module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.Complex.UpperHalfPlane.Manifold | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 13
} | {
"line": 98,
"column": 14
} | [
{
"pp": "n : ℕ∞ω\ng : GL (Fin 2) ℝ\n⊢ ContMDiff 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) n fun τ ↦ (denom g ↑τ)⁻¹",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ∞ω\ng : GL (Fin 2) ℝ\n⊢ ContMDiff 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) n fun τ ↦ (denom g ↑τ)⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Manifold | {
"line": 134,
"column": 4
} | {
"line": 134,
"column": 15
} | {
"line": 134,
"column": 16
} | [
{
"pp": "f : ℍ → ℂ\nhf : DifferentiableOn ℂ (f ∘ ↑ofComplex) {z | 0 < z.im}\nτ : ℍ\nthis : AnalyticOnNhd ℂ (f ∘ ↑ofComplex) {z | 0 < z.im}\nw : ℍ\nhτ : ∀ᶠ (a : ℍ) in 𝓝 τ, ↑a ∈ {↑τ}ᶜ → (f ∘ ↑ofComplex) ↑a ≠ 0\na : ℍ\nha : ↑a ∈ {↑τ}ᶜ → (f ∘ ↑ofComplex) ↑a ≠ 0\n⊢ a ∈ {τ}ᶜ → f a ≠ 0",
"ppTerm": "?m.221",
"... | [
"f : ℍ → ℂ\nhf : DifferentiableOn ℂ (f ∘ ↑ofComplex) {z | 0 < z.im}\nτ : ℍ\nthis : AnalyticOnNhd ℂ (f ∘ ↑ofComplex) {z | 0 < z.im}\nw : ℍ\nhτ : ∀ᶠ (a : ℍ) in 𝓝 τ, ↑a ∈ {↑τ}ᶜ → (f ∘ ↑ofComplex) ↑a ≠ 0\na : ℍ\nha : ↑a ∈ {↑τ}ᶜ → (f ∘ ↑ofComplex) ↑a ≠ 0\n⊢ ¬a = τ → ¬f a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Manifold | {
"line": 137,
"column": 59
} | {
"line": 137,
"column": 70
} | {
"line": 137,
"column": 71
} | [
{
"pp": "f g : ℍ → ℂ\nhf : MDiff f\nhg : MDiff g\nhfg : f * g = 0\n⊢ ∀ (x : ℍ), f x = 0 ∨ g x = 0",
"ppTerm": "?m.35",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f g : ℍ → ℂ\nhf : MDiff f\nhg : MDiff g\nhfg : f * g = 0\n⊢ ∀ (x : ℍ), f x = 0 ∨ g x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Manifold | {
"line": 144,
"column": 64
} | {
"line": 144,
"column": 75
} | {
"line": 144,
"column": 76
} | [
{
"pp": "ι : Type u_1\nf : ι → ℍ → ℂ\ns : Finset ι\nhf : ∀ i ∈ s, MDiff (f i)\nh0 : ∏ i ∈ s, f i = 0\n⊢ ∀ (x : ℍ), ∏ i ∈ s, f i x = 0",
"ppTerm": "?m.82",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_1\nf : ι → ℍ → ℂ\ns : Finset ι\nhf : ∀ i ∈ s, MDiff (f i)\nh0 : ∏ i ∈ s, f i = 0\n⊢ ∀ (x : ℍ), ∏ i ∈ s, f i x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Manifold | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 23
} | {
"line": 165,
"column": 24
} | [
{
"pp": "g : GL (Fin 2) ℝ\nk : ℤ\nτ : ℍ\n⊢ HasDerivAt (fun x ↦ denom g x) ↑(↑g 1 0) ↑τ",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"IsModuleTopology.toContinuousSMul",
"Units.val",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCo... | [
"g : GL (Fin 2) ℝ\nk : ℤ\nτ : ℍ\n⊢ HasDerivAt (HMul.hMul ↑(↑g 1 0)) ↑(↑g 1 0) ↑τ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Manifold | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 54
} | {
"line": 167,
"column": 55
} | [
{
"pp": "g : GL (Fin 2) ℝ\nk : ℤ\nτ : ℍ\nhd : HasDerivAt (fun x ↦ denom g x) ↑(↑g 1 0) ↑τ\nthis : HasDerivAt ((fun x ↦ x ^ k) ∘ denom g) (↑k * denom g ↑τ ^ (k - 1) * ↑(↑g 1 0)) ↑τ\n⊢ HasDerivAt (fun z ↦ denom g z ^ k) (↑k * ↑(↑g 1 0) * denom g ↑τ ^ (k - 1)) ↑τ",
"ppTerm": "?m.109",
"assigned": true,
... | [
