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.Topology.Order | {
"line": 971,
"column": 2
} | {
"line": 971,
"column": 25
} | {
"line": 973,
"column": 0
} | [
{
"pp": "α : Type u_1\nl : Filter α\np : α → Prop\nq : Prop\n⊢ Tendsto p l (𝓝 q) ↔ q → ∀ᶠ (x : α) in l, p x",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Pure.pure",
"False",
"eq_false",
"nhds_false",
"congrArg",
"Filter.Eventually",
"nhds_true"... | [] | by_cases q <;> simp [*] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Maps.Basic | {
"line": 577,
"column": 6
} | {
"line": 577,
"column": 18
} | {
"line": 577,
"column": 19
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nf : X → Y\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\n⊢ IsClosedMap f ↔ ∀ {u : Set X}, IsOpen[inst✝¹] u → IsOpen[inst✝] (kernImage f u)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.kernImage",
"congrArg"... | [
"X : Type u_1\nY : Type u_2\nf : X → Y\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\n⊢ (∀ (U : Set X), IsClosed[inst✝¹] U → IsClosed[inst✝] (f '' U)) ↔\n ∀ {u : Set X}, IsOpen[inst✝¹] u → IsOpen[inst✝] (kernImage f u)"
] | IsClosedMap, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Constructions.SumProd | {
"line": 953,
"column": 2
} | {
"line": 953,
"column": 78
} | {
"line": 954,
"column": 2
} | [
{
"pp": "X : Type u\nY : Type v\nZ : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nf : X → Z\ng : Y → Z\nhf : IsInducing f\nhg : IsInducing g\nhFg : Disjoint (𝓟 (range f)) (𝓝ˢ (range g))\nhfG : Disjoint (𝓝ˢ (range f)) (𝓟 (range g))\nx : X ⊕ Y\n⊢ 𝓝 x = comap... | [
"X : Type u\nY : Type v\nZ : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nf : X → Z\ng : Y → Z\nhf : IsInducing f\nhg : IsInducing g\nhFg : Disjoint (𝓟 (range f)) (𝓝ˢ (range g))\nhfG : Disjoint (𝓝ˢ (range f)) (𝓟 (range g))\nx : X ⊕ Y\n⊢ comap (Sum.elim f g) (𝓝... | apply le_antisymm ((hf.continuous.sumElim hg.continuous).tendsto x).le_comap | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Order.Filter.CountablyGenerated | {
"line": 166,
"column": 2
} | {
"line": 168,
"column": 82
} | {
"line": 170,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Sort u_5\nf g : Filter α\ninst✝¹ : f.IsCountablyGenerated\ninst✝ : g.IsCountablyGenerated\n⊢ (f ⊔ g).IsCountablyGenerated",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Filter.HasCountableBasis.mk",
"Filter.... | [] | rcases f.exists_antitone_basis with ⟨s, hs⟩
rcases g.exists_antitone_basis with ⟨t, ht⟩
exact HasCountableBasis.isCountablyGenerated ⟨hs.1.sup ht.1, Set.to_countable _⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.CountablyGenerated | {
"line": 166,
"column": 2
} | {
"line": 168,
"column": 82
} | {
"line": 170,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Sort u_5\nf g : Filter α\ninst✝¹ : f.IsCountablyGenerated\ninst✝ : g.IsCountablyGenerated\n⊢ (f ⊔ g).IsCountablyGenerated",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Filter.HasCountableBasis.mk",
"Filter.... | [] | rcases f.exists_antitone_basis with ⟨s, hs⟩
rcases g.exists_antitone_basis with ⟨t, ht⟩
exact HasCountableBasis.isCountablyGenerated ⟨hs.1.sup ht.1, Set.to_countable _⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.Pi | {
"line": 322,
"column": 2
} | {
"line": 322,
"column": 55
} | {
"line": 324,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\nf : (i : ι) → Filter (α i)\nβ : ι → Type u_3\nm : (i : ι) → α i → β i\ns : Set ((i : ι) → β i)\nh : ∀ (i : ι), ∃ t₁, m i ⁻¹' t₁ ∈ f i ∧ eval i ⁻¹' t₁ ⊆ s\ni : ι\nt : Set (β i)\nH : m i ⁻¹' t ∈ f i\nhH : eval i ⁻¹' t ⊆ s\n⊢ ∃ t₁ ∈ f i, eval i ⁻¹' t₁ ⊆ (fun k i ↦ m i (k i)... | [] | exact ⟨{ x : α i | m i x ∈ t }, H, fun x hx => hH hx⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Filter.Cofinite | {
"line": 333,
"column": 4
} | {
"line": 334,
"column": 33
} | {
"line": 336,
"column": 0
} | [
{
"pp": "case refine_4.refine_3\nα : Type u_2\nf : Filter α\nq : Set α × Filter α\nhq : (fun p ↦ p.2 ≤ cofinite ∧ Disjoint (𝓟 p.1) p.2 ∧ f = 𝓟 p.1 ⊔ p.2) q\nhqk : f.ker = q.1\n⊢ Coheyting.boundary f ≤ q.2",
"ppTerm": "?refine_4.refine_3",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | · grw [hq.2.2, Coheyting.boundary_sup_le, boundary_principal, bot_sup_eq]
exact Coheyting.boundary_le | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Constructions | {
"line": 318,
"column": 4
} | {
"line": 318,
"column": 68
} | {
"line": 319,
"column": 2
} | [
{
"pp": "case mp\nX : Type u\nx : CofiniteTopology X\nU V : Set (CofiniteTopology X)\nhVU : V ⊆ U\nV_op : V.Nonempty → Vᶜ.Finite\nhaV : x ∈ V\n⊢ U ∈ pure x ⊔ cofinite",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Pure.pure",
"Filter.instMembership",
"Iff.mpr",
"Mem... | [] | exact mem_sup.mpr ⟨hVU haV, mem_of_superset (V_op ⟨_, haV⟩) hVU⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Constructions | {
"line": 663,
"column": 2
} | {
"line": 663,
"column": 66
} | {
"line": 665,
"column": 0
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nt : Set ↑s\nht : IsOpen[instTopologicalSpaceSubtype] t\n⊢ ∃ c, IsOpen[inst✝] c ∧ Subtype.val '' t = c ∩ s",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"congrArg",
"Membership.mem",
"Exists",
"Eq.mp",
... | [] | simpa using IsInducing.subtypeVal.image_eq_isOpen_inter_range ht | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.Constructions | {
"line": 663,
"column": 2
} | {
"line": 663,
"column": 66
} | {
"line": 665,
"column": 0
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nt : Set ↑s\nht : IsOpen[instTopologicalSpaceSubtype] t\n⊢ ∃ c, IsOpen[inst✝] c ∧ Subtype.val '' t = c ∩ s",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"congrArg",
"Membership.mem",
"Exists",
"Eq.mp",
... | [] | simpa using IsInducing.subtypeVal.image_eq_isOpen_inter_range ht | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Constructions | {
"line": 663,
"column": 2
} | {
"line": 663,
"column": 66
} | {
"line": 665,
"column": 0
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nt : Set ↑s\nht : IsOpen[instTopologicalSpaceSubtype] t\n⊢ ∃ c, IsOpen[inst✝] c ∧ Subtype.val '' t = c ∩ s",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"congrArg",
"Membership.mem",
"Exists",
"Eq.mp",
... | [] | simpa using IsInducing.subtypeVal.image_eq_isOpen_inter_range ht | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.NhdsWithin | {
"line": 529,
"column": 2
} | {
"line": 529,
"column": 60
} | {
"line": 530,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\ns t : Set α\nhst : s ⊆ t\nJ : Set ↑s\na : ↑s\nx : α\n⊢ (∃ a, ↑a = x ∧ a ∈ J) ↔ ∃ a, ↑a = x ∧ ∃ (h : ↑a ∈ s), ⟨↑a, h⟩ ∈ J",
"ppTerm": "?m.87",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"_private.Mathlib.Topology.Nhds... | [
"α : Type u_1\ninst✝ : TopologicalSpace α\ns t : Set α\nhst : s ⊆ t\nJ : Set ↑s\na : ↑s\nx : α\n⊢ (∃ (x_1 : x ∈ s), ⟨x, ⋯⟩ ∈ J) ↔ ∃ (x_1 : x ∈ t) (h : x ∈ s), ⟨x, ⋯⟩ ∈ J"
] | simp only [SetCoe.exists, exists_and_left, exists_eq_left] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.NhdsWithin | {
"line": 587,
"column": 78
} | {
"line": 588,
"column": 82
} | {
"line": 590,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\ns t : Set α\n⊢ 𝓟 (s ∩ t) ≤ 𝓝ˢ[t] s",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"and_true",
"Set.inter_subset_right._simp_1",
"congrArg",
"Filter.inf_principal",
... | [] | by
simpa [nhdsSetWithin] using inf_le_of_left_le (b := 𝓟 t) <| principal_le_nhdsSet | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.UniformSpace.Basic | {
"line": 774,
"column": 2
} | {
"line": 774,
"column": 91
} | {
"line": 776,
"column": 0
} | [
{
