module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Algebra.Homology.HomologicalComplex | {
"line": 793,
"column": 2
} | {
"line": 794,
"column": 35
} | [
{
"pp": "V : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nX₀ X₁ : V\nd₀ : X₁ ⟶ X₀\nsucc' : {X₀ X₁ : V} → (f : X₁ ⟶ X₀) → (X₂ : V) ×' (d : X₂ ⟶ X₁) ×' d ≫ f = 0\n⊢ (mk' X₀ X₁ d₀ fun {X₀ X₁} ↦ succ').d 1 0 = d₀",
"usedConstants": [
"Eq.mpr",
"Nat.instOne",
"CategoryTheory.... | change ite (1 = 0 + 1) (𝟙 X₁ ≫ d₀) 0 = d₀
rw [if_pos rfl, Category.id_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.Opposite | {
"line": 44,
"column": 23
} | {
"line": 44,
"column": 35
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nX Y : C\nf : X ⟶ Y\nA B : Cᵒᵖ\ng : A ⟶ B\n⊢ f.op.unop ≫ (kernel.ι f.op.unop.op).unop = 0",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"Eq.mpr",
"Opposite",
"CategoryTheory.CategoryStruct.toQuive... | ← unop_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.Opposite | {
"line": 44,
"column": 36
} | {
"line": 44,
"column": 45
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nX Y : C\nf : X ⟶ Y\nA B : Cᵒᵖ\ng : A ⟶ B\n⊢ (kernel.ι f.op.unop.op ≫ f.op).unop = 0",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"Eq.mpr",
"Opposite",
"Quiver.Hom.unop_op",
"CategoryTheory... | f.unop_op | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Preadditive.Projective.Basic | {
"line": 271,
"column": 90
} | {
"line": 275,
"column": 11
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX Y : C\nh : Retract X Y\np : Projective Y\n⊢ Projective X",
"usedConstants": [
"CategoryTheory.Category.assoc",
"CategoryTheory.Epi",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"Exists",
"Cate... | by
refine Projective.mk (fun {A B} f e _ ↦ ?_)
rcases p.factors (h.r ≫ f) e with ⟨g, hg⟩
use h.i ≫ g
simp [hg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Abelian.Opposite | {
"line": 62,
"column": 23
} | {
"line": 62,
"column": 35
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nX Y : C\nf : X ⟶ Y\nA B : Cᵒᵖ\ng : A ⟶ B\n⊢ (cokernel.π f.op.unop.op).unop ≫ f.op.unop = 0",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"Eq.mpr",
"Opposite",
"CategoryTheory.CategoryStruct.toQui... | ← unop_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.Opposite | {
"line": 62,
"column": 36
} | {
"line": 62,
"column": 45
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nX Y : C\nf : X ⟶ Y\nA B : Cᵒᵖ\ng : A ⟶ B\n⊢ (f.op ≫ cokernel.π f.op.unop.op).unop = 0",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"Eq.mpr",
"Opposite",
"Quiver.Hom.unop_op",
"CategoryTheo... | f.unop_op | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Preadditive.Injective.Basic | {
"line": 328,
"column": 4
} | {
"line": 329,
"column": 57
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u_1\ninst✝³ : Category.{v_1, u_1} D\nF : C ⥤ D\ninst✝² : F.Full\ninst✝¹ : F.Faithful\ninst✝ : F.PreservesMonomorphisms\nI : C\nhI : Injective (F.obj I)\nX✝ Y✝ : C\ng : X✝ ⟶ I\nf : X✝ ⟶ Y✝\nx✝ : Mono f\n⊢ ∃ h, f ≫ h = g",
"usedConstants": [
"... | obtain ⟨h, fac⟩ := hI.factors (F.map g) (F.map f)
exact ⟨F.preimage h, F.map_injective (by simp [fac])⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Preadditive.Injective.Basic | {
"line": 328,
"column": 4
} | {
"line": 329,
"column": 57
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u_1\ninst✝³ : Category.{v_1, u_1} D\nF : C ⥤ D\ninst✝² : F.Full\ninst✝¹ : F.Faithful\ninst✝ : F.PreservesMonomorphisms\nI : C\nhI : Injective (F.obj I)\nX✝ Y✝ : C\ng : X✝ ⟶ I\nf : X✝ ⟶ Y✝\nx✝ : Mono f\n⊢ ∃ h, f ≫ h = g",
"usedConstants": [
"... | obtain ⟨h, fac⟩ := hI.factors (F.map g) (F.map f)
exact ⟨F.preimage h, F.map_injective (by simp [fac])⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 554,
"column": 17
} | {
"line": 555,
"column": 45
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : Preadditive C\ninst✝¹ : Preadditive D\nS : ShortComplex C\ninst✝ : HasZeroObject C\ns : S.Splitting\nW'✝ : C\nx : W'✝ ⟶ S.X₂\nhx : x ≫ S.g = 0\n⊢ (fun {W'} x x_1 ↦ x ≫ s.r) x hx ≫ S.f = x",
"usedCon... | by simp only [assoc, s.r_f, comp_sub, comp_id,
sub_eq_self, reassoc_of% hx, zero_comp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex | {
"line": 165,
"column": 2
} | {
"line": 165,
"column": 81
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nι : Type u_2\nc : ComplexShape ι\nK : HomologicalComplex C c\ni j k : ι\ninst✝ : K.HasHomology k\n⊢ K.d i j ≫ K.toCycles j k = 0",
"usedConstants": [
"CategoryTheory.Category.assoc",
"HomologicalComplex.toCycles_... | simp only [← cancel_mono (K.iCycles k), assoc, toCycles_i, d_comp_d, zero_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex | {
"line": 165,
"column": 2
} | {
"line": 165,
"column": 81
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nι : Type u_2\nc : ComplexShape ι\nK : HomologicalComplex C c\ni j k : ι\ninst✝ : K.HasHomology k\n⊢ K.d i j ≫ K.toCycles j k = 0",
"usedConstants": [
"CategoryTheory.Category.assoc",
"HomologicalComplex.toCycles_... | simp only [← cancel_mono (K.iCycles k), assoc, toCycles_i, d_comp_d, zero_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex | {
"line": 165,
"column": 2
} | {
"line": 165,
"column": 81
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nι : Type u_2\nc : ComplexShape ι\nK : HomologicalComplex C c\ni j k : ι\ninst✝ : K.HasHomology k\n⊢ K.d i j ≫ K.toCycles j k = 0",
"usedConstants": [
"CategoryTheory.Category.assoc",
"HomologicalComplex.toCycles_... | simp only [← cancel_mono (K.iCycles k), assoc, toCycles_i, d_comp_d, zero_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex | {
"line": 209,
"column": 2
} | {
"line": 209,
"column": 35
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nι : Type u_2\nc : ComplexShape ι\nK L M : HomologicalComplex C c\nφ : K ⟶ L\niso : K ≅ L\nψ : L ⟶ M\nj k : ι\ninst✝ : K.HasHomology j\n⊢ IsColimit (CokernelCofork.ofπ (K.homologyπ j) ⋯)",
"usedConstants": [
"Homologica... | exact (K.sc j).homologyIsCokernel | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Quiver.SingleObj | {
"line": 113,
"column": 2
} | {
"line": 115,
"column": 74
} | [
{
"pp": "α : Type u_1\nl : List α\n⊢ pathToList (listToPath l) = l",
"usedConstants": [
"Eq.mpr",