"g : GL (Fin 2) ℝ\nk : ℤ\nτ : ℍ\nhd : HasDerivAt (fun x ↦ denom g x) ↑(↑g 1 0) ↑τ\nthis : HasDerivAt ((fun x ↦ x ^ k) ∘ denom g) (↑k * denom g ↑τ ^ (k - 1) * ↑(↑g 1 0)) ↑τ\n⊢ HasDerivAt (fun z ↦ denom g z ^ k) (↑k * denom g ↑τ ^ (k - 1) * ↑(↑g 1 0)) ↑τ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Manifold | {
"line": 178,
"column": 4
} | {
"line": 180,
"column": 52
} | {
"line": 181,
"column": 2
} | [
{
"pp": "g : GL (Fin 2) ℝ\nhg : 0 < (↑g).det\nτ : ℍ\nthis : HasStrictDerivAt (num g / denom g) (↑(↑g).det / denom g ↑τ ^ 2) ↑τ\n⊢ HasStrictDerivAt (fun z ↦ ↑(g • ↑ofComplex z)) (↑(↑g).det / denom g ↑τ ^ 2) ↑τ",
"ppTerm": "?m.87",
"assigned": true,
"usedConstants": [
"UpperHalfPlane.glAction",
... | [] | refine this.congr_of_eventuallyEq ?_
rw [← isOpenEmbedding_coe.map_nhds_eq, eventuallyEq_map]
simp [Function.comp_def, coe_smul_of_det_pos hg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.UpperHalfPlane.Manifold | {
"line": 178,
"column": 4
} | {
"line": 180,
"column": 52
} | {
"line": 181,
"column": 2
} | [
{
"pp": "g : GL (Fin 2) ℝ\nhg : 0 < (↑g).det\nτ : ℍ\nthis : HasStrictDerivAt (num g / denom g) (↑(↑g).det / denom g ↑τ ^ 2) ↑τ\n⊢ HasStrictDerivAt (fun z ↦ ↑(g • ↑ofComplex z)) (↑(↑g).det / denom g ↑τ ^ 2) ↑τ",
"ppTerm": "?m.87",
"assigned": true,
"usedConstants": [
"UpperHalfPlane.glAction",
... | [] | refine this.congr_of_eventuallyEq ?_
rw [← isOpenEmbedding_coe.map_nhds_eq, eventuallyEq_map]
simp [Function.comp_def, coe_smul_of_det_pos hg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Norm.Transitivity | {
"line": 217,
"column": 4
} | {
"line": 217,
"column": 52
} | {
"line": 217,
"column": 53
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝⁶ : CommRing R\nL : Type u_6\nK : Type u_7\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : Algebra K L\ninst✝² : Algebra R L\ninst✝¹ : Algebra R K\ninst✝ : IsScalarTower R K L\nx : L\nhx : IsIntegral R x\nh : ¬FiniteDimensional K L\n⊢ IsIntegral R ((norm K) x)",
"ppTerm":... | [
"case neg\nR : Type u_1\ninst✝⁶ : CommRing R\nL : Type u_6\nK : Type u_7\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : Algebra K L\ninst✝² : Algebra R L\ninst✝¹ : Algebra R K\ninst✝ : IsScalarTower R K L\nx : L\nhx : IsIntegral R x\nh : ¬FiniteDimensional K L\n⊢ IsIntegral R 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Norm.Transitivity | {
"line": 245,
"column": 2
} | {
"line": 245,
"column": 26
} | {
"line": 246,
"column": 2
} | [
{
"pp": "case pos\nL : Type u_6\nK : Type u_7\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\nh : FiniteDimensional K L\nF : IntermediateField K L := K⟮x⟯\n⊢ (norm K) x = (norm K) (gen K x) ^ finrank (↥K⟮x⟯) L",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [
"case pos\nL : Type u_6\nK : Type u_7\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\nh : FiniteDimensional K L\nF : IntermediateField K L := K⟮x⟯\n⊢ (norm K) ↑(gen K x) = (norm K) (gen K x) ^ finrank (↥K⟮x⟯) L"
] | nth_rw 1 [← coe_gen K x] | Mathlib.Tactic._aux_Mathlib_Tactic_NthRewrite___macroRules_Mathlib_Tactic_tacticNth_rw______1 | Mathlib.Tactic.tacticNth_rw_____ |
