⟶ B\n⊢ lift f h ≫ μ = g ↔ g = lift f h ≫ μ"
] | eq_lift_inv_left, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monoidal.Grp | {
"line": 316,
"column": 8
} | {
"line": 316,
"column": 89
} | {
"line": 317,
"column": 8
} | [
{
"pp": "case refine_2\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : CartesianMonoidalCategory C\nA : C\ninst✝ : GrpObj A\ns : PullbackCone μ μ\nm : s.pt ⟶ (A ⊗ A) ⊗ A\nhm₁ : m ≫ μ ▷ A = s.fst\nhm₂ : m ≫ (α_ A A A).hom ≫ A ◁ μ = s.snd\n⊢ (m ≫ fst (A ⊗ A) A) ≫ snd A A =\n (lift (lift (s.snd ≫ fst A A) ... | [
"case refine_2\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : CartesianMonoidalCategory C\nA : C\ninst✝ : GrpObj A\ns : PullbackCone μ μ\nm : s.pt ⟶ (A ⊗ A) ⊗ A\nhm₁ : m ≫ μ ▷ A = s.fst\nhm₂ : m ≫ (α_ A A A).hom ≫ A ◁ μ = s.snd\nh : m ≫ fst (A ⊗ A) A ≫ fst A A = s.snd ≫ fst A A\n⊢ (m ≫ fst (A ⊗ A) A) ≫ snd A ... | have h : m ≫ fst _ _ ≫ fst _ _ = s.snd ≫ fst _ _ := by simpa using hm₂ =≫ fst _ _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Limits.Constructions.Pullbacks | {
"line": 46,
"column": 12
} | {
"line": 46,
"column": 42
} | {
"line": 46,
"column": 42
} | [
{
"pp": "C : Type u\n𝒞 : Category.{v, u} C\nX Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝¹ : HasLimit (pair X Y)\ninst✝ : HasLimit (parallelPair (prod.fst ≫ f) (prod.snd ≫ g))\nπ₁ : X ⨯ Y ⟶ X := prod.fst\nπ₂ : X ⨯ Y ⟶ Y := prod.snd\ne : equalizer (π₁ ≫ f) (π₂ ≫ g) ⟶ X ⨯ Y := equalizer.ι (π₁ ≫ f) (π₂ ≫ g)\ns : Pullbac... | [] | exact PullbackCone.condition _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Category.ModuleCat.Kernels | {
"line": 80,
"column": 4
} | {
"line": 82,
"column": 9
} | {
"line": 84,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nM N P : ModuleCat R\nf✝ f : M ⟶ N\ng : N ⟶ P\nH : Function.Exact ⇑(Hom.hom f) ⇑(Hom.hom g)\nH₂ : Function.Surjective ⇑(Hom.hom g)\n⊢ ∀ (j : WalkingParallelPair),\n (cokernelCocone f).ι.app j ≫\n (((Hom.hom g).ker.quotEquivOfEq (Hom.hom f).range ⋯).toModuleIso.symm ... | [] | rintro ⟨⟩ <;> ext x
· simpa using! (Function.Exact.apply_apply_eq_zero H x).symm
· rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Kernels | {
"line": 80,
"column": 4
} | {
"line": 82,
"column": 9
} | {
"line": 84,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nM N P : ModuleCat R\nf✝ f : M ⟶ N\ng : N ⟶ P\nH : Function.Exact ⇑(Hom.hom f) ⇑(Hom.hom g)\nH₂ : Function.Surjective ⇑(Hom.hom g)\n⊢ ∀ (j : WalkingParallelPair),\n (cokernelCocone f).ι.app j ≫\n (((Hom.hom g).ker.quotEquivOfEq (Hom.hom f).range ⋯).toModuleIso.symm ... | [] | rintro ⟨⟩ <;> ext x
· simpa using! (Function.Exact.apply_apply_eq_zero H x).symm
· rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Constructions.Equalizers | {
"line": 179,
"column": 6
} | {
"line": 180,
"column": 14
} | {
"line": 182,
"column": 0
} | [
{
"pp": "case h₁\nC : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\nG : C ⥤ D\ninst✝¹ : HasBinaryCoproducts C\ninst✝ : HasPushouts C\nF : WalkingParallelPair ⥤ C\nc : Cocone F\nm : (coequalizerCocone F).pt ⟶ c.pt\nJ : ∀ (j : WalkingParallelPair), (coequalizerCocone F).ι.app j ≫ ... | [] | rw [colimit.ι_desc, ← pushoutInl_eq_pushout_inr]
exact J1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Constructions.Equalizers | {
"line": 179,
"column": 6
} | {
"line": 180,
"column": 14
} | {
"line": 182,
"column": 0
} | [
{