"Quiver.SingleObj.pathToList",
"congrArg",
"id",
"List.rec",
"List.cons",
"List",
"Quiver.SingleObj.star",
"Quiver.SingleObj.listToPath",
"Eq.refl",
... | induction l with
| nil => rfl
| cons a l ih => change a :: pathToList (listToPath l) = a :: l; rw [ih] | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.Quiver.SingleObj | {
"line": 113,
"column": 2
} | {
"line": 115,
"column": 74
} | [
{
"pp": "α : Type u_1\nl : List α\n⊢ pathToList (listToPath l) = l",
"usedConstants": [
"Eq.mpr",
"Quiver.SingleObj.pathToList",
"congrArg",
"id",
"List.rec",
"List.cons",
"List",
"Quiver.SingleObj.star",
"Quiver.SingleObj.listToPath",
"Eq.refl",
... | induction l with
| nil => rfl
| cons a l ih => change a :: pathToList (listToPath l) = a :: l; rw [ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Quiver.SingleObj | {
"line": 113,
"column": 2
} | {
"line": 115,
"column": 74
} | [
{
"pp": "α : Type u_1\nl : List α\n⊢ pathToList (listToPath l) = l",
"usedConstants": [
"Eq.mpr",
"Quiver.SingleObj.pathToList",
"congrArg",
"id",
"List.rec",
"List.cons",
"List",
"Quiver.SingleObj.star",
"Quiver.SingleObj.listToPath",
"Eq.refl",
... | induction l with
| nil => rfl
| cons a l ih => change a :: pathToList (listToPath l) = a :: l; rw [ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.IsBounded | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 27
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : Preorder β\ninst✝¹ : NoMaxOrder β\nf : α → β\nl : Filter α\ninst✝ : l.NeBot\nhf : Tendsto f l atTop\nb : β\nhb : ∀ᶠ (x : β) in map f l, (fun x1 x2 ↦ x1 ≤ x2) x b\n⊢ False",
"usedConstants": [
"congrArg",
"Filter.map",
"Filter.Eventually",
... | rw [eventually_map] at hb | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.Homotopy | {
"line": 520,
"column": 6
} | {
"line": 526,
"column": 20
} | [
{
"pp": "case e_a.e_a\nι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g : C ⟶ D\nh k : D ⟶ E\ni✝ : ι\nP Q : ChainComplex V ℕ\ne : P ⟶ Q\nzero : P.X 0 ⟶ Q.X 1\ncomm_zero : e.f 0 = zero ≫ Q.d 1 0\none : P.X 1 ⟶ Q.X 2\ncomm_one : e... | cases i
· dsimp [fromNext, mkInductiveAux₂]
· dsimp [fromNext]
simp only [ChainComplex.next_nat_succ, dite_true]
rw [mkInductiveAux₃ e zero comm_zero one comm_one succ]
dsimp [xNextIso]
rw [id_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Homotopy | {
"line": 520,
"column": 6
} | {
"line": 526,
"column": 20
} | [
{
"pp": "case e_a.e_a\nι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g : C ⟶ D\nh k : D ⟶ E\ni✝ : ι\nP Q : ChainComplex V ℕ\ne : P ⟶ Q\nzero : P.X 0 ⟶ Q.X 1\ncomm_zero : e.f 0 = zero ≫ Q.d 1 0\none : P.X 1 ⟶ Q.X 2\ncomm_one : e... | cases i
· dsimp [fromNext, mkInductiveAux₂]
· dsimp [fromNext]
simp only [ChainComplex.next_nat_succ, dite_true]
rw [mkInductiveAux₃ e zero comm_zero one comm_one succ]
dsimp [xNextIso]
rw [id_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.LiminfLimsup | {
"line": 254,
"column": 45
} | {
"line": 254,
"column": 82
} | [
{
"pp": "α : Type u_1\ninst✝ : ConditionallyCompleteLattice α\ns : Set α\nh : BddAbove s\nhs : s.Nonempty\n⊢ sInf {a | ∀ x ∈ s, x ≤ a} = sSup s",
"usedConstants": [
"csInf_upperBounds_eq_csSup"
]
}
] | exact csInf_upperBounds_eq_csSup h hs | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.LiminfLimsup | {
"line": 367,
"column": 87
} | {
"line": 368,
"column": 19
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : CompleteLattice α\nf : Filter β\nu : β → α\n⊢ (blimsup u f fun x ↦ False) = ⊥",
"usedConstants": [
"sInf_univ",
"False",
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
"congrArg",
"Set.univ",
"Filter.Eventually"... | by
simp [blimsup_eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.LiminfLimsup | {
"line": 378,
"column": 2
} | {
"line": 378,
"column": 54
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : CompleteLattice α\nf : Filter β\n⊢ sInf {a | ∀ᶠ (n : β) in f, ⊥ ≤ a} ≤ ⊥",
"usedConstants": [
"Lattice.toSemilatticeSup",
"le_rfl",
"CompleteLattice.toLattice",
"Filter.Eventually",
"OrderBot.toBot",
"PartialOrder.toPreorder",
... | exact sInf_le (Eventually.of_forall fun _ => le_rfl) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.LiminfLimsup | {
"line": 424,
"column": 2
} | {
"line": 424,
"column": 77
} | [
{
"pp": "α : Type u_1\ninst✝ : CompleteLattice α\ns : Set α\n⊢ (𝓟 s).limsSup = sSup s",
"usedConstants": [
"Filter.limsSup",
"Eq.mpr",
"congrArg",
"Filter.Eventually",
"PartialOrder.toPreorder",
"setOf",
"Preorder.toLE",
"Membership.mem",
"CompleteLatti... | simpa only [limsSup, eventually_principal] using sInf_upperBounds_eq_sSup s | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Order.LiminfLimsup | {
"line": 424,
"column": 2
} | {
"line": 424,
"column": 77
} | [
{
"pp": "α : Type u_1\ninst✝ : CompleteLattice α\ns : Set α\n⊢ (𝓟 s).limsSup = sSup s",
"usedConstants": [
"Filter.limsSup",
"Eq.mpr",
"congrArg",
"Filter.Eventually",
"PartialOrder.toPreorder",
"setOf",
"Preorder.toLE",
"Membership.mem",
"CompleteLatti... | simpa only [limsSup, eventually_principal] using sInf_upperBounds_eq_sSup s | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.LiminfLimsup | {
"line": 424,
"column": 2
} | {
"line": 424,
"column": 77
} | [
{
"pp": "α : Type u_1\ninst✝ : CompleteLattice α\ns : Set α\n⊢ (𝓟 s).limsSup = sSup s",
"usedConstants": [
"Filter.limsSup",
"Eq.mpr",
"congrArg",
"Filter.Eventually",
"PartialOrder.toPreorder",
"setOf",
"Preorder.toLE",
"Membership.mem",
"CompleteLatti... | simpa only [limsSup, eventually_principal] using sInf_upperBounds_eq_sSup s | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.LiminfLimsup | {
"line": 706,
"column": 63
} | {
"line": 707,
"column": 99
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : CompleteBooleanAlgebra α\nf : Filter β\nu : β → α\n⊢ (limsup u f)ᶜ = liminf (compl ∘ u) f",
"usedConstants": [
"Filter.instMembership",
"iInf",
"Filter.liminf",
"Iff.of_eq",
"congrArg",
"iSup",
"Compl.compl",
"Funct... | by
simp only [limsup_eq_iInf_iSup, compl_iInf, compl_iSup, liminf_eq_iSup_iInf, Function.comp_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.LiminfLimsup | {
"line": 709,
"column": 63
} | {
"line": 710,
"column": 99