Mathlib.Analysis.Complex.UpperHalfPlane.Metric | {
"line": 119,
"column": 4
} | {
"line": 119,
"column": 94
} | {
"line": 119,
"column": 95
} | [
{
"pp": "z✝ w✝ : ℍ\nr : ℝ\nz w : ℍ\nh : dist z w = 0\n⊢ z = w",
"ppTerm": "?m.43",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"z✝ w✝ : ℍ\nr : ℝ\nz w : ℍ\nh : dist z w = 0\n⊢ z = w"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Group.Matrix | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 29
} | {
"line": 154,
"column": 30
} | [
{
"pp": "n : Type u_1\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\nA : Type u_4\ninst✝ : CommRing A\nx : GL n A\n⊢ x ∈ Set.range ⇑toGL ↔ x ∈ ⇑GeneralLinearGroup.det ⁻¹' {1}",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"MulOne.toOne",
"Matri... | [
"n : Type u_1\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\nA : Type u_4\ninst✝ : CommRing A\nx : GL n A\n⊢ (∃ y, ↑y = ↑x) ↔ (↑x).det = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Group.Matrix | {
"line": 158,
"column": 24
} | {
"line": 158,
"column": 48
} | {
"line": 158,
"column": 49
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝⁵ : Fintype n\ninst✝⁴ : DecidableEq n\ninst✝³ : CommRing R\ninst✝² : TopologicalSpace R\ninst✝¹ : IsTopologicalRing R\ninst✝ : T0Space R\n⊢ IsClosed[Units.instTopologicalSpaceUnits] (Set.range ⇑toGL)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
... | [
"n : Type u_1\nR : Type u_2\ninst✝⁵ : Fintype n\ninst✝⁴ : DecidableEq n\ninst✝³ : CommRing R\ninst✝² : TopologicalSpace R\ninst✝¹ : IsTopologicalRing R\ninst✝ : T0Space R\n⊢ IsClosed[Units.instTopologicalSpaceUnits] (⇑GeneralLinearGroup.det ⁻¹' {1})"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Metric | {
"line": 223,
"column": 6
} | {
"line": 223,
"column": 50
} | {
"line": 223,
"column": 51
} | [
{
"pp": "case hab\nz w : ℍ\n⊢ dist { re := 0, im := z.im } { re := 0, im := w.im } ≤ dist ↑z ↑w",
"ppTerm": "?hab",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"False",
"Real.partialOrder",
"Real",
"Real.lattice",
"C... | [
"case hab\nz w : ℍ\n⊢ |z.im - w.im| ≤ dist ↑z ↑w"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.ProperAction.Basic | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 59
} | {
"line": 143,
"column": 2
} | [
{
"pp": "G : Type u_1\nX : Type u_2\ninst✝⁴ : Group G\ninst✝³ : MulAction G X\ninst✝² : TopologicalSpace G\ninst✝¹ : TopologicalSpace X\nh_proper : ProperSMul G X\ninst✝ : T1Space G\nf : X → G × X := fun x ↦ (1, x)\nproper_f : IsProperMap f\ng : G × X → X × X := fun gx ↦ (gx.1 • gx.2, gx.2)\nproper_g : IsProper... | [
"G : Type u_1\nX : Type u_2\ninst✝⁴ : Group G\ninst✝³ : MulAction G X\ninst✝² : TopologicalSpace G\ninst✝¹ : TopologicalSpace X\nh_proper : ProperSMul G X\ninst✝ : T1Space G\nf : X → G × X := fun x ↦ (1, x)\nproper_f : IsProperMap f\ng : G × X → X × X := fun gx ↦ (gx.1 • gx.2, gx.2)\nproper_g : IsProperMap g\nthis ... | have : g ∘ f = fun x ↦ (x, x) := by ext x <;> simp [f, g] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 32
} | {
"line": 152,
"column": 33
} | [
{
