module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Topology.Compactness.LocallyCompact | {
"line": 118,
"column": 4
} | {
"line": 118,
"column": 50
} | {
"line": 119,
"column": 4
} | [
{
"pp": "case refine_2\nX✝ : Type u_1\nY : Type u_2\nι : Type u_3\ninst✝⁴ : TopologicalSpace X✝\ninst✝³ : TopologicalSpace Y\ns✝ t✝ : Set X✝\nX : ι → Type u_4\ninst✝² : (i : ι) → TopologicalSpace (X i)\ninst✝¹ : ∀ (i : ι), LocallyCompactSpace (X i)\ninst✝ : ∀ (i : ι), CompactSpace (X i)\nt : (i : ι) → X i\nn : ... | [
"case refine_3\nX✝ : Type u_1\nY : Type u_2\nι : Type u_3\ninst✝⁴ : TopologicalSpace X✝\ninst✝³ : TopologicalSpace Y\ns✝ t✝ : Set X✝\nX : ι → Type u_4\ninst✝² : (i : ι) → TopologicalSpace (X i)\ninst✝¹ : ∀ (i : ι), LocallyCompactSpace (X i)\ninst✝ : ∀ (i : ι), CompactSpace (X i)\nt : (i : ι) → X i\nn : Set ((i : ι)... | · exact forall₂_imp fun i _ hi' => hsub' i hi' | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Compactness.LocallyCompact | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 45
} | {
"line": 208,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : LocallyCompactSpace Y\nf : X → Y\nhf : IsInducing f\nU Z : Set Y\nhU : IsOpen[inst✝¹] U\nhZ : IsClosed[inst✝¹] Z\nhUZ : range f = U ∩ Z\nthis : ∀ (x : X), (𝓝 x).HasBasis (fun s ↦ (s ∈ 𝓝 (f x) ∧ IsCompact s) ... | [] | exacts [hs.inter_right hZ, hUZ ▸ by gcongr] | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_exacts_1 | Batteries.Tactic.exacts |
Mathlib.Topology.Bases | {
"line": 1010,
"column": 2
} | {
"line": 1038,
"column": 13
} | {
"line": 1040,
"column": 0
} | [
{
"pp": "α : Type u_1\nts : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nt : Set (Set α)\nht : ts = generateFrom t\n⊢ ∃ s ⊆ t, s.Countable ∧ ts = generateFrom s",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"congrArg",
"Set.ofPred",
... | [] | let t' := (fun f => ⋂₀ f) '' { f : Set (Set α) | f.Finite ∧ f ⊆ t }
have : IsTopologicalBasis t' := TopologicalSpace.isTopologicalBasis_of_subbasis ht
obtain ⟨s', s't', s'_count, hs'⟩ : ∃ s' ⊆ t', s'.Countable ∧ IsTopologicalBasis s' :=
this.exists_countable
have A : ∀ u ∈ s', ∃ (f : Set (Set α)), f.Finite ∧ ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Bases | {
"line": 1010,
"column": 2
} | {
"line": 1038,
"column": 13
} | {
"line": 1040,
"column": 0
} | [
{
"pp": "α : Type u_1\nts : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nt : Set (Set α)\nht : ts = generateFrom t\n⊢ ∃ s ⊆ t, s.Countable ∧ ts = generateFrom s",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"congrArg",
"Set.ofPred",
... | [] | let t' := (fun f => ⋂₀ f) '' { f : Set (Set α) | f.Finite ∧ f ⊆ t }
have : IsTopologicalBasis t' := TopologicalSpace.isTopologicalBasis_of_subbasis ht
obtain ⟨s', s't', s'_count, hs'⟩ : ∃ s' ⊆ t', s'.Countable ∧ IsTopologicalBasis s' :=
this.exists_countable
have A : ∀ u ∈ s', ∃ (f : Set (Set α)), f.Finite ∧ ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactness.LocallyFinite | {
"line": 29,
"column": 2
} | {
"line": 29,
"column": 60
} | {
"line": 30,
"column": 2
} | [
{
"pp": "X : Type u_1\nι : Type u_2\ninst✝ : TopologicalSpace X\ns : Set X\nf : ι → Set X\nhs : IsCompact s\nU : X → Set X\nhxU : ∀ (x : X), U x ∈ nhds x\nhUf : ∀ (x : X), {i | (f i ∩ U x).Nonempty}.Finite\nt : Finset X\nhsU : s ⊆ ⋃ x ∈ t, U x\n⊢ {i | (f i ∩ s).Nonempty}.Finite",
"ppTerm": "?m.47",
"ass... | [
"X : Type u_1\nι : Type u_2\ninst✝ : TopologicalSpace X\ns : Set X\nf : ι → Set X\nhs : IsCompact s\nU : X → Set X\nhxU : ∀ (x : X), U x ∈ nhds x\nhUf : ∀ (x : X), {i | (f i ∩ U x).Nonempty}.Finite\nt : Finset X\nhsU : s ⊆ ⋃ x ∈ t, U x\n⊢ {i | (f i ∩ s).Nonempty} ⊆ ⋃ i ∈ ↑t, {i_1 | (f i_1 ∩ U i).Nonempty}"
] | refine (t.finite_toSet.biUnion fun x _ => hUf x).subset ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Compactness.SigmaCompact | {
"line": 124,
"column": 60
} | {
"line": 124,
"column": 72
} | {
"line": 124,
"column": 72
} | [
{
"pp": "X : 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\n⊢ ⋃ n, f ⁻¹' L n ∩ s = (⋃ i, f ⁻¹' L i) ∩ s",
"ppTerm": "?m.158",
"assigned": true,
"use... | [
"X : 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\n⊢ ⋃ n, f ⁻¹' L n ∩ s = ⋃ i, f ⁻¹' L i ∩ s"
] | iUnion_inter | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Set.Subset | {
"line": 58,
"column": 2
} | {
"line": 59,
"column": 46
} | {
"line": 61,
"column": 0
} | [
{
"pp": "α : Type u_2\nA : Set α\nS : Set (Set α)\n⊢ A ↓∩ ⋃₀ S = ⋃₀ {x | ∃ B ∈ S, A ↓∩ B = x}",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.ofPred",
"Set.sUnion",
"Membership.mem",
"Exists",
"Set.Elem",
"id",
... | [] | rw [← Set.image, sUnion_image]
simp_rw [sUnion_eq_biUnion, preimage_iUnion] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Subset | {
"line": 58,
"column": 2
} | {
"line": 59,
"column": 46
} | {
"line": 61,
"column": 0
} | [
{
"pp": "α : Type u_2\nA : Set α\nS : Set (Set α)\n⊢ A ↓∩ ⋃₀ S = ⋃₀ {x | ∃ B ∈ S, A ↓∩ B = x}",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.ofPred",
"Set.sUnion",
"Membership.mem",
"Exists",
"Set.Elem",
"id",
... | [] | rw [← Set.image, sUnion_image]
simp_rw [sUnion_eq_biUnion, preimage_iUnion] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Subset | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 43
} | {
"line": 71,
"column": 0
} | [
{
"pp": "α : Type u_2\nA : Set α\nS : Set (Set α)\n⊢ A ↓∩ ⋂₀ S = ⋂₀ ((fun x ↦ A ↓∩ x) '' S)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"congrArg",
"Set.iInter",
"Membership.mem",
"Set.Elem",
"Set.preimage",
"congr",
"True",
"Set.preimage... | [] | simp only [preimage_sInter, sInter_image] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Set.Subset | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 43
} | {
"line": 71,
"column": 0
} | [
{
"pp": "α : Type u_2\nA : Set α\nS : Set (Set α)\n⊢ A ↓∩ ⋂₀ S = ⋂₀ ((fun x ↦ A ↓∩ x) '' S)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"congrArg",
"Set.iInter",
"Membership.mem",
"Set.Elem",
"Set.preimage",
"congr",
"True",
"Set.preimage... | [] | simp only [preimage_sInter, sInter_image] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Subset | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 43
} | {
"line": 71,
"column": 0
} | [
{
"pp": "α : Type u_2\nA : Set α\nS : Set (Set α)\n⊢ A ↓∩ ⋂₀ S = ⋂₀ ((fun x ↦ A ↓∩ x) '' S)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"congrArg",
"Set.iInter",
"Membership.mem",
"Set.Elem",
"Set.preimage",
"congr",
"True",
"Set.preimage... | [] | simp only [preimage_sInter, sInter_image] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactness.Compact | {
"line": 526,
"column": 2
} | {
"line": 526,
"column": 22
} | {
"line": 526,
"column": 22
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns t : Set X\nhs : IsCompact s\nht : IsCompact t\n⊢ IsCompact (s ∪ t)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"cond",
"Eq.mpr",
"congrArg",
"Set.instUnion",
"id",
"Bool",
"Union.union",
"... | [
"X : Type u\ninst✝ : TopologicalSpace X\ns t : Set X\nhs : IsCompact s\nht : IsCompact t\n⊢ IsCompact (⋃ b, bif b then s else t)"
] | rw [union_eq_iUnion] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Compactness.Compact | {
"line": 569,
"column": 2
} | {
"line": 570,
"column": 60
} | {
"line": 571,
"column": 2
} | [
{
"pp": "X : Type u\nT : TopologicalSpace X\nS : Set (Set X)\nhTS : T = generateFrom S\ns : Set X\nh : ∀ P ⊆ S, s ⊆ ⋃₀ P → ∃ Q ⊆ P, Q.Finite ∧ s ⊆ ⋃₀ Q\nF : Ultrafilter X\nhsF : s ∈ F\nhF : ¬∃ x ∈ s, ↑F ≤ 𝓝 x\nU : X → Set X\nhxU : ∀ x ∈ s, x ∈ U x\nhSU : ∀ x ∈ s, U x ∈ S\nhUF : ∀ x ∈ s, U x ∉ F\n⊢ False",
... | [
"X : Type u\nT : TopologicalSpace X\nS : Set (Set X)\nhTS : T = generateFrom S\ns : Set X\nh : ∀ P ⊆ S, s ⊆ ⋃₀ P → ∃ Q ⊆ P, Q.Finite ∧ s ⊆ ⋃₀ Q\nF : Ultrafilter X\nhsF : s ∈ F\nhF : ¬∃ x ∈ s, ↑F ≤ 𝓝 x\nU : X → Set X\nhxU : ∀ x ∈ s, x ∈ U x\nhSU : ∀ x ∈ s, U x ∈ S\nhUF : ∀ x ∈ s, U x ∉ F\nQ : Set (Set X)\nhQU : Q ⊆... | obtain ⟨Q, hQU, hQ, hsQ⟩ := h (U '' s) (by simpa [Set.subset_def])
(fun x hx ↦ Set.mem_sUnion_of_mem (hxU _ hx) (by grind)) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.Compactness.Compact | {