"pp": "α : Type ua\nβ : Type ub\nt₁ : UniformSpace α\nt₂ : UniformSpace β\nu : SetRel α α\nv : SetRel β β\nhu : u ∈ 𝓤 α\nhv : v ∈ 𝓤 β\n⊢ entourageProd u v ∈ 𝓤 (α × β)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"entourageProd",
... | [] | rw [uniformity_prod]; exact inter_mem_inf (preimage_mem_comap hu) (preimage_mem_comap hv) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.Basic | {
"line": 774,
"column": 2
} | {
"line": 774,
"column": 91
} | {
"line": 776,
"column": 0
} | [
{
"pp": "α : Type ua\nβ : Type ub\nt₁ : UniformSpace α\nt₂ : UniformSpace β\nu : SetRel α α\nv : SetRel β β\nhu : u ∈ 𝓤 α\nhv : v ∈ 𝓤 β\n⊢ entourageProd u v ∈ 𝓤 (α × β)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"entourageProd",
... | [] | rw [uniformity_prod]; exact inter_mem_inf (preimage_mem_comap hu) (preimage_mem_comap hv) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 871,
"column": 72
} | {
"line": 875,
"column": 51
} | {
"line": 877,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : α\nha : a < 0\n⊢ (Ioo a 0)⁻¹ = Iio a⁻¹",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Set.ext",
"GroupWithZero.toMonoidWithZero",
"Preorder.toLT",
... | [] | by
ext x
refine ⟨fun h ↦ (lt_inv_of_neg (inv_neg''.1 h.2) ha).2 h.1, fun h ↦ ?_⟩
have h' := (h.trans (inv_neg''.2 ha))
exact ⟨(lt_inv_of_neg ha h').2 h, inv_neg''.2 h'⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Filter.Pointwise | {
"line": 592,
"column": 7
} | {
"line": 592,
"column": 48
} | {
"line": 592,
"column": 49
} | [
{
"pp": "α : Type u_2\ninst✝ : Monoid α\nf : Filter α\nhf : 1 ≤ f\ns t : Set α\nht : t ∈ f\nhs : t * univ ⊆ s\n⊢ s = univ",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"MulOne.toOne",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
... | [
"α : Type u_2\ninst✝ : Monoid α\nf : Filter α\nhf : 1 ≤ f\ns t : Set α\nht : t ∈ f\nhs : univ ⊆ s\n⊢ s = univ"
] | mul_univ_of_one_mem (mem_one.1 <| hf ht), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Filter.Pointwise | {
"line": 637,
"column": 2
} | {
"line": 638,
"column": 35
} | {
"line": 640,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_2\ninst✝ : DivisionMonoid α\nf g : Filter α\n⊢ (∃ a b, f = pure a ∧ g = pure b ∧ a * b = 1) → f * g = 1",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Pure.pure",
"Eq.mpr",
"InvOneClass.toOne",
"HMul.hMul",
"DivInvOneMo... | [] | · rintro ⟨a, b, rfl, rfl, h⟩
rw [pure_mul_pure, h, pure_one] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.Filter.Pointwise | {
"line": 724,
"column": 64
} | {
"line": 729,
"column": 88
} | {
"line": 731,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝ : Group α\nf g : Filter α\n⊢ 1 ≤ f / g ↔ ¬Disjoint f g",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Iff.mpr",
"Disjoint.le_bot",
"False",
"instHDiv",
"Filter.instDiv",
"InvOneClass.toOne",
... | [] | by
refine ⟨fun h hfg => ?_, ?_⟩
· obtain ⟨s, hs, t, ht, hst⟩ := hfg.le_bot (mem_bot : ∅ ∈ ⊥)
exact Set.one_mem_div_iff.1 (h <| div_mem_div hs ht) (disjoint_iff.2 hst.symm)
· rintro h s ⟨t₁, h₁, t₂, h₂, hs⟩
exact hs (Set.one_mem_div_iff.2 fun ht => h <| disjoint_of_disjoint_of_mem ht h₁ h₂) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Compactness.SigmaCompact | {
"line": 119,
"column": 6
} | {
"line": 121,
"column": 47
} | {
"line": 122,
"column": 6
} | [
{
"pp": "case mpr.refine_1\nX : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\ns : Set X\nhf : IsInducing f\nL : ℕ → Set Y\nhcomp : ∀ (n : ℕ), IsCompact (L n)\nhcov : ⋃ n, L n = f '' s\nn : ℕ\n⊢ IsCompact ((fun n ↦ f ⁻¹' L n ∩ s) n)",
"ppTerm": "?mpr.refine_1",
... | [
"case mpr.refine_1\nX : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\ns : Set X\nhf : IsInducing f\nL : ℕ → Set Y\nhcomp : ∀ (n : ℕ), IsCompact (L n)\nhcov : ⋃ n, L n = f '' s\nn : ℕ\nthis : f '' (f ⁻¹' L n ∩ s) = L n\n⊢ IsCompact ((fun n ↦ f ⁻¹' L n ∩ s) n)"
] | have : f '' (f ⁻¹' (L n) ∩ s) = L n := by
rw [image_preimage_inter, inter_eq_left.mpr]
exact (subset_iUnion _ n).trans hcov.le | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Bases | {
"line": 496,
"column": 2
} | {
"line": 496,
"column": 56
} | {
"line": 498,
"column": 0
} | [
{
"pp": "α : Type u\nt : TopologicalSpace α\nι : Sort u_2\ninst✝ : Countable ι\ns c : ι → Set α\nhc : ∀ (i : ι), (c i).Countable\nh'c : ∀ (i : ι), s i ⊆ closure[t] (c i)\ni : ι\n⊢ s i ⊆ closure[t] (⋃ i, c i)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"HasSubset.Subset.trans",
... | [] | exact (h'c i).trans (closure_mono (subset_iUnion _ i)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Compactness.Compact | {
"line": 621,
"column": 4
} | {
"line": 621,
"column": 63
} | {
"line": 622,
"column": 4
} | [
{
"pp": "X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\ny : Y\nhf : Tendsto f (cocompact X) (𝓝 y)\nhfc : Continuous[inst✝¹, inst✝] f\nl : Filter Y\nhne : l.NeBot\nhle : l ≤ 𝓟 (insert y (range f))\ns : Set Y\nhsy : s ∈ 𝓝 y\nt : Set Y\nhtl : t ∈ l\nhd : Disjoint s t... | [
"X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\ny✝ : Y\nhf : Tendsto f (cocompact X) (𝓝 y✝)\nhfc : Continuous[inst✝¹, inst✝] f\nl : Filter Y\nhne : l.NeBot\nhle : l ≤ 𝓟 (insert y✝ (range f))\ns : Set Y\nhsy : s ∈ 𝓝 y✝\nt : Set Y\nhtl : t ∈ l\nhd : Disjoint s t\nK : Se... | filter_upwards [htl, le_principal_iff.1 hle] with y hyt hyf | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Topology.GDelta.Basic | {
"line": 191,
"column": 37
} | {
"line": 191,
"column": 94
} | {
"line": 193,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\n⊢ (∃ S ⊆ {t | IsOpen[inst✝] t ∧ Dense t}, S.Countable ∧ ⋂₀ S ⊆ s) ↔\n ∃ S, (∀ t ∈ S, IsOpen[inst✝] t) ∧ (∀ t ∈ S, Dense t) ∧ S.Countable ∧ ⋂₀ S ⊆ s",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_priva... | [] | by simp_rw [subset_def, mem_setOf, forall_and, and_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Bases | {
"line": 1049,
"column": 2
} | {
"line": 1052,
"column": 7
} | {
"line": 1054,
"column": 0
} | [
{
"pp": "ι : Type u_1\nE : ι → Type u_2\ninst✝ : (i : ι) → TopologicalSpace (E i)\ns : (i : ι) → Set (Set (E i))\nhs : ∀ (i : ι), IsTopologicalBasis (s i)\n⊢ IsTopologicalBasis (⋃ i, (fun u ↦ Sigma.mk i '' u) '' s i)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"TopologicalSpace.I... | [] | refine .of_hasBasis_nhds fun a ↦ ?_
rw [Sigma.nhds_eq]
convert! (((hs a.1).nhds_hasBasis).map _).to_image_id
aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Bases | {
"line": 1049,
"column": 2
} | {
"line": 1052,
"column": 7
} | {
"line": 1054,
"column": 0
} | [
{
"pp": "ι : Type u_1\nE : ι → Type u_2\ninst✝ : (i : ι) → TopologicalSpace (E i)\ns : (i : ι) → Set (Set (E i))\nhs : ∀ (i : ι), IsTopologicalBasis (s i)\n⊢ IsTopologicalBasis (⋃ i, (fun u ↦ Sigma.mk i '' u) '' s i)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"TopologicalSpace.I... | [] | refine .of_hasBasis_nhds fun a ↦ ?_
rw [Sigma.nhds_eq]
convert! (((hs a.1).nhds_hasBasis).map _).to_image_id
aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactness.Compact | {
"line": 1168,
"column": 6
} | {
"line": 1168,
"column": 74
} | {
"line": 1168,
"column": 74
} | [
{
"pp": "X✝ : Type u\nY : Type v\nι : Type u_1\ninst✝³ : TopologicalSpace X✝\ninst✝² : TopologicalSpace Y\ns t : Set X✝\nf : X✝ → Y\nX : ι → Type u_2\ninst✝¹ : (i : ι) → TopologicalSpace (X i)\ninst✝ : ∀ (i : ι), CompactSpace (X i)\n⊢ IsCompact univ",
"ppTerm": "?m.8",
"assigned": true,
"usedConstan... | [] | rw [← pi_univ univ]; exact isCompact_univ_pi fun i => isCompact_univ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Compactness.Compact | {
"line": 1168,
"column": 6
