module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Topology.UniformSpace.LocallyUniformConvergence | {
"line": 246,
"column": 59
} | {
"line": 246,
"column": 70
} | {
"line": 246,
"column": 71
} | [
{
"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\nh : ∀ (x : α), ∃ t ∈ 𝓝 x, TendstoUniformlyOn F f p t\n⊢ ∀ x ∈ univ, ∃ t ∈ 𝓝[univ] x, TendstoUniformlyOn F f p t",
"ppTerm": "?m.39",
"assigned": true,
"us... | [
"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\nh : ∀ (x : α), ∃ t ∈ 𝓝 x, TendstoUniformlyOn F f p t\n⊢ ∀ (x : α), ∃ t ∈ 𝓝 x, TendstoUniformlyOn F f p t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.LocallyUniformConvergence | {
"line": 289,
"column": 2
} | {
"line": 289,
"column": 67
} | {
"line": 290,
"column": 4
} | [
{
"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\n⊢ TendstoLocallyUniformly F f p ↔ ∀ (x : α), TendstoUniformlyOnFilter F f p (𝓝 x)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [
"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\n⊢ TendstoLocallyUniformlyOn F f p univ ↔ ∀ (x : α), TendstoUniformlyOnFilter F f p (𝓝[univ] x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.LocallyUniformConvergence | {
"line": 295,
"column": 2
} | {
"line": 295,
"column": 47
} | {
"line": 295,
"column": 48
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns : Set α\np : Filter ι\nhf : TendstoLocallyUniformlyOn F f p s\na : α\nha : a ∈ s\n⊢ 𝓟 {a} ≤ 𝓝[s] a",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Pu... | [
"α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns : Set α\np : Filter ι\nhf : TendstoLocallyUniformlyOn F f p s\na : α\nha : a ∈ s\n⊢ pure a ≤ 𝓝[s] a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.LocallyUniformConvergence | {
"line": 305,
"column": 4
} | {
"line": 305,
"column": 15
} | {
"line": 305,
"column": 16
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns : Set α\np : Filter ι\nG : ι → α → β\nhf : TendstoLocallyUniformlyOn F f p s\nhg : ∀ᶠ (n : ι) in p, ∀ x ∈ s, Inseparable (F n x) (G n x)\n⊢ ∀ᶠ (x : ι × α) in p ×ˢ 𝓟 s, Inseparable... | [
"α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns : Set α\np : Filter ι\nG : ι → α → β\nhf : TendstoLocallyUniformlyOn F f p s\nhg : ∀ᶠ (n : ι) in p, ∀ x ∈ s, Inseparable (F n x) (G n x)\n⊢ ∀ᶠ (x : ι × α) in p ×ˢ 𝓟 s, Inseparable (F x.1 x.2)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.LocallyUniformConvergence | {
"line": 336,
"column": 56
} | {
"line": 336,
"column": 67
} | {
"line": 336,
"column": 68
} | [
{
"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\nG : ι → α → β\nhf : TendstoLocallyUniformly F f p\nhg : ∀ᶠ (n : ι) in p, ∀ (x : α), Inseparable (F n x) (G n x)\n⊢ ∀ᶠ (n : ι) in p, ∀ x ∈ univ, Inseparable (F n x) (G n... | [
"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\nG : ι → α → β\nhf : TendstoLocallyUniformly F f p\nhg : ∀ᶠ (n : ι) in p, ∀ (x : α), Inseparable (F n x) (G n x)\n⊢ ∀ᶠ (n : ι) in p, ∀ (x : α), Inseparable (F n x) (G n x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.LocallyUniformConvergence | {
"line": 346,
"column": 62
} | {
"line": 346,
"column": 73
} | {
"line": 346,
"column": 74
} | [
{
"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\ng : α → β\nhf : TendstoLocallyUniformly F f p\nhg : ∀ (x : α), Inseparable (f x) (g x)\n⊢ ∀ x ∈ univ, Inseparable (f x) (g x)",
"ppTerm": "?m.43",
"assigned": t... | [
"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\ng : α → β\nhf : TendstoLocallyUniformly F f p\nhg : ∀ (x : α), Inseparable (f x) (g x)\n⊢ ∀ (x : α), Inseparable (f x) (g x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.MinMax | {
"line": 60,
"column": 2
} | {
"line": 60,
"column": 35
} | {
"line": 60,
"column": 36
} | [
{
"pp": "α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedMonoid α\na b c : α\n⊢ min (a / c) (b / c) = min a b / c",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMu... | [
"α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedMonoid α\na b c : α\n⊢ min (a * c⁻¹) (b * c⁻¹) = min a b * c⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.MinMax | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 35
} | {
"line": 64,
"column": 36
} | [
{
"pp": "α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedMonoid α\na b c : α\n⊢ max (a / c) (b / c) = max a b / c",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"Lattice.toSemilatticeSup",
"instHDiv",
... | [
"α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedMonoid α\na b c : α\n⊢ max (a * c⁻¹) (b * c⁻¹) = max a b * c⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.MinMax | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 59
} | {
"line": 91,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝² : AddCommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedAddMonoid α\na b c d : α\n⊢ |min a b - min c d| ≤ max |a - c| |b - d|",
"ppTerm": "?m.32",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝² : AddCommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedAddMonoid α\na b c d : α\n⊢ |min a b - min c d| ≤ max |a - c| |b - d|"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.MinMax | {
"line": 94,
"column": 2
} | {
"line": 95,
"column": 9
} | {
"line": 95,
"column": 10
} | [
{
"pp": "α : Type u_1\ninst✝² : AddCommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedAddMonoid α\na b c : α\n⊢ |max a c - max b c| ≤ |a - b|",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝² : AddCommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedAddMonoid α\na b c : α\n⊢ |max a c - max b c| ≤ |a - b|"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Monoid.Defs | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 13
} | {
"line": 86,
"column": 14
} | [
{
"pp": "α : Type u_2\nE : Type u_3\ninst✝² : CommGroup E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousMul E\nf g : α → E\nm : E\nx : Filter α\nhf : Tendsto f x (𝓝 m)\nhfg : Tendsto (g / f) x (𝓝 1)\n⊢ Tendsto g x (𝓝 m)",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars... | [
"α : Type u_2\nE : Type u_3\ninst✝² : CommGroup E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousMul E\nf g : α → E\nm : E\nx : Filter α\nhf : Tendsto f x (𝓝 m)\nhfg : Tendsto (g / f) x (𝓝 1)\n⊢ Tendsto g x (𝓝 m)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 698,
"column": 4
} | {
"line": 700,
"column": 71
} | {
"line": 701,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\ninst✝² : UniformSpace β\n𝔖 : Set (Set α)\nι : Type u_5\ninst✝¹ : Preorder ι\ninst✝ : IsDirectedOrder ι\nt : ι → Set α\nV : ι → Set (β × β)\nht : ∀ (n : ι), t n ∈ 𝔖\nhmono : Monotone t\nhex : ∀ s ∈ 𝔖, ∃ n, s ⊆ t n\nhb : (𝓤 β).HasAntitoneBasis V\nthis : None... | [] | rintro ⟨k, l⟩ -
rcases directed_of (· ≤ ·) k l with ⟨n, hkn, hln⟩
exact ⟨n, trivial, UniformOnFun.gen_mono (hmono hkn) (hb.2 <| hln)⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 698,
"column": 4
} | {
"line": 700,
"column": 71
} | {
"line": 701,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\ninst✝² : UniformSpace β\n𝔖 : Set (Set α)\nι : Type u_5\ninst✝¹ : Preorder ι\ninst✝ : IsDirectedOrder ι\nt : ι → Set α\nV : ι → Set (β × β)\nht : ∀ (n : ι), t n ∈ 𝔖\nhmono : Monotone t\nhex : ∀ s ∈ 𝔖, ∃ n, s ⊆ t n\nhb : (𝓤 β).HasAntitoneBasis V\nthis : None... | [] | rintro ⟨k, l⟩ -
rcases directed_of (· ≤ ·) k l with ⟨n, hkn, hln⟩
exact ⟨n, trivial, UniformOnFun.gen_mono (hmono hkn) (hb.2 <| hln)⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.Equicontinuity | {
"line": 561,