"line": 1009,
"column": 2
} | {
"line": 1010,
"column": 55
} | {
"line": 1012,
"column": 0
} | [
{
"pp": "X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\nhf : IsInducing f\nhf' : IsClosed[inst✝] (range f)\nK : Set Y\nhK : IsCompact K\n⊢ IsCompact (f ⁻¹' K)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [] | replace hK := hK.inter_right hf'
rwa [hf.isCompact_iff, image_preimage_eq_inter_range] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Compactness.Compact | {
"line": 1009,
"column": 2
} | {
"line": 1010,
"column": 55
} | {
"line": 1012,
"column": 0
} | [
{
"pp": "X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\nhf : IsInducing f\nhf' : IsClosed[inst✝] (range f)\nK : Set Y\nhK : IsCompact K\n⊢ IsCompact (f ⁻¹' K)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [] | replace hK := hK.inter_right hf'
rwa [hf.isCompact_iff, image_preimage_eq_inter_range] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Separation.Hausdorff | {
"line": 388,
"column": 2
} | {
"line": 388,
"column": 65
} | {
"line": 390,
"column": 0
} | [
{
"pp": "case inr.inr\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : T2Space X\ninst✝¹ : TopologicalSpace Y\ninst✝ : T2Space Y\nx y : Y\nh : Sum.inr x ≠ Sum.inr y\n⊢ ∃ u v,\n IsOpen[instTopologicalSpaceSum] u ∧ IsOpen[instTopologicalSpaceSum] v ∧ Sum.inr x ∈ u ∧ Sum.inr y ∈ v ∧ Disjoint u... | [] | · exact separated_by_isOpenEmbedding .inr <| ne_of_apply_ne _ h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.DiscreteSubset | {
"line": 324,
"column": 4
} | {
"line": 324,
"column": 27
} | {
"line": 325,
"column": 4
} | [
{
"pp": "case right\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : T1Space X\ns U : Set X\nh : ∀ z ∈ U, ∃ t ∈ 𝓝[≠] z, t ∩ (U \\ s) = ∅\nz : X\nh₁z : z ∈ U\nt : Set X\nh₁t : t ∈ 𝓝[≠] z\nh₂t : t ∩ (U \\ s) = ∅\n⊢ (insert z t ∩ (U \\ s)).Finite",
"ppTerm": "?right",
"assigned": true,
"usedConsta... | [
"case pos\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : T1Space X\ns U : Set X\nh : ∀ z ∈ U, ∃ t ∈ 𝓝[≠] z, t ∩ (U \\ s) = ∅\nz : X\nh₁z : z ∈ U\nt : Set X\nh₁t : t ∈ 𝓝[≠] z\nh₂t : t ∩ (U \\ s) = ∅\nhz : z ∈ U \\ s\n⊢ (insert z t ∩ (U \\ s)).Finite",
"case neg\nX : Type u_1\ninst✝¹ : TopologicalSpace X\nin... | by_cases hz : z ∈ U \ s | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Topology.Separation.Basic | {
"line": 759,
"column": 59
} | {
"line": 761,
"column": 60
} | {
"line": 763,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : T1Space X\nx : X\ns t : Set X\nf : X → Y\ny : X\nh : s =ᶠ[𝓝[{y}ᶜ] x] t\n⊢ ContinuousWithinAt f s x ↔ ContinuousWithinAt f t x",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mp... | [] | by
rw [← continuousWithinAt_insert_self (s := s), ← continuousWithinAt_insert_self (s := t)]
exact continuousWithinAt_congr_set (eventuallyEq_insert h) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.DiscreteSubset | {
"line": 502,
"column": 2
} | {
"line": 502,
"column": 22
} | {
"line": 503,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\ns t : Set X\nhs : IsDiscrete s\nht✝ : IsDiscrete t\nhsc : IsClosed[inst✝] s\nht : IsClosed[inst✝] t\n⊢ IsDiscrete (s ∪ t)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"cond",
"Eq.mpr",
"congrArg",
"Set.instUnion"... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\ns t : Set X\nhs : IsDiscrete s\nht✝ : IsDiscrete t\nhsc : IsClosed[inst✝] s\nht : IsClosed[inst✝] t\n⊢ IsDiscrete (⋃ b, bif b then s else t)"
] | rw [union_eq_iUnion] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Separation.Basic | {
"line": 1055,
"column": 6
} | {
"line": 1058,
"column": 40
} | {
"line": 1059,
"column": 4
} | [
{
"pp": "case insert.refine_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : R1Space X\nι : Type u_3\nx : ι\nt : Finset ι\nhx : x ∉ t\nih :\n ∀ {s : Set X},\n IsCompact s →\n ∀ (U : ι → Set X),\n (∀ i ∈ t, IsOpen[inst✝¹] (U i)) →\n s ⊆ ⋃ i ∈ t, U i → ∃ K, (∀ (i : ι), IsCompact (K i... | [] | intro i
rcases eq_or_ne i x with rfl | hi
· simp only [update_self, h2K₁]
· simp only [update_of_ne hi, h2K] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Separation.Basic | {
"line": 1055,
"column": 6
} | {
"line": 1058,
"column": 40
} | {
"line": 1059,
"column": 4
} | [
{
"pp": "case insert.refine_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : R1Space X\nι : Type u_3\nx : ι\nt : Finset ι\nhx : x ∉ t\nih :\n ∀ {s : Set X},\n IsCompact s →\n ∀ (U : ι → Set X),\n (∀ i ∈ t, IsOpen[inst✝¹] (U i)) →\n s ⊆ ⋃ i ∈ t, U i → ∃ K, (∀ (i : ι), IsCompact (K i... | [] | intro i
rcases eq_or_ne i x with rfl | hi
· simp only [update_self, h2K₁]
· simp only [update_of_ne hi, h2K] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Connected.Clopen | {
"line": 78,
"column": 8
} | {
"line": 78,
"column": 72
} | {
"line": 79,
"column": 6
} | [
{
"pp": "case refine_1.inr.refine_1\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns : Set (α ⊕ β)\nhs : IsConnected s\nx : β\nhx : inr x ∈ s\nh : s ⊆ range inr\n⊢ IsConnected (inr ⁻¹' s)",
"ppTerm": "?refine_1.inr.refine_1",
"assigned": true,
"usedConstants": [
... | [] | exact hs.preimage_of_isOpenMap Sum.inr_injective isOpenMap_inr h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Connected.Clopen | {
"line": 78,
"column": 8
} | {
"line": 78,
"column": 72
} | {
"line": 79,
"column": 6
} | [
{
"pp": "case refine_1.inr.refine_1\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns : Set (α ⊕ β)\nhs : IsConnected s\nx : β\nhx : inr x ∈ s\nh : s ⊆ range inr\n⊢ IsConnected (inr ⁻¹' s)",
"ppTerm": "?refine_1.inr.refine_1",
"assigned": true,
"usedConstants": [
... | [] | exact hs.preimage_of_isOpenMap Sum.inr_injective isOpenMap_inr h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Connected.Clopen | {
"line": 78,
"column": 8
} | {
"line": 78,
"column": 72
} | {
"line": 79,
"column": 6
} | [
{
"pp": "case refine_1.inr.refine_1\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns : Set (α ⊕ β)\nhs : IsConnected s\nx : β\nhx : inr x ∈ s\nh : s ⊆ range inr\n⊢ IsConnected (inr ⁻¹' s)",
"ppTerm": "?refine_1.inr.refine_1",
"assigned": true,
"usedConstants": [
... | [] | exact hs.preimage_of_isOpenMap Sum.inr_injective isOpenMap_inr h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Connected.Clopen | {
"line": 286,
"column": 8
} | {
"line": 287,
"column": 78
} | {
"line": 288,
"column": 6
} | [
{
"pp": "case refine_1.insert.refine_1\nα : Type u\ninst✝ : TopologicalSpace α\ns : Set α\nhne : s.Nonempty\nu : Set α\nU : Finset (Set α)\nuU : u ∉ U\nx✝ :\n s.Nonempty ∧\n ∀ (a : Set α), IsOpen[inst✝] a → ∀ (b : Set α), IsOpen[inst✝] b → s ⊆ a ∪ b → s ∩ (a ∩ b) = ∅ → s ⊆ a ∨ s ⊆ b\nh : ∀ (a : Set α), IsOp... | [] | refine subset_empty_iff.1 fun x ⟨hxs, hxu, v, hvU, hxv⟩ => ?_
exact ne_of_mem_of_not_mem hvU uU (hU.1 v hvU ⟨x, hxs, hxu, hxv⟩).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Connected.Clopen | {
"line": 286,
"column": 8
} | {
"line": 287,
"column": 78
} | {
"line": 288,
"column": 6
} | [
{
"pp": "case refine_1.insert.refine_1\nα : Type u\ninst✝ : TopologicalSpace α\ns : Set α\nhne : s.Nonempty\nu : Set α\nU : Finset (Set α)\nuU : u ∉ U\nx✝ :\n s.Nonempty ∧\n ∀ (a : Set α), IsOpen[inst✝] a → ∀ (b : Set α), IsOpen[inst✝] b → s ⊆ a ∪ b → s ∩ (a ∩ b) = ∅ → s ⊆ a ∨ s ⊆ b\nh : ∀ (a : Set α), IsOp... | [] | refine subset_empty_iff.1 fun x ⟨hxs, hxu, v, hvU, hxv⟩ => ?_
exact ne_of_mem_of_not_mem hvU uU (hU.1 v hvU ⟨x, hxs, hxu, hxv⟩).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Connected.Basic | {
"line": 440,
"column": 6
} | {
"line": 442,
"column": 37
} | {
"line": 443,
"column": 2
} | [
{
"pp": "case neg\nα : Type u\ninst✝ : TopologicalSpace α\ns u : Set α\nhs : IsPreconnected s\nhu : IsOpen[inst✝] u\nh'u : (s ∩ u).Nonempty\nh : closure[inst✝] u ∩ s ⊆ u\nx : α\nhx : x ∈ s\nxu : x ∉ u\n⊢ x ∈ u ∪ (closure[inst✝] u)ᶜ",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Com... | [] | right
intro h'x
exact xu (h (mem_inter h'x hx)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Connected.Basic | {
"line": 440,
"column": 6
} | {
"line": 442,
"column": 37