} | {
"line": 1168,
"column": 74
} | {
"line": 1168,
"column": 74
} | [
{
"pp": "X✝ : Type u\nY : Type v\nι : Type u_1\ninst✝³ : TopologicalSpace X✝\ninst✝² : TopologicalSpace Y\ns t : Set X✝\nf : X✝ → Y\nX : ι → Type u_2\ninst✝¹ : (i : ι) → TopologicalSpace (X i)\ninst✝ : ∀ (i : ι), CompactSpace (X i)\n⊢ IsCompact univ",
"ppTerm": "?m.8",
"assigned": true,
"usedConstan... | [] | rw [← pi_univ univ]; exact isCompact_univ_pi fun i => isCompact_univ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactness.Compact | {
"line": 1240,
"column": 22
} | {
"line": 1240,
"column": 27
} | {
"line": 1241,
"column": 2
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nS : Set X\nhS : IsClosed[inst✝¹] S\nhne : S.Nonempty\nopens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen[inst✝¹] U ∧ Uᶜ.Nonempty}\nU : Set X\nh : Maximal (fun x ↦ x ∈ opens) U\nUc : Sᶜ ⊆ U\nUo : IsOpen[inst✝¹] U\nUcne : Uᶜ.Nonempty\nV' : Set X\n... | [
"X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nS : Set X\nhS : IsClosed[inst✝¹] S\nhne : S.Nonempty\nopens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen[inst✝¹] U ∧ Uᶜ.Nonempty}\nU : Set X\nh : Maximal (fun x ↦ x ∈ opens) U\nUc : Sᶜ ⊆ U\nUo : IsOpen[inst✝¹] U\nUcne : Uᶜ.Nonempty\nV' : Set X\nV'sub : V' ⊆... | V'cls | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.Separation.Basic | {
"line": 210,
"column": 4
} | {
"line": 210,
"column": 54
} | {
"line": 211,
"column": 4
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : T0Space X\ns : Finset X\nihs : ∀ t ⊂ s, (↑t).Nonempty → IsOpen[inst✝¹] ↑t → ∃ x ∈ ↑t, IsOpen[inst✝¹] {x}\nhne : (↑s).Nonempty\nho : IsOpen[inst✝¹] ↑s\nht : ¬∃ t ⊂ s, t.Nonempty ∧ IsOpen[inst✝¹] ↑t\nt : Set X\nhts : t ⊆ ↑s\nhtne : t.Nonempty\nhto : IsOp... | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : T0Space X\ns : Finset X\nihs : ∀ t ⊂ s, (↑t).Nonempty → IsOpen[inst✝¹] ↑t → ∃ x ∈ ↑t, IsOpen[inst✝¹] {x}\nhne : (↑s).Nonempty\nho : IsOpen[inst✝¹] ↑s\nht : ¬∃ t ⊂ s, t.Nonempty ∧ IsOpen[inst✝¹] ↑t\nt : Finset X\nhts : ↑t ⊆ ↑s\nhtne : (↑t).Nonempty\nhto : IsOpen[in... | lift t to Finset X using s.finite_toSet.subset hts | Mathlib.Tactic._aux_Mathlib_Tactic_Lift___elabRules_Mathlib_Tactic_lift_1 | Mathlib.Tactic.lift |
Mathlib.Topology.Separation.Basic | {
"line": 314,
"column": 2
} | {
"line": 318,
"column": 36
} | {
"line": 320,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : R0Space X\nx : X\n⊢ IsCompact (closure[inst✝¹] {x})",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Specializes",
"Iff.of_eq",
"congrArg",
"subset_closure",
"Specializes.mem_open",
... | [] | refine isCompact_of_finite_subcover fun U hUo hxU ↦ ?_
obtain ⟨i, hi⟩ : ∃ i, x ∈ U i := mem_iUnion.1 <| hxU <| subset_closure rfl
refine ⟨{i}, fun y hy ↦ ?_⟩
rw [← specializes_iff_mem_closure, specializes_comm] at hy
simpa using hy.mem_open (hUo i) hi | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Separation.Basic | {
"line": 314,
"column": 2
} | {
"line": 318,
"column": 36
} | {
"line": 320,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : R0Space X\nx : X\n⊢ IsCompact (closure[inst✝¹] {x})",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Specializes",
"Iff.of_eq",
"congrArg",
"subset_closure",
"Specializes.mem_open",
... | [] | refine isCompact_of_finite_subcover fun U hUo hxU ↦ ?_
obtain ⟨i, hi⟩ : ∃ i, x ∈ U i := mem_iUnion.1 <| hxU <| subset_closure rfl
refine ⟨{i}, fun y hy ↦ ?_⟩
rw [← specializes_iff_mem_closure, specializes_comm] at hy
simpa using hy.mem_open (hUo i) hi | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.DiscreteSubset | {
"line": 265,
"column": 6
} | {
"line": 265,
"column": 15
} | {
"line": 266,
"column": 4
} | [
{
"pp": "case pos\nX : Type u_1\ninst✝ : TopologicalSpace X\ns U : Set X\nhs : ∀ x ∈ U, Disjoint (𝓝[≠] x) (𝓟 (U \\ s))\nhU : IsClosed[inst✝] U\nx : X\nhx : x ∈ (U \\ s)ᶜ\nh₁x : x ∈ U\na : X\nha : a ∈ {x}ᶜ → a ∉ U \\ s\nh₂a : a = x\n⊢ a ∈ (U \\ s)ᶜ",
"ppTerm": "?pos✝",
"assigned": true,
"usedConsta... | [] | tauto_set | Mathlib.Tactic.TautoSet._aux_Mathlib_Tactic_TautoSet___macroRules_Mathlib_Tactic_TautoSet_tacticTauto_set_1 | Mathlib.Tactic.TautoSet.tacticTauto_set |
Mathlib.Topology.DiscreteSubset | {
"line": 265,
"column": 6
} | {
"line": 265,
"column": 15
} | {
"line": 266,
"column": 4
} | [
{
"pp": "case pos\nX : Type u_1\ninst✝ : TopologicalSpace X\ns U : Set X\nhs : ∀ x ∈ U, Disjoint (𝓝[≠] x) (𝓟 (U \\ s))\nhU : IsClosed[inst✝] U\nx : X\nhx : x ∈ (U \\ s)ᶜ\nh₁x : x ∈ U\na : X\nha : a ∈ {x}ᶜ → a ∉ U \\ s\nh₂a : a = x\n⊢ a ∈ (U \\ s)ᶜ",
"ppTerm": "?pos✝",
"assigned": true,
"usedConsta... | [] | tauto_set | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.DiscreteSubset | {
"line": 265,
"column": 6
} | {
"line": 265,
"column": 15
} | {
"line": 266,
"column": 4
} | [
{
"pp": "case pos\nX : Type u_1\ninst✝ : TopologicalSpace X\ns U : Set X\nhs : ∀ x ∈ U, Disjoint (𝓝[≠] x) (𝓟 (U \\ s))\nhU : IsClosed[inst✝] U\nx : X\nhx : x ∈ (U \\ s)ᶜ\nh₁x : x ∈ U\na : X\nha : a ∈ {x}ᶜ → a ∉ U \\ s\nh₂a : a = x\n⊢ a ∈ (U \\ s)ᶜ",
"ppTerm": "?pos✝",
"assigned": true,
"usedConsta... | [] | tauto_set | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.DiscreteSubset | {
"line": 267,
"column": 6
} | {
"line": 267,
"column": 15
} | {
"line": 268,
"column": 2
} | [
{
"pp": "case neg\nX : Type u_1\ninst✝ : TopologicalSpace X\ns U : Set X\nhs : ∀ x ∈ U, Disjoint (𝓝[≠] x) (𝓟 (U \\ s))\nhU : IsClosed[inst✝] U\nx : X\nhx : x ∈ (U \\ s)ᶜ\nh₁x : x ∈ U\na : X\nha : a ∉ U \\ s\nh₂a : ¬a = x\n⊢ a ∈ (U \\ s)ᶜ",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
... | [] | tauto_set | Mathlib.Tactic.TautoSet._aux_Mathlib_Tactic_TautoSet___macroRules_Mathlib_Tactic_TautoSet_tacticTauto_set_1 | Mathlib.Tactic.TautoSet.tacticTauto_set |
Mathlib.Topology.DiscreteSubset | {
"line": 271,
"column": 4
} | {
"line": 271,
"column": 13
} | {
"line": 273,
"column": 0
} | [
{
"pp": "case right\nX : Type u_1\ninst✝ : TopologicalSpace X\ns U : Set X\nhs : s ∈ codiscreteWithin U\nhU : IsClosed[inst✝] U\nx : X\nhx : x ∈ (U \\ s)ᶜ\nh₁x : x ∉ U\ny : X\nhy : y ∈ Uᶜ\n⊢ y ∈ (U \\ s)ᶜ",
"ppTerm": "?right",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instDecidable... | [] | tauto_set | Mathlib.Tactic.TautoSet._aux_Mathlib_Tactic_TautoSet___macroRules_Mathlib_Tactic_TautoSet_tacticTauto_set_1 | Mathlib.Tactic.TautoSet.tacticTauto_set |
Mathlib.Topology.DiscreteSubset | {
"line": 283,
"column": 4
} | {
"line": 283,
"column": 13
} | {
"line": 285,
"column": 0
} | [
{
"pp": "case h.right\nX : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : T1Space X\nx : X\nU s : Set X\nhs : Finite ↑s\nt : Set X\nht : IsOpen[inst✝¹] t\nh₁ts : x ∈ t\nh₂ts : t ∩ {x}ᶜ ⊆ U\n⊢ x ∈ t \\ (s \\ {x}) ∧ t \\ (s \\ {x}) ∩ {x}ᶜ ⊆ U \\ s",
"ppTerm": "?h.right",
"assigned": true,
"usedConstan... | [] | tauto_set | Mathlib.Tactic.TautoSet._aux_Mathlib_Tactic_TautoSet___macroRules_Mathlib_Tactic_TautoSet_tacticTauto_set_1 | Mathlib.Tactic.TautoSet.tacticTauto_set |
Mathlib.Topology.DiscreteSubset | {
"line": 283,
"column": 4
} | {
"line": 283,
"column": 13
} | {
"line": 285,
"column": 0
} | [
{
"pp": "case h.right\nX : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : T1Space X\nx : X\nU s : Set X\nhs : Finite ↑s\nt : Set X\nht : IsOpen[inst✝¹] t\nh₁ts : x ∈ t\nh₂ts : t ∩ {x}ᶜ ⊆ U\n⊢ x ∈ t \\ (s \\ {x}) ∧ t \\ (s \\ {x}) ∩ {x}ᶜ ⊆ U \\ s",
"ppTerm": "?h.right",
"assigned": true,
"usedConstan... | [] | tauto_set | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.DiscreteSubset | {
"line": 283,
"column": 4
} | {
"line": 283,
"column": 13
} | {
"line": 285,
"column": 0
} | [
{