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : CompleteBooleanAlgebra α\nf : Filter β\nu : β → α\n⊢ (liminf u f)ᶜ = limsup (compl ∘ u) f",
"usedConstants": [
"Filter.instMembership",
"iInf",
"Filter.liminf",
"Iff.of_eq",
"congrArg",
"iSup",
"Compl.compl",
"Funct... | by
simp only [limsup_eq_iInf_iSup, compl_iInf, compl_iSup, liminf_eq_iSup_iInf, Function.comp_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.LiminfLimsup | {
"line": 988,
"column": 4
} | {
"line": 990,
"column": 26
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\ninst✝³ : ConditionallyCompleteLinearOrder β\nf✝¹ : Filter α\nu : α → β\ninst✝² : ConditionallyCompleteLinearOrder α\nf✝ : Filter α\nb : α\nf : ι → α\ns : ι' → Set ι\np : ι' → Prop\ninst✝¹ : Countable (Subtype p)\ninst✝ : N... | · rcases H with ⟨j, hj⟩
rcases (exists_surjective_nat (Subtype p)).choose_spec j with ⟨n, rfl⟩
exact ⟨n, Or.inl hj⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.LiminfLimsup | {
"line": 1106,
"column": 2
} | {
"line": 1106,
"column": 34
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : ConditionallyCompleteLattice β\ninst✝ : ConditionallyCompleteLattice γ\nf : Filter α\nv : α → β\nl : β → γ\nu : γ → β\ngc : GaloisConnection l u\nhlv : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f fun x ↦ l (v x)\nhv_co : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x... | rw [Filter.eventually_map] at hc | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.UniformSpace.LocallyUniformConvergence | {
"line": 73,
"column": 4
} | {
"line": 73,
"column": 28
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns : Set α\np : Filter ι\nhs : IsOpen[inst✝¹] s\nx : α\nhx : x ∈ s\n⊢ Tendsto (fun y ↦ (f y.2, F y.1 y.2)) (p ×ˢ 𝓝[s] x) (𝓤 β) ↔ Tendsto (fun y ↦ (f y.2, F y.1 y.2)) (p ×ˢ 𝓝 x) (𝓤... | rw [hs.nhdsWithin_eq hx] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.UniformSpace.LocallyUniformConvergence | {
"line": 73,
"column": 4
} | {
"line": 73,
"column": 28
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns : Set α\np : Filter ι\nhs : IsOpen[inst✝¹] s\nx : α\nhx : x ∈ s\n⊢ Tendsto (fun y ↦ (f y.2, F y.1 y.2)) (p ×ˢ 𝓝[s] x) (𝓤 β) ↔ Tendsto (fun y ↦ (f y.2, F y.1 y.2)) (p ×ˢ 𝓝 x) (𝓤... | rw [hs.nhdsWithin_eq hx] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.LocallyUniformConvergence | {
"line": 73,
"column": 4
} | {
"line": 73,
"column": 28
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns : Set α\np : Filter ι\nhs : IsOpen[inst✝¹] s\nx : α\nhx : x ∈ s\n⊢ Tendsto (fun y ↦ (f y.2, F y.1 y.2)) (p ×ˢ 𝓝[s] x) (𝓤 β) ↔ Tendsto (fun y ↦ (f y.2, F y.1 y.2)) (p ×ˢ 𝓝 x) (𝓤... | rw [hs.nhdsWithin_eq hx] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.LocallyUniformConvergence | {
"line": 177,
"column": 6
} | {
"line": 177,
"column": 50
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ninst✝² : TopologicalSpace α\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns : Set α\np : Filter ι\ninst✝ : UniformSpace γ\ng : β → γ\nt : Set β\nhg : UniformContinuousOn g t\nhf : TendstoLocallyUniformlyOn F f p s\nhfs : MapsTo f s t\nhFs : ... | tendstoLocallyUniformlyOn_iff_forall_tendsto | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UniformSpace.LocallyUniformConvergence | {
"line": 204,
"column": 6
} | {
"line": 204,
"column": 50
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ninst✝² : TopologicalSpace α\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns : Set α\np : Filter ι\ninst✝ : UniformSpace γ\nG : ι → α → γ\ng : α → γ\nhF : TendstoLocallyUniformlyOn F f p s\nhG : TendstoLocallyUniformlyOn G g p s\n⊢ TendstoLoc... | tendstoLocallyUniformlyOn_iff_forall_tendsto | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UniformSpace.LocallyUniformConvergence | {
"line": 306,
"column": 6
} | {
"line": 306,
"column": 50
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns : Set α\np : Filter ι\nG : ι → α → β\nhf : TendstoLocallyUniformlyOn F f p s\nhg✝ : ∀ᶠ (n : ι) in p, ∀ x ∈ s, Inseparable (F n x) (G n x)\nhg : ∀ᶠ (x : ι × α) in p ×ˢ 𝓟 s, Insepar... | tendstoLocallyUniformlyOn_iff_forall_tendsto | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UniformSpace.LocallyUniformConvergence | {
"line": 322,
"column": 6
} | {
"line": 322,
"column": 50
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns : Set α\np : Filter ι\ng : α → β\nhf : TendstoLocallyUniformlyOn F f p s\nhg✝ : ∀ x ∈ s, Inseparable (f x) (g x)\nhg : ∀ᶠ (x : ι × α) in p ×ˢ 𝓟 s, Inseparable (f x.2) (g x.2)\n⊢ T... | tendstoLocallyUniformlyOn_iff_forall_tendsto | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 260,
"column": 6
} | {
"line": 260,
"column": 69
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\n𝓐 : Filter ((α →ᵤ β) × (α →ᵤ β))\n𝓕 : Filter (β × β)\n⊢ 𝓐 ≤ UniformFun.filter α β 𝓕 ↔ (UniformFun.basis α β 𝓕).sets ⊆ 𝓐.sets",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
"UniformFun.filter.eq_1",
... | rw [UniformFun.filter, ← FilterBasis.generate, le_generate_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 260,
"column": 6
} | {
"line": 260,
"column": 69
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\n𝓐 : Filter ((α →ᵤ β) × (α →ᵤ β))\n𝓕 : Filter (β × β)\n⊢ 𝓐 ≤ UniformFun.filter α β 𝓕 ↔ (UniformFun.basis α β 𝓕).sets ⊆ 𝓐.sets",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
"UniformFun.filter.eq_1",
... | rw [UniformFun.filter, ← FilterBasis.generate, le_generate_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 260,
"column": 6
} | {
"line": 260,
"column": 69
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\n𝓐 : Filter ((α →ᵤ β) × (α →ᵤ β))\n𝓕 : Filter (β × β)\n⊢ 𝓐 ≤ UniformFun.filter α β 𝓕 ↔ (UniformFun.basis α β 𝓕).sets ⊆ 𝓐.sets",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
"UniformFun.filter.eq_1",
... | rw [UniformFun.filter, ← FilterBasis.generate, le_generate_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.CompactConvergence | {
"line": 334,
"column": 2
} | {
"line": 335,
"column": 63
} | [
{
"pp": "α : Type u₁\nβ : Type u₂\ninst✝² : TopologicalSpace α\ninst✝¹ : UniformSpace β\nf : C(α, β)\nι : Type u₃\np : Filter ι\nF : ι → C(α, β)\ninst✝ : CompactSpace α\n⊢ Tendsto F p (𝓝 f) ↔ TendstoUniformly (fun i a ↦ (F i) a) (⇑f) p",
"usedConstants": [
"UniformSpace",
"Equiv.instEquivLike",... | simp [isUniformEmbedding_uniformFunOfFun.isInducing.tendsto_nhds_iff,