} | {
"line": 443,
"column": 2
} | [
{
"pp": "case neg\nα : Type u\ninst✝ : TopologicalSpace α\ns u : Set α\nhs : IsPreconnected s\nhu : IsOpen[inst✝] u\nh'u : (s ∩ u).Nonempty\nh : closure[inst✝] u ∩ s ⊆ u\nx : α\nhx : x ∈ s\nxu : x ∉ u\n⊢ x ∈ u ∪ (closure[inst✝] u)ᶜ",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Com... | [] | right
intro h'x
exact xu (h (mem_inter h'x hx)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactness.Lindelof | {
"line": 255,
"column": 2
} | {
"line": 255,
"column": 74
} | {
"line": 256,
"column": 2
} | [
{
"pp": "X : Type u\nι : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\nb : Set ι\nc : ι → Set X\nhs : IsLindelof s\nhc₁ : ∀ (x : { a // a ∈ b }), IsOpen[inst✝] (c ↑x)\nhc₂ : s ⊆ ⋃ x, c ↑x\nd : Set ↑b\nhd : d.Countable ∧ s ⊆ ⋃ i ∈ d, c ↑i\n⊢ ∃ b' ⊆ b, b'.Countable ∧ s ⊆ ⋃ i ∈ b', c i",
"ppTerm": "?m.54",
... | [
"X : Type u\nι : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\nb : Set ι\nc : ι → Set X\nhs : IsLindelof s\nhc₁ : ∀ (x : { a // a ∈ b }), IsOpen[inst✝] (c ↑x)\nhc₂ : s ⊆ ⋃ x, c ↑x\nd : Set ↑b\nhd : d.Countable ∧ s ⊆ ⋃ i ∈ d, c ↑i\n⊢ s ⊆ ⋃ i ∈ Subtype.val '' d, c i"
] | refine ⟨Subtype.val '' d, by simp, Countable.image hd.1 Subtype.val, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Connected.Clopen | {
"line": 346,
"column": 4
} | {
"line": 346,
"column": 15
} | {
"line": 347,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u\ninst✝ : TopologicalSpace α\ns : Set α\nhs : IsClopen s\n⊢ IsPreconnected s → ∀ (a b : Set α), IsClopen a → IsClopen b → a.Nonempty → b.Nonempty → Disjoint a b → s ≠ a ∪ b",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_... | [
"case refine_1\nα : Type u\ninst✝ : TopologicalSpace α\ns : Set α\nhs : IsClopen s\n⊢ (∃ a b, IsClopen a ∧ IsClopen b ∧ a.Nonempty ∧ b.Nonempty ∧ Disjoint a b ∧ s = a ∪ b) → ¬IsPreconnected s"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Topology.Compactness.Lindelof | {
"line": 373,
"column": 2
} | {
"line": 373,
"column": 37
} | {
"line": 374,
"column": 2
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ns : Set X\ninst✝ : DiscreteTopology X\nhs : IsLindelof s\nthis : ∀ (x : X), {x} ∈ 𝓝 x\nt : Set X\nht : t.Countable\nleft✝ : ∀ x ∈ t, x ∈ s\nhssubt : s ⊆ ⋃ x ∈ t, {x}\n⊢ s.Countable",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"con... | [
"X : Type u\ninst✝¹ : TopologicalSpace X\ns : Set X\ninst✝ : DiscreteTopology X\nhs : IsLindelof s\nthis : ∀ (x : X), {x} ∈ 𝓝 x\nt : Set X\nht : t.Countable\nleft✝ : ∀ x ∈ t, x ∈ s\nhssubt : s ⊆ t\n⊢ s.Countable"
] | rw [biUnion_of_singleton] at hssubt | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Compactness.Lindelof | {
"line": 380,
"column": 2
} | {
"line": 380,
"column": 22
} | {
"line": 380,
"column": 22
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns t : Set X\nhs : IsLindelof s\nht : IsLindelof t\n⊢ IsLindelof (s ∪ t)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"cond",
"Eq.mpr",
"congrArg",
"Set.instUnion",
"id",
"Bool",
"IsLindelof",
... | [
"X : Type u\ninst✝ : TopologicalSpace X\ns t : Set X\nhs : IsLindelof s\nht : IsLindelof t\n⊢ IsLindelof (⋃ b, bif b then s else t)"
] | rw [union_eq_iUnion] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.DenseEmbedding | {
"line": 194,
"column": 2
} | {
"line": 194,
"column": 22
} | {
"line": 195,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ni : α → β\ninst✝¹ : TopologicalSpace γ\ninst✝ : T3Space γ\nb : β\nf : α → γ\ndi : IsDenseInducing i\nhf : ∀ᶠ (x : β) in 𝓝 b, ∃ c, Tendsto f (comap i (𝓝 x)) (𝓝 c)\n⊢ ContinuousAt (di.extend f) b",
... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ni : α → β\ninst✝¹ : TopologicalSpace γ\ninst✝ : T3Space γ\nb : β\nf : α → γ\ndi : IsDenseInducing i\nhf : ∀ᶠ (x : β) in 𝓝 b, ∃ c, Tendsto f (comap i (𝓝 x)) (𝓝 c)\nφ : β → γ := di.extend f\n⊢ ContinuousAt φ b"
] | set φ := di.extend f | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.Data.Rel.Cover | {
"line": 78,
"column": 56
} | {
"line": 78,
"column": 92
} | {
"line": 78,
"column": 92
} | [
{
"pp": "X : Type u_1\nU : SetRel X X\ns N : Set X\ninst✝¹ : U.IsRefl\ninst✝ : U.IsSymm\nhN : Maximal (fun N ↦ N ⊆ s ∧ U.IsSeparated N) N\nx : X\nhx : x ∈ s\nh : ∀ (y : X), y ∈ N → ¬(x, y) ∈ U\n⊢ insert x N ⊆ s",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"congrArg",
"and_se... | [] | simp [insert_subset_iff, hx, hN.1.1] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Rel.Cover | {
"line": 78,
"column": 56
} | {
"line": 78,
"column": 92
} | {
"line": 78,
"column": 92
} | [
{
"pp": "X : Type u_1\nU : SetRel X X\ns N : Set X\ninst✝¹ : U.IsRefl\ninst✝ : U.IsSymm\nhN : Maximal (fun N ↦ N ⊆ s ∧ U.IsSeparated N) N\nx : X\nhx : x ∈ s\nh : ∀ (y : X), y ∈ N → ¬(x, y) ∈ U\n⊢ insert x N ⊆ s",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"congrArg",
"and_se... | [] | simp [insert_subset_iff, hx, hN.1.1] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rel.Cover | {
"line": 78,
"column": 56
} | {
"line": 78,
"column": 92
} | {
"line": 78,
"column": 92
} | [
{
"pp": "X : Type u_1\nU : SetRel X X\ns N : Set X\ninst✝¹ : U.IsRefl\ninst✝ : U.IsSymm\nhN : Maximal (fun N ↦ N ⊆ s ∧ U.IsSeparated N) N\nx : X\nhx : x ∈ s\nh : ∀ (y : X), y ∈ N → ¬(x, y) ∈ U\n⊢ insert x N ⊆ s",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"congrArg",
"and_se... | [] | simp [insert_subset_iff, hx, hN.1.1] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Homeomorph.Lemmas | {
"line": 542,
"column": 6
} | {
"line": 542,
"column": 55
} | {
"line": 542,
"column": 55
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\nf : X → Y\ninst✝¹ : CompactSpace X\ninst✝ : T2Space Y\n⊢ IsHomeomorph f ↔ Continuous[inst✝³, inst✝²] f ∧ Bijective f",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Continu... | [
"X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\nf : X → Y\ninst✝¹ : CompactSpace X\ninst✝ : T2Space Y\n⊢ Continuous[inst✝³, inst✝²] f ∧ IsClosedMap f ∧ Bijective f ↔ Continuous[inst✝³, inst✝²] f ∧ Bijective f"
] | isHomeomorph_iff_continuous_isClosedMap_bijective | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Connected.Clopen | {
"line": 706,
"column": 2
} | {
"line": 707,
"column": 66
} | {
"line": 708,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝ : TopologicalSpace α\nh : ∀ (s : Set α), IsClopen s → s = ∅ ∨ s = univ\nf : α → Bool\nhf : Continuous[inst✝, _] f\nx y : α\n⊢ f x = f y",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Bool.not",
"congrArg",
"Compl.compl",
... | [
"α : Type u\ninst✝ : TopologicalSpace α\nh : ∀ (s : Set α), IsClopen s → s = ∅ ∨ s = univ\nf : α → Bool\nhf : Continuous[inst✝, _] f\nx y : α\nthis : f ⁻¹' {false} = (f ⁻¹' {true})ᶜ\n⊢ f x = f y"
] | have : f ⁻¹' {false} = (f ⁻¹' {true})ᶜ := by
rw [← Set.preimage_compl, Bool.compl_singleton, Bool.not_true] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.UniformSpace.Defs | {
"line": 242,
"column": 36
} | {
"line": 242,
"column": 68
} | {
"line": 242,
"column": 68
} | [
{
"pp": "α : Type ua\nu : UniformSpace α\na : α\n⊢ 𝓝 a = comap (Prod.mk a) UniformSpace.uniformity",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"nhds",
"id",
"Prod.mk",
"UniformSpace.toCore",
"UniformSpace.Core.uniformity"... | [
"α : Type ua\nu : UniformSpace α\na : α\n⊢ comap (Prod.mk a) u.toCore.uniformity = comap (Prod.mk a) UniformSpace.uniformity"
] | u.toCore.nhds_toTopologicalSpace | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UniformSpace.Defs | {
"line": 269,
"column": 42
} | {
"line": 269,
"column": 68
} | {
"line": 269,
"column": 68
} | [
{
"pp": "α✝ : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\nα : Type u_2\ni : TopologicalSpace α\nu : UniformSpace α\nh : i = u.toTopologicalSpace\nx : α\n⊢ 𝓝 x = comap (Prod.mk x) UniformSpace.uniformity",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"α✝ : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\nα : Type u_2\ni : TopologicalSpace α\nu : UniformSpace α\nh : i = u.toTopologicalSpace\nx : α\n⊢ comap (Prod.mk x) UniformSpace.uniformity = comap (Prod.mk x) UniformSpace.uniformity"
