module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Data.Nat.Totient | {
"line": 303,
"column": 2
} | {
"line": 303,
"column": 82
} | {
"line": 305,
"column": 0
} | [
{
"pp": "case neg\nn : ℕ\nhn : ¬n = 0\np : ℕ\nhp : 0 < n.factorization p\n⊢ p ^ (n.factorization p - 1) * (p - 1) * p = p ^ n.factorization p * (p - 1)",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Nat.pow_succ'",
"instPowNat",
"Finsupp.instFunLike",
"Eq.mpr",
... | [] | rw [mul_comm, ← mul_assoc, ← pow_succ', Nat.sub_one, Nat.succ_pred_eq_of_pos hp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Filter.Lift | {
"line": 49,
"column": 4
} | {
"line": 50,
"column": 49
} | {
"line": 52,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\nγ : Type u_3\nι : Sort u_6\np : ι → Prop\ns✝ : ι → Set α\nf : Filter α\nhf : f.HasBasis p s✝\nβ : ι → Type u_5\npg : (i : ι) → β i → Prop\nsg : (i : ι) → β i → Set γ\ng : Set α → Filter γ\nhg : ∀ (i : ι), (g (s✝ i)).HasBasis (pg i) (sg i)\ngm : Monotone g\ns : Set γ\n⊢ (∃ i... | [] | simp only [← (hg _).mem_iff]
exact hf.exists_iff fun t₁ t₂ ht H => gm ht H | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.Lift | {
"line": 49,
"column": 4
} | {
"line": 50,
"column": 49
} | {
"line": 52,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\nγ : Type u_3\nι : Sort u_6\np : ι → Prop\ns✝ : ι → Set α\nf : Filter α\nhf : f.HasBasis p s✝\nβ : ι → Type u_5\npg : (i : ι) → β i → Prop\nsg : (i : ι) → β i → Set γ\ng : Set α → Filter γ\nhg : ∀ (i : ι), (g (s✝ i)).HasBasis (pg i) (sg i)\ngm : Monotone g\ns : Set γ\n⊢ (∃ i... | [] | simp only [← (hg _).mem_iff]
exact hf.exists_iff fun t₁ t₂ ht H => gm ht H | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Totient | {
"line": 435,
"column": 2
} | {
"line": 435,
"column": 84
} | {
"line": 436,
"column": 2
} | [
{
"pp": "n : ℕ\nhn : 0 < n\nthis : ∏ p ∈ n.primeFactors, p ∣ n\n⊢ ∏ p ∈ n.primeFactors, p ^ (φ n / (p - 1)) ∣ n ^ φ n",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"instPowNat",
"Dvd.dvd",
"congrArg",
"Eq.mp",
"instOfNatNat",
"Nat.toti... | [
"n : ℕ\nhn : 0 < n\nthis : ∏ x ∈ n.primeFactors, x ^ φ n ∣ n ^ φ n\n⊢ ∏ p ∈ n.primeFactors, p ^ (φ n / (p - 1)) ∣ n ^ φ n"
] | rw [← Nat.pow_dvd_pow_iff (Nat.totient_pos.mpr hn).ne', ← Finset.prod_pow] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Neighborhoods | {
"line": 242,
"column": 2
} | {
"line": 242,
"column": 49
} | {
"line": 244,
"column": 0
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns V : Set X\n⊢ s ⊆ interior V ↔ ∀ x ∈ s, V ∈ 𝓝 x",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"congrArg",
"Membership.mem",
"nhds",
"id",
"LE.le",
"iff_self",
"Iff"... | [] | simp_rw [subset_def, mem_interior_iff_mem_nhds] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Topology.Neighborhoods | {
"line": 242,
"column": 2
} | {
"line": 242,
"column": 49
} | {
"line": 244,
"column": 0
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns V : Set X\n⊢ s ⊆ interior V ↔ ∀ x ∈ s, V ∈ 𝓝 x",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"congrArg",
"Membership.mem",
"nhds",
"id",
"LE.le",
"iff_self",
"Iff"... | [] | simp_rw [subset_def, mem_interior_iff_mem_nhds] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Neighborhoods | {
"line": 242,
"column": 2
} | {
"line": 242,
"column": 49
} | {
"line": 244,
"column": 0
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns V : Set X\n⊢ s ⊆ interior V ↔ ∀ x ∈ s, V ∈ 𝓝 x",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"congrArg",
"Membership.mem",
"nhds",
"id",
"LE.le",
"iff_self",
"Iff"... | [] | simp_rw [subset_def, mem_interior_iff_mem_nhds] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Continuous | {
"line": 264,
"column": 2
} | {
"line": 264,
"column": 20
} | {
"line": 266,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nα : Type u_4\nf : α → X\nhf : Surjective f\nx : X\n⊢ x ∈ closure[inst✝] (range f)",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
"IsClosed.closure_eq",
"Membersh... | [] | simp [hf.range_eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Continuous | {
"line": 264,
"column": 2
} | {
"line": 264,
"column": 20
} | {
"line": 266,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nα : Type u_4\nf : α → X\nhf : Surjective f\nx : X\n⊢ x ∈ closure[inst✝] (range f)",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
"IsClosed.closure_eq",
"Membersh... | [] | simp [hf.range_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Continuous | {
"line": 264,
"column": 2
} | {
"line": 264,
"column": 20
} | {
"line": 266,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nα : Type u_4\nf : α → X\nhf : Surjective f\nx : X\n⊢ x ∈ closure[inst✝] (range f)",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
"IsClosed.closure_eq",
"Membersh... | [] | simp [hf.range_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.ToIntervalMod | {
"line": 235,
"column": 2
} | {
"line": 235,
"column": 47
} | {
"line": 237,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : AddCommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedAddMonoid α\nhα : Archimedean α\np : α\nhp : 0 < p\na b : α\nm : ℤ\n⊢ b - toIocDiv hp a b • p + m • p ∈ Set.Ioc (a + m • p) (a + p + m • p)",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"IsRigh... | [] | simpa using sub_toIocDiv_zsmul_mem_Ioc hp a b | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Order.ToIntervalMod | {
"line": 319,
"column": 16
} | {
"line": 319,
"column": 35
} | {
"line": 319,
"column": 35
} | [
{
"pp": "α : Type u_1\ninst✝² : AddCommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedAddMonoid α\nhα : Archimedean α\np : α\nhp : 0 < p\na b : α\n⊢ toIcoDiv hp (a + p) b = toIcoDiv hp a b - 1",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"AddCo... | [
"α : Type u_1\ninst✝² : AddCommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedAddMonoid α\nhα : Archimedean α\np : α\nhp : 0 < p\na b : α\n⊢ toIcoDiv hp a b - 1 = toIcoDiv hp a b - 1"
] | toIcoDiv_add_right' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Constructions.SumProd | {
"line": 460,
"column": 46
} | {
"line": 460,
"column": 63
} | {
"line": 460,
"column": 63
} | [
{
"pp": "X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ns : Set X\nx : X\ny : Y\nhs : {x_1 | x_1 ∈ Prod.snd ⁻¹' {(x, y).2} → x_1 ∈ Prod.fst ⁻¹' s} ∈ 𝓝 (x, y)\n⊢ s ∈ 𝓝 (x, y).1",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [
"Filter.instMembership",... | [
"X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ns : Set X\nx : X\ny : Y\nhs : ∃ u ∈ 𝓝 x, ∃ v ∈ 𝓝 y, u ×ˢ v ⊆ {x_1 | x_1 ∈ Prod.snd ⁻¹' {(x, y).2} → x_1 ∈ Prod.fst ⁻¹' s}\n⊢ s ∈ 𝓝 (x, y).1"
] | mem_nhds_prod_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Constructions.SumProd | {
"line": 478,
"column": 46
} | {
"line": 478,
"column": 63
} | {
"line": 478,
"column": 63
} | [
{