UniformFun.tendsto_iff_tendstoUniformly, Function.comp_def] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.UniformSpace.CompactConvergence | {
"line": 334,
"column": 2
} | {
"line": 335,
"column": 63
} | [
{
"pp": "α : Type u₁\nβ : Type u₂\ninst✝² : TopologicalSpace α\ninst✝¹ : UniformSpace β\nf : C(α, β)\nι : Type u₃\np : Filter ι\nF : ι → C(α, β)\ninst✝ : CompactSpace α\n⊢ Tendsto F p (𝓝 f) ↔ TendstoUniformly (fun i a ↦ (F i) a) (⇑f) p",
"usedConstants": [
"UniformSpace",
"Equiv.instEquivLike",... | simp [isUniformEmbedding_uniformFunOfFun.isInducing.tendsto_nhds_iff,
UniformFun.tendsto_iff_tendstoUniformly, Function.comp_def] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.CompactConvergence | {
"line": 334,
"column": 2
} | {
"line": 335,
"column": 63
} | [
{
"pp": "α : Type u₁\nβ : Type u₂\ninst✝² : TopologicalSpace α\ninst✝¹ : UniformSpace β\nf : C(α, β)\nι : Type u₃\np : Filter ι\nF : ι → C(α, β)\ninst✝ : CompactSpace α\n⊢ Tendsto F p (𝓝 f) ↔ TendstoUniformly (fun i a ↦ (F i) a) (⇑f) p",
"usedConstants": [
"UniformSpace",
"Equiv.instEquivLike",... | simp [isUniformEmbedding_uniformFunOfFun.isInducing.tendsto_nhds_iff,
UniformFun.tendsto_iff_tendstoUniformly, Function.comp_def] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 577,
"column": 2
} | {
"line": 577,
"column": 58
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\n𝔖 : Set (Set α)\nS : Set α\nV : Set (β × β)\nuv : (α →ᵤ[𝔖] β) × (α →ᵤ[𝔖] β)\n⊢ uv ∈ UniformOnFun.gen 𝔖 S V ↔\n uv ∈ Prod.map (S.restrict ∘ ⇑UniformFun.toFun) (S.restrict ∘ ⇑UniformFun.toFun) ⁻¹' UniformFun.gen (↑S) β V",
"usedConstants": [
"Equiv.ins... | exact ⟨fun h ⟨x, hx⟩ => h x hx, fun h x hx => h ⟨x, hx⟩⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 676,
"column": 2
} | {
"line": 676,
"column": 74
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : UniformSpace β\n𝔖 : Set (Set α)\nι : Type u_5\nι' : Type u_6\ninst✝ : Nonempty ι\nt : ι → Set α\np : ι' → Prop\nV : ι' → Set (β × β)\nht : ∀ (i : ι), t i ∈ 𝔖\nhdir : Directed (fun x1 x2 ↦ x1 ⊆ x2) t\nhex : ∀ s ∈ 𝔖, ∃ i, s ⊆ t i\nhb : (𝓤 β).HasBasis p V\n⊢ (𝓤 (α... | have hne : 𝔖.Nonempty := (range_nonempty t).mono (range_subset_iff.2 ht) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 864,
"column": 6
} | {
"line": 864,
"column": 15
} | [
{
"pp": "α : Type u_1\nγ : Type u_3\nι : Type u_4\n𝔖 : Set (Set α)\nu : ι → UniformSpace γ\n⊢ ⨅ s ∈ 𝔖, ⨅ i, UniformSpace.comap (⇑UniformFun.ofFun ∘ s.restrict ∘ ⇑(toFun 𝔖)) (UniformFun.uniformSpace (↑s) γ) =\n ⨅ i, ⨅ s ∈ 𝔖, UniformSpace.comap (⇑UniformFun.ofFun ∘ s.restrict ∘ ⇑(toFun 𝔖)) (UniformFun.uni... | iInf_comm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 977,
"column": 95
} | {
"line": 980,
"column": 5
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\n⊢ IsUniformInducing fun f s ↦ UniformFun.ofFun ((↑s).restrict ((toFun 𝔖) f))",
"usedConstants": [
"Pi.uniformSpace_eq",
"Iff.mpr",
"Pi.uniformSpace",
"UniformSpace",
"Eq.mpr",
"iInf",
"E... | by
simp_rw [isUniformInducing_iff_uniformSpace, Pi.uniformSpace_eq, UniformSpace.comap_iInf,
← UniformSpace.comap_comap, iInf_subtype]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 1189,
"column": 8
} | {
"line": 1189,
"column": 35
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\np : Filter ι\ns : Set β\nF : ι → α → β\nf : α → β\ng : β → γ\nhF : ∀ᶠ (i : ι) in p, ∀ (x : α), F i x ∈ s\nhf : ∀ (x : α), f x ∈ s\nhg : UniformContinuousOn g s\nh : TendstoUniformly F f p\ns' : Set ... | eventuallyEq_iff_exists_mem | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 1195,
"column": 6
} | {
"line": 1195,
"column": 33
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\np : Filter ι\ns : Set β\nF : ι → α → β\nf : α → β\ng : β → γ\nhF✝ : ∀ᶠ (i : ι) in p, ∀ (x : α), F i x ∈ s\nhf : ∀ (x : α), f x ∈ s\nhg : UniformContinuousOn g s\nh : TendstoUniformly F f p\ns' : Set... | eventuallyEq_iff_exists_mem | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monoidal.Closed.Basic | {
"line": 531,
"column": 2
} | {
"line": 533,
"column": 42
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : MonoidalCategory C\nX Y Z : C\ninst✝¹ : Closed X\ninst✝ : Closed Y\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ curry' (f ≫ g) = (λ_ (𝟙_ C)).inv ≫ (curry' f ⊗ₘ curry' g) ≫ comp X Y Z",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Category.assoc",
"Cat... | rw [tensorHom_def_assoc, whiskerLeft_curry'_comp, MonoidalCategory.whiskerRight_id,
Category.assoc, Category.assoc, Iso.inv_hom_id_assoc, ← unitors_equal,
Iso.inv_hom_id_assoc, curry'_ihom_map] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Monoidal.Closed.Basic | {
"line": 531,
"column": 2
} | {
"line": 533,
"column": 42
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : MonoidalCategory C\nX Y Z : C\ninst✝¹ : Closed X\ninst✝ : Closed Y\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ curry' (f ≫ g) = (λ_ (𝟙_ C)).inv ≫ (curry' f ⊗ₘ curry' g) ≫ comp X Y Z",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Category.assoc",
"Cat... | rw [tensorHom_def_assoc, whiskerLeft_curry'_comp, MonoidalCategory.whiskerRight_id,
Category.assoc, Category.assoc, Iso.inv_hom_id_assoc, ← unitors_equal,
Iso.inv_hom_id_assoc, curry'_ihom_map] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Closed.Basic | {
"line": 531,
"column": 2
} | {
"line": 533,
"column": 42
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : MonoidalCategory C\nX Y Z : C\ninst✝¹ : Closed X\ninst✝ : Closed Y\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ curry' (f ≫ g) = (λ_ (𝟙_ C)).inv ≫ (curry' f ⊗ₘ curry' g) ≫ comp X Y Z",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Category.assoc",
"Cat... | rw [tensorHom_def_assoc, whiskerLeft_curry'_comp, MonoidalCategory.whiskerRight_id,
Category.assoc, Category.assoc, Iso.inv_hom_id_assoc, ← unitors_equal,
Iso.inv_hom_id_assoc, curry'_ihom_map] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.FiniteDimensional.Lemmas | {
"line": 457,
"column": 83
} | {
"line": 457,
"column": 94
} | [
{