] | u.nhds_eq_comap_uniformity | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 120,
"column": 2
} | {
"line": 121,
"column": 46
} | {
"line": 123,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nuniformSpace : UniformSpace α\ninst✝ : UniformSpace β\nf : Filter α\ng : Filter β\nhf : Cauchy f\nhg : Cauchy g\n⊢ Cauchy (f ×ˢ g)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Filter.map_snd_prod",
"instUniformSpaceProd",
... | [] | have := hf.1; have := hg.1
simpa [cauchy_prod_iff, hf.1] using ⟨hf, hg⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 120,
"column": 2
} | {
"line": 121,
"column": 46
} | {
"line": 123,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nuniformSpace : UniformSpace α\ninst✝ : UniformSpace β\nf : Filter α\ng : Filter β\nhf : Cauchy f\nhg : Cauchy g\n⊢ Cauchy (f ×ˢ g)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Filter.map_snd_prod",
"instUniformSpaceProd",
... | [] | have := hf.1; have := hg.1
simpa [cauchy_prod_iff, hf.1] using ⟨hf, hg⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.Basic | {
"line": 79,
"column": 2
} | {
"line": 80,
"column": 69
} | {
"line": 82,
"column": 0
} | [
{
"pp": "α : Type ua\nβ : Type ub\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns : SetRel α β\nhs : IsClosed[instTopologicalSpaceProd] s\nt : Set α\nht : t.Finite\n⊢ IsClosed[inst✝] (s.image t)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetRel",
... | [] | simp_rw [SetRel.image, ← exists_prop, Set.ofPred_exists]
exact ht.isClosed_biUnion fun _ _ => hs.preimage <| .prodMk_right _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.Basic | {
"line": 79,
"column": 2
} | {
"line": 80,
"column": 69
} | {
"line": 82,
"column": 0
} | [
{
"pp": "α : Type ua\nβ : Type ub\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns : SetRel α β\nhs : IsClosed[instTopologicalSpaceProd] s\nt : Set α\nht : t.Finite\n⊢ IsClosed[inst✝] (s.image t)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetRel",
... | [] | simp_rw [SetRel.image, ← exists_prop, Set.ofPred_exists]
exact ht.isClosed_biUnion fun _ _ => hs.preimage <| .prodMk_right _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.Defs | {
"line": 498,
"column": 2
} | {
"line": 498,
"column": 56
} | {
"line": 500,
"column": 0
} | [
{
"pp": "α : Type ua\ninst✝ : UniformSpace α\nx : α\ns : Set α\n⊢ s ∈ 𝓝 x ↔ {p | p.1 = x → p.2 ∈ s} ∈ 𝓤 α",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"congrArg",
"Set.ofPred",
"_private.Mathlib.Topology.UniformSpace.Defs.0.mem_nhds_uni... | [] | simp only [nhds_eq_comap_uniformity, mem_comap_prodMk] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.UniformSpace.Defs | {
"line": 498,
"column": 2
} | {
"line": 498,
"column": 56
} | {
"line": 500,
"column": 0
} | [
{
"pp": "α : Type ua\ninst✝ : UniformSpace α\nx : α\ns : Set α\n⊢ s ∈ 𝓝 x ↔ {p | p.1 = x → p.2 ∈ s} ∈ 𝓤 α",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"congrArg",
"Set.ofPred",
"_private.Mathlib.Topology.UniformSpace.Defs.0.mem_nhds_uni... | [] | simp only [nhds_eq_comap_uniformity, mem_comap_prodMk] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.Defs | {
"line": 498,
"column": 2
} | {
"line": 498,
"column": 56
} | {
"line": 500,
"column": 0
} | [
{
"pp": "α : Type ua\ninst✝ : UniformSpace α\nx : α\ns : Set α\n⊢ s ∈ 𝓝 x ↔ {p | p.1 = x → p.2 ∈ s} ∈ 𝓤 α",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"congrArg",
"Set.ofPred",
"_private.Mathlib.Topology.UniformSpace.Defs.0.mem_nhds_uni... | [] | simp only [nhds_eq_comap_uniformity, mem_comap_prodMk] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.Basic | {
"line": 972,
"column": 4
} | {
"line": 976,
"column": 65
} | {
"line": 977,
"column": 2
} | [
{
"pp": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\ns : Set ((α ⊕ β) × (α ⊕ β))\nhs : s ∈ map (fun p ↦ (inl p.1, inl p.2)) (𝓤 α) ⊔ map (fun p ↦ (inr p.1, inr p.2)) (𝓤 β)\n⊢ s ∈ (map (fun p ↦ (inl p.1, inl p.2)) (𝓤 α) ⊔ map (fun p ↦ (inr ... | [] | rcases comp_mem_uniformity_sets hs.1 with ⟨tα, htα, Htα⟩
rcases comp_mem_uniformity_sets hs.2 with ⟨tβ, htβ, Htβ⟩
filter_upwards [mem_lift' (union_mem_sup (image_mem_map htα) (image_mem_map htβ))]
rintro ⟨_, _⟩ ⟨z, ⟨⟨a, b⟩, hab, ⟨⟩⟩ | ⟨⟨a, b⟩, hab, ⟨⟩⟩, ⟨⟨_, c⟩, hbc, ⟨⟩⟩ | ⟨⟨_, c⟩, hbc, ⟨⟩⟩⟩
exacts [@Ht... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.Basic | {
"line": 972,
"column": 4
} | {
"line": 976,
"column": 65
} | {
"line": 977,
"column": 2
} | [
{
"pp": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\ns : Set ((α ⊕ β) × (α ⊕ β))\nhs : s ∈ map (fun p ↦ (inl p.1, inl p.2)) (𝓤 α) ⊔ map (fun p ↦ (inr p.1, inr p.2)) (𝓤 β)\n⊢ s ∈ (map (fun p ↦ (inl p.1, inl p.2)) (𝓤 α) ⊔ map (fun p ↦ (inr ... | [] | rcases comp_mem_uniformity_sets hs.1 with ⟨tα, htα, Htα⟩
rcases comp_mem_uniformity_sets hs.2 with ⟨tβ, htβ, Htβ⟩
filter_upwards [mem_lift' (union_mem_sup (image_mem_map htα) (image_mem_map htβ))]
rintro ⟨_, _⟩ ⟨z, ⟨⟨a, b⟩, hab, ⟨⟩⟩ | ⟨⟨a, b⟩, hab, ⟨⟩⟩, ⟨⟨_, c⟩, hbc, ⟨⟩⟩ | ⟨⟨_, c⟩, hbc, ⟨⟩⟩⟩
exacts [@Ht... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.Compact | {
"line": 191,
"column": 53
} | {
"line": 191,
"column": 61
} | {
"line": 192,
"column": 2
} | [
{
"pp": "γ : Type uc\nt : TopologicalSpace γ\ninst✝ : CompactSpace γ\nu u' : UniformSpace γ\nh : u.toTopologicalSpace = t\nh' : u'.toTopologicalSpace = t\nthis : CompactSpace γ\n⊢ CompactSpace γ",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
... | [] | rwa [h'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Topology.UniformSpace.UniformEmbedding | {
"line": 584,
"column": 2
} | {
"line": 587,
"column": 49
} | {
"line": 588,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : UniformSpace α\ninst✝³ : UniformSpace β\nγ : Type u_3\ninst✝² : UniformSpace γ\ninst✝¹ : CompleteSpace β\ninst✝ : CompleteSpace γ\ni : α → β\nf : α → γ\nhid : IsDenseInducing i\nhi : IsUniformInducing i\nh : IsUniformInducing f\nsf : α → SeparationQuotient γ := Sepa... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁴ : UniformSpace α\ninst✝³ : UniformSpace β\nγ : Type u_3\ninst✝² : UniformSpace γ\ninst✝¹ : CompleteSpace β\ninst✝ : CompleteSpace γ\ni : α → β\nf : α → γ\nhid : IsDenseInducing i\nhi : IsUniformInducing i\nh : IsUniformInducing f\nsf : α → SeparationQuotient γ := SeparationQuotie... | have hfu : IsUniformInducing fwd := by
refine IsUniformInducing.of_comp hfwd (SeparationQuotient.uniformContinuous_mk.comp hg') ?_
rw [Function.comp_assoc, key]
exact SeparationQuotient.isUniformInducing_mk | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.UniformSpace.Pi | {
"line": 130,
"column": 43
} | {
"line": 130,
"column": 52
} | {
"line": 130,
"column": 53
} | [
{
"pp": "ι : Type u_4\nX : Type u_5\nu : ι → UniformSpace X\nhu : ∀ (i : ι), CompleteSpace X\na✝ : Nontrivial X\nt : TopologicalSpace X\nht : T2Space X\nhut : ∀ (i : ι), (u i).toTopologicalSpace ≤ t\n⊢ ⨅ i, Filter.comap (fun x ↦ (x.1, x.2)) (𝓤 X) = ⨅ i, 𝓤 X",
"ppTerm": "?m.51",
"assigned": true,
"... | [
"ι : Type u_4\nX : Type u_5\nu : ι → UniformSpace X\nhu : ∀ (i : ι), CompleteSpace X\na✝ : Nontrivial X\nt : TopologicalSpace X\nht : T2Space X\nhut : ∀ (i : ι), (u i).toTopologicalSpace ≤ t\n⊢ ⨅ i, Filter.comap (fun x ↦ x) (𝓤 X) = ⨅ i, 𝓤 X"
] | Prod.eta, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.UniformSpace.LocallyUniformConvergence | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 46
} | {
"line": 117,
"column": 0
} | [
{
"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\np : Filter ι\nS : Set (Set α)\nhS : ∀ s ∈ S, IsOpen[inst✝¹] s\nh : ∀ s ∈ S, TendstoLocallyUniformlyOn F f p s\n⊢ TendstoLocallyUniformlyOn F f p (⋃ i ∈ S, i)",
"ppTerm": "?m.25",... | [] | exact tendstoLocallyUniformlyOn_biUnion hS h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.UniformSpace.UniformApproximation | {
"line": 146,
"column": 2
} | {
"line": 151,
"column": 86
} | {
"line": 153,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns : Set α\nx : α\np : Filter ι\ng : ι → α\nh : ContinuousWithinAt f s x\nhg : Tendsto g p (𝓝[s] x)\nhunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝[s] x, ∀ᶠ (n : ι) in p, ∀ y ∈ t, (f y, F n y) ∈ u\n⊢ ... | [] | refine Uniform.tendsto_nhds_right.2 fun u₀ hu₀ => ?_