"pp": "case h.right\nX : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : T1Space X\nx : X\nU s : Set X\nhs : Finite ↑s\nt : Set X\nht : IsOpen[inst✝¹] t\nh₁ts : x ∈ t\nh₂ts : t ∩ {x}ᶜ ⊆ U\n⊢ x ∈ t \\ (s \\ {x}) ∧ t \\ (s \\ {x}) ∩ {x}ᶜ ⊆ U \\ s",
"ppTerm": "?h.right",
"assigned": true,
"usedConstan... | [] | tauto_set | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Irreducible | {
"line": 242,
"column": 2
} | {
"line": 242,
"column": 53
} | {
"line": 243,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ns : Set X\ninst✝ : PreirreducibleSpace ↑s\n⊢ IsPreirreducible s",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"IsPreirreducible",
"Eq.mpr",
"Set.image_univ",
"congrArg",
"Set.univ",
"Membership.mem",
... | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\ns : Set X\ninst✝ : PreirreducibleSpace ↑s\n⊢ IsPreirreducible (Subtype.val '' univ)"
] | rw [← Subtype.range_coe (s := s), ← Set.image_univ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Separation.Basic | {
"line": 687,
"column": 2
} | {
"line": 687,
"column": 36
} | {
"line": 689,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : T1Space X\ninst✝ : ∀ (x : X), (𝓝[≠] x).NeBot\ns : Set X\nhs : Dense s\nt : Set X\nht : t.Finite\n⊢ t = ↑ht.toFinset",
"ppTerm": "?m.133",
"assigned": true,
"usedConstants": [
"Finset",
"Set.Finite.coe_toFinset",
"SetLike... | [] | exact (Finite.coe_toFinset _).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Separation.Hausdorff | {
"line": 688,
"column": 4
} | {
"line": 688,
"column": 91
} | {
"line": 689,
"column": 2
} | [
{
"pp": "case mp\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : R1Space X\nS : Set X\nh : IsPreirreducible S\nx : X\nhx : x ∈ S\ny : X\nhy : y ∈ S\ne : y ∉ closure[inst✝¹] {x}\nU V : Set X\nhU : IsOpen[inst✝¹] U\nhV : IsOpen[inst✝¹] V\nhxU : x ∈ U\nhyV : y ∈ V\nh' : Disjoint U V\n⊢ False",
"ppTerm": "?... | [] | exact ((h U V hU hV ⟨x, hx, hxU⟩ ⟨y, hy, hyV⟩).mono inter_subset_right).not_disjoint h' | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Connected.Basic | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 29
} | {
"line": 130,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : TopologicalSpace α\ns t : Set α\nhs : IsPreconnected s\nht : IsPreconnected t\nx : α\nhxs : x ∈ s\nhxt : x ∈ t\n⊢ IsPreconnected (s ∪ t)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"IsPreconnected.union"
],
"usedFVars": [
"α",
"ins... | [] | exact hs.union x hxs hxt ht | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Order.OrderClosed | {
"line": 394,
"column": 2
} | {
"line": 394,
"column": 60
} | {
"line": 396,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : ClosedIicTopology α\ninst✝ : TopologicalSpace β\na b : α\nf : α → β\nh : a < b\n⊢ ContinuousWithinAt f (Ioc a b) b ↔ ContinuousWithinAt f (Iic b) b",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [... | [] | simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsLE h] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Order.OrderClosed | {
"line": 394,
"column": 2
} | {
"line": 394,
"column": 60
} | {
"line": 396,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : ClosedIicTopology α\ninst✝ : TopologicalSpace β\na b : α\nf : α → β\nh : a < b\n⊢ ContinuousWithinAt f (Ioc a b) b ↔ ContinuousWithinAt f (Iic b) b",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [... | [] | simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsLE h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.OrderClosed | {
"line": 394,
"column": 2
} | {
"line": 394,
"column": 60
} | {
"line": 396,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : ClosedIicTopology α\ninst✝ : TopologicalSpace β\na b : α\nf : α → β\nh : a < b\n⊢ ContinuousWithinAt f (Ioc a b) b ↔ ContinuousWithinAt f (Iic b) b",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [... | [] | simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsLE h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Connected.LocallyConnected | {
"line": 107,
"column": 4
} | {
"line": 107,
"column": 60
} | {
"line": 108,
"column": 4
} | [
{
"pp": "case mpr\nα : Type u\ninst✝ : TopologicalSpace α\n⊢ (∀ (x : α), ∀ U ∈ 𝓝 x, ∃ V ∈ 𝓝 x, IsPreconnected V ∧ V ⊆ U) → LocallyConnectedSpace α",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"congrArg",
"Membership.mem",
... | [
"case mpr\nα : Type u\ninst✝ : TopologicalSpace α\n⊢ (∀ (x : α), ∀ U ∈ 𝓝 x, ∃ V ∈ 𝓝 x, IsPreconnected V ∧ V ⊆ U) →\n ∀ (F : Set α), IsOpen[inst✝] F → ∀ x ∈ F, IsOpen[inst✝] (connectedComponentIn F x)"
] | rw [locallyConnectedSpace_iff_connectedComponentIn_open] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Connected.Basic | {
"line": 423,
"column": 10
} | {
"line": 423,
"column": 41
} | {
"line": 423,
"column": 41
} | [
{
"pp": "α : Type u\ninst✝ : TopologicalSpace α\ns u v : Set α\nhu : IsOpen[inst✝] u\nhv : IsOpen[inst✝] v\nhuv : Disjoint u v\nhsuv : s ⊆ u ∪ v\nhsu : (s ∩ u).Nonempty\nhs : IsPreconnected s\nhsv : ¬Disjoint s v\n⊢ False",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"congrArg",
... | [
"α : Type u\ninst✝ : TopologicalSpace α\ns u v : Set α\nhu : IsOpen[inst✝] u\nhv : IsOpen[inst✝] v\nhuv : Disjoint u v\nhsuv : s ⊆ u ∪ v\nhsu : (s ∩ u).Nonempty\nhs : IsPreconnected s\nhsv : (s ∩ v).Nonempty\n⊢ False"
] | not_disjoint_iff_nonempty_inter | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Connected.Clopen | {
"line": 58,
"column": 2
} | {
"line": 61,
"column": 42
} | {
"line": 62,
"column": 2
} | [
{
"pp": "case refine_1\nι : Type u_1\nX : ι → Type u_2\nhι : Nonempty ι\ninst✝ : (i : ι) → TopologicalSpace (X i)\ns : Set ((i : ι) × X i)\nhs : IsPreconnected s\n⊢ ∃ i t, IsPreconnected t ∧ s = mk i '' t",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"IsConnected",
"Class... | [
"case refine_2\nι : Type u_1\nX : ι → Type u_2\nhι : Nonempty ι\ninst✝ : (i : ι) → TopologicalSpace (X i)\ns : Set ((i : ι) × X i)\n⊢ (∃ i t, IsPreconnected t ∧ s = mk i '' t) → IsPreconnected s"
] | · obtain rfl | h := s.eq_empty_or_nonempty
· exact ⟨Classical.choice hι, ∅, isPreconnected_empty, (Set.image_empty _).symm⟩
· obtain ⟨a, t, ht, rfl⟩ := Sigma.isConnected_iff.1 ⟨h, hs⟩
exact ⟨a, t, ht.isPreconnected, rfl⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Compactness.Lindelof | {
"line": 331,
"column": 4
} | {
"line": 331,
"column": 32
} | {
"line": 332,
"column": 2
} | [
{
"pp": "case h.left.a\nX : Type u\nι : Type u_1\ninst✝ : TopologicalSpace X\ns : Set ι\nf : ι → Set X\nhs : s.Countable\nhf : ∀ i ∈ s, IsLindelof (f i)\ni : Type u\nU : i → Set X\nhU : ∀ (i : i), IsOpen[inst✝] (U i)\nhUcover : ⋃ i ∈ s, f i ⊆ ⋃ i, U i\nhiU : ∀ i_1 ∈ s, f i_1 ⊆ ⋃ i, U i\nr : ι → Set i\nhr : ∀ i_... | [] | exact fun s hs ↦ (hr s hs).1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Compactness.Lindelof | {
"line": 446,
"column": 4
} | {
"line": 446,
"column": 63
} | {
"line": 447,
"column": 4
} | [
{
"pp": "X : Type u\nY : Type v\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nf : X → Y\ny : Y\nhf : Tendsto f (coLindelof X) (𝓝 y)\nhfc : Continuous[inst✝², inst✝¹] f\nl : Filter Y\nhne : l.NeBot\ninst✝ : CountableInterFilter l\nhle : l ≤ 𝓟 (insert y (range f))\ns : Set Y\nhsy : s ∈ 𝓝 y\nt : Se... | [
"X : Type u\nY : Type v\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nf : X → Y\ny✝ : Y\nhf : Tendsto f (coLindelof X) (𝓝 y✝)\nhfc : Continuous[inst✝², inst✝¹] f\nl : Filter Y\nhne : l.NeBot\ninst✝ : CountableInterFilter l\nhle : l ≤ 𝓟 (insert y✝ (range f))\ns : Set Y\nhsy : s ∈ 𝓝 y✝\nt : Set Y\nhtl... | filter_upwards [htl, le_principal_iff.1 hle] with y hyt hyf | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Topology.Compactness.Lindelof | {