"pp": "K : Type u\nV : Type v\nW : Type v'\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup W\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\ninst✝² : Module K W\nf : V →ₗ[K] W\np : Submodule K W\ninst✝¹ : FiniteDimensional K ↥p\ninst✝ : FiniteDimensional K ↥f.ker\n⊢ lift.{v, v'} (Module.rank K ↥p) + lift.{v', ... | lift_aleph0 | Mathlib.Tactic.evalGRewriteSeq | null |
Mathlib.LinearAlgebra.FiniteDimensional.Lemmas | {
"line": 462,
"column": 86
} | {
"line": 462,
"column": 97
} | [
{
"pp": "K : Type u\nV : Type v\nW : Type v'\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup W\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\ninst✝² : Module K W\nf : V →ₗ[K] W\np : Submodule K V\ninst✝¹ : FiniteDimensional K (V ⧸ p)\ninst✝ : FiniteDimensional K (W ⧸ f.range)\n⊢ lift.{v, v'} (Module.rank K (W ⧸... | lift_aleph0 | Mathlib.Tactic.evalGRewriteSeq | null |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 790,
"column": 6
} | {
"line": 790,
"column": 42
} | [
{
"pp": "R : Type u_1\nM : Type u_5\nM₁ : Type u_6\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M₁\ninst✝ : Module R M₁\nB : M →ₗ[R] M →ₗ[R] M₁\nhB : B.IsRefl\nhB' : B.SeparatingLeft\n⊢ B.SeparatingRight",
"usedConstants": [
"Eq.mpr",
"Submodule",
... | separatingRight_iff_flip_ker_eq_bot, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 995,
"column": 2
} | {
"line": 995,
"column": 88
} | [
{
"pp": "R : Type u_1\nM : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : LinearMap.BilinForm R M\nhs : ∀ (x : M), 0 ≤ (B x) x\nhB : IsSymm B\n⊢ (∃ x, (B x) x = 0 ∧ x ≠ 0 ∨ (B x) x ≠ 0 ∧ x = 0) ↔ ∃ x, x ≠ 0 ∧ (B x) x ≤ 0",
... | exact exists_congr fun x ↦ ⟨by aesop, fun ⟨h₀, h⟩ ↦ Or.inl ⟨le_antisymm h (hs x), h₀⟩⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Category.MonCat.FilteredColimits | {
"line": 149,
"column": 6
} | {
"line": 152,
"column": 13
} | [
{
"pp": "case refine_1\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nx y : M F\n⊢ ∀ (a b₁ b₂ : (j : J) × ↑(F.obj j)),\n (F ⋙ forget MonCat).ColimitTypeRel b₁ b₂ → colimitMulAux F a b₁ = colimitMulAux F a b₂",
"usedConstants": [
"MonoidHom.instFunLike",
"MonoidHo... | intro x y y' h
apply colimitMulAux_eq_of_rel_right
apply Types.FilteredColimit.rel_of_colimitTypeRel
exact h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.MonCat.FilteredColimits | {
"line": 149,
"column": 6
} | {
"line": 152,
"column": 13
} | [
{
"pp": "case refine_1\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nx y : M F\n⊢ ∀ (a b₁ b₂ : (j : J) × ↑(F.obj j)),\n (F ⋙ forget MonCat).ColimitTypeRel b₁ b₂ → colimitMulAux F a b₁ = colimitMulAux F a b₂",
"usedConstants": [
"MonoidHom.instFunLike",
"MonoidHo... | intro x y y' h
apply colimitMulAux_eq_of_rel_right
apply Types.FilteredColimit.rel_of_colimitTypeRel
exact h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.MonCat.FilteredColimits | {
"line": 153,
"column": 6
} | {
"line": 156,
"column": 13
} | [
{
"pp": "case refine_2\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nx y : M F\n⊢ ∀ (a₁ a₂ b : (j : J) × ↑(F.obj j)),\n (F ⋙ forget MonCat).ColimitTypeRel a₁ a₂ → colimitMulAux F a₁ b = colimitMulAux F a₂ b",
"usedConstants": [
"MonoidHom.instFunLike",
"MonCat.F... | intro x x' y h
apply colimitMulAux_eq_of_rel_left
apply Types.FilteredColimit.rel_of_colimitTypeRel
exact h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.MonCat.FilteredColimits | {
"line": 153,
"column": 6
} | {
"line": 156,
"column": 13
} | [
{
"pp": "case refine_2\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nx y : M F\n⊢ ∀ (a₁ a₂ b : (j : J) × ↑(F.obj j)),\n (F ⋙ forget MonCat).ColimitTypeRel a₁ a₂ → colimitMulAux F a₁ b = colimitMulAux F a₂ b",
"usedConstants": [
"MonoidHom.instFunLike",
"MonCat.F... | intro x x' y h
apply colimitMulAux_eq_of_rel_left
apply Types.FilteredColimit.rel_of_colimitTypeRel
exact h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.Ab | {
"line": 89,
"column": 43
} | {
"line": 92,
"column": 5
} | [
{
"pp": "S : ShortComplex Ab\nx : ↥(AddCommGrpCat.Hom.hom S.g).ker\n⊢ (ConcreteCategory.hom S.iCycles) ((ConcreteCategory.hom S.abCyclesIso.inv) x) = ↑x",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"CategoryTheory.ShortComplex.abCyclesIso",
"AddCommGrpCat.... | by
dsimp only [abCyclesIso]
rw [← ConcreteCategory.comp_apply, S.abLeftHomologyData.cyclesIso_inv_comp_iCycles]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ShortComplex.Ab | {
"line": 114,
"column": 15
} | {
"line": 114,
"column": 78
} | [
{
"pp": "S : ShortComplex Ab\nh : Function.Surjective ⇑S.abToCycles\nx₂ : ↑S.X₂\nhx₂ : (ConcreteCategory.hom S.g) x₂ = 0\nx₁ : ↑S.X₁\nhx₁ : S.abToCycles x₁ = ⟨x₂, hx₂⟩\n⊢ (ConcreteCategory.hom S.f) x₁ = x₂",
"usedConstants": [
"_private.Mathlib.Algebra.Homology.ShortComplex.Ab.0.CategoryTheory.ShortCo... | by simpa only [Subtype.ext_iff, abToCycles_apply_coe] using hx₁ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 292,
"column": 4
} | {
"line": 293,
"column": 90
} | [
{
"pp": "K✝ : Type uK\nV✝ : Type uV\ninst✝¹³ : CommSemiring K✝\ninst✝¹² : AddCommMonoid V✝\ninst✝¹¹ : Module K✝ V✝\ninst✝¹⁰ : Projective K✝ V✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁹ : CommSemiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : Module R M\ninst✝⁵ : Module R N\ninst✝⁴ ... | have heq := lift_rank_eq_of_equiv_equiv (R := K) (R' := K) (M := V) (M' := Dual K (Dual K V))
(ZeroHom.id K) (evalEquiv K V) bijective_id (fun r v ↦ (evalEquiv K V).map_smul _ _) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Limits.Filtered | {
"line": 59,
"column": 4
} | {
"line": 60,
"column": 61
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝ : Category.{v, u} C\nh : ∀ {J : Type v} [inst : SmallCategory J] [FinCategory J] (F : J ⥤ C), Nonempty (Cone F)\nJ : Type v\nx✝¹ : SmallCategory J\nx✝ : FinCategory J\nF : J ⥤ C\n⊢ ∃ X, Nonempty (limit (F ⋙ coyoneda.obj (op X)))",
"usedConstants": [
"CategoryT... | obtain ⟨c⟩ := h F
exact ⟨c.pt, ⟨(limitCompCoyonedaIsoCone F c.pt).inv c.π⟩⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Filtered | {
"line": 59,
"column": 4
} | {
"line": 60,
"column": 61
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝ : Category.{v, u} C\nh : ∀ {J : Type v} [inst : SmallCategory J] [FinCategory J] (F : J ⥤ C), Nonempty (Cone F)\nJ : Type v\nx✝¹ : SmallCategory J\nx✝ : FinCategory J\nF : J ⥤ C\n⊢ ∃ X, Nonempty (limit (F ⋙ coyoneda.obj (op X)))",