obtain ⟨u₁, h₁, u₁₀⟩ : ∃ u ∈ 𝓤 β, u ○ u ⊆ u₀ := comp_mem_uniformity_sets hu₀
rcases hunif u₁ h₁ with ⟨s, sx, hs⟩
have A : ∀ᶠ n in p, g n ∈ s := hg sx
have B : ∀ᶠ n in p, (f x, f (g n)) ∈ u₁ := hg (Uniform.continuousWithinAt_iff'_right.1 h h₁)
exact B.mp <|... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.UniformApproximation | {
"line": 146,
"column": 2
} | {
"line": 151,
"column": 86
} | {
"line": 153,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns : Set α\nx : α\np : Filter ι\ng : ι → α\nh : ContinuousWithinAt f s x\nhg : Tendsto g p (𝓝[s] x)\nhunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝[s] x, ∀ᶠ (n : ι) in p, ∀ y ∈ t, (f y, F n y) ∈ u\n⊢ ... | [] | refine Uniform.tendsto_nhds_right.2 fun u₀ hu₀ => ?_
obtain ⟨u₁, h₁, u₁₀⟩ : ∃ u ∈ 𝓤 β, u ○ u ⊆ u₀ := comp_mem_uniformity_sets hu₀
rcases hunif u₁ h₁ with ⟨s, sx, hs⟩
have A : ∀ᶠ n in p, g n ∈ s := hg sx
have B : ∀ᶠ n in p, (f x, f (g n)) ∈ u₁ := hg (Uniform.continuousWithinAt_iff'_right.1 h h₁)
exact B.mp <|... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.HeineCantor | {
"line": 128,
"column": 4
} | {
"line": 128,
"column": 27
} | {
"line": 129,
"column": 4
} | [
{
"pp": "case hunion.inl\nα : Type u_4\nβ : Type u_5\nE : Type u_6\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : UniformSpace E\nf : α → β → E\ns : Set α\nk : Set β\nq : α\nu : Set (E × E)\nhk : IsCompact k\nhf : ContinuousOn (Function.uncurry f) (s ×ˢ k)\nhq : q ∈ s\nhu : u ∈ 𝓤 E\nt t' : ... | [
"case hunion.inr\nα : Type u_4\nβ : Type u_5\nE : Type u_6\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : UniformSpace E\nf : α → β → E\ns : Set α\nk : Set β\nq : α\nu : Set (E × E)\nhk : IsCompact k\nhf : ContinuousOn (Function.uncurry f) (s ×ˢ k)\nhq : q ∈ s\nhu : u ∈ 𝓤 E\nt t' : Set β\nv : S... | · exact hv p hp.1 x h'x | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.UniformSpace.Equicontinuity | {
"line": 451,
"column": 91
} | {
"line": 452,
"column": 62
} | {
"line": 454,
"column": 0
} | [
{
"pp": "ι : Type u_1\nX : Type u_3\nα : Type u_6\ntX : TopologicalSpace X\nuα : UniformSpace α\nF : ι → X → α\nS : Set X\nx₀ : X\n⊢ EquicontinuousWithinAt F S x₀ ↔ EquicontinuousWithinAt Subtype.val S x₀",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
... | [] | by
simp only [EquicontinuousWithinAt, forall_subtype_range_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Filter.NAry | {
"line": 44,
"column": 4
} | {
"line": 44,
"column": 78
} | {
"line": 46,
"column": 0
} | [
{
"pp": "α : Type u_1\nα' : Type u_2\nβ : Type u_3\nβ' : Type u_4\nγ : Type u_5\nγ' : Type u_6\nδ : Type u_7\nδ' : Type u_8\nε : Type u_9\nε' : Type u_10\nm✝ : α → β → γ\nf✝ f₁ f₂ : Filter α\ng✝ g₁ g₂ : Filter β\nh : Filter γ\ns : Set α\nt : Set β\nu : Set γ\na : α\nb : β\nm : α → β → γ\nf : Filter α\ng : Filte... | [] | simp only [mem_map, mem_prod_iff, image2_subset_iff, prod_subset_iff]; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.NAry | {
"line": 44,
"column": 4
} | {
"line": 44,
"column": 78
} | {
"line": 46,
"column": 0
} | [
{
"pp": "α : Type u_1\nα' : Type u_2\nβ : Type u_3\nβ' : Type u_4\nγ : Type u_5\nγ' : Type u_6\nδ : Type u_7\nδ' : Type u_8\nε : Type u_9\nε' : Type u_10\nm✝ : α → β → γ\nf✝ f₁ f₂ : Filter α\ng✝ g₁ g₂ : Filter β\nh : Filter γ\ns : Set α\nt : Set β\nu : Set γ\na : α\nb : β\nm : α → β → γ\nf : Filter α\ng : Filte... | [] | simp only [mem_map, mem_prod_iff, image2_subset_iff, prod_subset_iff]; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.Equicontinuity | {
"line": 659,
"column": 2
} | {
"line": 662,
"column": 5
} | {
"line": 664,
"column": 0
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nα : Type u_6\nβ : Type u_8\nuα : UniformSpace α\nuβ : UniformSpace β\np : κ → Prop\ns : κ → Set (β × β)\nF : ι → β → α\nhβ : (𝓤 β).HasBasis p s\n⊢ UniformEquicontinuous F ↔ ∀ U ∈ 𝓤 α, ∃ k, p k ∧ ∀ (x y : β), (x, y) ∈ s k → ∀ (i : ι), (F i x, F i y) ∈ U",
"ppTerm": "?m.... | [] | rw [uniformEquicontinuous_iff_uniformContinuous, UniformContinuous,
hβ.tendsto_iff (UniformFun.hasBasis_uniformity ι α)]
simp only [Prod.forall]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.Equicontinuity | {
"line": 659,
"column": 2
} | {
"line": 662,
"column": 5
} | {
"line": 664,
"column": 0
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nα : Type u_6\nβ : Type u_8\nuα : UniformSpace α\nuβ : UniformSpace β\np : κ → Prop\ns : κ → Set (β × β)\nF : ι → β → α\nhβ : (𝓤 β).HasBasis p s\n⊢ UniformEquicontinuous F ↔ ∀ U ∈ 𝓤 α, ∃ k, p k ∧ ∀ (x y : β), (x, y) ∈ s k → ∀ (i : ι), (F i x, F i y) ∈ U",
"ppTerm": "?m.... | [] | rw [uniformEquicontinuous_iff_uniformContinuous, UniformContinuous,
hβ.tendsto_iff (UniformFun.hasBasis_uniformity ι α)]
simp only [Prod.forall]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.MulAction | {
"line": 299,
"column": 6
} | {
"line": 299,
"column": 78
} | {
"line": 300,
"column": 6
} | [
{
"pp": "case pos\nM : Type u_1\nX : Type u_2\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : TopologicalSpace X\ninst✝³ : Group M\ninst✝² : IsTopologicalGroup M\ninst✝¹ : MulAction M X\ninst✝ : DiscreteTopology X\nh : ∀ (x : X), IsOpen[inst✝⁵] ↑(MulAction.stabilizer M x)\ny x : X\nU : Set M := {m' | m' • y = x}\nhU' : ... | [
"M : Type u_1\nX : Type u_2\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : TopologicalSpace X\ninst✝³ : Group M\ninst✝² : IsTopologicalGroup M\ninst✝¹ : MulAction M X\ninst✝ : DiscreteTopology X\nh : ∀ (x : X), IsOpen[inst✝⁵] ↑(MulAction.stabilizer M x)\ny x : X\nU : Set M := {m' | m' • y = x}\nhU' : U ≠ ∅\nm : M\nhm : m •... | convert! (h x).preimage (by fun_prop : Continuous fun m' : M ↦ m' * m⁻¹) | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Topology.Algebra.MulAction | {
"line": 311,
"column": 95
} | {
"line": 319,
"column": 46
} | {
"line": 321,
"column": 0
} | [
{
"pp": "G₀ : Type u_5\nX : Type u_6\ninst✝⁶ : GroupWithZero G₀\ninst✝⁵ : Zero X\ninst✝⁴ : MulActionWithZero G₀ X\ninst✝³ : TopologicalSpace G₀\ninst✝² : (𝓝[≠] 0).NeBot\ninst✝¹ : TopologicalSpace X\ninst✝ : ContinuousSMul G₀ X\ns : Set X\nhs : s ∈ 𝓝 0\n⊢ univ • s = univ",
"ppTerm": "?m.32",
"assigned"... | [] | by
refine Set.eq_univ_of_forall fun x ↦ ?_
have : Tendsto (· • x) (𝓝 (0 : G₀)) (𝓝 0) :=
zero_smul G₀ x ▸ tendsto_id.smul tendsto_const_nhds
rcases Filter.nonempty_of_mem (inter_mem_nhdsWithin {0}ᶜ <| mem_map.1 <| this hs)
with ⟨c, hc₀, hc⟩
simp only [mem_compl_iff, mem_singleton_iff] at hc₀
simp onl... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.ConstMulAction | {
"line": 400,
"column": 6
} | {
"line": 400,
"column": 29
} | {
"line": 401,
"column": 2
} | [
{
"pp": "case inl.inr\nG₀ : Type u_4\ninst✝⁵ : GroupWithZero G₀\nE : Type u_5\ninst✝⁴ : Zero E\ninst✝³ : MulActionWithZero G₀ E\ninst✝² : TopologicalSpace E\ninst✝¹ : T1Space E\ninst✝ : ContinuousConstSMul G₀ E\ns : Set E\nhs : s.Nonempty\n⊢ closure[inst✝²] 0 = 0",
"ppTerm": "?inl.inr",
"assigned": true... | [] | exact closure_singleton | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Maps.Proper.Basic | {
"line": 183,
"column": 4
} | {
"line": 183,
"column": 18
} | {
"line": 185,
"column": 0
} | [
{
"pp": "case right\nX : Type u_1\nY : Type u_2\nZ : Type u_3\nW : Type u_4\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : TopologicalSpace W\nf : X → Y\ng : Z → W\nhf : Continuous[inst✝³, inst✝²] f ∧ ∀ ⦃𝒰 : Ultrafilter X⦄ ⦃y : Y⦄, Tendsto f (↑𝒰) (𝓝 y) → ∃ x, ... | [] | exact ⟨hx, hz⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Maps.Proper.Basic | {
"line": 170,
"column": 2
} | {
"line": 183,
"column": 18
} | {
"line": 185,
"column": 0
} | [
{
"pp": "case right\nX : Type u_1\nY : Type u_2\nZ : Type u_3\nW : Type u_4\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : TopologicalSpace W\nf : X → Y\ng : Z → W\nhf : Continuous[inst✝³, inst✝²] f ∧ ∀ ⦃𝒰 : Ultrafilter X⦄ ⦃y : Y⦄, Tendsto f (↑𝒰) (𝓝 y) → ∃ x, ... | [] | · intro 𝒰 ⟨y, w⟩ hyw
-- That means that `f` tends to `y` along `map fst 𝒰` and `g` tends to `w` along `map snd 𝒰`.