"column": 2
} | {
"line": 562,
"column": 53
} | {
"line": 564,
"column": 0
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nα' : Type u_7\nβ : Type u_8\nuβ : UniformSpace β\nu : κ → UniformSpace α'\nF : ι → β → α'\n⊢ UniformEquicontinuous F ↔ ∀ (k : κ), UniformEquicontinuous F",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"UniformContinuous",
"UniformSpace",
... | [] | simp_rw [uniformEquicontinuous_iff_uniformContinuous (uα := _)]
rw [UniformFun.iInf_eq, uniformContinuous_iInf_rng] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.Equicontinuity | {
"line": 561,
"column": 2
} | {
"line": 562,
"column": 53
} | {
"line": 564,
"column": 0
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nα' : Type u_7\nβ : Type u_8\nuβ : UniformSpace β\nu : κ → UniformSpace α'\nF : ι → β → α'\n⊢ UniformEquicontinuous F ↔ ∀ (k : κ), UniformEquicontinuous F",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"UniformContinuous",
"UniformSpace",
... | [] | simp_rw [uniformEquicontinuous_iff_uniformContinuous (uα := _)]
rw [UniformFun.iInf_eq, uniformContinuous_iInf_rng] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.NAry | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 53
} | {
"line": 68,
"column": 54
} | [
{
"pp": "α : Type u_1\nβ : Type u_3\nγ : Type u_5\nf : Filter α\ng : Filter β\nι : Type u_11\nι' : Type u_12\np : ι → Prop\nq : ι' → Prop\ns : ι → Set α\nt : ι' → Set β\nm : α → β → γ\nhf : f.HasBasis p s\nhg : g.HasBasis q t\n⊢ (map₂ m f g).HasBasis (fun i ↦ p i.1 ∧ q i.2) fun i ↦ image2 m (s i.1) (t i.2)",
... | [
"α : Type u_1\nβ : Type u_3\nγ : Type u_5\nf : Filter α\ng : Filter β\nι : Type u_11\nι' : Type u_12\np : ι → Prop\nq : ι' → Prop\ns : ι → Set α\nt : ι' → Set β\nm : α → β → γ\nhf : f.HasBasis p s\nhg : g.HasBasis q t\n⊢ (Filter.map (fun p ↦ m p.1 p.2) (f ×ˢ g)).HasBasis (fun i ↦ p i.1 ∧ q i.2) fun i ↦\n (fun p ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 1127,
"column": 2
} | {
"line": 1127,
"column": 69
} | {
"line": 1128,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nδ₁ : Type u_5\nδ₂ : Type u_6\nφ₁ : δ₁ → α\nφ₂ : δ₂ → α\n𝔗₁ : Set (Set δ₁)\n𝔗₂ : Set (Set δ₂)\nh_image₁ : MapsTo (fun x ↦ φ₁ '' x) 𝔗₁ 𝔖\nh_image₂ : MapsTo (fun x ↦ φ₂ '' x) 𝔗₂ 𝔖\nh_preimage₁ : MapsTo (fun x ↦ φ₁ ⁻¹' x) 𝔖 𝔗₁\nh... | [
"α : Type u_1\nβ : Type u_2\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nδ₁ : Type u_5\nδ₂ : Type u_6\nφ₁ : δ₁ → α\nφ₂ : δ₂ → α\n𝔗₁ : Set (Set δ₁)\n𝔗₂ : Set (Set δ₂)\nh_image₁ : MapsTo (fun x ↦ φ₁ '' x) 𝔗₁ 𝔖\nh_image₂ : MapsTo (fun x ↦ φ₂ '' x) 𝔗₂ 𝔖\nh_preimage₁ : MapsTo (fun x ↦ φ₁ ⁻¹' x) 𝔖 𝔗₁\nh_preimage₂ :... | set ψ₁ : Π S : Set α, φ₁ ⁻¹' S → S := fun S ↦ S.restrictPreimage φ₁ | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 1131,
"column": 4
} | {
"line": 1132,
"column": 62
} | {
"line": 1132,
"column": 63
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nδ₁ : Type u_5\nδ₂ : Type u_6\nφ₁ : δ₁ → α\nφ₂ : δ₂ → α\n𝔗₁ : Set (Set δ₁)\n𝔗₂ : Set (Set δ₂)\nh_image₁ : MapsTo (fun x ↦ φ₁ '' x) 𝔗₁ 𝔖\nh_image₂ : MapsTo (fun x ↦ φ₂ '' x) 𝔗₂ 𝔖\nh_preimage₁ : MapsTo (fun x ↦ φ₁ ⁻¹' x) 𝔖 𝔗₁\nh... | [
"α : Type u_1\nβ : Type u_2\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nδ₁ : Type u_5\nδ₂ : Type u_6\nφ₁ : δ₁ → α\nφ₂ : δ₂ → α\n𝔗₁ : Set (Set δ₁)\n𝔗₂ : Set (Set δ₂)\nh_image₁ : MapsTo (fun x ↦ φ₁ '' x) 𝔗₁ 𝔖\nh_image₂ : MapsTo (fun x ↦ φ₂ '' x) 𝔗₂ 𝔖\nh_preimage₁ : MapsTo (fun x ↦ φ₁ ⁻¹' x) 𝔖 𝔗₁\nh_preimage₂ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 1154,
"column": 4
} | {
"line": 1155,
"column": 62
} | {
"line": 1155,
"column": 63
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nδ : ι → Type u_5\nφ : (i : ι) → δ i → α\n𝔗 : (i : ι) → Set (Set (δ i))\nh_image : ∀ (i : ι), MapsTo (fun x ↦ φ i '' x) (𝔗 i) 𝔖\nh_preimage : ∀ (i : ι), MapsTo (fun x ↦ φ i ⁻¹' x) 𝔖 (𝔗 i)\nh_cover : ∀ S ∈ 𝔖, ∃ I, I... | [
"α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nδ : ι → Type u_5\nφ : (i : ι) → δ i → α\n𝔗 : (i : ι) → Set (Set (δ i))\nh_image : ∀ (i : ι), MapsTo (fun x ↦ φ i '' x) (𝔗 i) 𝔖\nh_preimage : ∀ (i : ι), MapsTo (fun x ↦ φ i ⁻¹' x) 𝔖 (𝔗 i)\nh_cover : ∀ S ∈ 𝔖, ∃ I, I.Finite ∧ S ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Pointwise | {
"line": 236,
"column": 2
} | {
"line": 236,
"column": 13
} | {
"line": 236,
"column": 14
} | [
{
"pp": "α : Type u_2\ninst✝ : InvolutiveInv α\nf : Filter α\nι : Sort u_7\np : ι → Prop\ns : ι → Set α\nh : f.HasBasis p s\n⊢ f⁻¹.HasBasis p fun i ↦ (s i)⁻¹",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\ninst✝ : InvolutiveInv α\nf : Filter α\nι : Sort u_7\np : ι → Prop\ns : ι → Set α\nh : f.HasBasis p s\n⊢ f⁻¹.HasBasis p fun i ↦ (s i)⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 1214,
"column": 65
} | {
"line": 1214,
"column": 76
} | {
"line": 1214,
"column": 77
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\np : Filter ι\ns : Set β\nF : ι → α → β\nf : α → β\nt : Set α\nhF : ∀ᶠ (i : ι) in p, ∀ x ∈ t, F i x ∈ s\nhf : ∀ x ∈ t, f x ∈ s\ng : β → γ\nhg : UniformContinuousOn g s\nh : TendstoUniformlyOn F f p t... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\np : Filter ι\ns : Set β\nF : ι → α → β\nf : α → β\nt : Set α\nhF : ∀ᶠ (i : ι) in p, ∀ x ∈ t, F i x ∈ s\nhf : ∀ x ∈ t, f x ∈ s\ng : β → γ\nhg : UniformContinuousOn g s\nh : TendstoUniformlyOn F f p t\n⊢ ∀ᶠ (i : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 1215,
"column": 9
} | {
"line": 1215,
"column": 20
} | {
"line": 1215,
"column": 21
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\np : Filter ι\ns : Set β\nF : ι → α → β\nf : α → β\nt : Set α\nhF : ∀ᶠ (i : ι) in p, ∀ x ∈ t, F i x ∈ s\nhf : ∀ x ∈ t, f x ∈ s\ng : β → γ\nhg : UniformContinuousOn g s\nh : TendstoUniformlyOn F f p t... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\np : Filter ι\ns : Set β\nF : ι → α → β\nf : α → β\nt : Set α\nhF : ∀ᶠ (i : ι) in p, ∀ x ∈ t, F i x ∈ s\nhf : ∀ x ∈ t, f x ∈ s\ng : β → γ\nhg : UniformContinuousOn g s\nh : TendstoUniformlyOn F f p t\n⊢ ∀ a ∈ t,... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.MulAction | {
"line": 145,
"column": 4
} | {
"line": 145,
"column": 38
} | {
"line": 145,
"column": 39
} | [
{
"pp": "M : Type u_1\nX : Type u_2\nY : Type u_3\nα : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : SMul M X\ninst✝² : ContinuousSMul M X\nf : Y → M\ng : Y → X\nb : Y\ns : Set Y\ninst✝¹ : SMul Mᵐᵒᵖ X\ninst✝ : IsCentralScalar M X\n⊢ Continuous[instTopo... | [
"M : Type u_1\nX : Type u_2\nY : Type u_3\nα : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : SMul M X\ninst✝² : ContinuousSMul M X\nf : Y → M\ng : Y → X\nb : Y\ns : Set Y\ninst✝¹ : SMul Mᵐᵒᵖ X\ninst✝ : IsCentralScalar M X\n⊢ Continuous[instTopologicalSpace... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.MulAction | {
"line": 193,
"column": 4
} | {
"line": 194,
"column": 11
} | {
"line": 194,
"column": 12
} | [
{
"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... | [
"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✝, inst✝⁶] f\nhs... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Pointwise | {
"line": 737,
"column": 2
} | {
"line": 737,
"column": 13
} | {
"line": 737,
"column": 14
} | [
{
"pp": "α : Type u_2\ninst✝ : Group α\nf : Filter α\nh : f.NeBot\n⊢ 1 ≤ f / f",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"Filter.instDiv",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
"Filter.instCom... | [
"α : Type u_2\ninst✝ : Group α\nf : Filter α\nh : f.NeBot\n⊢ ¬f = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Pointwise | {