"pp": "X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ns : Set Y\nx : X\ny : Y\nhs : {x_1 | x_1 ∈ Prod.fst ⁻¹' {(x, y).1} → x_1 ∈ Prod.snd ⁻¹' s} ∈ 𝓝 (x, y)\n⊢ s ∈ 𝓝 (x, y).2",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [
"Filter.instMembership",... | [
"X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ns : Set Y\nx : X\ny : Y\nhs : ∃ u ∈ 𝓝 x, ∃ v ∈ 𝓝 y, u ×ˢ v ⊆ {x_1 | x_1 ∈ Prod.fst ⁻¹' {(x, y).1} → x_1 ∈ Prod.snd ⁻¹' s}\n⊢ s ∈ 𝓝 (x, y).2"
] | mem_nhds_prod_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Constructions.SumProd | {
"line": 603,
"column": 2
} | {
"line": 606,
"column": 54
} | {
"line": 608,
"column": 0
} | [
{
"pp": "X : Type u\nY : Type v\nW : Type u_1\nZ : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : TopologicalSpace W\nf : X → Y\ng : Z → W\nhf : IsOpenMap f\nhg : IsOpenMap g\n⊢ IsOpenMap (Prod.map f g)",
"ppTerm": "?m.17",
"assigned": true,
... | [] | rw [isOpenMap_iff_nhds_le]
rintro ⟨a, b⟩
rw [nhds_prod_eq, nhds_prod_eq, ← Filter.prod_map_map_eq']
exact Filter.prod_mono (hf.nhds_le a) (hg.nhds_le b) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Constructions.SumProd | {
"line": 603,
"column": 2
} | {
"line": 606,
"column": 54
} | {
"line": 608,
"column": 0
} | [
{
"pp": "X : Type u\nY : Type v\nW : Type u_1\nZ : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : TopologicalSpace W\nf : X → Y\ng : Z → W\nhf : IsOpenMap f\nhg : IsOpenMap g\n⊢ IsOpenMap (Prod.map f g)",
"ppTerm": "?m.17",
"assigned": true,
... | [] | rw [isOpenMap_iff_nhds_le]
rintro ⟨a, b⟩
rw [nhds_prod_eq, nhds_prod_eq, ← Filter.prod_map_map_eq']
exact Filter.prod_mono (hf.nhds_le a) (hg.nhds_le b) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Constructions | {
"line": 573,
"column": 8
} | {
"line": 573,
"column": 29
} | {
"line": 573,
"column": 29
} | [
{
"pp": "X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\nhf : IsOpenEmbedding f\ns : Set X\nt : Set Y\nH : MapsTo f s t\nhs : IsOpen[inst✝¹] s\n⊢ IsOpen[instTopologicalSpaceSubtype] (range (MapsTo.restrict f s t H))",
"ppTerm": "?m.35",
"assigned": true,
"... | [
"X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\nhf : IsOpenEmbedding f\ns : Set X\nt : Set Y\nH : MapsTo f s t\nhs : IsOpen[inst✝¹] s\n⊢ IsOpen[instTopologicalSpaceSubtype] (Subtype.val ⁻¹' f '' s)"
] | MapsTo.range_restrict | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Bases | {
"line": 299,
"column": 2
} | {
"line": 299,
"column": 54
} | {
"line": 301,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type u_1\nt : TopologicalSpace α\ninst✝ : TopologicalSpace β\nB : Set (Set β)\nhB : IsTopologicalBasis B\nf : α → β\n⊢ Continuous[t, inst✝] f ↔ ∀ s ∈ B, IsOpen[t] (f ⁻¹' s)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Continuous",
"c... | [] | rw [hB.eq_generateFrom, continuous_generateFrom_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Bases | {
"line": 299,
"column": 2
} | {
"line": 299,
"column": 54
} | {
"line": 301,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type u_1\nt : TopologicalSpace α\ninst✝ : TopologicalSpace β\nB : Set (Set β)\nhB : IsTopologicalBasis B\nf : α → β\n⊢ Continuous[t, inst✝] f ↔ ∀ s ∈ B, IsOpen[t] (f ⁻¹' s)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Continuous",
"c... | [] | rw [hB.eq_generateFrom, continuous_generateFrom_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Bases | {
"line": 299,
"column": 2
} | {
"line": 299,
"column": 54
} | {
"line": 301,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type u_1\nt : TopologicalSpace α\ninst✝ : TopologicalSpace β\nB : Set (Set β)\nhB : IsTopologicalBasis B\nf : α → β\n⊢ Continuous[t, inst✝] f ↔ ∀ s ∈ B, IsOpen[t] (f ⁻¹' s)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Continuous",
"c... | [] | rw [hB.eq_generateFrom, continuous_generateFrom_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Bases | {
"line": 316,
"column": 10
} | {
"line": 316,
"column": 38
} | {
"line": 317,
"column": 2
} | [
{
"pp": "α : Type u\nt : TopologicalSpace α\nh : IsTopologicalBasis {∅}\n⊢ IsEmpty α",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"congrArg",
"sdiff_self",
"Eq.mp",
"Set.instSingletonSet",
"IsEmpty",
... | [] | by simpa using h.sdiff_empty | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Compactness.SigmaCompact | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 27
} | {
"line": 96,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\ns t : Set X\nhs : IsClosed[inst✝] s\nh : s ⊆ t\nK : ℕ → Set X\nhcompact : ∀ (n : ℕ), IsCompact (K n)\nhcov : ⋃ n, K n = t\n⊢ s ∩ t = s",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Set.inter_eq_left",
"Iff.mpr",
"LE.le... | [] | exact inter_eq_left.mpr h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.UpperLower.Closure | {
"line": 116,
"column": 6
} | {
"line": 116,
"column": 17
} | {
"line": 116,
"column": 18
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\ns : Set α\n⊢ upperClosure s = ⊤ ↔ s = ∅",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"UpperSet",
"eq_top_iff",
"congrArg",
"upperClosure",
"PartialOrder.toPreorder",
"UpperSet.completelyDistrib... | [
"α : Type u_1\ninst✝ : Preorder α\ns : Set α\n⊢ ⊤ ≤ upperClosure s ↔ s = ∅"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.UpperLower.CompleteLattice | {
"line": 207,
"column": 2
} | {
"line": 207,
"column": 34
} | {
"line": 208,
"column": 2
} | [
{
"pp": "α : Type u_1\nι : Sort u_4\ninst✝ : LE α\na : α\nf : ι → UpperSet α\n⊢ a ∈ ⨅ i, f i ↔ ∃ i, a ∈ f i",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"iInf",
"UpperSet",
"congrArg",
"UpperSet.instSetLike",
"Membership.mem",
"Exists"... | [
"α : Type u_1\nι : Sort u_4\ninst✝ : LE α\na : α\nf : ι → UpperSet α\n⊢ a ∈ ⋃ i, ↑(f i) ↔ ∃ i, a ∈ f i"
] | rw [← SetLike.mem_coe, coe_iInf] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.DiscreteSubset | {
"line": 135,
"column": 2
} | {
"line": 135,
"column": 20
} | {
"line": 137,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\ns : Set X\nhs : IsDiscrete s\nhf : IsOpenMap f\nhs' : IsOpen[inst✝¹] s\nx : X\nhx : x ∈ s\n⊢ 𝓝 (f x) ≤ map f (𝓝 x)",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [
"IsOpenMap.nhd... | [] | exact hf.nhds_le x | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Irreducible | {
"line": 288,
"column": 4
} | {
"line": 293,
"column": 80
} | {
"line": 294,
"column": 2
} | [
{
"pp": "case refine_1\nX : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\nh : IsIrreducible s\nU : Finset (Set X)\nhu : ∀ u ∈ U, IsOpen[inst✝] u\nhU : ∀ u ∈ U, (s ∩ u).Nonempty\n⊢ (s ∩ ⋂₀ ↑U).Nonempty",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"IsPreirreducible",
"E... | [] | induction U using Finset.induction_on with
| empty => simpa using h.nonempty
| insert u U _ IH =>
rw [Finset.coe_insert, sInter_insert]
rw [Finset.forall_mem_insert] at hu hU
exact h.2 _ _ hu.1 (U.finite_toSet.isOpen_sInter hu.2) hU.1 (IH hu.2 hU.2) | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Topology.Irreducible | {
"line": 288,
"column": 4
} | {
"line": 293,
"column": 80
} | {