simp_rw [nhds_prod_eq, tendsto_prod_iff'] at hyw
-- Thus, by properness of `f` and `g`, we get some `x : X` and `z : Z` such that `f x = y`,
-- `g z = w`, `map fst 𝒰` tends to `x`, and `map ... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 99,
"column": 2
} | {
"line": 100,
"column": 6
} | {
"line": 102,
"column": 0
} | [
{
"pp": "G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : SeparatelyContinuousMul G\nc a : G\n⊢ map (fun x ↦ c * x) (𝓝[≠] a) = 𝓝[≠] (c * a)",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"H... | [] | convert! (Homeomorph.mulLeft c).isEmbedding.map_nhdsWithin_eq .. using 2
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 99,
"column": 2
} | {
"line": 100,
"column": 6
} | {
"line": 102,
"column": 0
} | [
{
"pp": "G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : SeparatelyContinuousMul G\nc a : G\n⊢ map (fun x ↦ c * x) (𝓝[≠] a) = 𝓝[≠] (c * a)",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"H... | [] | convert! (Homeomorph.mulLeft c).isEmbedding.map_nhdsWithin_eq .. using 2
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Group.Pointwise | {
"line": 369,
"column": 2
} | {
"line": 369,
"column": 34
} | {
"line": 370,
"column": 2
} | [
{
"pp": "G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : IsTopologicalGroup G\nS : Subgroup G\nhS : IsDiscrete ↑S\nV : Set G\nhV : V ∈ 𝓝 1 ∧ V ∩ ↑S = {1}\nU : Set G\nhU : U ∈ 𝓝 1\nhUinv : U⁻¹ = U\nhUV : U * U ⊆ V\ng : G\nhgS : g ∈ S\nx : G\nhx : x ∈ U\nhgx : (fun x ↦ g * x) x ∈ U\n⊢ g = 1",... | [
"G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : IsTopologicalGroup G\nS : Subgroup G\nhS : IsDiscrete ↑S\nV : Set G\nhV : V ∈ 𝓝 1 ∧ V ∩ ↑S = {1}\nU : Set G\nhU : U ∈ 𝓝 1\nhUinv : U⁻¹ = U\nhUV : U * U ⊆ V\ng : G\nhgS : g ∈ S\nx : G\nhx : x ∈ U\nhgx : (fun x ↦ g * x) x ∈ U\n⊢ g ∈ U * U"
] | refine hV.2.subset ⟨hUV ?_, hgS⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Algebra.IsUniformGroup.Defs | {
"line": 378,
"column": 4
} | {
"line": 379,
"column": 84
} | {
"line": 380,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : IsUniformGroup α\n𝓕 : Filter (α × α)\n⊢ 𝓕 ≤ 𝓤 α ↔ 𝓕 ≤ comap (fun x ↦ x.2 * x.1⁻¹) (𝓝 1)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InvOneClass.toOne",
"HMul.hMul",... | [
"α : Type u_1\nβ : Type u_2\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : IsUniformGroup α\n𝓕 : Filter (α × α)\n⊢ Tendsto id 𝓕 (𝓤 α) ↔ Tendsto ((Prod.mk 1 ∘ fun x ↦ x.2 * x.1⁻¹) * fun x ↦ (x.1, x.1)) 𝓕 (𝓤 α)"
] | rw [nhds_eq_comap_uniformity, comap_comap, ← tendsto_iff_comap,
← (tendsto_diag_uniformity Prod.fst 𝓕).uniformity_mul_iff_left, ← tendsto_id'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Filter.AtTopBot.Field | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 89
} | {
"line": 123,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\nl : Filter β\nf : β → α\nr : α\nhr : 0 < r\n⊢ Tendsto (fun x ↦ r * f x) l atBot ↔ Tendsto f l atBot",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroCl... | [] | simpa only [← mul_neg, ← tendsto_neg_atTop_iff] using tendsto_const_mul_atTop_of_pos hr | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Order.Filter.AtTopBot.Field | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 89
} | {
"line": 123,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\nl : Filter β\nf : β → α\nr : α\nhr : 0 < r\n⊢ Tendsto (fun x ↦ r * f x) l atBot ↔ Tendsto f l atBot",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroCl... | [] | simpa only [← mul_neg, ← tendsto_neg_atTop_iff] using tendsto_const_mul_atTop_of_pos hr | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.AtTopBot.Field | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 89
} | {
"line": 123,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\nl : Filter β\nf : β → α\nr : α\nhr : 0 < r\n⊢ Tendsto (fun x ↦ r * f x) l atBot ↔ Tendsto f l atBot",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroCl... | [] | simpa only [← mul_neg, ← tendsto_neg_atTop_iff] using tendsto_const_mul_atTop_of_pos hr | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Field | {
"line": 196,
"column": 2
} | {
"line": 199,
"column": 73
} | {
"line": 201,
"column": 0
} | [
{
"pp": "α : Type u_2\n𝕜 : Type u_3\nf g : α → 𝕜\nS : Set α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : TopologicalSpace 𝕜\ninst✝³ : T1Space 𝕜\ninst✝² : Field 𝕜\ninst✝¹ : ContinuousInv₀ 𝕜\ninst✝ : ContinuousMul 𝕜\nhS : IsPreconnected S\nhf : ContinuousOn f S\nhg : ContinuousOn g S\nhsq : EqOn (f ^ 2) (g ^ 2) ... | [] | have hsq : EqOn ((f / g) ^ 2) 1 S := fun x hx => by
simpa [div_eq_one_iff_eq (pow_ne_zero _ (hg_ne hx)), div_pow] using hsq hx
simpa +contextual [EqOn, div_eq_iff (hg_ne _)]
using hS.eq_one_or_eq_neg_one_of_sq_eq (hf.div hg fun z => hg_ne) hsq | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Field | {
"line": 196,
"column": 2
} | {
"line": 199,
"column": 73
} | {
"line": 201,
"column": 0
} | [
{
"pp": "α : Type u_2\n𝕜 : Type u_3\nf g : α → 𝕜\nS : Set α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : TopologicalSpace 𝕜\ninst✝³ : T1Space 𝕜\ninst✝² : Field 𝕜\ninst✝¹ : ContinuousInv₀ 𝕜\ninst✝ : ContinuousMul 𝕜\nhS : IsPreconnected S\nhf : ContinuousOn f S\nhg : ContinuousOn g S\nhsq : EqOn (f ^ 2) (g ^ 2) ... | [] | have hsq : EqOn ((f / g) ^ 2) 1 S := fun x hx => by
simpa [div_eq_one_iff_eq (pow_ne_zero _ (hg_ne hx)), div_pow] using hsq hx
simpa +contextual [EqOn, div_eq_iff (hg_ne _)]
using hS.eq_one_or_eq_neg_one_of_sq_eq (hf.div hg fun z => hg_ne) hsq | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.IsUniformGroup.Basic | {
"line": 550,
"column": 2
} | {
"line": 550,
"column": 48
} | {
"line": 551,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nG : Type u_5\ninst✝¹² : TopologicalSpace α\ninst✝¹¹ : AddCommGroup α\ninst✝¹⁰ : IsTopologicalAddGroup α\ninst✝⁹ : TopologicalSpace β\ninst✝⁸ : AddCommGroup β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : AddCommGroup γ\ninst✝⁵ : IsTopologicalAddGroup γ\n... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nG : Type u_5\ninst✝¹² : TopologicalSpace α\ninst✝¹¹ : AddCommGroup α\ninst✝¹⁰ : IsTopologicalAddGroup α\ninst✝⁹ : TopologicalSpace β\ninst✝⁸ : AddCommGroup β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : AddCommGroup γ\ninst✝⁵ : IsTopologicalAddGroup γ\ninst✝⁴ : Top... | rcases this with ⟨U₁, U₁_nhds, V₁, V₁_nhds, H⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.Algebra.IsUniformGroup.Basic | {
"line": 580,
"column": 4
} | {
"line": 580,
"column": 19
} | {
"line": 581,
"column": 4
} | [
{
"pp": "case left\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nG : Type u_5\ninst✝¹⁴ : TopologicalSpace α\ninst✝¹³ : AddCommGroup α\ninst✝¹² : IsTopologicalAddGroup α\ninst✝¹¹ : TopologicalSpace β\ninst✝¹⁰ : AddCommGroup β\ninst✝⁹ : TopologicalSpace γ\ninst✝⁸ : AddCommGroup γ\ninst✝⁷ : IsTopologica... | [
"case left\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nG : Type u_5\ninst✝¹⁴ : TopologicalSpace α\ninst✝¹³ : AddCommGroup α\ninst✝¹² : IsTopologicalAddGroup α\ninst✝¹¹ : TopologicalSpace β\ninst✝¹⁰ : AddCommGroup β\ninst✝⁹ : TopologicalSpace γ\ninst✝⁸ : AddCommGroup γ\ninst✝⁷ : IsTopologicalAddGroup γ\... | apply NeBot.map | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Algebra.Order.Group | {
"line": 47,
"column": 6
} | {
"line": 50,
"column": 15
} | {
"line": 51,
"column": 2
} | [
{