"line": 521,
"column": 2
} | {
"line": 521,
"column": 27
} | {
"line": 522,
"column": 2
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nhs : IsSigmaCompact s\n⊢ IsLindelof s",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"congrArg",
"IsSigmaCompact",
"Exists",
"Eq.mp",
"And",
"Nat",
"IsSigmaCompact.eq_1",
"Eq",
... | [
"X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nhs : ∃ K, (∀ (n : ℕ), IsCompact (K n)) ∧ ⋃ n, K n = s\n⊢ IsLindelof s"
] | rw [IsSigmaCompact] at hs | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Compactness.Lindelof | {
"line": 521,
"column": 2
} | {
"line": 525,
"column": 28
} | {
"line": 527,
"column": 0
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nhs : IsSigmaCompact s\n⊢ IsLindelof s",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"IsSigmaCompact",
"Exists",
"Eq.mp",
"id",
"And.casesOn",
"And",
"Exi... | [] | rw [IsSigmaCompact] at hs
rcases hs with ⟨K, ⟨hc, huniv⟩⟩
rw [← huniv]
have hl : ∀ n, IsLindelof (K n) := fun n ↦ IsCompact.isLindelof (hc n)
exact isLindelof_iUnion hl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Compactness.Lindelof | {
"line": 521,
"column": 2
} | {
"line": 525,
"column": 28
} | {
"line": 527,
"column": 0
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nhs : IsSigmaCompact s\n⊢ IsLindelof s",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"IsSigmaCompact",
"Exists",
"Eq.mp",
"id",
"And.casesOn",
"And",
"Exi... | [] | rw [IsSigmaCompact] at hs
rcases hs with ⟨K, ⟨hc, huniv⟩⟩
rw [← huniv]
have hl : ∀ n, IsLindelof (K n) := fun n ↦ IsCompact.isLindelof (hc n)
exact isLindelof_iUnion hl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactness.Lindelof | {
"line": 742,
"column": 4
} | {
"line": 742,
"column": 11
} | {
"line": 742,
"column": 12
} | [
{
"pp": "X : Type u\nY : Type v\nι : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ns t✝ : Set X\ninst✝ : SecondCountableTopology X\nt : Set X\nx✝¹ : t ⊆ univ\nx✝ : Filter X\n⊢ ∀ {ι : Type u} (U : ι → Set X),\n (∀ (i : ι), IsOpen[inst✝²] (U i)) → t ⊆ ⋃ i, U i → ∃ t_1, t_1.Countable ∧ t ⊆... | [
"X : Type u\nY : Type v\nι✝ : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ns t✝ : Set X\ninst✝ : SecondCountableTopology X\nt : Set X\nx✝¹ : t ⊆ univ\nx✝ : Filter X\nι : Type u\n⊢ ∀ (U : ι → Set X), (∀ (i : ι), IsOpen[inst✝²] (U i)) → t ⊆ ⋃ i, U i → ∃ t_1, t_1.Countable ∧ t ⊆ ⋃ i ∈ t_1, U i"
... | intro ι | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Topology.Separation.Regular | {
"line": 770,
"column": 6
} | {
"line": 771,
"column": 95
} | {
"line": 772,
"column": 6
} | [
{
"pp": "case neg\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : T2Space X\ninst✝ : CompactSpace X\nx : X\nhs : IsClosed[inst✝²] (⋂ s, ↑s)\na b : Set X\nha : IsClosed[inst✝²] a\nhb : IsClosed[inst✝²] b\nhab : ⋂ s, ↑s ⊆ a ∪ b\nab_disj : Disjoint a b\nu v : Set X\nhu : IsOpen[inst✝²] u\nhv : IsOpen[inst✝²] ... | [
"case neg\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : T2Space X\ninst✝ : CompactSpace X\nx : X\nhs : IsClosed[inst✝²] (⋂ s, ↑s)\na b : Set X\nha : IsClosed[inst✝²] a\nhb : IsClosed[inst✝²] b\nhab : ⋂ s, ↑s ⊆ a ∪ b\nab_disj : Disjoint a b\nu v : Set X\nhu : IsOpen[inst✝²] u\nhv : IsOpen[inst✝²] v\nhau : a ⊆... | have h1 : x ∈ v :=
(hab.trans (union_subset_union hau hbv) (mem_iInter.2 fun i => i.2.2)).resolve_left hxu | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Separation.Regular | {
"line": 782,
"column": 2
} | {
"line": 801,
"column": 77
} | {
"line": 802,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : T2Space X\ninst✝ : CompactSpace X\n⊢ T2Space (ConnectedComponents X)",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"False",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"con... | [] | refine ⟨ConnectedComponents.surjective_coe.forall₂.2 fun a b ne => ?_⟩
rw [ConnectedComponents.coe_ne_coe] at ne
have h := connectedComponent_disjoint ne
-- write ↑b as the intersection of all clopen subsets containing it
rw [connectedComponent_eq_iInter_isClopen b, disjoint_iff_inter_eq_empty] at h
-- Now we... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Separation.Regular | {
"line": 782,
"column": 2
} | {
"line": 801,
"column": 77
} | {
"line": 802,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : T2Space X\ninst✝ : CompactSpace X\n⊢ T2Space (ConnectedComponents X)",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"False",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"con... | [] | refine ⟨ConnectedComponents.surjective_coe.forall₂.2 fun a b ne => ?_⟩
rw [ConnectedComponents.coe_ne_coe] at ne
have h := connectedComponent_disjoint ne
-- write ↑b as the intersection of all clopen subsets containing it
rw [connectedComponent_eq_iInter_isClopen b, disjoint_iff_inter_eq_empty] at h
-- Now we... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Connected.Clopen | {
"line": 694,
"column": 2
} | {
"line": 695,
"column": 31
} | {
"line": 697,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : TopologicalSpace α\ns : Set α\nhs : ∀ (f : α → Bool), ContinuousOn f s → ∀ x ∈ s, ∀ y ∈ s, f x = f y\nu v : Set α\nu_op : IsOpen[inst✝] u\nv_op : IsOpen[inst✝] v\nhsuv : s ⊆ u ∪ v\nx : α\nx_in_s : x ∈ s\nx_in_u : x ∈ u\nH : s ∩ (u ∩ v) = ∅\ny : α\ny_in_s : y ∈ s\ny_in_v : y ∈ v\nhy ... | [] | simpa [(u.mem_iff_boolIndicator _).mp x_in_u, (u.notMem_iff_boolIndicator _).mp hy] using
hs _ this x x_in_s y y_in_s | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.Algebra.IsUniformGroup.Defs | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 26
} | {
"line": 145,
"column": 2
} | [
{
"pp": "Gᵣ : Type u_3\ninst✝² : UniformSpace Gᵣ\ninst✝¹ : Group Gᵣ\ninst✝ : IsRightUniformGroup Gᵣ\n⊢ 𝓤 Gᵣ = comap (fun x ↦ x.2 / x.1) (𝓝 1)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"InvOneClass.toOne",
"HM... | [
"Gᵣ : Type u_3\ninst✝² : UniformSpace Gᵣ\ninst✝¹ : Group Gᵣ\ninst✝ : IsRightUniformGroup Gᵣ\n⊢ 𝓤 Gᵣ = comap (fun x ↦ x.2 * x.1⁻¹) (𝓝 1)"
] | simp_rw [div_eq_mul_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 800,
"column": 2
} | {
"line": 800,
"column": 26
} | {
"line": 801,
"column": 2
} | [
{
"pp": "G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : IsTopologicalGroup G\nι : Sort u_1\np : ι → Prop\ns : ι → Set G\nhb : (𝓝 1).HasBasis p s\nx : G\n⊢ (Filter.comap (fun x_1 ↦ x_1 * x⁻¹) (𝓝 1)).HasBasis p fun i ↦ {y | y / x ∈ s i}",
"ppTerm": "?m.29",
"assigned": true,
"use... | [
"G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : IsTopologicalGroup G\nι : Sort u_1\np : ι → Prop\ns : ι → Set G\nhb : (𝓝 1).HasBasis p s\nx : G\n⊢ (Filter.comap (fun x_1 ↦ x_1 * x⁻¹) (𝓝 1)).HasBasis p fun i ↦ {y | y * x⁻¹ ∈ s i}"
] | simp_rw [div_eq_mul_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 1120,
"column": 14
} | {
"line": 1120,
"column": 15
} | {
"line": 1120,
"column": 16
} | [
{
"pp": "G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : IsTopologicalGroup G\nS : Subgroup G\nhS : Tendsto (⇑S.subtype) cofinite (cocompact G)\nK : Set G\n⊢ ∀ {L : Set G}, IsCompact K → IsCompact L → {γ | ((fun x ↦ γ • x) '' K ∩ L).Nonempty}.Finite",
"ppTerm": "?m.23",
"assigned": tr... | [
"G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : IsTopologicalGroup G\nS : Subgroup G\nhS : Tendsto (⇑S.subtype) cofinite (cocompact G)\nK L : Set G\n⊢ IsCompact K → IsCompact L → {γ | ((fun x ↦ γ • x) '' K ∩ L).Nonempty}.Finite"
] | L | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 1144,
"column": 14
} | {
"line": 1144,
"column": 15
} | {
"line": 1144,