"line": 1157,
"column": 2
} | {
"line": 1157,
"column": 48
} | {
"line": 1157,
"column": 49
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝¹ : Monoid γ\ninst✝ : MulAction γ β\nm : α → β\nc : γ\nf : Filter α\ng : Filter β\nhc : IsUnit c\nd : γ\nhd : d * c = 1\nH : Tendsto (c • m) f (c • g)\n⊢ Tendsto m f g",
"ppTerm": "?m.33",
"assigned": false,
"usedConstants": [],
"usedFVars"... | [
"α : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝¹ : Monoid γ\ninst✝ : MulAction γ β\nm : α → β\nc : γ\nf : Filter α\ng : Filter β\nhc : IsUnit c\nd : γ\nhd : d * c = 1\nH : Tendsto (c • m) f (c • g)\n⊢ Tendsto m f g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Basic | {
"line": 370,
"column": 4
} | {
"line": 371,
"column": 11
} | {
"line": 371,
"column": 12
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf g : C(α, β)\nι : Type u_5\nS : ι → Set α\nφ : (i : ι) → C(↑(S i), β)\nhφ : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), (φ... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf g : C(α, β)\nι : Type u_5\nS : ι → Set α\nφ : (i : ι) → C(↑(S i), β)\nhφ : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), (φ i) ⟨x, hxi⟩... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Basic | {
"line": 474,
"column": 2
} | {
"line": 474,
"column": 13
} | {
"line": 474,
"column": 14
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nf : C(X, Y)\nhf : IsQuotientMap ⇑f\ng : C(X, Z)\nh : FactorsThrough ⇑g ⇑f\na✝ : X\n⊢ ((hf.lift g h).comp f) a✝ = g a✝",
"ppTerm": "?m.37",
"assigned": true,
"usedC... | [
"X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nf : C(X, Y)\nhf : IsQuotientMap ⇑f\ng : C(X, Z)\nh : FactorsThrough ⇑g ⇑f\na✝ : X\n⊢ g (surjInv ⋯ (f a✝)) = g a✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Basic | {
"line": 485,
"column": 4
} | {
"line": 485,
"column": 15
} | {
"line": 485,
"column": 16
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nf : C(X, Y)\nhf : IsQuotientMap ⇑f\ng✝ : C(X, Z)\nh : FactorsThrough ⇑g✝ ⇑f\ng : C(Y, Z)\na : Y\n⊢ ((fun g ↦ hf.lift ↑g ⋯) ((fun g ↦ ⟨g.comp f, ⋯⟩) g)) a = g a",
"ppTerm":... | [
"X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nf : C(X, Y)\nhf : IsQuotientMap ⇑f\ng✝ : C(X, Z)\nh : FactorsThrough ⇑g✝ ⇑f\ng : C(Y, Z)\na : Y\n⊢ g (f (surjInv ⋯ a)) = g a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.ConstMulAction | {
"line": 118,
"column": 32
} | {
"line": 118,
"column": 66
} | {
"line": 118,
"column": 67
} | [
{
"pp": "M : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : SMul M α\ninst✝³ : ContinuousConstSMul M α\ninst✝² : TopologicalSpace β\ng : β → α\nb : β\ns : Set β\ninst✝¹ : SMul Mᵐᵒᵖ α\ninst✝ : IsCentralScalar M α\nc : M\n⊢ Continuous[inst✝⁵, inst✝⁵] fun x ↦ MulOpposite.op c • x",
... | [
"M : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : SMul M α\ninst✝³ : ContinuousConstSMul M α\ninst✝² : TopologicalSpace β\ng : β → α\nb : β\ns : Set β\ninst✝¹ : SMul Mᵐᵒᵖ α\ninst✝ : IsCentralScalar M α\nc : M\n⊢ Continuous[inst✝⁵, inst✝⁵] fun x ↦ c • x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.ConstMulAction | {
"line": 160,
"column": 4
} | {
"line": 160,
"column": 63
} | {
"line": 160,
"column": 64
} | [
{
"pp": "M : Type u_1\nα : Type u_2\ninst✝⁴ : TopologicalSpace α\ninst✝³ : SMul M α\ninst✝² : ContinuousConstSMul M α\nN : Type u_4\nβ : Type u_5\ninst✝¹ : SMul N β\ninst✝ : TopologicalSpace β\ng : β → α\nhg : IsInducing g\nf : N → M\nhf : ∀ {c : N} {x : β}, g (c • x) = f c • g x\nc : N\n⊢ Continuous[inst✝, ins... | [
"M : Type u_1\nα : Type u_2\ninst✝⁴ : TopologicalSpace α\ninst✝³ : SMul M α\ninst✝² : ContinuousConstSMul M α\nN : Type u_4\nβ : Type u_5\ninst✝¹ : SMul N β\ninst✝ : TopologicalSpace β\ng : β → α\nhg : IsInducing g\nf : N → M\nhf : ∀ {c : N} {x : β}, g (c • x) = f c • g x\nc : N\n⊢ Continuous[inst✝, inst✝⁴] fun x ↦... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.ConstMulAction | {
"line": 204,
"column": 15
} | {
"line": 204,
"column": 47
} | {
"line": 204,
"column": 48
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nG : Type u_4\ninst✝³ : TopologicalSpace α\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : ContinuousConstSMul G α\nf : β → α\nl : Filter β\na : α\nc : G\nh : Tendsto (fun x ↦ c • f x) l (𝓝 (c • a))\n⊢ Tendsto f l (𝓝 a)",
"ppTerm": "?m.24",
"assigned": false,
... | [
"α : Type u_2\nβ : Type u_3\nG : Type u_4\ninst✝³ : TopologicalSpace α\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : ContinuousConstSMul G α\nf : β → α\nl : Filter β\na : α\nc : G\nh : Tendsto (fun x ↦ c • f x) l (𝓝 (c • a))\n⊢ Tendsto f l (𝓝 a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.ConstMulAction | {
"line": 308,
"column": 2
} | {
"line": 308,
"column": 36
} | {
"line": 309,
"column": 2
} | [
{
"pp": "α : Type u_2\nG : Type u_4\ninst✝⁴ : TopologicalSpace α\ninst✝³ : Group G\ninst✝² : MulAction G α\ninst✝¹ : ContinuousConstSMul G α\nx : α\ninst✝ : IsPretransitive G α\nhx : IsClosed[inst✝⁴] {x}\ng : G\n⊢ IsClosed[inst✝⁴] {g • x}",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_2\nG : Type u_4\ninst✝⁴ : TopologicalSpace α\ninst✝³ : Group G\ninst✝² : MulAction G α\ninst✝¹ : ContinuousConstSMul G α\nx : α\ninst✝ : IsPretransitive G α\nhx : IsClosed[inst✝⁴] {x}\ng : G\n⊢ IsClosed[inst✝⁴] (g • {x})"
] | rw [← image_singleton, image_smul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.ConstMulAction | {
"line": 317,
"column": 2
} | {
"line": 317,
"column": 36
} | {
"line": 318,
"column": 2
} | [
{
"pp": "α : Type u_2\nG : Type u_4\ninst✝⁴ : TopologicalSpace α\ninst✝³ : Group G\ninst✝² : MulAction G α\ninst✝¹ : ContinuousConstSMul G α\nx : α\ninst✝ : IsPretransitive G α\nhx : IsOpen[inst✝⁴] {x}\ng : G\n⊢ IsOpen[inst✝⁴] {g • x}",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"... | [
"α : Type u_2\nG : Type u_4\ninst✝⁴ : TopologicalSpace α\ninst✝³ : Group G\ninst✝² : MulAction G α\ninst✝¹ : ContinuousConstSMul G α\nx : α\ninst✝ : IsPretransitive G α\nhx : IsOpen[inst✝⁴] {x}\ng : G\n⊢ IsOpen[inst✝⁴] (g • {x})"
] | rw [← image_singleton, image_smul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 482,
"column": 2
} | {
"line": 482,
"column": 29
} | {
"line": 482,
"column": 30
} | [
{
"pp": "α : Type u_1\ninst✝² : AddCommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedAddMonoid α\na b c : α\n⊢ (fun x ↦ x + a) ⁻¹' [[b, c]] = [[b - a, c - a]]",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"AddCommGroup.toAddCommMonoid",
"... | [
"α : Type u_1\ninst✝² : AddCommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedAddMonoid α\na b c : α\n⊢ (fun x ↦ a + x) ⁻¹' [[b, c]] = [[b - a, c - a]]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 786,
"column": 2
} | {
"line": 786,
"column": 29
} | {
"line": 786,
"column": 30
} | [
{
"pp": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na c : α\nh : c < 0\n⊢ (fun x ↦ c * x) ⁻¹' Iio a = Ioi (a / c)",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na c : α\nh : c < 0\n⊢ (fun x ↦ c * x) ⁻¹' Iio a = Ioi (a / c)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 791,
"column": 2
} | {
"line": 791,
"column": 29
} | {
"line": 791,
"column": 30
} | [
{
"pp": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na c : α\nh : c < 0\n⊢ (fun x ↦ c * x) ⁻¹' Ioi a = Iio (a / c)",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na c : α\nh : c < 0\n⊢ (fun x ↦ c * x) ⁻¹' Ioi a = Iio (a / c)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 796,
"column": 2
} | {