"pp": "case inr\nG : Type u_1\ninst✝⁴ : TopologicalSpace G\ninst✝³ : CommGroup G\ninst✝² : LinearOrder G\ninst✝¹ : IsOrderedMonoid G\ninst✝ : OrderTopology G\na b ε : G\nhε : ε > 1\nhε_min : ∀ a < ε, a ≤ 1\n⊢ ∀ᶠ (b_1 : G × G) in 𝓝 (a, b), |b_1.1 * b_1.2 / (a * b)|ₘ < ε",
"ppTerm": "?inr",
"assigned":... | [] | have (x : G) : ∀ᶠ y in 𝓝 x, y = x :=
(eventually_mabs_div_lt _ hε).mono fun y hy ↦ mabs_div_le_one.mp <| hε_min _ hy
filter_upwards [(this _).prod_nhds (this _)]
simp [hε] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Order.Group | {
"line": 47,
"column": 6
} | {
"line": 50,
"column": 15
} | {
"line": 51,
"column": 2
} | [
{
"pp": "case inr\nG : Type u_1\ninst✝⁴ : TopologicalSpace G\ninst✝³ : CommGroup G\ninst✝² : LinearOrder G\ninst✝¹ : IsOrderedMonoid G\ninst✝ : OrderTopology G\na b ε : G\nhε : ε > 1\nhε_min : ∀ a < ε, a ≤ 1\n⊢ ∀ᶠ (b_1 : G × G) in 𝓝 (a, b), |b_1.1 * b_1.2 / (a * b)|ₘ < ε",
"ppTerm": "?inr",
"assigned":... | [] | have (x : G) : ∀ᶠ y in 𝓝 x, y = x :=
(eventually_mabs_div_lt _ hε).mono fun y hy ↦ mabs_div_le_one.mp <| hε_min _ hy
filter_upwards [(this _).prod_nhds (this _)]
simp [hε] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Order.Field | {
"line": 39,
"column": 2
} | {
"line": 39,
"column": 68
} | {
"line": 40,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nα : Type u_2\ninst✝⁴ : Semifield 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nl : Filter α\nf g : α → 𝕜\nC : 𝕜\nhC : 0 < C\nhf : Tendsto f l atTop\nhg : Tendsto g l (𝓝 C)\n⊢ Tendsto (fun x ↦ f x * g x) l atTop",
... | [
"𝕜 : Type u_1\nα : Type u_2\ninst✝⁴ : Semifield 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nl : Filter α\nf g : α → 𝕜\nC : 𝕜\nhC : 0 < C\nhf : Tendsto f l atTop\nhg : Tendsto g l (𝓝 C)\n⊢ (fun x ↦ f x * (C / 2)) ≤ᶠ[l] fun x ↦ f x * g x"
] | refine tendsto_atTop_mono' _ ?_ (hf.atTop_mul_const (half_pos hC)) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Algebra.Order.Field | {
"line": 243,
"column": 23
} | {
"line": 243,
"column": 31
} | {
"line": 243,
"column": 31
} | [
{
"pp": "𝕜 : Type u_1\nα : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nl : Filter α\nf g : α → 𝕜\nC : 𝕜\nhf : ∀ᶠ (x : α) in l, |f x| ≤ C\nhg : Tendsto g l (𝓝 0)\n⊢ 0 = |0|",
"ppTerm": "?m.186",
"assign... | [] | abs_zero | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Algebra.Order.Field | {
"line": 295,
"column": 6
} | {
"line": 295,
"column": 55
} | {
"line": 296,
"column": 4
} | [
{
"pp": "case refine_1.ofNat\n𝕜 : Type u_1\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nc d : 𝕜\nhc : c ≠ 0\nn : ℕ\nh : Tendsto (fun x ↦ c * x ^ Int.ofNat n) atTop (𝓝 d)\n⊢ Int.ofNat n = 0 ∧ c = d",
"ppTerm": "?refin... | [] | simpa [tendsto_const_mul_pow_nhds_iff hc] using h | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.Algebra.Order.Field | {
"line": 290,
"column": 84
} | {
"line": 302,
"column": 60
} | {
"line": 304,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nn : ℤ\nc d : 𝕜\nhc : c ≠ 0\n⊢ Tendsto (fun x ↦ c * x ^ n) atTop (𝓝 d) ↔ n = 0 ∧ c = d ∨ n < 0 ∧ d = 0",
"ppTerm": "?m.33",
"assigned": true,
... | [] | by
refine ⟨fun h => ?_, fun h => ?_⟩
· cases n with
| ofNat n =>
left
simpa [tendsto_const_mul_pow_nhds_iff hc] using h
| negSucc n =>
have hn := Int.negSucc_lt_zero n
exact Or.inr ⟨hn, tendsto_nhds_unique h (tendsto_const_mul_zpow_atTop_zero hn)⟩
· rcases h with h | h
· simp o... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Instances.AddCircle.Defs | {
"line": 138,
"column": 27
} | {
"line": 138,
"column": 47
} | {
"line": 138,
"column": 48
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : AddCommGroup 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsOrderedAddMonoid 𝕜\ninst✝² : Archimedean 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np : 𝕜\nhp : 0 < p\na x : 𝕜\nhx : ¬x ≡ a [PMOD p]\n⊢ toIcoDiv hp a =ᶠ[𝓝[<] x ⊔ 𝓝[≥] x] fun x_1 ↦ toIcoDiv hp a x",
"p... | [
"𝕜 : Type u_1\ninst✝⁵ : AddCommGroup 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsOrderedAddMonoid 𝕜\ninst✝² : Archimedean 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np : 𝕜\nhp : 0 < p\na x : 𝕜\nhx : ¬x ≡ a [PMOD p]\n⊢ ∀ᶠ (x_1 : 𝕜) in 𝓝[<] x ⊔ 𝓝[≥] x, toIcoDiv hp a x_1 = toIcoDiv hp a x"
] | Filter.EventuallyEq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Instances.AddCircle.Defs | {
"line": 157,
"column": 27
} | {
"line": 157,
"column": 47
} | {
"line": 157,
"column": 48
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : AddCommGroup 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsOrderedAddMonoid 𝕜\ninst✝² : Archimedean 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np : 𝕜\nhp : 0 < p\na x : 𝕜\nhx : ¬x ≡ a [PMOD p]\n⊢ toIocDiv hp a =ᶠ[𝓝[≤] x ⊔ 𝓝[>] x] fun x_1 ↦ toIocDiv hp a x",
"p... | [
"𝕜 : Type u_1\ninst✝⁵ : AddCommGroup 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsOrderedAddMonoid 𝕜\ninst✝² : Archimedean 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np : 𝕜\nhp : 0 < p\na x : 𝕜\nhx : ¬x ≡ a [PMOD p]\n⊢ ∀ᶠ (x_1 : 𝕜) in 𝓝[≤] x ⊔ 𝓝[>] x, toIocDiv hp a x_1 = toIocDiv hp a x"
] | Filter.EventuallyEq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.Basic | {
"line": 462,
"column": 2
} | {
"line": 471,
"column": 50
} | {
"line": 473,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : Nontrivial α\ns : Set α\nhs : IsOpen[inst✝³] s\nh : s.Nonempty\n⊢ ∃ a b, a < b ∧ Ioo a b ⊆ s",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Set.Ioc",
"Preorder.toLT",
... | [] | obtain ⟨x, hx⟩ : ∃ x, x ∈ s := h
obtain ⟨y, hy⟩ : ∃ y, y ≠ x := exists_ne x
rcases lt_trichotomy x y with (H | rfl | H)
· obtain ⟨u, xu, hu⟩ : ∃ u, x < u ∧ Ico x u ⊆ s :=
exists_Ico_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩
exact ⟨x, u, xu, Ioo_subset_Ico_self.trans hu⟩
· exact (hy rfl).elim
· obta... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.Basic | {
"line": 462,
"column": 2
} | {
"line": 471,
"column": 50
} | {
"line": 473,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : Nontrivial α\ns : Set α\nhs : IsOpen[inst✝³] s\nh : s.Nonempty\n⊢ ∃ a b, a < b ∧ Ioo a b ⊆ s",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Set.Ioc",
"Preorder.toLT",
... | [] | obtain ⟨x, hx⟩ : ∃ x, x ∈ s := h
obtain ⟨y, hy⟩ : ∃ y, y ≠ x := exists_ne x
rcases lt_trichotomy x y with (H | rfl | H)
· obtain ⟨u, xu, hu⟩ : ∃ u, x < u ∧ Ico x u ⊆ s :=
exists_Ico_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩
exact ⟨x, u, xu, Ioo_subset_Ico_self.trans hu⟩
· exact (hy rfl).elim
· obta... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Internal.Types.Grp | {
"line": 51,
"column": 12
} | {
"line": 52,
"column": 43
} | {
"line": 52,
"column": 44
} | [
{
"pp": "A : GrpCat\n⊢ CartesianMonoidalCategory.lift (𝟙 (MonTypeEquivalenceMon.inverse.obj ((forget₂ GrpCat MonCat).obj A)).X)\n (↾fun x ↦ x⁻¹) ≫\n μ =\n SemiCartesianMonoidalCategory.toUnit (MonTypeEquivalenceMon.inverse.obj ((forget₂ GrpCat MonCat).obj A)).X ≫ η",
"ppTerm": "?m.106",
... | [] | ext x
exact mul_inv_cancel (G := A) x | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Internal.Types.Grp | {
"line": 51,
"column": 12
} | {
"line": 52,
"column": 43
} | {
"line": 52,
"column": 44
} | [
{
"pp": "A : GrpCat\n⊢ CartesianMonoidalCategory.lift (𝟙 (MonTypeEquivalenceMon.inverse.obj ((forget₂ GrpCat MonCat).obj A)).X)\n (↾fun x ↦ x⁻¹) ≫\n μ =\n SemiCartesianMonoidalCategory.toUnit (MonTypeEquivalenceMon.inverse.obj ((forget₂ GrpCat MonCat).obj A)).X ≫ η",
"ppTerm": "?m.106",
... | [] | ext x
exact mul_inv_cancel (G := A) x | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.AddCircle.Defs | {
"line": 757,
"column": 2