"pp": "case h₁\nC : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\nG : C ⥤ D\ninst✝¹ : HasBinaryCoproducts C\ninst✝ : HasPushouts C\nF : WalkingParallelPair ⥤ C\nc : Cocone F\nm : (coequalizerCocone F).pt ⟶ c.pt\nJ : ∀ (j : WalkingParallelPair), (coequalizerCocone F).ι.app j ≫ ... | [] | rw [colimit.ι_desc, ← pushoutInl_eq_pushout_inr]
exact J1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers | {
"line": 154,
"column": 55
} | {
"line": 155,
"column": 58
} | {
"line": 155,
"column": 58
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nG : C ⥤ D\nX Y Z : C\nf g : X ⟶ Y\nh : Y ⟶ Z\nw : f ≫ h = g ≫ h\ninst✝¹ : HasCoequalizer f g\ninst✝ : PreservesColimit (parallelPair f g) G\n⊢ G.map f ≫ G.map (coequalizer.π f g) = G.map g ≫ G.map (coequalizer.π f g)"... | [] | by
simp only [← G.map_comp]; rw [coequalizer.condition] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers | {
"line": 193,
"column": 6
} | {
"line": 193,
"column": 20
} | {
"line": 194,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nG : C ⥤ D\nX Y Z : C\nf g : X ⟶ Y\nh✝ : Y ⟶ Z\nw : f ≫ h✝ = g ≫ h✝\ninst✝² : HasCoequalizer f g\ninst✝¹ : HasCoequalizer (G.map f) (G.map g)\ninst✝ : PreservesColimit (parallelPair f g) G\nW : D\nh k : G.obj (coequali... | [] | apply epi_comp | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers | {
"line": 193,
"column": 6
} | {
"line": 193,
"column": 20
} | {
"line": 194,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nG : C ⥤ D\nX Y Z : C\nf g : X ⟶ Y\nh✝ : Y ⟶ Z\nw : f ≫ h✝ = g ≫ h✝\ninst✝² : HasCoequalizer f g\ninst✝¹ : HasCoequalizer (G.map f) (G.map g)\ninst✝ : PreservesColimit (parallelPair f g) G\nW : D\nh k : G.obj (coequali... | [] | apply epi_comp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers | {
"line": 193,
"column": 6
} | {
"line": 193,
"column": 20
} | {
"line": 194,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nG : C ⥤ D\nX Y Z : C\nf g : X ⟶ Y\nh✝ : Y ⟶ Z\nw : f ≫ h✝ = g ≫ h✝\ninst✝² : HasCoequalizer f g\ninst✝¹ : HasCoequalizer (G.map f) (G.map g)\ninst✝ : PreservesColimit (parallelPair f g) G\nW : D\nh k : G.obj (coequali... | [] | apply epi_comp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Basic | {
"line": 328,
"column": 25
} | {
"line": 330,
"column": 18
} | {
"line": 332,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nX Y : C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : IsNormalEpiCategory C\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nx✝ : Epi f\n⊢ IsRegularEpi f",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"CategoryTheory.IsRegularEpi",
"inferInstance",
"Ca... | [] | by
haveI := normalEpiOfEpi f
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Abelian.NonPreadditive | {
"line": 358,
"column": 41
} | {
"line": 358,
"column": 53
} | {
"line": 358,
"column": 54
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na b : X ⟶ Y\n| 0 - a - (b - 0)",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"HSub.hSub",
"Category... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na b : X ⟶ Y\n| 0 - b - (a - 0)"
] | sub_sub_sub, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.Abelian.NonPreadditive | {
"line": 364,
"column": 6
} | {
"line": 364,
"column": 18
} | {
"line": 364,
"column": 19