"column": 16
} | [
{
"pp": "G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : IsTopologicalGroup G\nS : Subgroup G\nhS : Tendsto (⇑S.subtype) cofinite (cocompact G)\nK : Set G\n⊢ ∀ {L : Set G}, IsCompact K → IsCompact L → {γ | ((fun x ↦ γ • x) '' K ∩ L).Nonempty}.Finite",
"ppTerm": "?m.25",
"assigned": tr... | [
"G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : IsTopologicalGroup G\nS : Subgroup G\nhS : Tendsto (⇑S.subtype) cofinite (cocompact G)\nK L : Set G\n⊢ IsCompact K → IsCompact L → {γ | ((fun x ↦ γ • x) '' K ∩ L).Nonempty}.Finite"
] | L | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 1239,
"column": 6
} | {
"line": 1239,
"column": 61
} | {
"line": 1240,
"column": 6
} | [
{
"pp": "G : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝⁴ : TopologicalSpace G\ninst✝³ : Group G\ninst✝² : IsTopologicalGroup G\ninst✝¹ : SeparableSpace G\ninst✝ : WeaklyLocallyCompactSpace G\nL : Set G\nhLc : IsCompact L\nhL1 : L ∈ 𝓝 1\nx : G\n⊢ (range (denseSeq G) ∩ (fun y ↦ x * y) ⁻¹' L).Nonempty",
... | [
"G : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝⁴ : TopologicalSpace G\ninst✝³ : Group G\ninst✝² : IsTopologicalGroup G\ninst✝¹ : SeparableSpace G\ninst✝ : WeaklyLocallyCompactSpace G\nL : Set G\nhLc : IsCompact L\nx : G\nhL1 : L ∈ 𝓝 ((Homeomorph.mulLeft x) ((Homeomorph.mulLeft x).symm 1))\n⊢ (range (denseSe... | rw [← (Homeomorph.mulLeft x).apply_symm_apply 1] at hL1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 358,
"column": 4
} | {
"line": 362,
"column": 53
} | {
"line": 363,
"column": 2
} | [
{
"pp": "α : Type u\nuniformSpace : UniformSpace α\nι : Sort u_1\ns : ι → Set α\nhs : ∀ (i : ι), IsComplete (s i)\nU : SetRel α α\nhU : U ∈ 𝓤 α\nhd : ∀ (i j : ι), ∀ x ∈ s i, ∀ y ∈ s j, (x, y) ∈ U → i = j\nS : Set α := ⋃ i, s i\nl : Filter α\nhl : Cauchy l\nhls : S ∈ l\nhl_ne : l.NeBot\nhl' : ∀ s ∈ 𝓤 α, ∃ t ∈ ... | [] | rcases Filter.nonempty_of_mem htl with ⟨x, hx⟩
rcases mem_iUnion.1 (htS hx) with ⟨i, hi⟩
refine ⟨i, fun y hy => ?_⟩
rcases mem_iUnion.1 (htS hy) with ⟨j, hj⟩
rwa [hd i j x hi y hj (htU <| mk_mem_prod hx hy)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 358,
"column": 4
} | {
"line": 362,
"column": 53
} | {
"line": 363,
"column": 2
} | [
{
"pp": "α : Type u\nuniformSpace : UniformSpace α\nι : Sort u_1\ns : ι → Set α\nhs : ∀ (i : ι), IsComplete (s i)\nU : SetRel α α\nhU : U ∈ 𝓤 α\nhd : ∀ (i j : ι), ∀ x ∈ s i, ∀ y ∈ s j, (x, y) ∈ U → i = j\nS : Set α := ⋃ i, s i\nl : Filter α\nhl : Cauchy l\nhls : S ∈ l\nhl_ne : l.NeBot\nhl' : ∀ s ∈ 𝓤 α, ∃ t ∈ ... | [] | rcases Filter.nonempty_of_mem htl with ⟨x, hx⟩
rcases mem_iUnion.1 (htS hx) with ⟨i, hi⟩
refine ⟨i, fun y hy => ?_⟩
rcases mem_iUnion.1 (htS hy) with ⟨j, hj⟩
rwa [hd i j x hi y hj (htU <| mk_mem_prod hx hy)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 980,
"column": 4
} | {
"line": 981,
"column": 83
} | {
"line": 982,
"column": 4
} | [
{
"pp": "α : Type u\nuniformSpace : UniformSpace α\ninst✝ : (𝓤 α).IsCountablyGenerated\ns : Set α\nh : TotallyBounded s\nU : Set (α × α)\nhU : U ∈ 𝓤 α\n⊢ ∃ t, t.Countable ∧ s ⊆ ⋃ x ∈ t, ball x U",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"SetRel.symmetrize",
"SetRel.inv"... | [
"α : Type u\nuniformSpace : UniformSpace α\ninst✝ : (𝓤 α).IsCountablyGenerated\ns : Set α\nh : TotallyBounded s\nU : Set (α × α)\nhU : U ∈ 𝓤 α\nt : Set α\nht : t.Finite\nhst : s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ SetRel.inv U}\n⊢ ∃ t, t.Countable ∧ s ⊆ ⋃ x ∈ t, ball x U"
] | obtain ⟨t, ht, hst⟩ := h (SetRel.inv U)
(mem_of_superset (symmetrize_mem_uniformity hU) SetRel.symmetrize_subset_inv) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.Algebra.Module.ModuleTopology | {
"line": 229,
"column": 4
} | {
"line": 234,
"column": 10
} | {
"line": 236,
"column": 0
} | [
{
"pp": "case mpr\nR : Type u_1\nS : Type u_2\nτR : TopologicalSpace R\nτS : TopologicalSpace S\ninst✝⁸ : Semiring R\ninst✝⁷ : Semiring S\nσ : R →+* S\nσ' : S →+* R\ninst✝⁶ : RingHomInvPair σ σ'\ninst✝⁵ : RingHomInvPair σ' σ\nA : Type u_3\ninst✝⁴ : AddCommMonoid A\ninst✝³ : Module R A\nτA : TopologicalSpace A\n... | [] | · rintro ⟨h1, h2⟩
use τ.induced e
rw [induced_compose]
refine ⟨⟨continuousSMul_inducedₛₗ g hσ, continuousAdd_induced h⟩, ?_⟩
nth_rw 2 [← induced_id (t := τ)]
simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Homology.ShortComplex.LeftHomology | {
"line": 336,
"column": 35
} | {
"line": 336,
"column": 57
} | {
"line": 336,
"column": 58
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh₁ : S₁.LeftHomologyData\nh₂ : S₂.LeftHomologyData\nφK : h₁.K ⟶ h₂.K := h₂.liftK (h₁.i ≫ φ.τ₂) ⋯\n⊢ S₁.f ≫ φ.τ₂ = φ.τ₁ ≫ h₂.f' ≫ h₂.i",
"ppTerm": "?m.165",
"assigned": true,
"... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh₁ : S₁.LeftHomologyData\nh₂ : S₂.LeftHomologyData\nφK : h₁.K ⟶ h₂.K := h₂.liftK (h₁.i ≫ φ.τ₂) ⋯\n⊢ S₁.f ≫ φ.τ₂ = φ.τ₁ ≫ S₂.f"
] | LeftHomologyData.f'_i, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.ShortComplex.LeftHomology | {
"line": 863,
"column": 20
} | {
"line": 863,
"column": 28
} | {
"line": 863,
"column": 28
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₁.LeftHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\ni : h.K ⟶ S₂.X₂ := h.i ≫ φ.τ₂\nwi : i ≫ S₂.g = 0\nW'✝ : C\nx : W'✝ ⟶ S₂.X₂\nhx : x ≫ S₂.g = 0\n⊢ h.lif... | [] | simp [i] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.ShortComplex.LeftHomology | {
"line": 863,
"column": 20
} | {
"line": 863,
"column": 28
} | {
"line": 863,
"column": 28
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₁.LeftHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\ni : h.K ⟶ S₂.X₂ := h.i ≫ φ.τ₂\nwi : i ≫ S₂.g = 0\nW'✝ : C\nx : W'✝ ⟶ S₂.X₂\nhx : x ≫ S₂.g = 0\n⊢ h.lif... | [] | simp [i] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.LeftHomology | {
"line": 863,
"column": 20
} | {
"line": 863,
"column": 28
} | {
"line": 863,
"column": 28
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₁.LeftHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\ni : h.K ⟶ S₂.X₂ := h.i ≫ φ.τ₂\nwi : i ≫ S₂.g = 0\nW'✝ : C\nx : W'✝ ⟶ S₂.X₂\nhx : x ≫ S₂.g = 0\n⊢ h.lif... | [] | simp [i] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.LeftHomology | {
"line": 899,
"column": 20
} | {
"line": 899,
"column": 28
} | {
"line": 899,
"column": 28
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₂.LeftHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\ni : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂\nwi : i ≫ S₁.g = 0\nW'✝ : C\nx : W'✝ ⟶ S₁.X₂\nhx : x ≫ S₁.g = 0\n⊢ h... | [] | simp [i] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.ShortComplex.LeftHomology | {
"line": 899,
"column": 20
} | {
"line": 899,
"column": 28
} | {
"line": 899,
"column": 28
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₂.LeftHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\ni : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂\nwi : i ≫ S₁.g = 0\nW'✝ : C\nx : W'✝ ⟶ S₁.X₂\nhx : x ≫ S₁.g = 0\n⊢ h... | [] | simp [i] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.LeftHomology | {
"line": 899,
"column": 20
} | {
"line": 899,
"column": 28
} | {
"line": 899,
"column": 28
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₂.LeftHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\ni : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂\nwi : i ≫ S₁.g = 0\nW'✝ : C\nx : W'✝ ⟶ S₁.X₂\nhx : x ≫ S₁.g = 0\n⊢ h... | [] | simp [i] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Module.Equiv | {
"line": 1094,
"column": 4
} | {
"line": 1096,
"column": 31
} | {