"line": 294,
"column": 2
} | [
{
"pp": "case refine_1\nX : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\nh : IsIrreducible s\nU : Finset (Set X)\nhu : ∀ u ∈ U, IsOpen[inst✝] u\nhU : ∀ u ∈ U, (s ∩ u).Nonempty\n⊢ (s ∩ ⋂₀ ↑U).Nonempty",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"IsPreirreducible",
"E... | [] | induction U using Finset.induction_on with
| empty => simpa using h.nonempty
| insert u U _ IH =>
rw [Finset.coe_insert, sInter_insert]
rw [Finset.forall_mem_insert] at hu hU
exact h.2 _ _ hu.1 (U.finite_toSet.isOpen_sInter hu.2) hU.1 (IH hu.2 hU.2) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Irreducible | {
"line": 288,
"column": 4
} | {
"line": 293,
"column": 80
} | {
"line": 294,
"column": 2
} | [
{
"pp": "case refine_1\nX : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\nh : IsIrreducible s\nU : Finset (Set X)\nhu : ∀ u ∈ U, IsOpen[inst✝] u\nhU : ∀ u ∈ U, (s ∩ u).Nonempty\n⊢ (s ∩ ⋂₀ ↑U).Nonempty",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"IsPreirreducible",
"E... | [] | induction U using Finset.induction_on with
| empty => simpa using h.nonempty
| insert u U _ IH =>
rw [Finset.coe_insert, sInter_insert]
rw [Finset.forall_mem_insert] at hu hU
exact h.2 _ _ hu.1 (U.finite_toSet.isOpen_sInter hu.2) hU.1 (IH hu.2 hU.2) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactness.Compact | {
"line": 1234,
"column": 8
} | {
"line": 1234,
"column": 14
} | {
"line": 1235,
"column": 8
} | [
{
"pp": "case neg\nX : 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}\nc : Set (Set X)\nhc : c ⊆ opens\nhz : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c\nhcne : ¬c.Nonempty\n⊢ ∃ ub ∈ open... | [
"case h\nX : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nS : Set X\nhS : IsClosed[inst✝¹] S\nhne : S.Nonempty\nopens : Set (Set X) := ⋯\nc : Set (Set X)\nhc : c ⊆ opens\nhz : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c\nhcne : ¬c.Nonempty\n⊢ Sᶜ ∈ opens ∧ ∀ s ∈ c, s ⊆ Sᶜ"
] | use Sᶜ | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Topology.Connected.Basic | {
"line": 309,
"column": 4
} | {
"line": 309,
"column": 47
} | {
"line": 310,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns : Set α\nH : IsPreconnected s\nf : α → β\nhf : ContinuousOn f s\nu v : Set β\nhu : IsOpen[inst✝] u\nhv : IsOpen[inst✝] v\nx : α\nxs : x ∈ s\nxu : f x ∈ u\ny : α\nys : y ∈ s\nyv : f y ∈ v\nu' : Set α\nhu' :... | [] | exacts [u'_eq ▸ ⟨xu, xs⟩, v'_eq ▸ ⟨yv, ys⟩] | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_exacts_1 | Batteries.Tactic.exacts |
Mathlib.Topology.Connected.Basic | {
"line": 324,
"column": 45
} | {
"line": 324,
"column": 79
} | {
"line": 324,
"column": 80
} | [
{
"pp": "α : Type u\ninst✝ : TopologicalSpace α\ns : Set α\nh : IsPreconnected s\nt t' : Set α\nht : IsClosed[inst✝] t\nht' : IsClosed[inst✝] t'\nhtt' : s ⊆ t ∪ t'\nx : α\nxs : x ∈ s\nxt : x ∈ t\ny : α\nys : y ∈ s\nyt' : y ∈ t'\n⊢ ¬Disjoint s (t ∩ t')",
"ppTerm": "?m.79",
"assigned": true,
"usedCons... | [
"α : Type u\ninst✝ : TopologicalSpace α\ns : Set α\nh : IsPreconnected s\nt t' : Set α\nht : IsClosed[inst✝] t\nht' : IsClosed[inst✝] t'\nhtt' : s ⊆ t ∪ t'\nx : α\nxs : x ∈ s\nxt : x ∈ t\ny : α\nys : y ∈ s\nyt' : y ∈ t'\n⊢ ¬s ⊆ (t ∩ t')ᶜ"
] | ← subset_compl_iff_disjoint_right, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Connected.Basic | {
"line": 333,
"column": 45
} | {
"line": 333,
"column": 79
} | {
"line": 333,
"column": 80
} | [
{
"pp": "α : Type u\ninst✝ : TopologicalSpace α\ns : Set α\nh :\n ∀ (t t' : Set α),\n IsClosed[inst✝] t → IsClosed[inst✝] t' → s ⊆ t ∪ t' → (s ∩ t).Nonempty → (s ∩ t').Nonempty → (s ∩ (t ∩ t')).Nonempty\nu v : Set α\nhu : IsOpen[inst✝] u\nhv : IsOpen[inst✝] v\nhuv : s ⊆ u ∪ v\nx : α\nxs : x ∈ s\nxu : x ∈ u\... | [
"α : Type u\ninst✝ : TopologicalSpace α\ns : Set α\nh :\n ∀ (t t' : Set α),\n IsClosed[inst✝] t → IsClosed[inst✝] t' → s ⊆ t ∪ t' → (s ∩ t).Nonempty → (s ∩ t').Nonempty → (s ∩ (t ∩ t')).Nonempty\nu v : Set α\nhu : IsOpen[inst✝] u\nhv : IsOpen[inst✝] v\nhuv : s ⊆ u ∪ v\nx : α\nxs : x ∈ s\nxu : x ∈ u\ny : α\nys :... | ← subset_compl_iff_disjoint_right, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Compactness.Lindelof | {
"line": 57,
"column": 45
} | {
"line": 60,
"column": 23
} | {
"line": 62,
"column": 0
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ns : Set X\nhs : IsLindelof s\nf : Filter X\ninst✝ : CountableInterFilter f\nhf : ∀ x ∈ s, sᶜ ∈ 𝓝 x ⊓ f\n⊢ sᶜ ∈ f",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_forall_eq",
"Filter.instMembership",
... | [] | by
contrapose! hf
simp only [notMem_iff_inf_principal_compl, compl_compl, inf_assoc] at hf ⊢
exact hs inf_le_right | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Connected.Basic | {
"line": 715,
"column": 16
} | {
"line": 715,
"column": 87
} | {
"line": 715,
"column": 87
} | [
{
"pp": "α : Type u\ninst✝ : TopologicalSpace α\nh : ∀ (x : α), connectedComponent x = univ\nhα : Nonempty α\n⊢ IsPreconnected univ",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.univ",
"Classical.choice",
"id",
"connectedComp... | [] | rw [← h (Classical.choice hα)]; exact isPreconnected_connectedComponent | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Connected.Basic | {
"line": 715,
"column": 16
} | {
"line": 715,
"column": 87
} | {
"line": 715,
"column": 87
} | [
{
"pp": "α : Type u\ninst✝ : TopologicalSpace α\nh : ∀ (x : α), connectedComponent x = univ\nhα : Nonempty α\n⊢ IsPreconnected univ",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.univ",
"Classical.choice",
"id",
"connectedComp... | [] | rw [← h (Classical.choice hα)]; exact isPreconnected_connectedComponent | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactness.Lindelof | {
"line": 225,
"column": 86
} | {
"line": 225,
"column": 95
} | {
"line": 226,
"column": 2
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nι : Type v\nhs : IsLindelof s\nt : ι → Set X\nhtc : ∀ (i : ι), IsClosed[inst✝] (t i)\nhst : s ∩ ⋂ i, t i = ∅\nU : ι → Set X := tᶜ\n⊢ ∀ (i : ι), IsClosed[inst✝] (t i)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [],
"usedFVars... | [] | exact htc | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Separation.Regular | {
"line": 605,
"column": 9
} | {
"line": 605,
"column": 43
} | {
"line": 605,
"column": 44
} | [
{
"pp": "case refine_2\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : CompletelyNormalSpace Y\ne : X → Y\nhe : IsInducing e\ns t : Set X\nhd₁ : Disjoint (closure[inst✝²] s) t\nhd₂ : Disjoint s (closure[inst✝²] t)\n⊢ Disjoint (e '' s) (closure[inst✝¹] (e '' t))",
... | [
"case refine_2\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : CompletelyNormalSpace Y\ne : X → Y\nhe : IsInducing e\ns t : Set X\nhd₁ : Disjoint (closure[inst✝²] s) t\nhd₂ : Disjoint s (closure[inst✝²] t)\n⊢ e '' s ⊆ (closure[inst✝¹] (e '' t))ᶜ"
] | ← subset_compl_iff_disjoint_right, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Connected.TotallyDisconnected | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 67