"pp": "X : Type u_1\nx : OnePoint X\n⊢ x ≠ ∞ ↔ ∃ y, ↑y = x",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"OnePoint.infty",
"OnePoint.some",
"Exists",
"Ne",
"OnePoint.rec",
"Iff",
"Eq",
"OnePoint"
],
"usedFVars": [
"X",
... | [
"case infty\nX : Type u_1\n⊢ ∞ ≠ ∞ ↔ ∃ y, ↑y = ∞",
"case coe\nX : Type u_1\nx✝ : X\n⊢ ↑x✝ ≠ ∞ ↔ ∃ y, ↑y = ↑x✝"
] | induction x using OnePoint.rec | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 157,
"column": 74
} | {
"line": 158,
"column": 58
} | {
"line": 160,
"column": 0
} | [
{
"pp": "X : Type u_1\nx : OnePoint X\n⊢ x ∉ range some ↔ x = ∞",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"OnePoint.infty",
"congrArg",
"Compl.compl",
"Iff.rfl",
"OnePoint.compl_range_coe",
"OnePoint.some",
"Membership.mem",
... | [] | by
rw [← mem_compl_iff, compl_range_coe, mem_singleton_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 208,
"column": 4
} | {
"line": 208,
"column": 29
} | {
"line": 208,
"column": 30
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝ : TopologicalSpace X\ns t : Set (OnePoint X)\nhms : ∞ ∈ s → IsCompact (some ⁻¹' s)ᶜ\nhs : IsOpen[inst✝] (some ⁻¹' s)\nhmt : ∞ ∈ t → IsCompact (some ⁻¹' t)ᶜ\nht : IsOpen[inst✝] (some ⁻¹' t)\nhms' : ∞ ∈ s\nhmt' : ∞ ∈ t\n⊢ IsCompact (some ⁻¹' (s ∩ t))ᶜ",
"ppTerm": "?m... | [
"X : Type u_1\nY : Type u_2\ninst✝ : TopologicalSpace X\ns t : Set (OnePoint X)\nhms : ∞ ∈ s → IsCompact (some ⁻¹' s)ᶜ\nhs : IsOpen[inst✝] (some ⁻¹' s)\nhmt : ∞ ∈ t → IsCompact (some ⁻¹' t)ᶜ\nht : IsOpen[inst✝] (some ⁻¹' t)\nhms' : ∞ ∈ s\nhmt' : ∞ ∈ t\n⊢ IsCompact ((some ⁻¹' s)ᶜ ∪ (some ⁻¹' t)ᶜ)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 245,
"column": 6
} | {
"line": 245,
"column": 50
} | {
"line": 245,
"column": 51
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\n⊢ IsOpen[instTopologicalSpace] (some '' s) ↔ IsOpen[inst✝] s",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"OnePoint.infty_notMem_image_coe",
"OnePoint.some",
"id",
"One... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\n⊢ IsOpen[inst✝] (some ⁻¹' some '' s) ↔ IsOpen[inst✝] s"
] | isOpen_iff_of_notMem infty_notMem_image_coe, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 472,
"column": 2
} | {
"line": 472,
"column": 32
} | {
"line": 472,
"column": 33
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nx y : OnePoint X\n⊢ Inseparable x y ↔ x = ∞ ∧ y = ∞ ∨ ∃ x', x = ↑x' ∧ ∃ y', y = ↑y' ∧ Inseparable x' y'",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"OnePoint.infty",
"OnePoint.some",
"Exists",
"OnePoint.rec",
... | [
"case infty\nX : Type u_1\ninst✝ : TopologicalSpace X\ny : OnePoint X\n⊢ Inseparable ∞ y ↔ ∞ = ∞ ∧ y = ∞ ∨ ∃ x', ∞ = ↑x' ∧ ∃ y', y = ↑y' ∧ Inseparable x' y'",
"case coe\nX : Type u_1\ninst✝ : TopologicalSpace X\ny : OnePoint X\nx✝ : X\n⊢ Inseparable (↑x✝) y ↔ ↑x✝ = ∞ ∧ y = ∞ ∨ ∃ x', ↑x✝ = ↑x' ∧ ∃ y', y = ↑y' ∧ In... | induction x using OnePoint.rec | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Topology.Compactness.CompactlyGeneratedSpace | {
"line": 300,
"column": 2
} | {
"line": 302,
"column": 45
} | {
"line": 304,
"column": 0