"line": 1097,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_1\nM : Type u_2\nM₂ : Type u_3\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : TopologicalSpace M₂\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R M₂\nh : IsInvertible 0\n⊢ inverse 0 = 0",
"ppTerm": "?pos✝",
"assigned"... | [] | rcases isInvertible_zero_iff.1 h with ⟨hM, hM₂⟩
ext x
exact Subsingleton.elim _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Module.Equiv | {
"line": 1094,
"column": 4
} | {
"line": 1096,
"column": 31
} | {
"line": 1097,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_1\nM : Type u_2\nM₂ : Type u_3\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : TopologicalSpace M₂\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R M₂\nh : IsInvertible 0\n⊢ inverse 0 = 0",
"ppTerm": "?pos✝",
"assigned"... | [] | rcases isInvertible_zero_iff.1 h with ⟨hM, hM₂⟩
ext x
exact Subsingleton.elim _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.RightHomology | {
"line": 418,
"column": 16
} | {
"line": 423,
"column": 9
} | {
"line": 423,
"column": 9
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh₁ : S₁.RightHomologyData\nh₂ : S₂.RightHomologyData\nψ₁ ψ₂ : RightHomologyMapData φ h₁ h₂\n⊢ ψ₁ = ψ₂",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [] | by
have hQ : ψ₁.φQ = ψ₂.φQ := by rw [← cancel_epi h₁.p, commp, commp]
have hH : ψ₁.φH = ψ₂.φH := by rw [← cancel_mono h₂.ι, commι, commι, hQ]
cases ψ₁
cases ψ₂
congr | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ShortComplex.RightHomology | {
"line": 1030,
"column": 2
} | {
"line": 1030,
"column": 33
} | {
"line": 1031,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nS : ShortComplex C\ninst✝ : S.HasLeftHomology\n⊢ S.opcyclesOpIso.hom ≫ S.toCycles.op = S.op.fromOpcycles",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"CategoryTheory.ShortComplex.opcycles",
"... | [
"C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nS : ShortComplex C\ninst✝ : S.HasLeftHomology\n⊢ S.leftHomologyData.op.opcyclesIso.hom ≫ S.leftHomologyData.f'.op = S.op.fromOpcycles"
] | dsimp [opcyclesOpIso, toCycles] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.Algebra.Homology.ShortComplex.RightHomology | {
"line": 1033,
"column": 38
} | {
"line": 1033,
"column": 60
} | {
"line": 1033,
"column": 61
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nS : ShortComplex C\ninst✝ : S.HasLeftHomology\n⊢ (S.leftHomologyData.f' ≫ S.leftHomologyData.i).op = S.op.g",
"ppTerm": "?m.88",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.ShortComplex... | [
"C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nS : ShortComplex C\ninst✝ : S.HasLeftHomology\n⊢ S.f.op = S.op.g"
] | LeftHomologyData.f'_i, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.ShortComplex.Homology | {
"line": 626,
"column": 39
} | {
"line": 626,
"column": 60
} | {
"line": 626,
"column": 61
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nS : ShortComplex C\nh₁ h₁' : S.LeftHomologyData\nh₂ h₂' : S.RightHomologyData\n⊢ leftRightHomologyComparison' h₁ h₂ = leftRightHomologyComparison' h₁ h₂ ≫ rightHomologyMap' (𝟙 S) h₂ h₂",
"ppTerm": "?m.95",
"assigned": true,
... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nS : ShortComplex C\nh₁ h₁' : S.LeftHomologyData\nh₂ h₂' : S.RightHomologyData\n⊢ leftRightHomologyComparison' h₁ h₂ = leftRightHomologyComparison' h₁ h₂ ≫ 𝟙 h₂.H"
] | rightHomologyMap'_id, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.ShortComplex.Homology | {
"line": 1139,
"column": 8
} | {
"line": 1139,
"column": 35
} | {
"line": 1139,
"column": 36
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\ninst✝¹ : S₁.HasHomology\ninst✝ : S₂.HasHomology\nh₁ : IsIso (opcyclesMap φ)\nh₂ : Mono φ.τ₃\nh : (S₂.homologyι ≫ inv (opcyclesMap φ)) ≫ S₁.fromOpcycles = 0\nz : S₂.homology ⟶ (Kernel... | [
"case refine_2\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\ninst✝¹ : S₁.HasHomology\ninst✝ : S₂.HasHomology\nh₁ : IsIso (opcyclesMap φ)\nh₂ : Mono φ.τ₃\nh : (S₂.homologyι ≫ inv (opcyclesMap φ)) ≫ S₁.fromOpcycles = 0\nz : S₂.homology ⟶ (KernelFork.ofι S₁.... | ← cancel_mono S₂.homologyι, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Category.ModuleCat.Topology.Basic | {
"line": 310,
"column": 4
} | {
"line": 313,
"column": 29
} | {
"line": 315,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : Ring R\ninst✝¹ : TopologicalSpace R\nM : ModuleCat R\nI : Type u_1\nX : I → TopModuleCat R\nf : (i : I) → M ⟶ (X i).toModuleCat\nJ : Type u_2\ninst✝ : Category.{v_1, u_2} J\nF : J ⥤ TopModuleCat R\nc : Cone (F ⋙ forget₂ (TopModuleCat R) (ModuleCat R))\nhc : IsLimit c\ns : Cone F\nm... | [] | ext x
refine congr($(hc.uniq ((forget₂ _ _).mapCone s) ((forget₂ _ _).map m) fun j ↦ ?_).hom x)
ext y
exact congr($(H j).hom y) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Topology.Basic | {
"line": 310,
"column": 4
} | {
"line": 313,
"column": 29
} | {
"line": 315,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : Ring R\ninst✝¹ : TopologicalSpace R\nM : ModuleCat R\nI : Type u_1\nX : I → TopModuleCat R\nf : (i : I) → M ⟶ (X i).toModuleCat\nJ : Type u_2\ninst✝ : Category.{v_1, u_2} J\nF : J ⥤ TopModuleCat R\nc : Cone (F ⋙ forget₂ (TopModuleCat R) (ModuleCat R))\nhc : IsLimit c\ns : Cone F\nm... | [] | ext x
refine congr($(hc.uniq ((forget₂ _ _).mapCone s) ((forget₂ _ _).map m) fun j ↦ ?_).hom x)
ext y
exact congr($(H j).hom y) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.Abelian | {
"line": 96,
"column": 2
} | {
"line": 99,
"column": 12
} | {
"line": 100,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\nS : ShortComplex C\nD : Type u'\ninst✝¹ : Category.{v', u'} D\ninst✝ : HasZeroMorphisms D\nγ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f\nf' : S.X₁ ⟶ kernel S.g := kernel.lift S.g S.f ⋯\nhf' : f' = kernel.lift γ f' ⋯ ≫ kernel.... | [
"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\nS : ShortComplex C\nD : Type u'\ninst✝¹ : Category.{v', u'} D\ninst✝ : HasZeroMorphisms D\nγ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f\nf' : S.X₁ ⟶ kernel S.g := kernel.lift S.g S.f ⋯\nhf' : f' = kernel.lift γ f' ⋯ ≫ kernel.ι γ\nwπ : f'... | have fac : f' = Abelian.factorThruImage S.f ≫ e.hom ≫ kernel.ι γ := by
rw [hf', he]
simp only [γ, f', kernel.lift_ι, abelianImageToKernel, ← cancel_mono (kernel.ι S.g),
assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Homology.ShortComplex.Preadditive | {
"line": 583,
"column": 65
} | {
"line": 583,
"column": 87
} | {
"line": 584,
"column": 6
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nH₁ : S₁.LeftHomologyData\nH₂ : S₂.LeftHomologyData\nh₀ : S₁.X₁ ⟶ S₂.X₁\nh₀_f : h₀ ≫ S₂.f = 0\nh₁ : S₁.X₂ ⟶ S₂.X₁\nh₂ : S₁.X₃ ⟶ S₂.X₂\nh₃ : S₁.X₃ ⟶ S₂.X₃\ng_h₃ : S₁.g ≫ h₃ = 0\n⊢ S₁.f ≫... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nH₁ : S₁.LeftHomologyData\nH₂ : S₂.LeftHomologyData\nh₀ : S₁.X₁ ⟶ S₂.X₁\nh₀_f : h₀ ≫ S₂.f = 0\nh₁ : S₁.X₂ ⟶ S₂.X₁\nh₂ : S₁.X₃ ⟶ S₂.X₂\nh₃ : S₁.X₃ ⟶ S₂.X₃\ng_h₃ : S₁.g ≫ h₃ = 0\n⊢ S₁.f ≫ h₁ ≫ S₂.f =... | LeftHomologyData.f'_i, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Subobject.FactorThru | {
"line": 66,
"column": 8
} | {
"line": 67,
"column": 79
} | {
"line": 68,
"column": 6
} | [
{
"pp": "case mp\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX✝ Y✝ Z : C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nX Y : C\nP✝ : Subobject Y\nf : X ⟶ Y\nP Q : MonoOver Y\nh : P ≅ Q\n⊢ P.Factors f → Q.Factors f",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTh... | [] | rintro ⟨i, w⟩
exact ⟨i ≫ h.hom.hom.left, by rw [Category.assoc, Over.w h.hom.hom, w]⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Subobject.FactorThru | {