"line": 796,
"column": 29
} | {
"line": 796,
"column": 30
} | [
{
"pp": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na c : α\nh : c < 0\n⊢ (fun x ↦ c * x) ⁻¹' Iic a = Ici (a / c)",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na c : α\nh : c < 0\n⊢ (fun x ↦ c * x) ⁻¹' Iic a = Ici (a / c)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 801,
"column": 2
} | {
"line": 801,
"column": 29
} | {
"line": 801,
"column": 30
} | [
{
"pp": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na c : α\nh : c < 0\n⊢ (fun x ↦ c * x) ⁻¹' Ici a = Iic (a / c)",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na c : α\nh : c < 0\n⊢ (fun x ↦ c * x) ⁻¹' Ici a = Iic (a / c)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 806,
"column": 2
} | {
"line": 806,
"column": 29
} | {
"line": 806,
"column": 30
} | [
{
"pp": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na b c : α\nh : c < 0\n⊢ (fun x ↦ c * x) ⁻¹' Ioo a b = Ioo (b / c) (a / c)",
"ppTerm": "?m.27",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na b c : α\nh : c < 0\n⊢ (fun x ↦ c * x) ⁻¹' Ioo a b = Ioo (b / c) (a / c)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 811,
"column": 2
} | {
"line": 811,
"column": 29
} | {
"line": 811,
"column": 30
} | [
{
"pp": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na b c : α\nh : c < 0\n⊢ (fun x ↦ c * x) ⁻¹' Ioc a b = Ico (b / c) (a / c)",
"ppTerm": "?m.27",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na b c : α\nh : c < 0\n⊢ (fun x ↦ c * x) ⁻¹' Ioc a b = Ico (b / c) (a / c)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 816,
"column": 2
} | {
"line": 816,
"column": 29
} | {
"line": 816,
"column": 30
} | [
{
"pp": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na b c : α\nh : c < 0\n⊢ (fun x ↦ c * x) ⁻¹' Ico a b = Ioc (b / c) (a / c)",
"ppTerm": "?m.27",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na b c : α\nh : c < 0\n⊢ (fun x ↦ c * x) ⁻¹' Ico a b = Ioc (b / c) (a / c)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 821,
"column": 2
} | {
"line": 821,
"column": 29
} | {
"line": 821,
"column": 30
} | [
{
"pp": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na b c : α\nh : c < 0\n⊢ (fun x ↦ c * x) ⁻¹' Icc a b = Icc (b / c) (a / c)",
"ppTerm": "?m.27",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na b c : α\nh : c < 0\n⊢ (fun x ↦ c * x) ⁻¹' Icc a b = Icc (b / c) (a / c)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 864,
"column": 2
} | {
"line": 864,
"column": 29
} | {
"line": 864,
"column": 30
} | [
{
"pp": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na b c : α\n⊢ (fun x ↦ a * x) '' [[b, c]] = [[a * b, a * c]]",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na b c : α\n⊢ (fun x ↦ a * x) '' [[b, c]] = [[a * b, a * c]]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Monoid | {
"line": 185,
"column": 4
} | {
"line": 185,
"column": 31
} | {
"line": 185,
"column": 32
} | [
{
"pp": "M : Type u_6\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\na b : M\nh : a ⤳ b\nn x✝ : ℕ\nihn : (a ^ x✝) ⤳ (b ^ x✝)\n⊢ (a ^ x✝.succ) ⤳ (b ^ x✝.succ)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Specializes",
... | [
"M : Type u_6\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\na b : M\nh : a ⤳ b\nn x✝ : ℕ\nihn : (a ^ x✝) ⤳ (b ^ x✝)\n⊢ (a ^ x✝ * a) ⤳ (b ^ x✝ * b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Monoid | {
"line": 202,
"column": 4
} | {
"line": 202,
"column": 15
} | {
"line": 202,
"column": 16
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Monoid | {
"line": 205,
"column": 4
} | {
"line": 205,
"column": 15
} | {
"line": 205,
"column": 16
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Monoid | {
"line": 441,
"column": 4
} | {
"line": 441,
"column": 18
} | {
"line": 442,
"column": 4
} | [
{
"pp": "case right\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : MulZeroClass M\ninst✝ : ContinuousMul M\nK U : Set M\nhK : IsCompact K\nhU : U ∈ 𝓝 0\ns t V : Set M\nV_in : V ∈ 𝓝 0\nhV' : s * V ⊆ U\nW : Set M\nW_in : W ∈ 𝓝 0\nhW' : t * W ⊆ U\n⊢ (s ∪ t) * (V ∩ W) ⊆ U",
"ppTerm": "?right",
"ass... | [
"case right\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : MulZeroClass M\ninst✝ : ContinuousMul M\nK U : Set M\nhK : IsCompact K\nhU : U ∈ 𝓝 0\ns t V : Set M\nV_in : V ∈ 𝓝 0\nhV' : s * V ⊆ U\nW : Set M\nW_in : W ∈ 𝓝 0\nhW' : t * W ⊆ U\n⊢ s * (V ∩ W) ∪ t * (V ∩ W) ⊆ U"
] | rw [union_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.Monoid | {
"line": 446,
"column": 47
} | {
"line": 446,
"column": 58
} | {
"line": 446,
"column": 59
} | [
{
"pp": "M : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : MulZeroClass M\ninst✝ : ContinuousMul M\nK U : Set M\nhK : IsCompact K\nhU : U ∈ 𝓝 0\nx : M\nhx : x ∈ K\n⊢ U ∈ 𝓝 (x * 0)",
"ppTerm": "?m.200",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"HMu... | [
"M : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : MulZeroClass M\ninst✝ : ContinuousMul M\nK U : Set M\nhK : IsCompact K\nhU : U ∈ 𝓝 0\nx : M\nhx : x ∈ K\n⊢ U ∈ 𝓝 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Monoid | {
"line": 498,
"column": 2
} | {
"line": 498,
"column": 36
} | {
"line": 498,
"column": 37
} | [
{
"pp": "M : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : MulZeroClass M\ninst✝ : ContinuousMul M\nl : Filter (M × M)\nhl : Disjoint l (cocompact (M × M))\nh'l : l ≤ (𝓝 0).coprod (𝓝 0)\n⊢ Tendsto (fun x ↦ x.1 * x.2) l (𝓝 0)",
"ppTerm": "?m.33",
"assigned": false,
"usedConstants": [],
"used... | [
"M : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : MulZeroClass M\ninst✝ : ContinuousMul M\nl : Filter (M × M)\nhl : Disjoint l (cocompact (M × M))\nh'l : l ≤ (𝓝 0).coprod (𝓝 0)\n⊢ Tendsto (fun x ↦ x.1 * x.2) l (𝓝 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Maps.Proper.Basic | {
"line": 240,
"column": 4
} | {
"line": 240,
"column": 55
} | {
"line": 243,
"column": 4
} | [
{
"pp": "case mpr\nX : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\nH : Continuous[inst✝¹, inst✝] f ∧ IsClosedMap f ∧ ∀ (y : Y), IsCompact (f ⁻¹' {y})\nℱ : Filter X\ny : Y\nhy : MapClusterPt y ℱ f\n⊢ ∃ x, f x = y ∧ ClusterPt x ℱ",
"ppTerm": "?mpr",
"assigne... | [
"case mpr\nX : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\nH : Continuous[inst✝¹, inst✝] f ∧ IsClosedMap f ∧ ∀ (y : Y), IsCompact (f ⁻¹' {y})\nℱ : Filter X\ny : Y\nhy : (ℱ.lift' closure[inst✝¹] ⊓ 𝓟 (f ⁻¹' {y})).NeBot\n⊢ ∃ x, f x = y ∧ ClusterPt x ℱ"
] | rw [H.2.1.mapClusterPt_iff_lift'_closure H.1] at hy | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Maps.Proper.Basic | {
"line": 312,
"column": 14
} | {
"line": 312,
"column": 25
} | {
"line": 312,
"column": 26
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : T1Space Y\ny : Y\nh : ∀ (y_1 : Y), IsCompact ((fun x ↦ y) ⁻¹' {y_1})\n⊢ IsCompact univ",
"ppTerm": "?m.39",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : T1Space Y\ny : Y\nh : ∀ (y_1 : Y), IsCompact ((fun x ↦ y) ⁻¹' {y_1})\n⊢ IsCompact univ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Monoid | {
"line": 562,
"column": 23
} | {
"line": 562,
"column": 39
} | {
"line": 562,
"column": 40
} | [
{
"pp": "M : Type u_3\ninst✝⁴ : GroupWithZero M\ninst✝³ : TopologicalSpace M\ninst✝² : ContinuousMul M\ninst✝¹ : CompactSpace M\ninst✝ : T1Space M\nU : Set M\nhU : IsOpen[inst✝³] U\nh0U : 0 ∈ U\nh1U : 1 ∉ U\nW : Set M\nhW : W ∈ 𝓝 0\nhW' : univ * W ⊆ U\nx : M\nhx : x ≠ 0\nhxW : x ∈ W\n⊢ 1 ∈ univ * W",
"ppTe... | [