} | {
"line": 110,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝ : TopologicalSpace α\n⊢ TotallyDisconnectedSpace α ↔ ∀ (x : α), connectedComponent x = {x}",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.instSingletonSet",
"id",
"connectedComponent",
"TotallyDiscon... | [
"α : Type u\ninst✝ : TopologicalSpace α\n⊢ (∀ (x : α), (connectedComponent x).Subsingleton) ↔ ∀ (x : α), connectedComponent x = {x}"
] | rw [totallyDisconnectedSpace_iff_connectedComponent_subsingleton] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Compactness.Lindelof | {
"line": 326,
"column": 2
} | {
"line": 326,
"column": 76
} | {
"line": 327,
"column": 2
} | [
{
"pp": "X : 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\n⊢ ∃ t, t.Countable ∧ ⋃ i ∈ s, f i ⊆ ⋃ i... | [
"X : 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\niSets : ∀ i_1 ∈ s, ∃ r, r.Countable ∧ f i_1 ⊆ ⋃ i_2... | have iSets := fun i is ↦ (hf i is).elim_countable_subcover U hU (hiU i is) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Separation.Regular | {
"line": 715,
"column": 4
} | {
"line": 715,
"column": 91
} | {
"line": 716,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : CompletelyNormalSpace X\ninst✝ : R0Space X\n⊢ ∀ (x : SeparationQuotient X), IsClosed {x}",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Pure.pure",
"Filter.instMembership",
"Eq.mpr",
"Contin... | [] | rwa [((t1Space_TFAE (SeparationQuotient X)).out 1 0 :), SeparationQuotient.t1Space_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Topology.Separation.Regular | {
"line": 715,
"column": 4
} | {
"line": 715,
"column": 91
} | {
"line": 716,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : CompletelyNormalSpace X\ninst✝ : R0Space X\n⊢ ∀ (x : SeparationQuotient X), IsClosed {x}",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Pure.pure",
"Filter.instMembership",
"Eq.mpr",
"Contin... | [] | rwa [((t1Space_TFAE (SeparationQuotient X)).out 1 0 :), SeparationQuotient.t1Space_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Separation.Regular | {
"line": 715,
"column": 4
} | {
"line": 715,
"column": 91
} | {
"line": 716,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : CompletelyNormalSpace X\ninst✝ : R0Space X\n⊢ ∀ (x : SeparationQuotient X), IsClosed {x}",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Pure.pure",
"Filter.instMembership",
"Eq.mpr",
"Contin... | [] | rwa [((t1Space_TFAE (SeparationQuotient X)).out 1 0 :), SeparationQuotient.t1Space_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Connected.Clopen | {
"line": 591,
"column": 26
} | {
"line": 594,
"column": 94
} | {
"line": 595,
"column": 4
} | [
{
"pp": "α : Type u\nβ : Type v\nι✝ : Type u_1\nX : ι✝ → Type u_2\ninst✝ : TopologicalSpace α\ns t u v : Set α\nι : Type u_3\nU : ι → Set α\nhclopen : ∀ (i : ι), IsClopen (U i)\nhdisj : Pairwise (Disjoint on U)\nhunion : ⋃ i, U i = univ\nhconn : ∀ (i : ι), IsPreconnected (U i)\nheq : ∀ {x : α} {i : ι} (hx : x ∈... | [] | by
apply hdisj.eq
rw [Set.not_disjoint_iff]
exact ⟨x, x.2, (hclopen j).connectedComponent_subset y.2 (hxy ▸ mem_connectedComponent)⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.UniformSpace.Defs | {
"line": 382,
"column": 6
} | {
"line": 382,
"column": 45
} | {
"line": 383,
"column": 6
} | [
{
"pp": "case hg\nα : Type ua\nβ : Type ub\ninst✝ : UniformSpace α\nf : SetRel α α → Filter β\nh : Monotone f\n⊢ Monotone fun s ↦ s ○ s",
"ppTerm": "?hg",
"assigned": true,
"usedConstants": [
"SetRel",
"PartialOrder.toPreorder",
"monotone_id",
"CompleteLattice.toConditionally... | [
"case hh\nα : Type ua\nβ : Type ub\ninst✝ : UniformSpace α\nf : SetRel α α → Filter β\nh : Monotone f\n⊢ Monotone f"
] | · exact monotone_id.relComp monotone_id | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.UniformSpace.Defs | {
"line": 536,
"column": 6
} | {
"line": 536,
"column": 31
} | {
"line": 536,
"column": 31
} | [
{
"pp": "α : Type ua\ninst✝ : UniformSpace α\nx : α\nV : SetRel α α\nV_in : V ∈ 𝓤 α\n⊢ ball x V ∈ 𝓝 x",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"congrArg",
"uniformity",
"Membership.mem",
"Exists",
"nhds",... | [
"α : Type ua\ninst✝ : UniformSpace α\nx : α\nV : SetRel α α\nV_in : V ∈ 𝓤 α\n⊢ ∃ V_1 ∈ 𝓤 α, ball x V_1 ⊆ ball x V"
] | UniformSpace.mem_nhds_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UniformSpace.Defs | {
"line": 546,
"column": 6
} | {
"line": 546,
"column": 31
} | {
"line": 546,
"column": 31
} | [
{
"pp": "α : Type ua\ninst✝ : UniformSpace α\nx : α\ns : Set α\n⊢ s ∈ 𝓝 x ↔ ∃ V ∈ 𝓤 α, SetRel.IsSymm V ∧ ball x V ⊆ s",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"congrArg",
"uniformity",
"Membership.mem",
"Exists... | [
"α : Type ua\ninst✝ : UniformSpace α\nx : α\ns : Set α\n⊢ (∃ V ∈ 𝓤 α, ball x V ⊆ s) ↔ ∃ V ∈ 𝓤 α, SetRel.IsSymm V ∧ ball x V ⊆ s"
] | UniformSpace.mem_nhds_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 96,
"column": 31
} | {
"line": 96,
"column": 42
} | {
"line": 96,
"column": 43
} | [
{
"pp": "β : Type v\nu v : UniformSpace β\nF : Filter β\n⊢ F.NeBot ∧ F ×ˢ F ≤ 𝓤 β ⊓ 𝓤 β ↔ (F.NeBot ∧ F ×ˢ F ≤ 𝓤 β) ∧ F.NeBot ∧ F ×ˢ F ≤ 𝓤 β",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SProd.sprod",
"congrArg",
"Filter.NeBot",
"Filter.instCom... | [
"β : Type v\nu v : UniformSpace β\nF : Filter β\n⊢ F.NeBot ∧ F ×ˢ F ≤ 𝓤 β ∧ F ×ˢ F ≤ 𝓤 β ↔ (F.NeBot ∧ F ×ˢ F ≤ 𝓤 β) ∧ F.NeBot ∧ F ×ˢ F ≤ 𝓤 β"
] | le_inf_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UniformSpace.Separation | {
"line": 136,
"column": 10
} | {
"line": 136,
"column": 32
} | {
"line": 136,
"column": 32
} | [
{
"pp": "α : Type u\ninst✝¹ : UniformSpace α\nι : Type u_1\ninst✝ : LinearOrder ι\nB : ι → SetRel α α\nhB : (𝓤 α).HasAntitoneBasis B\ns t : Set α\nhSt : Disjoint (closure[inst✝¹.toTopologicalSpace] s) t\nhsT : Disjoint s (closure[inst✝¹.toTopologicalSpace] t)\nS : Bool → Set α := fun b ↦ Bool.casesOn b s t\nU ... | [
"α : Type u\ninst✝¹ : UniformSpace α\nι : Type u_1\ninst✝ : LinearOrder ι\nB : ι → SetRel α α\nhB : (𝓤 α).HasAntitoneBasis B\ns t : Set α\nhSt : Disjoint (closure[inst✝¹.toTopologicalSpace] s) t\nhsT : Disjoint s (closure[inst✝¹.toTopologicalSpace] t)\nS : Bool → Set α := fun b ↦ Bool.casesOn b s t\nU : (b : Bool)... | mem_nhdsSet_iff_forall | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 333,
"column": 2
} | {
"line": 333,
"column": 100
} | {
"line": 334,
"column": 2
} | [
{
"pp": "α : Type u\nuniformSpace : UniformSpace α\ns : Set α\nH : ∀ (l : Ultrafilter α), Cauchy ↑l → ↑l ≤ 𝓟 s → ∃ x ∈ s, ↑l ≤ 𝓝 x\nl : Filter α\nhl : Cauchy l\nhls : l ≤ 𝓟 s\nthis : l.NeBot\n⊢ ∃ x ∈ s, ClusterPt x l",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Ultrafilter.of_... | [
"α : Type u\nuniformSpace : UniformSpace α\ns : Set α\nH : ∀ (l : Ultrafilter α), Cauchy ↑l → ↑l ≤ 𝓟 s → ∃ x ∈ s, ↑l ≤ 𝓝 x\nl : Filter α\nhl : Cauchy l\nhls : l ≤ 𝓟 s\nthis : l.NeBot\nx : α\nhxs : x ∈ s\nhxl : ↑(Ultrafilter.of l) ≤ 𝓝 x\n⊢ ∃ x ∈ s, ClusterPt x l"