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na : X ⟶ Y\n⊢ a - a - (0 - a) = a",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"HSub.hSub... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na : X ⟶ Y\n⊢ a - 0 - (a - a) = a"
] | sub_sub_sub, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.NonPreadditive | {
"line": 374,
"column": 6
} | {
"line": 374,
"column": 18
} | {
"line": 374,
"column": 19
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na b : X ⟶ Y\n⊢ 0 - 0 - (-b - a) = b + a",
"ppTerm": "?m.90",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"HS... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na b : X ⟶ Y\n⊢ 0 - -b - (0 - a) = b + a"
] | sub_sub_sub, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.NonPreadditive | {
"line": 385,
"column": 6
} | {
"line": 385,
"column": 18
} | {
"line": 385,
"column": 19
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na b : X ⟶ Y\n⊢ 0 - 0 - (a - b) = 0 - a + b",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na b : X ⟶ Y\n⊢ 0 - a - (0 - b) = 0 - a + b"
] | sub_sub_sub, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.NonPreadditive | {
"line": 390,
"column": 24
} | {
"line": 390,
"column": 36
} | {
"line": 390,
"column": 37
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na b c : X ⟶ Y\n⊢ a - b - (0 - c) = a - (b - c)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na b c : X ⟶ Y\n⊢ a - 0 - (b - c) = a - (b - c)"
] | sub_sub_sub, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Category.Grp.Colimits | {
"line": 166,
"column": 56
} | {
"line": 166,
"column": 74
} | {
"line": 166,
"column": 75
} | [
{
"pp": "J : Type u\ninst✝¹ : Category.{v, u} J\nF : J ⥤ AddCommGrpCat\nc : Cocone F\ninst✝ : DecidableEq J\nx : Quot F\nj : J\na : ↑(F.obj j)\n⊢ (quotUliftToQuot F) ((quotToQuotUlift F) ((Quot.ι F j) a)) = (Quot.ι F j) a",
"ppTerm": "?m.98",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"J : Type u\ninst✝¹ : Category.{v, u} J\nF : J ⥤ AddCommGrpCat\nc : Cocone F\ninst✝ : DecidableEq J\nx : Quot F\nj : J\na : ↑(F.obj j)\n⊢ (quotUliftToQuot F) ((Quot.ι (F ⋙ uliftFunctor) j) { down := a }) = (Quot.ι F j) a"
] | quotToQuotUlift_ι, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Category.Grp.Colimits | {
"line": 235,
"column": 4
} | {
"line": 236,
"column": 20
} | {
"line": 237,
"column": 4
} | [
{
"pp": "J : Type u\ninst✝¹ : Category.{v, u} J\nF : J ⥤ AddCommGrpCat\nc : Cocone F\ninst✝ : DecidableEq J\nh : Function.Bijective ⇑(Quot.desc F c)\ns : Cocone F\nm : c.pt ⟶ s.pt\nhm : ∀ (j : J), c.ι.app j ≫ m = s.ι.app j\nx : Quot F\n⊢ (Hom.hom m) ((AddEquiv.ofBijective (Quot.desc F c) h) x) = (Quot.desc F s)... | [
"J : Type u\ninst✝¹ : Category.{v, u} J\nF : J ⥤ AddCommGrpCat\nc : Cocone F\ninst✝ : DecidableEq J\nh : Function.Bijective ⇑(Quot.desc F c)\ns : Cocone F\nm : c.pt ⟶ s.pt\nhm : ∀ (j : J), c.ι.app j ≫ m = s.ι.app j\nx : Quot F\n⊢ (Hom.hom m).comp ↑(AddEquiv.ofBijective (Quot.desc F c) h) = Quot.desc F s"
] | suffices eq : m.hom.comp (AddEquiv.ofBijective (Quot.desc F c) h) = Quot.desc F s by
rw [← eq]; rfl | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Algebra.Category.Grp.Colimits | {
"line": 309,
"column": 44
} | {
"line": 311,
"column": 12
} | {
"line": 312,
"column": 2
} | [
{
"pp": "G H : AddCommGrpCat\nf : G ⟶ H\n⊢ f ≫ ofHom (mk' (Hom.hom f).range) = 0",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"AddCommGrpCat.instCategory",
"Quiver.Hom",