"M : Type u_3\ninst✝⁴ : GroupWithZero M\ninst✝³ : TopologicalSpace M\ninst✝² : ContinuousMul M\ninst✝¹ : CompactSpace M\ninst✝ : T1Space M\nU : Set M\nhU : IsOpen[inst✝³] U\nh0U : 0 ∈ U\nh1U : 1 ∉ U\nW : Set M\nhW : W ∈ 𝓝 0\nhW' : univ * W ⊆ U\nx : M\nhx : x ≠ 0\nhxW : x ∈ W\n⊢ 1 ∈ univ * W"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Monoid | {
"line": 564,
"column": 10
} | {
"line": 564,
"column": 35
} | {
"line": 564,
"column": 36
} | [
{
"pp": "M : Type u_3\ninst✝⁴ : GroupWithZero M\ninst✝³ : TopologicalSpace M\ninst✝² : ContinuousMul M\ninst✝¹ : CompactSpace M\ninst✝ : T1Space M\nU : Set M\nhU : IsOpen[inst✝³] U\nh0U : 0 ∈ U\nh1U : 1 ∉ U\nW : Set M\nhW : W ∈ 𝓝 0\nhW' : univ * W ⊆ U\nH : ¬∃ x, x ≠ 0 ∧ x ∈ W\n⊢ W ⊆ {0}",
"ppTerm": "?m.153... | [
"M : Type u_3\ninst✝⁴ : GroupWithZero M\ninst✝³ : TopologicalSpace M\ninst✝² : ContinuousMul M\ninst✝¹ : CompactSpace M\ninst✝ : T1Space M\nU : Set M\nhU : IsOpen[inst✝³] U\nh0U : 0 ∈ U\nh1U : 1 ∉ U\nW : Set M\nhW : W ∈ 𝓝 0\nhW' : univ * W ⊆ U\nH : ¬∃ x, x ≠ 0 ∧ x ∈ W\n⊢ ∀ y ∈ W, y = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Monoid | {
"line": 564,
"column": 43
} | {
"line": 564,
"column": 54
} | {
"line": 564,
"column": 55
} | [
{
"pp": "M : Type u_3\ninst✝⁴ : GroupWithZero M\ninst✝³ : TopologicalSpace M\ninst✝² : ContinuousMul M\ninst✝¹ : CompactSpace M\ninst✝ : T1Space M\nU : Set M\nhU : IsOpen[inst✝³] U\nh0U : 0 ∈ U\nh1U : 1 ∉ U\nW : Set M\nhW : W ∈ 𝓝 0\nhW' : univ * W ⊆ U\nH : ¬∃ x, x ≠ 0 ∧ x ∈ W\n⊢ {0} ⊆ W",
"ppTerm": "?m.154... | [
"M : Type u_3\ninst✝⁴ : GroupWithZero M\ninst✝³ : TopologicalSpace M\ninst✝² : ContinuousMul M\ninst✝¹ : CompactSpace M\ninst✝ : T1Space M\nU : Set M\nhU : IsOpen[inst✝³] U\nh0U : 0 ∈ U\nh1U : 1 ∉ U\nW : Set M\nhW : W ∈ 𝓝 0\nhW' : univ * W ⊆ U\nH : ¬∃ x, x ≠ 0 ∧ x ∈ W\n⊢ 0 ∈ W"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Monoid | {
"line": 592,
"column": 2
} | {
"line": 592,
"column": 35
} | {
"line": 592,
"column": 36
} | [
{
"pp": "M : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : MulOneClass M\ninst✝ : ContinuousMul M\nU : Set M\nhU : U ∈ 𝓝 1\n⊢ ∃ V, IsOpen[inst✝²] V ∧ 1 ∈ V ∧ V * V ⊆ U",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"congrAr... | [
"M : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : MulOneClass M\ninst✝ : ContinuousMul M\nU : Set M\nhU : U ∈ 𝓝 1\n⊢ ∃ V, IsOpen[inst✝²] V ∧ 1 ∈ V ∧ ∀ x ∈ V, ∀ y ∈ V, x * y ∈ U"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Monoid | {
"line": 757,
"column": 2
} | {
"line": 757,
"column": 30
} | {
"line": 757,
"column": 31
} | [
{
"pp": "case right\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\nu : Set M\nhu : u ∈ 𝓝 1\nW : Set M\nW1 : W ∈ 𝓝 1\nh : ∀ v ∈ W, ∀ w ∈ W, v * w ∈ u\nV : Set M\nV1 : V ∈ 𝓝 1\nh' : ∀ v ∈ V, ∀ w ∈ V, v * w ∈ W\nv w s t : M\nv_in : v ∈ V\nw_in : w ∈ V\ns_in : s ∈ V\nt_in... | [
"case right\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\nu : Set M\nhu : u ∈ 𝓝 1\nW : Set M\nW1 : W ∈ 𝓝 1\nh : ∀ v ∈ W, ∀ w ∈ W, v * w ∈ u\nV : Set M\nV1 : V ∈ 𝓝 1\nh' : ∀ v ∈ V, ∀ w ∈ V, v * w ∈ W\nv w s t : M\nv_in : v ∈ V\nw_in : w ∈ V\ns_in : s ∈ V\nt_in : t ∈ V\n⊢ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Monoid | {
"line": 780,
"column": 61
} | {
"line": 786,
"column": 9
} | {
"line": 788,
"column": 0
} | [
{
"pp": "ι : Type u_1\nM : Type u_3\nX : Type u_5\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\nf : ι → X → M\nl : List ι\nt : Set X\nh : ∀ i ∈ l, ContinuousOn (f i) t\n⊢ ContinuousOn (fun a ↦ (List.map (fun i ↦ f i a) l).prod) t",
"ppTerm": "?m.22",
... | [] | by
intro x hx
rw [continuousWithinAt_iff_continuousAt_restrict _ hx]
refine tendsto_list_prod _ fun i hi => ?_
specialize h i hi x hx
rw [continuousWithinAt_iff_continuousAt_restrict _ hx] at h
exact h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Monoid | {
"line": 790,
"column": 12
} | {
"line": 790,
"column": 23
} | {
"line": 790,
"column": 24
} | [
{
"pp": "M : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\n⊢ Continuous[inst✝², inst✝²] fun a ↦ a ^ 0",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Continuous",
"Monoid.toMulOneClass",
"congrArg... | [
"M : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\n⊢ Continuous[inst✝², inst✝²] fun a ↦ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Monoid | {
"line": 914,
"column": 2
} | {
"line": 914,
"column": 13
} | {
"line": 914,
"column": 14
} | [
{
"pp": "case mk\nι : Type u_1\nα : Type u_2\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : CommMonoid M\ninst✝ : ContinuousMul M\nf : ι → α → M\nx : Filter α\na : ι → M\ns : Multiset ι\nl : List ι\n⊢ (∀ i ∈ Quot.mk (⇑(List.isSetoid ι)) l, Tendsto (f i) x (𝓝 (a i))) →\n Tendsto (fun b ↦ (Multiset.map ... | [
"case mk\nι : Type u_1\nα : Type u_2\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : CommMonoid M\ninst✝ : ContinuousMul M\nf : ι → α → M\nx : Filter α\na : ι → M\ns : Multiset ι\nl : List ι\n⊢ (∀ i ∈ l, Tendsto (f i) x (𝓝 (a i))) → Tendsto (fun b ↦ (List.map (fun c ↦ f c b) l).prod) x (𝓝 (List.map a l).prod... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Monoid | {
"line": 931,
"column": 2
} | {
"line": 931,
"column": 13
} | {
"line": 931,
"column": 14
} | [
{
"pp": "case mk\nι : Type u_1\nM : Type u_3\nX : Type u_5\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : CommMonoid M\ninst✝ : ContinuousMul M\nf : ι → X → M\ns : Multiset ι\nl : List ι\n⊢ (∀ i ∈ Quot.mk (⇑(List.isSetoid ι)) l, Continuous[inst✝³, inst✝²] (f i)) →\n Continuous[inst✝³, in... | [
"case mk\nι : Type u_1\nM : Type u_3\nX : Type u_5\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : CommMonoid M\ninst✝ : ContinuousMul M\nf : ι → X → M\ns : Multiset ι\nl : List ι\n⊢ (∀ i ∈ l, Continuous[inst✝³, inst✝²] (f i)) → Continuous[inst✝³, inst✝²] fun a ↦ (List.map (fun i ↦ f i a) l).pro... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Monoid | {
"line": 937,
"column": 2
} | {
"line": 937,
"column": 13
} | {
"line": 937,
"column": 14
} | [
{
"pp": "case mk\nι : Type u_1\nM : Type u_3\nX : Type u_5\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : CommMonoid M\ninst✝ : ContinuousMul M\nf : ι → X → M\ns : Multiset ι\nt : Set X\nl : List ι\n⊢ (∀ i ∈ Quot.mk (⇑(List.isSetoid ι)) l, ContinuousOn (f i) t) →\n ContinuousOn (fun a ↦ ... | [
"case mk\nι : Type u_1\nM : Type u_3\nX : Type u_5\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : CommMonoid M\ninst✝ : ContinuousMul M\nf : ι → X → M\ns : Multiset ι\nt : Set X\nl : List ι\n⊢ (∀ i ∈ l, ContinuousOn (f i) t) → ContinuousOn (fun a ↦ (List.map (fun i ↦ f i a) l).prod) t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Group.Pointwise | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 28
} | {
"line": 114,
"column": 29
} | [
{
"pp": "α : Type u\ninst✝² : TopologicalSpace α\ninst✝¹ : Group α\ninst✝ : ContinuousConstSMul α α\ns : Set α\na : α\nh : s ∈ 𝓝 1\n⊢ {a} * s ∈ 𝓝 a",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\ninst✝² : TopologicalSpace α\ninst✝¹ : Group α\ninst✝ : ContinuousConstSMul α α\ns : Set α\na : α\nh : s ∈ 𝓝 1\n⊢ {a} * s ∈ 𝓝 a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Group.Pointwise | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 28
} | {
"line": 142,
"column": 29
} | [
{
"pp": "α : Type u\ninst✝² : TopologicalSpace α\ninst✝¹ : Group α\ninst✝ : ContinuousConstSMul αᵐᵒᵖ α\ns : Set α\na : α\nh : s ∈ 𝓝 1\n⊢ s * {a} ∈ 𝓝 a",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\ninst✝² : TopologicalSpace α\ninst✝¹ : Group α\ninst✝ : ContinuousConstSMul αᵐᵒᵖ α\ns : Set α\na : α\nh : s ∈ 𝓝 1\n⊢ s * {a} ∈ 𝓝 a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Group.Pointwise | {