] | rcases H (Ultrafilter.of l) hl.ultrafilter_of ((Ultrafilter.of_le l).trans hls) with ⟨x, hxs, hxl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.UniformSpace.Separation | {
"line": 308,
"column": 4
} | {
"line": 308,
"column": 51
} | {
"line": 310,
"column": 0
} | [
{
"pp": "case neg\nα : Type u\nβ : Type v\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : T0Space β\nf : α → β\nhf : ¬UniformContinuous f\n⊢ UniformContinuous fun x ↦ f ⋯.some",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"SeparationQuotient.instUniformSpace",
"Non... | [] | exact uniformContinuous_of_const fun a _ => rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.UniformSpace.Separation | {
"line": 314,
"column": 2
} | {
"line": 314,
"column": 50
} | {
"line": 314,
"column": 50
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nh : UniformContinuous f\na : α\n⊢ map f (mk a) = mk (f a)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"SeparationQuotient.map._proof_1",
"Eq.mpr",
"SeparationQuotient.instUnif... | [
"α : Type u\nβ : Type v\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nh : UniformContinuous f\na : α\n⊢ (mk ∘ f) a = mk (f a)"
] | rw [map, lift'_mk (uniformContinuous_mk.comp h)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.UniformSpace.Basic | {
"line": 417,
"column": 6
} | {
"line": 417,
"column": 45
} | {
"line": 417,
"column": 45
} | [
{
"pp": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\nf : α → β\nu : UniformSpace β\n⊢ Monotone fun s ↦ s ○ s",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"SetRel",
"PartialOrder.toPreorder",
"monotone_id",
"CompleteLattice.toConditionallyCom... | [] | · exact monotone_id.relComp monotone_id | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.UniformSpace.UniformEmbedding | {
"line": 490,
"column": 6
} | {
"line": 490,
"column": 87
} | {
"line": 491,
"column": 6
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\ninst✝¹ : UniformSpace γ\ne : β → α\nh_e : IsUniformInducing e\nh_dense : DenseRange e\nf : β → γ\nh_f : UniformContinuous f\ninst✝ : CompleteSpace γ\nd : Set (γ × γ)\nhd : d ∈ 𝓤 γ\ns : Set (γ × γ)\nhs : s ∈ 𝓤 ... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\ninst✝¹ : UniformSpace γ\ne : β → α\nh_e : IsUniformInducing e\nh_dense : DenseRange e\nf : β → γ\nh_f : UniformContinuous f\ninst✝ : CompleteSpace γ\nd : Set (γ × γ)\nhd : d ∈ 𝓤 γ\ns : Set (γ × γ)\nhs : s ∈ 𝓤 γ\nhs_comp :... | let ⟨m₁, hm₁, m₂, hm₂, (hm : m₁ ×ˢ m₂ ⊆ interior t)⟩ := mem_nhds_prod_iff.mp this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Topology.UniformSpace.Compact | {
"line": 192,
"column": 2
} | {
"line": 192,
"column": 67
} | {
"line": 194,
"column": 0
} | [
{
"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\nthis : CompactSpace γ\n⊢ 𝓤 γ = 𝓤 γ",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fil... | [] | rw [@compactSpace_uniformity _ u, compactSpace_uniformity, h, h'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.UniformSpace.Compact | {
"line": 189,
"column": 2
} | {
"line": 192,
"column": 67
} | {
"line": 194,
"column": 0
} | [
{
"pp": "γ : Type uc\nt : TopologicalSpace γ\ninst✝ : CompactSpace γ\nu u' : UniformSpace γ\nh : u.toTopologicalSpace = t\nh' : u'.toTopologicalSpace = t\n⊢ u = u'",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Filter.instSupSet",
"congrArg",
"iSup",
... | [] | refine UniformSpace.ext ?_
have : @CompactSpace γ u.toTopologicalSpace := by rwa [h]
have : @CompactSpace γ u'.toTopologicalSpace := by rwa [h']
rw [@compactSpace_uniformity _ u, compactSpace_uniformity, h, h'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.Compact | {
"line": 189,
"column": 2
} | {
"line": 192,
"column": 67
} | {
"line": 194,
"column": 0
} | [
{
"pp": "γ : Type uc\nt : TopologicalSpace γ\ninst✝ : CompactSpace γ\nu u' : UniformSpace γ\nh : u.toTopologicalSpace = t\nh' : u'.toTopologicalSpace = t\n⊢ u = u'",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Filter.instSupSet",
"congrArg",
"iSup",
... | [] | refine UniformSpace.ext ?_
have : @CompactSpace γ u.toTopologicalSpace := by rwa [h]
have : @CompactSpace γ u'.toTopologicalSpace := by rwa [h']
rw [@compactSpace_uniformity _ u, compactSpace_uniformity, h, h'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.Pi | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 94
} | {
"line": 96,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u\nU : (i : ι) → UniformSpace (α i)\ninst✝ : Nonempty ι\nl : Filter ((i : ι) → α i)\n⊢ Cauchy l ↔ ∀ (i : ι), Cauchy (map (eval i) l)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Pi.uniformSpace_eq",
"Pi.uniformSpace",
"UniformSpace"... | [] | simp_rw +instances [Pi.uniformSpace_eq, cauchy_iInf_uniformSpace, cauchy_comap_uniformSpace] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Topology.UniformSpace.Pi | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 94
} | {
"line": 96,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u\nU : (i : ι) → UniformSpace (α i)\ninst✝ : Nonempty ι\nl : Filter ((i : ι) → α i)\n⊢ Cauchy l ↔ ∀ (i : ι), Cauchy (map (eval i) l)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Pi.uniformSpace_eq",
"Pi.uniformSpace",
"UniformSpace"... | [] | simp_rw +instances [Pi.uniformSpace_eq, cauchy_iInf_uniformSpace, cauchy_comap_uniformSpace] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.Pi | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 94
} | {
"line": 96,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u\nU : (i : ι) → UniformSpace (α i)\ninst✝ : Nonempty ι\nl : Filter ((i : ι) → α i)\n⊢ Cauchy l ↔ ∀ (i : ι), Cauchy (map (eval i) l)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Pi.uniformSpace_eq",
"Pi.uniformSpace",
"UniformSpace"... | [] | simp_rw +instances [Pi.uniformSpace_eq, cauchy_iInf_uniformSpace, cauchy_comap_uniformSpace] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.Pi | {
"line": 108,
"column": 29
} | {
"line": 108,
"column": 54
} | {
"line": 108,
"column": 54
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nβ : Type u_3\nα : ι → Type u\nU : (i : ι) → UniformSpace (α i)\ninst✝¹ : UniformSpace β\ninst✝ : ∀ (i : ι), CompleteSpace (α i)\nf : Filter ((i : ι) → α i)\nthis : f.NeBot\nhf : ∀ (i : ι), Cauchy (map (eval i) f)\n⊢ ∃ x, f ≤ 𝓝 x",
"ppTerm": "?m.20",
"assigned": tru... | [
"ι : Type u_1\nι' : Type u_2\nβ : Type u_3\nα : ι → Type u\nU : (i : ι) → UniformSpace (α i)\ninst✝¹ : UniformSpace β\ninst✝ : ∀ (i : ι), CompleteSpace (α i)\nf : Filter ((i : ι) → α i)\nthis : f.NeBot\nhf : ∀ (i : ι), ∃ x, map (eval i) f ≤ 𝓝 x\n⊢ ∃ x, f ≤ 𝓝 x"
] | cauchy_iff_exists_le_nhds | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 257,
"column": 2
} | {
"line": 257,
"column": 6
} | {
"line": 258,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\n𝓐 : Filter ((α →ᵤ β) × (α →ᵤ β))\n𝓕 : Filter (β × β)\n⊢ (fun 𝓐 ↦ map (UniformFun.phi α β) (𝓐 ×ˢ ⊤)) 𝓐 ≤ 𝓕 ↔ 𝓐 ≤ (fun 𝓕 ↦ UniformFun.filter α β 𝓕) 𝓕",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"SProd.sprod",