"usedConstants": [
"CategoryT... | obtain ⟨c⟩ := h F
exact ⟨c.pt, ⟨(limitCompCoyonedaIsoCone F c.pt).inv c.π⟩⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products | {
"line": 296,
"column": 2
} | {
"line": 299,
"column": 17
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : Type u₂\ninst✝¹ : Category.{v₂, u₂} J\nX : Type v₂\nA B : C\ninst✝ : HasBinaryProduct A B\n⊢ HasBinaryCoproduct (op A) (op B)",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.HasBinaryCoproduct",
"CategoryTheory.Limits.HasProduct... | have : HasProduct fun x ↦ (WalkingPair.casesOn x A B : C) := ‹_›
change HasCoproduct _
convert! (inferInstance : HasCoproduct fun x ↦ op (WalkingPair.casesOn x A B : C)) with x
cases x <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products | {
"line": 296,
"column": 2
} | {
"line": 299,
"column": 17
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : Type u₂\ninst✝¹ : Category.{v₂, u₂} J\nX : Type v₂\nA B : C\ninst✝ : HasBinaryProduct A B\n⊢ HasBinaryCoproduct (op A) (op B)",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.HasBinaryCoproduct",
"CategoryTheory.Limits.HasProduct... | have : HasProduct fun x ↦ (WalkingPair.casesOn x A B : C) := ‹_›
change HasCoproduct _
convert! (inferInstance : HasCoproduct fun x ↦ op (WalkingPair.casesOn x A B : C)) with x
cases x <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products | {
"line": 352,
"column": 6
} | {
"line": 352,
"column": 18
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nA B : C\ninst✝ : HasBinaryProduct A B\n⊢ (opProdIsoCoprod A B).inv.unop ≫ coprod.inl.unop = prod.fst",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.opProdIsoCoprod",
"Opposite",
"CategoryTheory.CategoryStruct.toQuiver",
... | ← unop_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products | {
"line": 356,
"column": 6
} | {
"line": 356,
"column": 18
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nA B : C\ninst✝ : HasBinaryProduct A B\n⊢ (opProdIsoCoprod A B).inv.unop ≫ coprod.inr.unop = prod.snd",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.opProdIsoCoprod",
"Opposite",
"CategoryTheory.CategoryStruct.toQuiver",
... | ← unop_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Comma.Presheaf.Basic | {
"line": 148,
"column": 2
} | {
"line": 148,
"column": 32
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nA : Cᵒᵖ ⥤ Type v\nY : C\nη : yoneda.obj Y ⟶ A\nX : C\ns : yoneda.obj X ⟶ A\np : OverArrows η s\n⊢ (ConcreteCategory.hom (η.app (op (unop (op X))))) p.val = yonedaEquiv s",
"usedConstants": [
"CategoryTheory.Functor",
"Opposite",
"Equiv.instEq... | simp only [unop_op, p.app_val] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Comma.Presheaf.Basic | {
"line": 300,
"column": 2
} | {
"line": 303,
"column": 52
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nA : Cᵒᵖ ⥤ Type v\nF : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v\nX : C\np q : YonedaCollection F X\nh : p.fst = q.fst\nh' : (ConcreteCategory.hom (F.map (eqToHom ⋯))) q.snd = p.snd\n⊢ p = q",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor",
... | rcases p with ⟨p, p'⟩
rcases q with ⟨q, q'⟩
obtain rfl : p = q := yonedaEquiv.symm.injective h
exact Sigma.ext rfl (by simpa [snd] using h'.symm) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Comma.Presheaf.Basic | {
"line": 300,
"column": 2
} | {
"line": 303,
"column": 52
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nA : Cᵒᵖ ⥤ Type v\nF : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v\nX : C\np q : YonedaCollection F X\nh : p.fst = q.fst\nh' : (ConcreteCategory.hom (F.map (eqToHom ⋯))) q.snd = p.snd\n⊢ p = q",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor",
... | rcases p with ⟨p, p'⟩
rcases q with ⟨q, q'⟩
obtain rfl : p = q := yonedaEquiv.symm.injective h
exact Sigma.ext rfl (by simpa [snd] using h'.symm) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Comma.Presheaf.Basic | {
"line": 340,
"column": 82
} | {
"line": 341,
"column": 82
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nA : Cᵒᵖ ⥤ Type v\nF : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v\nX Y : C\nf : X ⟶ Y\np : YonedaCollection F Y\n⊢ (map₂ F f p).yonedaEquivFst = (ConcreteCategory.hom (A.map f.op)) p.yonedaEquivFst",
"usedConstants": [
"CategoryTheory.Functor",
"Catego... | by
simp only [YonedaCollection.yonedaEquivFst_eq, map₂_fst, yonedaEquiv_naturality] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Preserves.FunctorCategory | {
"line": 75,
"column": 14
} | {
"line": 75,
"column": 71
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v₁, u} C\nD : Type u₂\ninst✝⁴ : Category.{u, u₂} D\ninst✝³ : HasBinaryProducts D\ninst✝² : HasColimits D\ninst✝¹ : ∀ (X : D), PreservesColimits (prod.functor.obj X)\nF : C ⥤ D\nJ : Type u\ninst✝ : Category.{u, u} J\nK : J ⥤ C ⥤ D\nc : Cocone K\nt : IsColimit c\nk : C\nthi... | apply asIso (prodComparison ((evaluation C D).obj k) F G) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Functor.KanExtension.Basic | {
"line": 598,
"column": 4
} | {
"line": 598,
"column": 69
} | [
{
"pp": "case mp\nC : Type u_1\nH : Type u_3\nD : Type u_4\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_3, u_3} H\ninst✝ : Category.{v_4, u_4} D\nL : C ⥤ D\nF₁ F₂ : C ⥤ H\nF₁' F₂' : D ⥤ H\nα₁ : L ⋙ F₁' ⟶ F₁\nα₂ : L ⋙ F₂' ⟶ F₂\ne : F₁ ≅ F₂\ne' : F₁' ≅ F₂'\nh : L.whiskerLeft e'.hom ≫ α₂ = α₁ ≫ e.hom\neq ... | exact fun _ => ⟨⟨eq.1 (isUniversalOfIsRightKanExtension F₁' α₁)⟩⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Functor.KanExtension.Basic | {
"line": 598,
"column": 4
} | {
"line": 598,
"column": 69
} | [
{
"pp": "case mp\nC : Type u_1\nH : Type u_3\nD : Type u_4\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_3, u_3} H\ninst✝ : Category.{v_4, u_4} D\nL : C ⥤ D\nF₁ F₂ : C ⥤ H\nF₁' F₂' : D ⥤ H\nα₁ : L ⋙ F₁' ⟶ F₁\nα₂ : L ⋙ F₂' ⟶ F₂\ne : F₁ ≅ F₂\ne' : F₁' ≅ F₂'\nh : L.whiskerLeft e'.hom ≫ α₂ = α₁ ≫ e.hom\neq ... | exact fun _ => ⟨⟨eq.1 (isUniversalOfIsRightKanExtension F₁' α₁)⟩⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Functor.KanExtension.Basic | {
"line": 598,
"column": 4
} | {
"line": 598,
"column": 69
} | [
{