"line": 157,
"column": 10
} | {
"line": 157,
"column": 21
} | {
"line": 157,
"column": 22
} | [
{
"pp": "G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : SeparatelyContinuousMul G\ns s' : Set G\nhs₀ : s ∈ 𝓝 1\nh_symm : ∀ x ∈ s, x⁻¹ ∈ s\ny : G\nhy : y ∈ closure[inst✝²] s'\n⊢ (fun x ↦ x * y) '' s ∈ 𝓝 y",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Filter.in... | [
"G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : SeparatelyContinuousMul G\ns s' : Set G\nhs₀ : s ∈ 𝓝 1\nh_symm : ∀ x ∈ s, x⁻¹ ∈ s\ny : G\nhy : y ∈ closure[inst✝²] s'\n⊢ (fun x ↦ x * y⁻¹) ⁻¹' s ∈ 𝓝 y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Group.Pointwise | {
"line": 158,
"column": 2
} | {
"line": 158,
"column": 13
} | {
"line": 158,
"column": 14
} | [
{
"pp": "G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : SeparatelyContinuousMul G\ns s' : Set G\nhs₀ : s ∈ 𝓝 1\nh_symm : ∀ x ∈ s, x⁻¹ ∈ s\ny : G\nhy : y ∈ closure[inst✝²] s'\nb : G\nhb : b ∈ s\nhc : (fun x ↦ x * y) b ∈ s'\n⊢ y ∈ s * s'",
"ppTerm": "?m.104",
"assigned": false,
"u... | [
"G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : SeparatelyContinuousMul G\ns s' : Set G\nhs₀ : s ∈ 𝓝 1\nh_symm : ∀ x ∈ s, x⁻¹ ∈ s\ny : G\nhy : y ∈ closure[inst✝²] s'\nb : G\nhb : b ∈ s\nhc : (fun x ↦ x * y) b ∈ s'\n⊢ y ∈ s * s'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Group.Pointwise | {
"line": 167,
"column": 10
} | {
"line": 167,
"column": 21
} | {
"line": 167,
"column": 22
} | [
{
"pp": "G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : SeparatelyContinuousMul G\ns s' : Set G\nhs'₀ : s' ∈ 𝓝 1\nh_symm : ∀ x ∈ s', x⁻¹ ∈ s'\ny : G\nhy : y ∈ closure[inst✝²] s\n⊢ (fun x ↦ y * x) '' s' ∈ 𝓝 y",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Filte... | [
"G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : SeparatelyContinuousMul G\ns s' : Set G\nhs'₀ : s' ∈ 𝓝 1\nh_symm : ∀ x ∈ s', x⁻¹ ∈ s'\ny : G\nhy : y ∈ closure[inst✝²] s\n⊢ (fun x ↦ y⁻¹ * x) ⁻¹' s' ∈ 𝓝 y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Group.Pointwise | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 13
} | {
"line": 168,
"column": 14
} | [
{
"pp": "G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : SeparatelyContinuousMul G\ns s' : Set G\nhs'₀ : s' ∈ 𝓝 1\nh_symm : ∀ x ∈ s', x⁻¹ ∈ s'\ny : G\nhy : y ∈ closure[inst✝²] s\nb : G\nhb : b ∈ s'\nhc : (fun x ↦ y * x) b ∈ s\n⊢ y ∈ s * s'",
"ppTerm": "?m.104",
"assigned": false,
... | [
"G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : SeparatelyContinuousMul G\ns s' : Set G\nhs'₀ : s' ∈ 𝓝 1\nh_symm : ∀ x ∈ s', x⁻¹ ∈ s'\ny : G\nhy : y ∈ closure[inst✝²] s\nb : G\nhb : b ∈ s'\nhc : (fun x ↦ y * x) b ∈ s\n⊢ y ∈ s * s'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Group.Pointwise | {
"line": 266,
"column": 67
} | {
"line": 266,
"column": 88
} | {
"line": 267,
"column": 2
} | [
{
"pp": "G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : IsTopologicalGroup G\nF : Set G\nhF : IsClosed[inst✝²] F\n⊢ F * closure[inst✝²] {1} = closure[inst✝²] F * closure[inst✝²] {1}",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InvOneClass.toOne... | [] | by rw [hF.closure_eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Group.Pointwise | {
"line": 279,
"column": 4
} | {
"line": 279,
"column": 57
} | {
"line": 280,
"column": 2
} | [
{
"pp": "G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : IsTopologicalGroup G\nt : Set G\nht : t * closure[inst✝²] {1} = t\nx : G\nhx : x ∈ tᶜ\ng : G\nhg : g ∈ ↑⊥.topologicalClosure\nH : x * g ∈ t\n⊢ x ∈ t * ↑⊥.topologicalClosure",
"ppTerm": "?m.117",
"assigned": true,
"usedConsta... | [] | exact ⟨x * g, H, g⁻¹, Subgroup.inv_mem _ hg, by simp⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Algebra.Group.Pointwise | {
"line": 286,
"column": 14
} | {
"line": 286,
"column": 25
} | {
"line": 286,
"column": 26
} | [
{
"pp": "G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : IsTopologicalGroup G\nt : Set G\nh : tᶜ * closure[inst✝²] {1} = tᶜ\n⊢ t * closure[inst✝²] {1} = t",
"ppTerm": "?m.34",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : IsTopologicalGroup G\nt : Set G\nh : tᶜ * closure[inst✝²] {1} = tᶜ\n⊢ t * closure[inst✝²] {1} = t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.IsUniformGroup.Defs | {
"line": 150,
"column": 31
} | {
"line": 150,
"column": 60
} | {
"line": 150,
"column": 61
} | [
{
"pp": "Gᵣ : Type u_3\ninst✝² : UniformSpace Gᵣ\ninst✝¹ : Group Gᵣ\ninst✝ : IsRightUniformGroup Gᵣ\n⊢ comap Prod.swap (𝓤 Gᵣ) = comap (fun x ↦ x.1 / x.2) (𝓝 1)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"InvOneClass.toOne",
"DivInvOneMon... | [
"Gᵣ : Type u_3\ninst✝² : UniformSpace Gᵣ\ninst✝¹ : Group Gᵣ\ninst✝ : IsRightUniformGroup Gᵣ\n⊢ comap Prod.swap (comap (fun x ↦ x.2 / x.1) (𝓝 1)) = comap (fun x ↦ x.1 / x.2) (𝓝 1)"
] | uniformity_eq_comap_nhds_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.IsUniformGroup.Defs | {
"line": 232,
"column": 13
} | {
"line": 232,
"column": 24
} | {
"line": 232,
"column": 25
} | [
{
"pp": "case empty\nα : Type u_3\nβ : Type u_4\nι : Type u_5\ninst✝³ : UniformSpace α\ninst✝² : CommGroup α\ninst✝¹ : IsUniformGroup α\ninst✝ : UniformSpace β\nf : ι → β → α\nh : ∀ i ∈ ∅, UniformContinuous (f i)\n⊢ UniformContinuous fun x ↦ ∏ i ∈ ∅, f i x",
"ppTerm": "?empty",
"assigned": true,
"us... | [
"case empty\nα : Type u_3\nβ : Type u_4\nι : Type u_5\ninst✝³ : UniformSpace α\ninst✝² : CommGroup α\ninst✝¹ : IsUniformGroup α\ninst✝ : UniformSpace β\nf : ι → β → α\nh : ∀ i ∈ ∅, UniformContinuous (f i)\n⊢ UniformContinuous fun x ↦ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.IsUniformGroup.Defs | {
"line": 235,
"column": 4
} | {
"line": 235,
"column": 34
} | {
"line": 235,
"column": 35
} | [
{
"pp": "case cons\nα : Type u_3\nβ : Type u_4\nι : Type u_5\ninst✝³ : UniformSpace α\ninst✝² : CommGroup α\ninst✝¹ : IsUniformGroup α\ninst✝ : UniformSpace β\nf : ι → β → α\na : ι\ns : Finset ι\nha : a ∉ s\nih : (∀ i ∈ s, UniformContinuous (f i)) → UniformContinuous fun x ↦ ∏ i ∈ s, f i x\nh : UniformContinuou... | [
"case cons\nα : Type u_3\nβ : Type u_4\nι : Type u_5\ninst✝³ : UniformSpace α\ninst✝² : CommGroup α\ninst✝¹ : IsUniformGroup α\ninst✝ : UniformSpace β\nf : ι → β → α\na : ι\ns : Finset ι\nha : a ∉ s\nih : (∀ i ∈ s, UniformContinuous (f i)) → UniformContinuous fun x ↦ ∏ i ∈ s, f i x\nh : UniformContinuous (f a) ∧ ∀ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.IsUniformGroup.Defs | {
"line": 304,
"column": 16
} | {
"line": 304,
"column": 27
} | {
"line": 304,
"column": 28
} | [
{
"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": false,
"usedConstants": [],
"usedFVars": [],
... | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.IsUniformGroup.Defs | {
"line": 315,
"column": 16
} | {
"line": 315,
"column": 27
} | {
"line": 315,
"column": 28
} | [
{
"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\nhg : Tendsto g l (𝓤 α)\nhfg : Tendsto (f * g) l (𝓤 α)\n⊢ Tendsto f l (𝓤 α)",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"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\nhg : Tendsto g l (𝓤 α)\nhfg : Tendsto (f * g) l (𝓤 α)\n⊢ Tendsto f l (𝓤 α)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Group.Pointwise | {
"line": 395,
"column": 71
} | {
"line": 395,
"column": 82
} | {
"line": 395,
"column": 83
} | [
{