"Filter.map",
... | [
"α : Type u_1\nβ : Type u_2\n𝓐 : Filter ((α →ᵤ β) × (α →ᵤ β))\n𝓕 : Filter (β × β)\n⊢ 𝓐 ≤ (fun 𝓕 ↦ UniformFun.filter α β 𝓕) 𝓕 ↔ (fun 𝓐 ↦ map (UniformFun.phi α β) (𝓐 ×ˢ ⊤)) 𝓐 ≤ 𝓕"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 428,
"column": 2
} | {
"line": 429,
"column": 92
} | {
"line": 430,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : UniformSpace β\nf : γ → α\n⊢ UniformContinuous fun g ↦ ofFun (toFun g ∘ f)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"UniformContinuous",
"Eq.mpr",
"Equiv.instEquivLike",
... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : UniformSpace β\nf : γ → α\n⊢ ∀ ib ∈ 𝓤 β,\n ∃ ia ∈ 𝓤 β, ∀ x ∈ UniformFun.gen α β ia, (ofFun (toFun x.1 ∘ f), ofFun (toFun x.2 ∘ f)) ∈ UniformFun.gen γ β ib"
] | rw [UniformContinuous,
(UniformFun.hasBasis_uniformity α β).tendsto_iff (UniformFun.hasBasis_uniformity γ β)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.UniformSpace.Equicontinuity | {
"line": 201,
"column": 2
} | {
"line": 201,
"column": 55
} | {
"line": 203,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_6\nβ : Type u_8\nuα : UniformSpace α\nuβ : UniformSpace β\nF : ι → β → α\n⊢ UniformEquicontinuousOn F univ ↔ UniformEquicontinuous F",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Set.instSProd",
"SProd.sprod",
... | [] | simp [UniformEquicontinuousOn, UniformEquicontinuous] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.UniformSpace.Equicontinuity | {
"line": 201,
"column": 2
} | {
"line": 201,
"column": 55
} | {
"line": 203,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_6\nβ : Type u_8\nuα : UniformSpace α\nuβ : UniformSpace β\nF : ι → β → α\n⊢ UniformEquicontinuousOn F univ ↔ UniformEquicontinuous F",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Set.instSProd",
"SProd.sprod",
... | [] | simp [UniformEquicontinuousOn, UniformEquicontinuous] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.Equicontinuity | {
"line": 201,
"column": 2
} | {
"line": 201,
"column": 55
} | {
"line": 203,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_6\nβ : Type u_8\nuα : UniformSpace α\nuβ : UniformSpace β\nF : ι → β → α\n⊢ UniformEquicontinuousOn F univ ↔ UniformEquicontinuous F",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Set.instSProd",
"SProd.sprod",
... | [] | simp [UniformEquicontinuousOn, UniformEquicontinuous] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 456,
"column": 68
} | {
"line": 458,
"column": 58
} | {
"line": 460,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\np : Filter ι\ninst✝ : UniformSpace β\nF : ι → α →ᵤ β\nf : α →ᵤ β\n⊢ Tendsto F p (𝓝 f) ↔ TendstoUniformly (⇑toFun ∘ F) (toFun f) p",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"Equiv.... | [] | by
rw [(UniformFun.hasBasis_nhds α β f).tendsto_right_iff, TendstoUniformly]
simp only [mem_setOf, UniformFun.gen, Function.comp_def] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 535,
"column": 4
} | {
"line": 535,
"column": 19
} | {
"line": 537,
"column": 0
} | [
{
"pp": "case refine_3.inr\nα : Type u_1\nβ : Type u_2\ninst✝ : UniformSpace β\nδ₁ : Type u_6\nδ₂ : Type u_7\nφ₁ : δ₁ → α\nφ₂ : δ₂ → α\nh_cover : range φ₁ ∪ range φ₂ = univ\nU : Set (β × β)\nhU : U ∈ 𝓤 β\nx✝ : (α →ᵤ β) × (α →ᵤ β)\nf g : α →ᵤ β\nhfg :\n (f, g) ∈\n (fun p ↦ ((⇑ofFun ∘ (fun x ↦ x ∘ φ₁) ∘ ⇑toF... | [] | · exact hfg.2 y | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Order.Group.MinMax | {
"line": 86,
"column": 2
} | {
"line": 87,
"column": 89
} | {
"line": 89,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝² : AddCommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedAddMonoid α\na b c d : α\n⊢ max c d - max a b ≤ max |a - c| |b - d|",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"le_abs_se... | [] | · rw [abs_sub_comm a c, abs_sub_comm b d]
exact (max_sub_max_le_max _ _ _ _).trans (max_le_max (le_abs_self _) (le_abs_self _)) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.UniformSpace.Equicontinuity | {
"line": 683,
"column": 87
} | {
"line": 686,
"column": 5
} | {
"line": 688,
"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 (α × α)\nF : ι → β → α\nS : Set β\nhα : (𝓤 α).HasBasis p s\n⊢ UniformEquicontinuousOn F S ↔\n ∀ (k : κ), p k → ∀ᶠ (xy : β × β) in 𝓤 β ⊓ 𝓟 (S ×ˢ S), ∀ (i : ι), (F i xy.1, F i... | [] | by
rw [uniformEquicontinuousOn_iff_uniformContinuousOn, UniformContinuousOn,
(UniformFun.hasBasis_uniformity_of_basis ι α hα).tendsto_right_iff]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Filter.NAry | {
"line": 44,
"column": 4
} | {
"line": 44,
"column": 73
} | {
"line": 44,
"column": 73
} | [
{
"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... | [
"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 : Filter β\nx✝ : Se... | simp only [mem_map, mem_prod_iff, image2_subset_iff, prod_subset_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.UniformSpace.Equicontinuity | {
"line": 740,
"column": 2
} | {
"line": 740,
"column": 36
} | {
"line": 742,
"column": 0
} | [
{
"pp": "ι : Type u_1\nX : Type u_3\nα : Type u_6\nβ : Type u_8\ntX : TopologicalSpace X\nuα : UniformSpace α\nuβ : UniformSpace β\nF : ι → X → α\nS : Set X\nu : α → β\nhu : IsUniformInducing u\nx : X\n⊢ EquicontinuousWithinAt F S x ↔ EquicontinuousWithinAt ((fun x ↦ u ∘ x) ∘ F) S x",
"ppTerm": "?m.34",
... | [] | rw [hu.equicontinuousWithinAt_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.MulAction | {
"line": 192,
"column": 21
} | {
"line": 194,
"column": 82
} | {
"line": 196,
"column": 0
} | [
{
"pp": "M : Type u_1\nX : Type u_2\nY : Type u_3\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : SMul M X\ninst✝² : ContinuousSMul M X\ng : Y → X\nN : Type u_5\ninst✝¹ : SMul N Y\ninst✝ : TopologicalSpace N\nf : N → M\nhg : IsInducing g\nhf : Continuous[inst✝, i... | [] | by
simpa only [hg.continuous_iff, Function.comp_def, hsmul]
using (hf.comp continuous_fst).fun_smul <| hg.continuous.comp continuous_snd | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 106,
"column": 2
} | {
"line": 108,
"column": 38
} | {
"line": 110,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝³ : Mul α\ninst✝² : PartialOrder α\ninst✝¹ : MulLeftStrictMono α\ninst✝ : MulRightStrictMono α\na b : α\n⊢ Iio a * Iic b ⊆ Iio (a * b)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"PartialOrder.toPreorder",
"Preorder.toLE",
... | [] | have := mulLeftMono_of_mulLeftStrictMono α
rintro x ⟨y, hya, z, hzb, rfl⟩
exact mul_lt_mul_of_lt_of_le hya hzb | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 106,
"column": 2
} | {
"line": 108,
"column": 38
} | {
"line": 110,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝³ : Mul α\ninst✝² : PartialOrder α\ninst✝¹ : MulLeftStrictMono α\ninst✝ : MulRightStrictMono α\na b : α\n⊢ Iio a * Iic b ⊆ Iio (a * b)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"PartialOrder.toPreorder",