"pp": "case mp\nC : Type u_1\nH : Type u_3\nD : Type u_4\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_3, u_3} H\ninst✝ : Category.{v_4, u_4} D\nL : C ⥤ D\nF₁ F₂ : C ⥤ H\nF₁' F₂' : D ⥤ H\nα₁ : L ⋙ F₁' ⟶ F₁\nα₂ : L ⋙ F₂' ⟶ F₂\ne : F₁ ≅ F₂\ne' : F₁' ≅ F₂'\nh : L.whiskerLeft e'.hom ≫ α₂ = α₁ ≫ e.hom\neq ... | exact fun _ => ⟨⟨eq.1 (isUniversalOfIsRightKanExtension F₁' α₁)⟩⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Functor.KanExtension.Basic | {
"line": 719,
"column": 4
} | {
"line": 724,
"column": 54
} | [
{
"pp": "case refine_1\nC : Type u_1\nH : Type u_3\nD : Type u_4\nD' : Type u_5\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_3, u_3} H\ninst✝² : Category.{v_4, u_4} D\ninst✝¹ : Category.{v_5, u_5} D'\nL : C ⥤ D\nL' : D ⥤ D'\nF₀ : C ⥤ H\nF₁ : D ⥤ H\nα : F₀ ⟶ L ⋙ F₁\ninst✝ : F₁.IsLeftKanExtension α\nF₂ :... | let i :
(LeftExtension.precomp₂ L' α).obj (LeftExtension.mk F₂ β) ≅
Ψ.inverse.obj (LeftExtension.mk F₂ γ) :=
StructuredArrow.isoMk (NatIso.ofComponents fun _ ↦ .refl _) <| by
ext x
simp [Ψ, ← congr_app hγ x, ← Functor.map_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Basic | {
"line": 168,
"column": 33
} | {
"line": 173,
"column": 88
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nJ : Type u_1\ninst✝³ : Category.{v_1, u_1} J\nD : Type u_2\ninst✝² : Category.{v_2, u_2} D\ne : C ≌ D\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : HasExactLimitsOfShape J C\n⊢ HasExactLimitsOfShape J D",
"usedConstants": [
"CategoryTheory.Limits.PreservesCol... | by
haveI : HasLimitsOfShape J D := Adjunction.hasLimitsOfShape_of_equivalence e.inverse
refine ⟨⟨fun _ _ _ => ⟨@fun K => ?_⟩⟩⟩
refine preservesColimit_of_natIso K (?_ : e.congrRight.inverse ⋙ lim ⋙ e.functor ≅ lim)
apply e.symm.congrRight.fullyFaithfulFunctor.preimageIso
exact isoWhiskerLeft (_ ⋙ lim) e.unitI... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 672,
"column": 6
} | {
"line": 672,
"column": 23
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nG : C\ninst✝ : ∀ (A : C), HasProduct fun x ↦ G\n⊢ IsCoseparator G ↔ ∀ (A : C), Mono (Pi.lift fun f ↦ f)",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Mono",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
... | isCoseparator_def | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Subobject.Comma | {
"line": 176,
"column": 6
} | {
"line": 176,
"column": 18
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nS : C ⥤ D\nT : D\nA : CostructuredArrow S T\nP : (CostructuredArrow S T)ᵒᵖ\nf : P ⟶ op A\ninst✝ : Mono f.unop.left.op\n⊢ f.unop.left.op.unop ≫ (Subobject.underlyingIso f.unop.left.op).hom.unop = (Subobject.mk f.unop.l... | ← unop_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 734,
"column": 6
} | {
"line": 734,
"column": 23
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : HasZeroMorphisms C\nβ : Type w\nf : β → C\nhf : (ObjectProperty.ofObj f).IsCoseparating\nc : Fan f\nhc : IsLimit c\n⊢ IsCoseparator c.pt",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | isCoseparator_def | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Presheaf | {
"line": 114,
"column": 53
} | {
"line": 114,
"column": 83
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nℰ : Type u₂\ninst✝ : Category.{v₂, u₂} ℰ\nA : C ⥤ ℰ\nP : Cᵒᵖ ⥤ Type (max w v₁ v₂)\nE : ℰ\ng : P ⟶ (restrictedULiftYoneda A).obj E\ny y' : CostructuredArrow uliftYoneda.{max w v₂, v₁, u₁} P\nf : y ⟶ y'\n⊢ A.map f.left ≫ (uliftYonedaEquiv (y'.hom ≫ g)).down =\n ... | map_comp_uliftYonedaEquiv_down | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Presheaf | {
"line": 495,
"column": 2
} | {
"line": 498,
"column": 87
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nG : (Cᵒᵖ ⥤ Type (max w v₁ v₂)) ⥤ Dᵒᵖ ⥤ Type (max w v₁ v₂)\nφ : F ⋙ uliftYoneda.{max w v₁, v₂, u₂} ⟶ uliftYoneda.{max w v₂, v₁, u₁} ⋙ G\nP : Cᵒᵖ ⥤ Type (max w v₁ v₂)\nx y : P.Elements\nf : x ⟶ y\n⊢ uliftYoned... | have eq₁ : uliftYoneda.map f.1.unop ≫ uliftYonedaEquiv.symm x.2 =
uliftYonedaEquiv.{max w v₂}.symm y.2 :=
uliftYonedaEquiv.injective
(by simpa only [Equiv.apply_symm_apply, ← uliftYonedaEquiv_naturality] using f.2) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Limits.Presheaf | {
"line": 569,
"column": 2
} | {
"line": 569,
"column": 93
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\nG : (Cᵒᵖ ⥤ Type (max w v₁ v₂)) ⥤ Dᵒᵖ ⥤ Type (max w v₁ v₂)\nφ : F ⋙ uliftYoneda.{max w v₁, v₂, u₂} ⟶ uliftYoneda.{max w v₂, v₁, u₁} ⋙ G\ninst✝ : ∀ (P : Cᵒᵖ ⥤ Type (max w v₁ v₂)), F.op.HasLeftKanExtension P\n... | apply (F.op.lan.obj (uliftYoneda.obj X)).hom_ext_of_isLeftKanExtension (F.op.lanUnit.app _) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Limits.Types.Coproducts | {
"line": 299,
"column": 12
} | {
"line": 299,
"column": 44
} | [
{
"pp": "X Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective ⇑(ConcreteCategory.hom c.inl)\nh₂ : Injective ⇑(ConcreteCategory.hom c.inr)\nh₃ : IsCompl (Set.range ⇑(ConcreteCategory.hom c.inl)) (Set.range ⇑(ConcreteCategory.hom c.inr))\n⊢ ∀ (x : (fun X ↦ X) (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { ... | eq_compl_iff_isCompl.mpr h₃.symm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Types.Coproducts | {
"line": 298,
"column": 6
} | {
"line": 300,
"column": 29
} | [
{
"pp": "case mpr\nX Y : Type u\nc : BinaryCofan X Y\nh₁ : Injective ⇑(ConcreteCategory.hom c.inl)\nh₂ : Injective ⇑(ConcreteCategory.hom c.inr)\nh₃ : IsCompl (Set.range ⇑(ConcreteCategory.hom c.inl)) (Set.range ⇑(ConcreteCategory.hom c.inr))\n⊢ Nonempty (IsColimit c)",
"usedConstants": [
"Iff.mpr",
... | have : ∀ x, x ∈ Set.range c.inl ∨ x ∈ Set.range c.inr := by
rw [eq_compl_iff_isCompl.mpr h₃.symm]
exact fun _ => or_not | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Module.Injective | {
"line": 287,
"column": 6
} | {
"line": 287,
"column": 68
} | [
{
"pp": "R : Type u\ninst✝⁷ : Ring R\nQ : Type v\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM : Type u_1\nN : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ⇑i)\nh : Baer R Q\ny : N\nr : R\n... | ExtensionOfMaxAdjoin.extendIdealTo_is_extension i f h y r this | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Interval.Set.Group | {
"line": 170,
"column": 2
} | {
"line": 170,
"column": 25
} | [
{