"pp": "G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : IsTopologicalGroup G\nK : Set G\nhK : IsCompact K\nx : G\nh : K ∈ 𝓝 x\ny : G\n⊢ K ∈ 𝓝 ((y * x⁻¹)⁻¹ • y)",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"DivInv... | [
"G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : IsTopologicalGroup G\nK : Set G\nhK : IsCompact K\nx : G\nh : K ∈ 𝓝 x\ny : G\n⊢ K ∈ 𝓝 x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.Monoid | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 28
} | {
"line": 88,
"column": 29
} | [
{
"pp": "α : Type u_1\nM : Type u_2\ninst✝² : CommMonoid M\ninst✝¹ : Preorder M\ninst✝ : IsOrderedMonoid M\nl : Filter α\nf : α → M\nhf : Tendsto f l atTop\nn : ℕ\nhn : 0 < n\nx : α\nhx : 1 ≤ f x\n⊢ f x ≤ (fun x ↦ f x ^ n) x",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Preorder.t... | [
"α : Type u_1\nM : Type u_2\ninst✝² : CommMonoid M\ninst✝¹ : Preorder M\ninst✝ : IsOrderedMonoid M\nl : Filter α\nf : α → M\nhf : Tendsto f l atTop\nn : ℕ\nhn : 0 < n\nx : α\nhx : 1 ≤ f x\n⊢ f x ≤ f x ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.Monoid | {
"line": 153,
"column": 2
} | {
"line": 153,
"column": 22
} | {
"line": 154,
"column": 2
} | [
{
"pp": "α : Type u_1\nM : Type u_2\ninst✝² : CommMonoid M\ninst✝¹ : Preorder M\ninst✝ : IsOrderedCancelMonoid M\nl : Filter α\nf g : α → M\nhg : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l g\nh : Tendsto (fun x ↦ f x * g x) l atTop\n⊢ Tendsto f l atTop",
"ppTerm": "?m.24",
"assigned": true,
"usedConstan... | [
"α : Type u_1\nM : Type u_2\ninst✝² : CommMonoid M\ninst✝¹ : Preorder M\ninst✝ : IsOrderedCancelMonoid M\nl : Filter α\nf g : α → M\nh : Tendsto (fun x ↦ f x * g x) l atTop\nC : M\nhC : ∀ᶠ (x : M) in map g l, (fun x1 x2 ↦ x1 ≤ x2) x C\n⊢ Tendsto f l atTop"
] | obtain ⟨C, hC⟩ := hg | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.Algebra.IsUniformGroup.Defs | {
"line": 489,
"column": 2
} | {
"line": 489,
"column": 13
} | {
"line": 489,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : IsUniformGroup α\np : α → Prop\nhp : ∀ᶠ (x : α) in 𝓝 1, p x\n⊢ ∀ᶠ (x : α) in 𝓝 1, ∀ (g : α), p (g * x * g⁻¹)",
"ppTerm": "?m.32",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : IsUniformGroup α\np : α → Prop\nhp : ∀ᶠ (x : α) in 𝓝 1, p x\n⊢ ∀ᶠ (x : α) in 𝓝 1, ∀ (g : α), p (g * x * g⁻¹)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.Ring | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 67
} | {
"line": 53,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : Ring α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedRing α\nl : Filter β\nf g : β → α\nhf : Tendsto f l atTop\nhg : Tendsto g l atBot\nthis : Tendsto (fun x ↦ f x * (Neg.neg ∘ g) x) l atTop\n⊢ Tendsto (fun x ↦ f x * g x) l atBot",
"ppTerm": "?m.42",
"assigned"... | [
"α : Type u_1\nβ : Type u_2\ninst✝² : Ring α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedRing α\nl : Filter β\nf g : β → α\nhf : Tendsto f l atTop\nhg : Tendsto g l atBot\nthis : Tendsto (fun x ↦ f x * (Neg.neg ∘ g) x) l atTop\n⊢ Tendsto (fun x ↦ f x * g x) l atBot"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.Ring | {
"line": 59,
"column": 2
} | {
"line": 59,
"column": 67
} | {
"line": 60,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : Ring α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedRing α\nl : Filter β\nf g : β → α\nhf : Tendsto f l atBot\nhg : Tendsto g l atTop\nthis : Tendsto (fun x ↦ -f x * g x) l atTop\n⊢ Tendsto (fun x ↦ f x * g x) l atBot",
"ppTerm": "?m.52",
"assigned": false,
... | [
"α : Type u_1\nβ : Type u_2\ninst✝² : Ring α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedRing α\nl : Filter β\nf g : β → α\nhf : Tendsto f l atBot\nhg : Tendsto g l atTop\nthis : Tendsto (fun x ↦ -f x * g x) l atTop\n⊢ Tendsto (fun x ↦ f x * g x) l atBot"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.Ring | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 32
} | {
"line": 66,
"column": 33
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : Ring α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedRing α\nl : Filter β\nf g : β → α\nhf : Tendsto f l atBot\nhg : Tendsto g l atBot\nthis : Tendsto (fun x ↦ -f x * -g x) l atTop\n⊢ Tendsto (fun x ↦ f x * g x) l atTop",
"ppTerm": "?m.66",
"assigned": false,
... | [
"α : Type u_1\nβ : Type u_2\ninst✝² : Ring α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedRing α\nl : Filter β\nf g : β → α\nhf : Tendsto f l atBot\nhg : Tendsto g l atBot\nthis : Tendsto (fun x ↦ -f x * -g x) l atTop\n⊢ Tendsto (fun x ↦ f x * g x) l atTop"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.IsUniformGroup.Defs | {
"line": 542,
"column": 39
} | {
"line": 542,
"column": 50
} | {
"line": 542,
"column": 51
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁷ : UniformSpace α\ninst✝⁶ : Group α\ninst✝⁵ : IsUniformGroup α\nhom : Type u_3\ninst✝⁴ : UniformSpace β\ninst✝³ : Group β\ninst✝² : IsUniformGroup β\ninst✝¹ : FunLike hom α β\ninst✝ : MonoidHomClass hom α β\nf : hom\nhf : ContinuousAt (⇑f) 1\n⊢ Tendsto (⇑f) (𝓝 1) (𝓝 ... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁷ : UniformSpace α\ninst✝⁶ : Group α\ninst✝⁵ : IsUniformGroup α\nhom : Type u_3\ninst✝⁴ : UniformSpace β\ninst✝³ : Group β\ninst✝² : IsUniformGroup β\ninst✝¹ : FunLike hom α β\ninst✝ : MonoidHomClass hom α β\nf : hom\nhf : ContinuousAt (⇑f) 1\n⊢ Tendsto (⇑f) (𝓝 1) (𝓝 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.Ring | {
"line": 92,
"column": 9
} | {
"line": 92,
"column": 42
} | {
"line": 93,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝² : Ring α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\n⊢ ¬Tendsto (fun x ↦ x ^ 0) atTop atBot",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"False",
"Lattice.toSemilatticeSup",
"instNoBotOrderOfNoMinOrder",
... | [] | by simp [not_tendsto_const_atBot] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 200,
"column": 19
} | {
"line": 200,
"column": 30
} | {
"line": 200,
"column": 31
} | [
{
"pp": "G : Type u_1\ninst✝³ : DivInvMonoid G\ninst✝² : TopologicalSpace G\ninst✝¹ : ContinuousMul G\ninst✝ : ContinuousInv G\nx y : G\nh : x ⤳ y\nn : ℕ\n⊢ (x ^ Int.ofNat n) ⤳ (y ^ Int.ofNat n)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"Sp... | [
"G : Type u_1\ninst✝³ : DivInvMonoid G\ninst✝² : TopologicalSpace G\ninst✝¹ : ContinuousMul G\ninst✝ : ContinuousInv G\nx y : G\nh : x ⤳ y\nn : ℕ\n⊢ (x ^ n) ⤳ (y ^ n)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 201,
"column": 21
} | {
"line": 201,
"column": 32
} | {
"line": 201,
"column": 33
} | [
{
"pp": "G : Type u_1\ninst✝³ : DivInvMonoid G\ninst✝² : TopologicalSpace G\ninst✝¹ : ContinuousMul G\ninst✝ : ContinuousInv G\nx y : G\nh : x ⤳ y\nn : ℕ\n⊢ (x ^ Int.negSucc n) ⤳ (y ^ Int.negSucc n)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",... | [
"G : Type u_1\ninst✝³ : DivInvMonoid G\ninst✝² : TopologicalSpace G\ninst✝¹ : ContinuousMul G\ninst✝ : ContinuousInv G\nx y : G\nh : x ⤳ y\nn : ℕ\n⊢ (x ^ (n + 1))⁻¹ ⤳ (y ^ (n + 1))⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 309,
"column": 15
} | {
"line": 309,
"column": 41
} | {
"line": 309,
"column": 42
} | [
{
"pp": "G : Type w\nα : Type u\ninst✝² : TopologicalSpace G\ninst✝¹ : InvolutiveInv G\ninst✝ : ContinuousInv G\nl : Filter α\nm : α → G\na : G\nh : Tendsto (fun x ↦ (m x)⁻¹) l (𝓝 a⁻¹)\n⊢ Tendsto m l (𝓝 a)",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedG... | [
"G : Type w\nα : Type u\ninst✝² : TopologicalSpace G\ninst✝¹ : InvolutiveInv G\ninst✝ : ContinuousInv G\nl : Filter α\nm : α → G\na : G\nh : Tendsto (fun x ↦ (m x)⁻¹) l (𝓝 a⁻¹)\n⊢ Tendsto m l (𝓝 a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.IsUniformGroup.Defs | {