"Preorder.toLE",
... | [] | have := mulLeftMono_of_mulLeftStrictMono α
rintro x ⟨y, hya, z, hzb, rfl⟩
exact mul_lt_mul_of_lt_of_le hya hzb | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 112,
"column": 2
} | {
"line": 114,
"column": 38
} | {
"line": 116,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝³ : Mul α\ninst✝² : PartialOrder α\ninst✝¹ : MulLeftStrictMono α\ninst✝ : MulRightStrictMono α\na b : α\n⊢ Ioi a * Ici b ⊆ Ioi (a * b)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Set.Ioi",
"HMul.hMul",
"Set.Ici",
"PartialOrder.toPreo... | [] | have := mulLeftMono_of_mulLeftStrictMono α
rintro x ⟨y, hya, z, hzb, rfl⟩
exact mul_lt_mul_of_lt_of_le hya hzb | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 112,
"column": 2
} | {
"line": 114,
"column": 38
} | {
"line": 116,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝³ : Mul α\ninst✝² : PartialOrder α\ninst✝¹ : MulLeftStrictMono α\ninst✝ : MulRightStrictMono α\na b : α\n⊢ Ioi a * Ici b ⊆ Ioi (a * b)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Set.Ioi",
"HMul.hMul",
"Set.Ici",
"PartialOrder.toPreo... | [] | have := mulLeftMono_of_mulLeftStrictMono α
rintro x ⟨y, hya, z, hzb, rfl⟩
exact mul_lt_mul_of_lt_of_le hya hzb | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 24
} | {
"line": 212,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\na b c : α\n⊢ (fun x ↦ a * x) ⁻¹' Ioo b c = Ioo (b / a) (c / a)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Set.Ioi",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass"... | [] | simp [← Ioi_inter_Iio] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 24
} | {
"line": 212,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\na b c : α\n⊢ (fun x ↦ a * x) ⁻¹' Ioo b c = Ioo (b / a) (c / a)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Set.Ioi",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass"... | [] | simp [← Ioi_inter_Iio] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 24
} | {
"line": 212,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\na b c : α\n⊢ (fun x ↦ a * x) ⁻¹' Ioo b c = Ioo (b / a) (c / a)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Set.Ioi",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass"... | [] | simp [← Ioi_inter_Iio] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 246,
"column": 2
} | {
"line": 246,
"column": 24
} | {
"line": 248,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\na b c : α\n⊢ (fun x ↦ x * a) ⁻¹' Ioo b c = Ioo (b / a) (c / a)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Set.Ioi",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass"... | [] | simp [← Ioi_inter_Iio] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 246,
"column": 2
} | {
"line": 246,
"column": 24
} | {
"line": 248,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\na b c : α\n⊢ (fun x ↦ x * a) ⁻¹' Ioo b c = Ioo (b / a) (c / a)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Set.Ioi",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass"... | [] | simp [← Ioi_inter_Iio] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 246,
"column": 2
} | {
"line": 246,
"column": 24
} | {
"line": 248,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\na b c : α\n⊢ (fun x ↦ x * a) ⁻¹' Ioo b c = Ioo (b / a) (c / a)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Set.Ioi",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass"... | [] | simp [← Ioi_inter_Iio] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 545,
"column": 90
} | {
"line": 546,
"column": 74
} | {
"line": 548,
"column": 0
} | [
{
"pp": "G₀ : Type u_2\ninst✝² : GroupWithZero G₀\ninst✝¹ : PartialOrder G₀\ninst✝ : MulPosReflectLT G₀\nc a : G₀\nh : 0 < c\n⊢ (fun x ↦ x * c) ⁻¹' Ioi a = Ioi (a / c)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Set.Ioi",
... | [] | by
simpa only [division_def] using! (OrderIso.mulRight₀ c h).preimage_Ioi a | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 616,
"column": 4
} | {
"line": 616,
"column": 43
} | {
"line": 617,
"column": 2
} | [
{
"pp": "case inl\nG₀ : Type u_2\ninst✝² : GroupWithZero G₀\ninst✝¹ : PartialOrder G₀\ninst✝ : PosMulReflectLT G₀\nb c : G₀\nhbc : b ≤ c\nha : 0 ≤ 0\n⊢ (fun x ↦ 0 * x) '' Icc b c = Icc (0 * b) (0 * c)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"GroupWithZero.toMo... | [] | simp [(nonempty_Icc.2 hbc).image_const] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 616,
"column": 4
} | {
"line": 616,
"column": 43
} | {
"line": 617,
"column": 2
} | [
{
"pp": "case inl\nG₀ : Type u_2\ninst✝² : GroupWithZero G₀\ninst✝¹ : PartialOrder G₀\ninst✝ : PosMulReflectLT G₀\nb c : G₀\nhbc : b ≤ c\nha : 0 ≤ 0\n⊢ (fun x ↦ 0 * x) '' Icc b c = Icc (0 * b) (0 * c)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"GroupWithZero.toMo... | [] | simp [(nonempty_Icc.2 hbc).image_const] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 616,
"column": 4
} | {
"line": 616,
"column": 43
} | {
"line": 617,
"column": 2
} | [
{
"pp": "case inl\nG₀ : Type u_2\ninst✝² : GroupWithZero G₀\ninst✝¹ : PartialOrder G₀\ninst✝ : PosMulReflectLT G₀\nb c : G₀\nhbc : b ≤ c\nha : 0 ≤ 0\n⊢ (fun x ↦ 0 * x) '' Icc b c = Icc (0 * b) (0 * c)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"GroupWithZero.toMo... | [] | simp [(nonempty_Icc.2 hbc).image_const] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Maps.Proper.Basic | {
"line": 104,
"column": 13
} | {
"line": 104,
"column": 40
} | {
"line": 104,
"column": 40
} | [
{
"pp": "case mp\nX : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\nx✝ : Continuous[inst✝¹, inst✝] f\nH : ∀ ⦃ℱ : Filter X⦄ ⦃y : Y⦄, MapClusterPt y ℱ f → ∃ x, f x = y ∧ ClusterPt x ℱ\n𝒰 : Ultrafilter X\ny : Y\nhY : ↑(Ultrafilter.map f 𝒰) ≤ 𝓝 y\n⊢ ∃ x, f x = y ∧ ↑�... | [
"case mp\nX : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\nx✝ : Continuous[inst✝¹, inst✝] f\nH : ∀ ⦃ℱ : Filter X⦄ ⦃y : Y⦄, MapClusterPt y ℱ f → ∃ x, f x = y ∧ ClusterPt x ℱ\n𝒰 : Ultrafilter X\ny : Y\nhY : ClusterPt y ↑(Ultrafilter.map f 𝒰)\n⊢ ∃ x, f x = y ∧ ClusterPt... | ← Ultrafilter.clusterPt_iff | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Algebra.Monoid | {
"line": 201,
"column": 4
} | {
"line": 201,
"column": 8
} | {
"line": 202,
"column": 4
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : Mul M\ninst✝⁵ : ContinuousMul M\ninst✝⁴ : TopologicalSpace N\ninst✝³ : Monoid N\ninst✝² : ContinuousMul N\ninst✝¹ : T2Space N\nf : ι → Nˣ\nr₁ r₂ : N\nl : Filter ι\nin... | [
"ι : Type u_1\nα : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : Mul M\ninst✝⁵ : ContinuousMul M\ninst✝⁴ : TopologicalSpace N\ninst✝³ : Monoid N\ninst✝² : ContinuousMul N\ninst✝¹ : T2Space N\nf : ι → Nˣ\nr₁ r₂ : N\nl : Filter ι\ninst✝ : l.NeBo... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Topology.Algebra.Monoid | {
"line": 204,
"column": 4
} | {
"line": 204,
"column": 8
} | {
"line": 205,
"column": 4