"line": 66,
"column": 8
} | {
"line": 67,
"column": 79
} | {
"line": 68,
"column": 6
} | [
{
"pp": "case mp\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX✝ Y✝ Z : C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nX Y : C\nP✝ : Subobject Y\nf : X ⟶ Y\nP Q : MonoOver Y\nh : P ≅ Q\n⊢ P.Factors f → Q.Factors f",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTh... | [] | rintro ⟨i, w⟩
exact ⟨i ≫ h.hom.hom.left, by rw [Category.assoc, Over.w h.hom.hom, w]⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Subobject.Basic | {
"line": 603,
"column": 62
} | {
"line": 609,
"column": 47
} | {
"line": 611,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX Y : C\ninst✝ : HasPullbacks C\nf : X ⟶ Y\ny : Subobject Y\n⊢ ∃ φ, IsPullback φ ((pullback f).obj y).arrow y.arrow f",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.pullback",
"CategoryT... | [] | by
obtain ⟨A, i, ⟨_, rfl⟩⟩ := mk_surjective y
rw [pullback_obj]
exists (underlyingIso (pullback.snd (mk i).arrow f)).hom ≫ pullback.fst (mk i).arrow f
exact IsPullback.of_iso (IsPullback.of_hasPullback (mk i).arrow f)
(underlyingIso (pullback.snd (mk i).arrow f)).symm (Iso.refl _) (Iso.refl _) (Iso.refl... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Subobject.MonoOver | {
"line": 368,
"column": 6
} | {
"line": 368,
"column": 15
} | {
"line": 370,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nX Y Z : C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nf : X ⟶ Y\ninst✝ : Mono f\ng h : MonoOver X\ne : (map f).obj g ⟶ (map f).obj h\n⊢ Over.Hom.left e.hom ≫ h.arrow ≫ f = g.arrow ≫ f",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
... | [] | apply w e | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Homology.HomologicalComplex | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 65
} | {
"line": 92,
"column": 2
} | [
{
"pp": "case pos\nι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nX₁ : ι → V\nd₁ : (i j : ι) → X₁ i ⟶ X₁ j\ns₁ : ∀ (i j : ι), ¬c.Rel i j → d₁ i j = 0\nh₁ : ∀ (i j k : ι), c.Rel i j → c.Rel j k → d₁ i j ≫ d₁ j k = 0\nd₂ : (i j : ι) → X₁ i ⟶ X₁ j\ns₂ : ∀ (i j... | [
"case neg\nι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nX₁ : ι → V\nd₁ : (i j : ι) → X₁ i ⟶ X₁ j\ns₁ : ∀ (i j : ι), ¬c.Rel i j → d₁ i j = 0\nh₁ : ∀ (i j k : ι), c.Rel i j → c.Rel j k → d₁ i j ≫ d₁ j k = 0\nd₂ : (i j : ι) → X₁ i ⟶ X₁ j\ns₂ : ∀ (i j : ι), ¬c.Re... | · simpa only [comp_id, id_comp, eqToHom_refl] using h_d i j hij | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Homology.HomologicalComplex | {
"line": 378,
"column": 73
} | {
"line": 380,
"column": 35
} | {
"line": 382,
"column": 0
} | [
{
"pp": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ni j j' : ι\nrij : c.Rel i j\nrij' : c.Rel i j'\n⊢ C.d i j' ≫ eqToHom ⋯ = C.d i j",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"CategoryTheory.... | [] | by
obtain rfl := c.next_eq rij rij'
simp only [eqToHom_refl, comp_id] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 95,
"column": 2
} | {
"line": 96,
"column": 41
} | {
"line": 98,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS : ShortComplex C\nh : S.HomologyData\n⊢ S.Exact ↔ IsZero h.left.H",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"CategoryTheory.ShortComplex.HomologyData.left",
"CategoryTheory.ShortComplex.H... | [] | haveI := HasHomology.mk' h
exact LeftHomologyData.exact_iff h.left | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 95,
"column": 2
} | {
"line": 96,
"column": 41
} | {
"line": 98,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS : ShortComplex C\nh : S.HomologyData\n⊢ S.Exact ↔ IsZero h.left.H",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"CategoryTheory.ShortComplex.HomologyData.left",
"CategoryTheory.ShortComplex.H... | [] | haveI := HasHomology.mk' h
exact LeftHomologyData.exact_iff h.left | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 100,
"column": 2
} | {
"line": 101,
"column": 43
} | {
"line": 103,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS : ShortComplex C\nh : S.HomologyData\n⊢ S.Exact ↔ IsZero h.right.H",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"CategoryTheory.ShortComplex.RightHomologyData.exact_iff",
"CategoryTheory.Sho... | [] | haveI := HasHomology.mk' h
exact RightHomologyData.exact_iff h.right | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 100,
"column": 2
} | {
"line": 101,
"column": 43
} | {
"line": 103,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS : ShortComplex C\nh : S.HomologyData\n⊢ S.Exact ↔ IsZero h.right.H",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"CategoryTheory.ShortComplex.RightHomologyData.exact_iff",
"CategoryTheory.Sho... | [] | haveI := HasHomology.mk' h
exact RightHomologyData.exact_iff h.right | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 223,
"column": 2
} | {
"line": 223,
"column": 64
} | {
"line": 224,
"column": 2
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Category.{v_2, u_2} D\ninst✝⁴ : HasZeroMorphisms C\ninst✝³ : HasZeroMorphisms D\nS : ShortComplex C\nh : S.Exact\nF : C ⥤ D\ninst✝² : F.PreservesZeroMorphisms\ninst✝¹ : F.PreservesRightHomologyOf S\ninst✝ : (S.map F).HasHomology\nthis... | [
"C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Category.{v_2, u_2} D\ninst✝⁴ : HasZeroMorphisms C\ninst✝³ : HasZeroMorphisms D\nS : ShortComplex C\nF : C ⥤ D\ninst✝² : F.PreservesZeroMorphisms\ninst✝¹ : F.PreservesRightHomologyOf S\ninst✝ : (S.map F).HasHomology\nthis : S.HasHomology\nh : 𝟙 ... | rw [S.rightHomologyData.exact_iff, IsZero.iff_id_eq_zero] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex | {
"line": 378,
"column": 24
} | {
"line": 378,
"column": 51
} | {
"line": 378,
"column": 51
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : HasZeroMorphisms C\nι : Type u_2\nc : ComplexShape ι\nK L M : HomologicalComplex C c\nφ : K ⟶ L\nψ : L ⟶ M\ni : ι\ninst✝² : K.HasHomology i\ninst✝¹ : L.HasHomology i\ninst✝ : M.HasHomology i\n⊢ ShortComplex.cyclesMap ((shortComplexFunctor C c i).ma... | [
"C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : HasZeroMorphisms C\nι : Type u_2\nc : ComplexShape ι\nK L M : HomologicalComplex C c\nφ : K ⟶ L\nψ : L ⟶ M\ni : ι\ninst✝² : K.HasHomology i\ninst✝¹ : L.HasHomology i\ninst✝ : M.HasHomology i\n⊢ ShortComplex.cyclesMap ((shortComplexFunctor C c i).map φ) ≫\n ... | ShortComplex.cyclesMap_comp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.Functor | {
"line": 107,
"column": 2
} | {
"line": 108,
"column": 5
} | {
"line": 110,
"column": 0
} | [
{
"pp": "T : Type u_1\ninst✝² : Category.{v_1, u_1} T\nV : Type u_2\ninst✝¹ : Category.{v_2, u_2} V\ninst✝ : Abelian V\nι : Type u_3\nc : ComplexShape ι\nK₁ K₂ : HomologicalComplex (T ⥤ V) c\nf : K₁ ⟶ K₂\ni : ι\n⊢ QuasiIsoAt f i ↔ ∀ (t : T), QuasiIsoAt ((((evaluation T V).obj t).mapHomologicalComplex c).map f) ... | [] | simp only [quasiIsoAt_iff, ShortComplex.quasiIso_iff_evaluation]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Functor | {
"line": 107,
"column": 2
} | {
"line": 108,
"column": 5
} | {
"line": 110,
"column": 0
} | [
{
"pp": "T : Type u_1\ninst✝² : Category.{v_1, u_1} T\nV : Type u_2\ninst✝¹ : Category.{v_2, u_2} V\ninst✝ : Abelian V\nι : Type u_3\nc : ComplexShape ι\nK₁ K₂ : HomologicalComplex (T ⥤ V) c\nf : K₁ ⟶ K₂\ni : ι\n⊢ QuasiIsoAt f i ↔ ∀ (t : T), QuasiIsoAt ((((evaluation T V).obj t).mapHomologicalComplex c).map f) ... | [] | simp only [quasiIsoAt_iff, ShortComplex.quasiIso_iff_evaluation]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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