"line": 617,
"column": 4
} | {
"line": 617,
"column": 15
} | {
"line": 617,
"column": 16
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nU : Set G\nH : U ∈ 𝓝 1\nV : Set G\nV_nhds : V ∈ 𝓝 1\nV_mul : ∀ v ∈ V, ∀ w ∈ V, v * w ∈ U\nx y z : G\nhz₁ : (x, z) ∈ (fun p ↦ p.2 * p.1⁻¹) ⁻¹' V\nhz₂ : (z, y) ∈ (fun p ↦ p.2 * p.1⁻¹) ⁻¹' V\n⊢ (x, y) ∈ (fun p ↦ p... | [
"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nU : Set G\nH : U ∈ 𝓝 1\nV : Set G\nV_nhds : V ∈ 𝓝 1\nV_mul : ∀ v ∈ V, ∀ w ∈ V, v * w ∈ U\nx y z : G\nhz₁ : (x, z) ∈ (fun p ↦ p.2 * p.1⁻¹) ⁻¹' V\nhz₂ : (z, y) ∈ (fun p ↦ p.2 * p.1⁻¹) ⁻¹' V\n⊢ y * x⁻¹ ∈ U"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 413,
"column": 29
} | {
"line": 413,
"column": 73
} | {
"line": 413,
"column": 74
} | [
{
"pp": "G : Type u_1\nH : Type u_2\ninst✝⁴ : Inv G\ninst✝³ : Inv H\ninst✝² : TopologicalSpace G\ninst✝¹ : TopologicalSpace H\ninst✝ : ContinuousInv H\nf : G → H\nhf : IsInducing f\nhf_inv : ∀ (x : G), f x⁻¹ = (f x)⁻¹\n⊢ Continuous[inst✝², inst✝¹] (f ∘ fun a ↦ a⁻¹)",
"ppTerm": "?m.28",
"assigned": true,... | [
"G : Type u_1\nH : Type u_2\ninst✝⁴ : Inv G\ninst✝³ : Inv H\ninst✝² : TopologicalSpace G\ninst✝¹ : TopologicalSpace H\ninst✝ : ContinuousInv H\nf : G → H\nhf : IsInducing f\nhf_inv : ∀ (x : G), f x⁻¹ = (f x)⁻¹\n⊢ Continuous[inst✝², inst✝¹] fun x ↦ (f x)⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 467,
"column": 22
} | {
"line": 467,
"column": 33
} | {
"line": 467,
"column": 34
} | [
{
"pp": "G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : IsTopologicalGroup G\nn : ℕ\n⊢ Continuous[inst✝², inst✝²] fun a ↦ a ^ Int.ofNat n",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"Continuous",
"congrArg",
"D... | [
"G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : IsTopologicalGroup G\nn : ℕ\n⊢ Continuous[inst✝², inst✝²] fun a ↦ a ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 468,
"column": 24
} | {
"line": 468,
"column": 35
} | {
"line": 468,
"column": 36
} | [
{
"pp": "G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : IsTopologicalGroup G\nn : ℕ\n⊢ Continuous[inst✝², inst✝²] fun a ↦ a ^ Int.negSucc n",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"Continuous",
"congrArg",
... | [
"G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : IsTopologicalGroup G\nn : ℕ\n⊢ Continuous[inst✝², inst✝²] fun a ↦ (a ^ (n + 1))⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.IsUniformGroup.Defs | {
"line": 660,
"column": 4
} | {
"line": 660,
"column": 15
} | {
"line": 660,
"column": 16
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nU : Set G\nH : U ∈ 𝓝 1\nV : Set G\nV_nhds : V ∈ 𝓝 1\nV_mul : ∀ v ∈ V, ∀ w ∈ V, v * w ∈ U\nx y z : G\nhz₁ : (x, z) ∈ (fun p ↦ p.1⁻¹ * p.2) ⁻¹' V\nhz₂ : (z, y) ∈ (fun p ↦ p.1⁻¹ * p.2) ⁻¹' V\n⊢ (x, y) ∈ (fun p ↦ p... | [
"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nU : Set G\nH : U ∈ 𝓝 1\nV : Set G\nV_nhds : V ∈ 𝓝 1\nV_mul : ∀ v ∈ V, ∀ w ∈ V, v * w ∈ U\nx y z : G\nhz₁ : (x, z) ∈ (fun p ↦ p.1⁻¹ * p.2) ⁻¹' V\nhz₂ : (z, y) ∈ (fun p ↦ p.1⁻¹ * p.2) ⁻¹' V\n⊢ x⁻¹ * y ∈ U"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 565,
"column": 2
} | {
"line": 565,
"column": 28
} | {
"line": 565,
"column": 29
} | [
{
"pp": "H : Type x\ninst✝⁴ : TopologicalSpace H\ninst✝³ : CommGroup H\ninst✝² : PartialOrder H\ninst✝¹ : IsOrderedMonoid H\ninst✝ : ContinuousInv H\na : H\n⊢ Tendsto Inv.inv (𝓝[>] a⁻¹) (𝓝[<] a)",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"H : Type x\ninst✝⁴ : TopologicalSpace H\ninst✝³ : CommGroup H\ninst✝² : PartialOrder H\ninst✝¹ : IsOrderedMonoid H\ninst✝ : ContinuousInv H\na : H\n⊢ Tendsto Inv.inv (𝓝[>] a⁻¹) (𝓝[<] a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 569,
"column": 2
} | {
"line": 569,
"column": 28
} | {
"line": 569,
"column": 29
} | [
{
"pp": "H : Type x\ninst✝⁴ : TopologicalSpace H\ninst✝³ : CommGroup H\ninst✝² : PartialOrder H\ninst✝¹ : IsOrderedMonoid H\ninst✝ : ContinuousInv H\na : H\n⊢ Tendsto Inv.inv (𝓝[<] a⁻¹) (𝓝[>] a)",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"H : Type x\ninst✝⁴ : TopologicalSpace H\ninst✝³ : CommGroup H\ninst✝² : PartialOrder H\ninst✝¹ : IsOrderedMonoid H\ninst✝ : ContinuousInv H\na : H\n⊢ Tendsto Inv.inv (𝓝[<] a⁻¹) (𝓝[>] a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 581,
"column": 2
} | {
"line": 581,
"column": 28
} | {
"line": 581,
"column": 29
} | [
{
"pp": "H : Type x\ninst✝⁴ : TopologicalSpace H\ninst✝³ : CommGroup H\ninst✝² : PartialOrder H\ninst✝¹ : IsOrderedMonoid H\ninst✝ : ContinuousInv H\na : H\n⊢ Tendsto Inv.inv (𝓝[≥] a⁻¹) (𝓝[≤] a)",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"H : Type x\ninst✝⁴ : TopologicalSpace H\ninst✝³ : CommGroup H\ninst✝² : PartialOrder H\ninst✝¹ : IsOrderedMonoid H\ninst✝ : ContinuousInv H\na : H\n⊢ Tendsto Inv.inv (𝓝[≥] a⁻¹) (𝓝[≤] a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 585,
"column": 2
} | {
"line": 585,
"column": 28
} | {
"line": 585,
"column": 29
} | [
{
"pp": "H : Type x\ninst✝⁴ : TopologicalSpace H\ninst✝³ : CommGroup H\ninst✝² : PartialOrder H\ninst✝¹ : IsOrderedMonoid H\ninst✝ : ContinuousInv H\na : H\n⊢ Tendsto Inv.inv (𝓝[≤] a⁻¹) (𝓝[≥] a)",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"H : Type x\ninst✝⁴ : TopologicalSpace H\ninst✝³ : CommGroup H\ninst✝² : PartialOrder H\ninst✝¹ : IsOrderedMonoid H\ninst✝ : ContinuousInv H\na : H\n⊢ Tendsto Inv.inv (𝓝[≤] a⁻¹) (𝓝[≥] a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 677,
"column": 33
} | {
"line": 677,
"column": 91
} | {
"line": 677,
"column": 92
} | [
{
"pp": "G : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : TopologicalSpace α\nf : α → G\ns✝ : Set α\nx : α\ns : Subgroup G\ng : G\nhg : g ∈ closure[inst✝³] ↑s\n⊢ g⁻¹ ∈ closure[inst✝³] ↑s",
"ppTerm": "?m.21",
"assigned":... | [
"G : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : TopologicalSpace α\nf : α → G\ns✝ : Set α\nx : α\ns : Subgroup G\ng : G\nhg : g ∈ closure[inst✝³] ↑s\n⊢ g ∈ closure[inst✝³] ↑s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 728,
"column": 2
} | {
"line": 728,
"column": 44
} | {
"line": 728,
"column": 45
} | [
{
"pp": "G : Type u_1\ninst✝² : TopologicalSpace G\ninst✝¹ : MulOneClass G\ninst✝ : ContinuousMul G\ng h : G\nhg : g ∈ connectedComponent 1\nhh : h ∈ connectedComponent 1\nhmul : g ∈ connectedComponent (g * h)\n⊢ g * h ∈ connectedComponent g",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
... | [
"G : Type u_1\ninst✝² : TopologicalSpace G\ninst✝¹ : MulOneClass G\ninst✝ : ContinuousMul G\ng h : G\nhg : g ∈ connectedComponent 1\nhh : h ∈ connectedComponent 1\nhmul : g ∈ connectedComponent (g * h)\n⊢ g * h ∈ connectedComponent (g * h)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 770,
"column": 2
} | {
"line": 770,
"column": 99
} | {
"line": 771,
"column": 4
} | [
{
"pp": "G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : IsTopologicalGroup G\ns : Set G\nhs : s ∈ 𝓝 1\nthis : (fun p ↦ p.1 * p.2⁻¹) ⁻¹' s ∈ 𝓝 (1, 1)\n⊢ ∃ V ∈ 𝓝 1, ∀ v ∈ V, ∀ w ∈ V, v / w ∈ s",
"ppTerm": "?m.87",
"assigned": true,
"usedConstants": [
"Filter.instMembership... | [
"G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : IsTopologicalGroup G\ns : Set G\nhs : s ∈ 𝓝 1\nthis : (fun p ↦ p.1 * p.2⁻¹) ⁻¹' s ∈ 𝓝 (1, 1)\n⊢ ∃ V ∈ 𝓝 1, ∀ v ∈ V, ∀ w ∈ V, v * w⁻¹ ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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