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : Mul M\ninst✝⁵ : ContinuousMul M\ninst✝⁴ : TopologicalSpace N\ninst✝³ : Monoid N\ninst✝² : ContinuousMul N\ninst✝¹ : T2Space N\nf : ι → Nˣ\nr₁ r₂ : N\nl : Filter ι\nin... | [
"ι : Type u_1\nα : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : Mul M\ninst✝⁵ : ContinuousMul M\ninst✝⁴ : TopologicalSpace N\ninst✝³ : Monoid N\ninst✝² : ContinuousMul N\ninst✝¹ : T2Space N\nf : ι → Nˣ\nr₁ r₂ : N\nl : Filter ι\ninst✝ : l.NeBo... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Topology.Algebra.Monoid | {
"line": 736,
"column": 2
} | {
"line": 736,
"column": 24
} | {
"line": 738,
"column": 0
} | [
{
"pp": "case e'_3\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : SeparatelyContinuousMul M\na b : M\nha : b * a = 1\nx : M\n⊢ ((fun x ↦ b * x) ∘ fun x ↦ a * x) x = id x",
"ppTerm": "?e'_3",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Semigroup.toMul",
... | [] | simp [← mul_assoc, ha] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Algebra.Monoid | {
"line": 962,
"column": 2
} | {
"line": 964,
"column": 57
} | {
"line": 966,
"column": 0
} | [
{
"pp": "ι : Type u_1\nX : Type u_6\nM : Type u_7\ninst✝ : CommMonoid M\ns : Finset ι\nl : Filter X\nf g : ι → X → M\nhs : ∀ i ∈ s, f i =ᶠ[l] g i\n⊢ ∏ i ∈ s, f i =ᶠ[l] ∏ i ∈ s, g i",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset.prod_apply",
... | [] | replace hs : ∀ᶠ x in l, ∀ i ∈ s, f i x = g i x := by rwa [eventually_all_finset]
filter_upwards [hs] with x hx
simp only [Finset.prod_apply, Finset.prod_congr rfl hx] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Monoid | {
"line": 962,
"column": 2
} | {
"line": 964,
"column": 57
} | {
"line": 966,
"column": 0
} | [
{
"pp": "ι : Type u_1\nX : Type u_6\nM : Type u_7\ninst✝ : CommMonoid M\ns : Finset ι\nl : Filter X\nf g : ι → X → M\nhs : ∀ i ∈ s, f i =ᶠ[l] g i\n⊢ ∏ i ∈ s, f i =ᶠ[l] ∏ i ∈ s, g i",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset.prod_apply",
... | [] | replace hs : ∀ᶠ x in l, ∀ i ∈ s, f i x = g i x := by rwa [eventually_all_finset]
filter_upwards [hs] with x hx
simp only [Finset.prod_apply, Finset.prod_congr rfl hx] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.IsUniformGroup.Defs | {
"line": 292,
"column": 2
} | {
"line": 292,
"column": 21
} | {
"line": 293,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : IsUniformGroup α\nι : Type u_3\nf g : ι → α × α\nl : Filter ι\nhf : Tendsto f l (𝓤 α)\nhg : Tendsto g l (𝓤 α)\n⊢ Tendsto (f / g) l (𝓤 α)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivIn... | [
"α : Type u_1\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : IsUniformGroup α\nι : Type u_3\nf g : ι → α × α\nl : Filter ι\nhf : Tendsto f l (𝓤 α)\nhg : Tendsto g l (𝓤 α)\n⊢ Tendsto (f * g⁻¹) l (𝓤 α)"
] | rw [div_eq_mul_inv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.IsUniformGroup.Defs | {
"line": 304,
"column": 16
} | {
"line": 304,
"column": 64
} | {
"line": 304,
"column": 64
} | [
{
"pp": "α : Type u_1\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : IsUniformGroup α\nι : Type u_3\nf g : ι → α × α\nl : Filter ι\nhf : Tendsto f l (𝓤 α)\nhfg : Tendsto (f * g) l (𝓤 α)\n⊢ Tendsto g l (𝓤 α)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Prod.instInv",
... | [] | simpa using hf.uniformity_inv.uniformity_mul hfg | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.Algebra.IsUniformGroup.Defs | {
"line": 304,
"column": 16
} | {
"line": 304,
"column": 64
} | {
"line": 304,
"column": 64
} | [
{
"pp": "α : Type u_1\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : IsUniformGroup α\nι : Type u_3\nf g : ι → α × α\nl : Filter ι\nhf : Tendsto f l (𝓤 α)\nhfg : Tendsto (f * g) l (𝓤 α)\n⊢ Tendsto g l (𝓤 α)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Prod.instInv",
... | [] | simpa using hf.uniformity_inv.uniformity_mul hfg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.IsUniformGroup.Defs | {
"line": 304,
"column": 16
} | {
"line": 304,
"column": 64
} | {
"line": 304,
"column": 64
} | [
{
"pp": "α : Type u_1\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : IsUniformGroup α\nι : Type u_3\nf g : ι → α × α\nl : Filter ι\nhf : Tendsto f l (𝓤 α)\nhfg : Tendsto (f * g) l (𝓤 α)\n⊢ Tendsto g l (𝓤 α)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Prod.instInv",
... | [] | simpa using hf.uniformity_inv.uniformity_mul hfg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 314,
"column": 2
} | {
"line": 314,
"column": 31
} | {
"line": 316,
"column": 0
} | [
{
"pp": "G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : InvolutiveInv G\ninst✝ : ContinuousInv G\ns : Set G\nhs : IsCompact s\n⊢ IsCompact ((fun x ↦ x⁻¹) '' s)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"InvolutiveInv.toInv",
"Inv.inv",
"ContinuousInv.continuous_i... | [] | exact hs.image continuous_inv | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Filter.AtTopBot.Field | {
"line": 54,
"column": 2
} | {
"line": 54,
"column": 93
} | {
"line": 55,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : Semifield α\ninst✝² : LinearOrder α\ninst✝¹ : IsStrictOrderedRing α\nl : Filter β\nf : β → α\nr : α\ninst✝ : l.NeBot\nh : Tendsto f l atTop\nhrf : Tendsto (fun x ↦ r * f x) l atTop\nhr : r ≤ 0\n⊢ False",
"ppTerm": "?m.32",
"assigned": true,
"usedConstant... | [
"α : Type u_1\nβ : Type u_2\ninst✝³ : Semifield α\ninst✝² : LinearOrder α\ninst✝¹ : IsStrictOrderedRing α\nl : Filter β\nf : β → α\nr : α\ninst✝ : l.NeBot\nh : Tendsto f l atTop\nhrf : Tendsto (fun x ↦ r * f x) l atTop\nhr : r ≤ 0\nx : β\nhx : 0 ≤ f x\nhrx : 0 < r * f x\n⊢ False"
] | rcases ((h.eventually_ge_atTop 0).and (hrf.eventually_gt_atTop 0)).exists with ⟨x, hx, hrx⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Order.Filter.AtTopBot.Field | {
"line": 132,
"column": 61
} | {
"line": 133,
"column": 59
} | {
"line": 135,
"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 ↦ f x / r) l atBot ↔ Tendsto f l atBot",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWit... | [] | by
simp [div_eq_mul_inv, tendsto_mul_const_atBot_of_pos, hr] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.GroupWithZero | {
"line": 331,
"column": 4
} | {
"line": 332,
"column": 42
} | {
"line": 333,
"column": 4
} | [
{
"pp": "G₀ : Type u_3\ninst✝² : TopologicalSpace G₀\ninst✝¹ : GroupWithZero G₀\ninst✝ : SeparatelyContinuousMul G₀\nh : Tendsto Inv.inv (𝓝 1) (𝓝 1)\nx : G₀\nhx : x ≠ 0\nhx' : x⁻¹ ≠ 0\n⊢ ContinuousAt Inv.inv x",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWi... | [
"G₀ : Type u_3\ninst✝² : TopologicalSpace G₀\ninst✝¹ : GroupWithZero G₀\ninst✝ : SeparatelyContinuousMul G₀\nh : Tendsto Inv.inv (𝓝 1) (𝓝 1)\nx : G₀\nhx : x ≠ 0\nhx' : x⁻¹ ≠ 0\n⊢ Tendsto ((fun x_1 ↦ x_1 * x⁻¹⁻¹) ∘ Inv.inv ∘ fun x_1 ↦ x * x_1) (𝓝 1) (𝓝 1)"
] | rw [ContinuousAt, ← map_mul_left_nhds_one₀ hx, ← nhds_translation_mul_inv₀ hx',
tendsto_map'_iff, tendsto_comap_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
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