} | {
"line": 757,
"column": 39
} | {
"line": 758,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : AddCommGroup 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsOrderedAddMonoid 𝕜\np a : 𝕜\nhp : Fact (0 < p)\ninst✝ : Archimedean 𝕜\n⊢ ⇑(equivIccQuot p a) ∘ Quotient.mk'' = fun x ↦ Quot.mk (EndpointIdent p a) ⟨toIocMod ⋯ a x, ⋯⟩",
"ppTerm": "?m.51",
"assigned": true,
"used... | [
"𝕜 : Type u_1\ninst✝³ : AddCommGroup 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsOrderedAddMonoid 𝕜\np a : 𝕜\nhp : Fact (0 < p)\ninst✝ : Archimedean 𝕜\n⊢ (fun x ↦ Quot.mk (EndpointIdent p a) ⟨toIcoMod ⋯ a x, ⋯⟩) = fun x ↦ Quot.mk (EndpointIdent p a) ⟨toIocMod ⋯ a x, ⋯⟩"
] | rw [equivIccQuot_comp_mk_eq_toIcoMod] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Category.ModuleCat.Subobject | {
"line": 37,
"column": 37
} | {
"line": 48,
"column": 13
} | {
"line": 48,
"column": 13
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nM : ModuleCat R\nS : Subobject M\n⊢ S = (fun N ↦ mk (↟N.subtype)) ((fun S ↦ (Hom.hom S.arrow).range) S)",
"ppTerm": "?m.356",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"CategoryTheory.Over",
"CategoryTheory.Subobject.arro... | [] | by
fapply eq_mk_of_comm
· apply LinearEquiv.toModuleIso
apply LinearEquiv.ofBijective (LinearMap.codRestrict
(LinearMap.range S.arrow.hom) S.arrow.hom _)
constructor
· simp [← LinearMap.ker_eq_bot, ker_eq_bot_of_mono]
· rw [← LinearMap.range_eq_top, Li... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Abelian.Refinements | {
"line": 157,
"column": 6
} | {
"line": 157,
"column": 20
} | {
"line": 157,
"column": 21
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nS : ShortComplex C\nA : C\nx₂ x₂' : A ⟶ S.X₂\nhx₂ : x₂ ≫ S.g = 0\nhx₂' : x₂' ≫ S.g = 0\n⊢ S.liftCycles x₂ hx₂ ≫ S.homologyπ = S.liftCycles x₂' hx₂' ≫ S.homologyπ ↔ S.liftCycles (x₂ - x₂') ⋯ ≫ S.homologyπ = 0",
"ppTerm": "?m.159",
... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nS : ShortComplex C\nA : C\nx₂ x₂' : A ⟶ S.X₂\nhx₂ : x₂ ≫ S.g = 0\nhx₂' : x₂' ≫ S.g = 0\n⊢ S.liftCycles x₂ hx₂ ≫ S.homologyπ - S.liftCycles x₂' hx₂' ≫ S.homologyπ = 0 ↔\n S.liftCycles (x₂ - x₂') ⋯ ≫ S.homologyπ = 0"
] | ← sub_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.Refinements | {
"line": 238,
"column": 4
} | {
"line": 238,
"column": 73
} | {
"line": 240,
"column": 0
} | [
{
"pp": "case mpr\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nh :\n ∀ ⦃A : C⦄ (y₂ : A ⟶ S₂.X₂),\n y₂ ≫ S₂.g = 0 → ∃ A' π, ∃ (_ : Epi π), ∃ x₂, ∃ (_ : x₂ ≫ S₁.g = 0), ∃ y₁, π ≫ y₂ = x₂ ≫ φ.τ₂ + y₁ ≫ S₂.f\nA : C\nγ : A ⟶ S₂.homology\nA₁ : C\nπ₁ : A₁ ⟶... | [] | exact ⟨A₂, 𝟙 _, inferInstance, y₁, by simpa only [id_comp] using hy₁⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Category.ModuleCat.Adjunctions | {
"line": 390,
"column": 10
} | {
"line": 390,
"column": 19
} | {
"line": 390,
"column": 19
} | [
{
"pp": "case single\nR : Type u_1\ninst✝⁸ : CommRing R\nC : Type u\ninst✝⁷ : Category.{v, u} C\nD : Type u\ninst✝⁶ : Category.{v, u} D\ninst✝⁵ : Preadditive D\ninst✝⁴ : Linear R D\nF G : Free R C ⥤ D\ninst✝³ : F.Additive\ninst✝² : Functor.Linear R F\ninst✝¹ : G.Additive\ninst✝ : Functor.Linear R G\nα : embeddi... | [
"case single\nR : Type u_1\ninst✝⁸ : CommRing R\nC : Type u\ninst✝⁷ : Category.{v, u} C\nD : Type u\ninst✝⁶ : Category.{v, u} D\ninst✝⁵ : Preadditive D\ninst✝⁴ : Linear R D\nF G : Free R C ⥤ D\ninst✝³ : F.Additive\ninst✝² : Functor.Linear R F\ninst✝¹ : G.Additive\ninst✝ : Functor.Linear R G\nα : embedding R C ⋙ F ≅... | comp_smul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Types.Pullbacks | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 87
} | {
"line": 158,
"column": 0
} | [
{
"pp": "X Y Z : Type u\nf : X ⟶ Z\ng : Y ⟶ Z\nx✝ : PullbackObj f g\n⊢ (hom ((pullbackIsoPullback f g).inv ≫ pullback.snd f g)).toFun x✝ = (hom (↾fun p ↦ (↑p).2)).toFun x✝",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.pullback",
"CategoryTheory.Limits.p... | [] | exact PullbackCone.IsLimit.equivPullbackObj_symm_apply_snd (pullbackIsPullback f g) _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory | {
"line": 298,
"column": 18
} | {
"line": 304,
"column": 74
} | {
"line": 304,
"column": 75
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type s\ninst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝ : ConcreteCategory C FC\nJ : MulticospanShape\nI : MulticospanIndex J C\nx : { x // ∀ (i : J.R), (hom (I.fst i)) (x (J.fst i)) = (hom (I.snd i)) (x (J.snd i)) }\n⊢ ... | [] | by
rintro (a | b) (a' | b') (f | f | f)
· simp only [WalkingMulticospan.Hom.id_eq_id, Functor.map_id]; rfl
· rfl
· dsimp
exact (x.2 b').symm
· simp only [WalkingMulticospan.Hom.id_eq_id, Functor.map_id]; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ShortComplex.ModuleCat | {
"line": 92,
"column": 34
} | {
"line": 92,
"column": 45
} | {
"line": 92,
"column": 45
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nS : ShortComplex (ModuleCat R)\nX₁ X₂ X₃ : ModuleCat R\nf : X₁ ⟶ X₂\ng : X₂ ⟶ X₃\nhfg : (ModuleCat.Hom.hom f).range = (ModuleCat.Hom.hom g).ker\n⊢ (ModuleCat.Hom.hom f).range ≤ (ModuleCat.Hom.hom g).ker",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
... | [] | by rw [hfg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monad.Limits | {
"line": 90,
"column": 8
} | {
"line": 90,
"column": 32
} | {
"line": 90,
"column": 33
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nT : Monad C\nJ : Type u\ninst✝ : Category.{v, u} J\nD : J ⥤ T.Algebra\nc : Cone (D ⋙ T.forget)\nt : IsLimit c\nX Y : J\nf : X ⟶ Y\n⊢ (((const J).obj (conePoint D c t)).map f ≫ { f := c.π.app Y, h := ⋯ }).f = ({ f := c.π.app X, h := ⋯ } ≫ D.map f).f",
"ppTe... | [] | simpa using (c.w f).symm | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Monad.Limits | {
"line": 192,
"column": 8
} | {
"line": 192,
"column": 32
} | {
"line": 192,
"column": 33
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nT : Monad C\nJ : Type u\ninst✝² : Category.{v, u} J\nD : J ⥤ T.Algebra\nc : Cocone (D ⋙ T.forget)\nt : IsColimit c\ninst✝¹ : PreservesColimit (D ⋙ T.forget) T.toFunctor\ninst✝ : PreservesColimit ((D ⋙ T.forget) ⋙ T.toFunctor) T.toFunctor\nj : J\n⊢ (T.mapCocone... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nT : Monad C\nJ : Type u\ninst✝² : Category.{v, u} J\nD : J ⥤ T.Algebra\nc : Cocone (D ⋙ T.forget)\nt : IsColimit c\ninst✝¹ : PreservesColimit (D ⋙ T.forget) T.toFunctor\ninst✝ : PreservesColimit ((D ⋙ T.forget) ⋙ T.toFunctor) T.toFunctor\nj : J\n⊢ T.map ((T.mapCocone c).ι... | Functor.mapCocone_ι_app, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monad.Limits | {
"line": 192,
"column": 33
} | {
"line": 192,
"column": 57
} | {
"line": 193,
"column": 6
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nT : Monad C\nJ : Type u\ninst✝² : Category.{v, u} J\nD : J ⥤ T.Algebra\nc : Cocone (D ⋙ T.forget)\nt : IsColimit c\ninst✝¹ : PreservesColimit (D ⋙ T.forget) T.toFunctor\ninst✝ : PreservesColimit ((D ⋙ T.forget) ⋙ T.toFunctor) T.toFunctor\nj : J\n⊢ T.map ((T.ma... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nT : Monad C\nJ : Type u\ninst✝² : Category.{v, u} J\nD : J ⥤ T.Algebra\nc : Cocone (D ⋙ T.forget)\nt : IsColimit c\ninst✝¹ : PreservesColimit (D ⋙ T.forget) T.toFunctor\ninst✝ : PreservesColimit ((D ⋙ T.forget) ⋙ T.toFunctor) T.toFunctor\nj : J\n⊢ T.map (T.map (c.ι.app j)... | Functor.mapCocone_ι_app, | Lean.Elab.Tactic.evalRewriteSeq | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.