"pp": "α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\na b : α\nm n : ℤ\nhmn : m ≠ n\nx : α\nhx : x ∈ Ioc (a * b ^ m) (a * b ^ (m + 1)) ∩ Ioc (a * b ^ n) (a * b ^ (n + 1))\nhb : 1 < b\ni1 : m ≤ n\ni2 : n ≤ m\n⊢ m = n",
"usedConstants": [
"SemilatticeInf.toPart... | exact le_antisymm i1 i2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Order.Interval.Set.Group | {
"line": 185,
"column": 2
} | {
"line": 185,
"column": 25
} | [
{
"pp": "α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\na b : α\nm n : ℤ\nhmn : m ≠ n\nx : α\nhx : x ∈ Ico (a * b ^ m) (a * b ^ (m + 1)) ∩ Ico (a * b ^ n) (a * b ^ (n + 1))\nhb : 1 < b\ni1 : m ≤ n\ni2 : n ≤ m\n⊢ m = n",
"usedConstants": [
"SemilatticeInf.toPart... | exact le_antisymm i1 i2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Module.Injective | {
"line": 408,
"column": 2
} | {
"line": 408,
"column": 34
} | [
{
"pp": "R : Type u\ninst✝³ : Ring R\nQ : Type v\ninst✝² : AddCommGroup Q\ninst✝¹ : Module R Q\ninst✝ : Small.{v, u} R\ninj : Injective R Q\nI : Ideal R\ng : ↥I →ₗ[R] Q\neI : Shrink.{v, u} ↥I ≃ₗ[R] ↥I := Shrink.linearEquiv R ↥I\n⊢ ∃ g', ∀ (x : R) (mem : x ∈ I), g' x = g ⟨x, mem⟩",
"usedConstants": [
"... | let eR := Shrink.linearEquiv R R | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Data.Nat.Factorization.Induction | {
"line": 44,
"column": 6
} | {
"line": 44,
"column": 33
} | [
{
"pp": "case convert_2\na✝ b m n✝ p✝ : ℕ\nmotive : ℕ → Sort u_1\nzero : motive 0\none : motive 1\nprime_pow_mul : (a p n : ℕ) → Prime p → ¬p ∣ a → 0 < n → motive a → motive (p ^ n * a)\na n k : ℕ\nhk : (m : ℕ) → m < k + 2 → motive m\np : ℕ := (k + 2).minFac\nhp : Prime p\nt : ℕ := (k + 2).factorization p\nhpt ... | · simp [htp.ne', hp.one_lt] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Nat.Totient | {
"line": 72,
"column": 67
} | {
"line": 72,
"column": 80
} | [
{
"pp": "n : ℕ\n⊢ (n + 1).gcd 1 = 1",
"usedConstants": [
"Nat.gcd",
"Eq.mpr",
"congrArg",
"Nat.gcd_one_right",
"id",
"instOfNatNat",
"instHAdd",
"HAdd.hAdd",
"Nat",
"instAddNat",
"OfNat.ofNat",
"Eq"
]
}
] | gcd_one_right | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.Totient | {
"line": 227,
"column": 4
} | {
"line": 227,
"column": 22
} | [
{
"pp": "p : ℕ\nh : #({a ∈ Ico (0 + 1) p | p.Coprime a}) = p - 1\nhp : 1 < p\n⊢ Prime p",
"usedConstants": [
"Nat.Coprime",
"Nat.instOne",
"congrArg",
"PartialOrder.toPreorder",
"Nat.instDecidableCoprime",
"HSub.hSub",
"Nat.instLocallyFiniteOrder",
"Semilattic... | ← Nat.card_Ico 1 p | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.Totient | {
"line": 372,
"column": 45
} | {
"line": 374,
"column": 46
} | [
{
"pp": "p n : ℕ\nhp : Prime p\nh : ¬p ∣ n\n⊢ φ (p * n) = (p - 1) * φ n",
"usedConstants": [
"Eq.mpr",
"Nat.Coprime",
"False",
"Dvd.dvd",
"HMul.hMul",
"eq_false",
"congrArg",
"HSub.hSub",
"Nat.totient_prime",
"Eq.mp",
"id",
"instSubNat"... | by
rw [totient_mul _, totient_prime hp]
simpa [h] using coprime_or_dvd_of_prime hp n | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Filter.AtTopBot.Monoid | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 81
} | [
{
"pp": "α : Type u_1\nM : Type u_2\ninst✝² : CommMonoid M\ninst✝¹ : Preorder M\ninst✝ : IsOrderedMonoid M\nl : Filter α\nf : α → M\nhf : Tendsto f l atTop\nn : ℕ\nhn : 0 < n\n⊢ Tendsto (fun x ↦ f x ^ n) l atTop",
"usedConstants": [
"MulOne.toOne",
"Monoid.toMulOneClass",
"Preorder.toLE",
... | refine tendsto_atTop_mono' _ ((hf.eventually_ge_atTop 1).mono fun x hx ↦ ?_) hf | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Order.Filter.AtTopBot.Field | {
"line": 101,
"column": 97
} | {
"line": 102,
"column": 61
} | [
{
"pp": "α : Type u_1\ninst✝² : Semifield α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\nn : ℤ\nhn : 0 < n\n⊢ Tendsto (fun x ↦ x ^ n) atTop atTop",
"usedConstants": [
"zpow_natCast",
"False",
"GroupWithZero.toDivInvMonoid",
"eq_false",
"congrArg",
"PartialOrder... | by
lift n to ℕ using hn.le; simp [(Int.natCast_pos.mp hn).ne'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Filter.AtTopBot.Field | {
"line": 226,
"column": 2
} | {
"line": 226,
"column": 58
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : Field α\ninst✝² : LinearOrder α\ninst✝¹ : IsStrictOrderedRing α\nl : Filter β\nf : β → α\nr : α\ninst✝ : l.NeBot\nh : Tendsto f l atBot\n⊢ Tendsto (fun x ↦ f x / r) l atTop ↔ r < 0",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"DivInvMonoid.... | simp [div_eq_mul_inv, tendsto_mul_const_atTop_iff_neg h] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.Filter.AtTopBot.Field | {
"line": 226,
"column": 2
} | {
"line": 226,
"column": 58
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : Field α\ninst✝² : LinearOrder α\ninst✝¹ : IsStrictOrderedRing α\nl : Filter β\nf : β → α\nr : α\ninst✝ : l.NeBot\nh : Tendsto f l atBot\n⊢ Tendsto (fun x ↦ f x / r) l atTop ↔ r < 0",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"DivInvMonoid.... | simp [div_eq_mul_inv, tendsto_mul_const_atTop_iff_neg h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.AtTopBot.Field | {
"line": 226,
"column": 2
} | {
"line": 226,
"column": 58
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : Field α\ninst✝² : LinearOrder α\ninst✝¹ : IsStrictOrderedRing α\nl : Filter β\nf : β → α\nr : α\ninst✝ : l.NeBot\nh : Tendsto f l atBot\n⊢ Tendsto (fun x ↦ f x / r) l atTop ↔ r < 0",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"DivInvMonoid.... | simp [div_eq_mul_inv, tendsto_mul_const_atTop_iff_neg h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.IsUniformGroup.Basic | {
"line": 337,
"column": 52
} | {
"line": 342,
"column": 89
} | [
{
"pp": "G✝ : Type u_1\ninst✝⁵ : Group G✝\ninst✝⁴ : TopologicalSpace G✝\ninst✝³ : IsTopologicalGroup G✝\nG : Type u_2\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\n⊢ G ≃ᵤ Gᵐᵒᵖ",
"usedConstants": [
"UniformContinuous",
"UniformSpace",
"MulOpposite.opEquiv",
... | by
letI : UniformSpace G := IsTopologicalGroup.rightUniformSpace G
letI : UniformSpace Gᵐᵒᵖ := IsTopologicalGroup.leftUniformSpace Gᵐᵒᵖ
refine ⟨MulOpposite.opEquiv, ?_, ?_⟩
· simp [uniformContinuous_iff, ← comap_op_leftUniformSpace]
· simp [uniformContinuous_iff, ← comap_op_leftUniformSpace, ← UniformSpace.co... | [anonymous] | Lean.Parser.Term.byTactic |
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