"congrArg",
"AddCommGroup.toAddCommMonoid",
... | [] | by
ext x
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Filter.Lift | {
"line": 182,
"column": 6
} | {
"line": 182,
"column": 60
} | {
"line": 183,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\nι : Sort u_4\ninst✝ : Nonempty ι\nf : ι → Filter α\ng : Set α → Filter β\nhg : ∀ (s t : Set α), g (s ∩ t) = g s ⊓ g t\ns : Set β\nt : Set α\nht : t ∈ iInf f\ninhabited_h : Inhabited ι\n⊢ ⨅ i, (f i).lift g ≤ g univ",
"ppTerm": "?refine_1",
"assigned": t... | [] | exact iInf₂_le_of_le default univ (iInf_le _ univ_mem) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Closure | {
"line": 544,
"column": 76
} | {
"line": 545,
"column": 78
} | {
"line": 547,
"column": 0
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns t : Set X\n⊢ frontier (s ∪ t) ⊆ frontier s ∩ closure tᶜ ∪ closure sᶜ ∩ frontier t",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"frontier_compl",
"frontier",
"congrArg",
"Compl.compl",
"Set.instUnion",
... | [] | by
simpa only [frontier_compl, ← compl_union] using frontier_inter_subset sᶜ tᶜ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Filter.Prod | {
"line": 360,
"column": 2
} | {
"line": 360,
"column": 86
} | {
"line": 361,
"column": 2
} | [
{
"pp": "case mp\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nm : α × β → γ\nf : Filter α\ng : Filter β\ns : Set γ\n⊢ (∃ t₁ ∈ f, ∃ t₂ ∈ g, t₁ ×ˢ t₂ ⊆ m ⁻¹' s) → ∃ t ∈ g, ∃ x ∈ f, ∀ x_1 ∈ x, ∀ y ∈ t, m (x_1, y) ∈ s",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
... | [
"case mpr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nm : α × β → γ\nf : Filter α\ng : Filter β\ns : Set γ\n⊢ (∃ t ∈ g, ∃ x ∈ f, ∀ x_1 ∈ x, ∀ y ∈ t, m (x_1, y) ∈ s) → ∃ t₁ ∈ f, ∃ t₂ ∈ g, t₁ ×ˢ t₂ ⊆ m ⁻¹' s"
] | · exact fun ⟨t, ht, s, hs, h⟩ => ⟨s, hs, t, ht, fun x hx y hy => @h ⟨x, y⟩ ⟨hx, hy⟩⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.Filter.Prod | {
"line": 526,
"column": 34
} | {
"line": 526,
"column": 47
} | {
"line": 526,
"column": 47
} | [
{
"pp": "α : Type u_6\nβ : Type u_7\nι : Type u_8\na : α\nb : β\ni : ι\n⊢ map (Prod.map (fun x ↦ b) id) (𝓟 ({a}ᶜ ×ˢ {i}ᶜ)ᶜ) = 𝓟 ({b} ×ˢ univ)",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Eq.mpr",
"SProd.sprod",
"congrArg",
"Filter.map",
... | [
"α : Type u_6\nβ : Type u_7\nι : Type u_8\na : α\nb : β\ni : ι\n⊢ 𝓟 (Prod.map (fun x ↦ b) id '' ({a}ᶜ ×ˢ {i}ᶜ)ᶜ) = 𝓟 ({b} ×ˢ univ)"
] | map_principal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 11
} | {
"line": 86,
"column": 12
} | [
{
"pp": "case univ\nα : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\ns : Set α\nha : a ∈ univ\n⊢ ⨅ s ∈ {s | a ∈ s ∧ s ∈ g}, 𝓟 s ≤ 𝓟 univ",
"ppTerm": "?univ",
"assigned": true,
"usedConstants": [
"iInf",
"LE.le.trans_eq",
"Filter.instInfSet",
... | [] | | univ => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Topology.Order | {
"line": 274,
"column": 4
} | {
"line": 274,
"column": 13
} | {
"line": 274,
"column": 14
} | [
{
"pp": "case refl.refine_1.univ\nα : Type u_1\nU : Set α\ninst✝ : IndiscreteTopology α\n⊢ univ = ∅ ∨ univ = univ",
"ppTerm": "?refl.refine_1.univ",
"assigned": true,
"usedConstants": [
"Set.univ",
"Set.instEmptyCollection",
"EmptyCollection.emptyCollection",
"Eq",
"Or.... | [] | | univ => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
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