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactlyGeneratedSpace X\ns : Set X\nhs : ∀ ⦃K : Set X⦄, IsCompact K → IsOpen[inst✝¹] (s ∩ K)\n⊢ IsOpen[inst✝¹] s",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Continuous",
"congrArg",
"Set.prei... | [] | refine isOpen' fun K _ _ _ f hf ↦ ?_
rw [← Set.preimage_inter_range]
exact (hs (isCompact_range hf)).preimage hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Compactness.CompactlyGeneratedSpace | {
"line": 300,
"column": 2
} | {
"line": 302,
"column": 45
} | {
"line": 304,
"column": 0
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactlyGeneratedSpace X\ns : Set X\nhs : ∀ ⦃K : Set X⦄, IsCompact K → IsOpen[inst✝¹] (s ∩ K)\n⊢ IsOpen[inst✝¹] s",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Continuous",
"congrArg",
"Set.prei... | [] | refine isOpen' fun K _ _ _ f hf ↦ ?_
rw [← Set.preimage_inter_range]
exact (hs (isCompact_range hf)).preimage hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 531,
"column": 2
} | {
"line": 531,
"column": 32
} | {
"line": 531,
"column": 33
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ns : Set (OnePoint X)\ninst✝¹ : WeaklyLocallyCompactSpace X\ninst✝ : R1Space X\nkey : ∀ (z : X), Disjoint (𝓝 ↑z) (𝓝 ∞)\nx y : OnePoint X\n⊢ x ⤳ y ∨ Disjoint (𝓝 x) (𝓝 y)",
"ppTerm": "?m.74",
"assigned": true,
"usedConstants": [
... | [
"case infty\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ns : Set (OnePoint X)\ninst✝¹ : WeaklyLocallyCompactSpace X\ninst✝ : R1Space X\nkey : ∀ (z : X), Disjoint (𝓝 ↑z) (𝓝 ∞)\ny : OnePoint X\n⊢ ∞ ⤳ y ∨ Disjoint (𝓝 ∞) (𝓝 y)",
"case coe\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ns : ... | induction x using OnePoint.rec | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Topology.Compactness.CompactlyGeneratedSpace | {
"line": 335,
"column": 6
} | {
"line": 335,
"column": 21
} | {
"line": 335,
"column": 22
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : T2Space X\nh : ∀ (s : Set X), (∀ (K : Set X), IsCompact K → IsClosed[inst✝¹] (s ∩ K)) → IsClosed[inst✝¹] s\ns : Set X\nhs :\n ∀ (K : Type u) [inst : TopologicalSpace K] [CompactSpace K] [T2Space K] (f : K → X),\n Continuous[_, inst✝¹] f → IsClosed (f... | [
"X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : T2Space X\nh : ∀ (s : Set X), (∀ (K : Set X), IsCompact K → IsClosed[inst✝¹] (s ∩ K)) → IsClosed[inst✝¹] s\ns : Set X\nhs :\n ∀ (K : Type u) [inst : TopologicalSpace K] [CompactSpace K] [T2Space K] (f : K → X),\n Continuous[_, inst✝¹] f → IsClosed (f ⁻¹' s)\nK :... | Set.inter_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 554,
"column": 2
} | {
"line": 555,
"column": 47
} | {
"line": 555,
"column": 48
} | [
{
"pp": "case h\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : Infinite X\ninst✝ : DiscreteTopology X\ninhabited_h : Inhabited X\n⊢ ¬Tendsto (⇑CofiniteTopology.of.symm) (𝓝 (CofiniteTopology.of ↑default))\n (𝓝 (CofiniteTopology.of.symm (CofiniteTopology.of ↑default)))",
"ppTerm": "?h",
"assi... | [
"case h\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : Infinite X\ninst✝ : DiscreteTopology X\ninhabited_h : Inhabited X\n⊢ {x | ¬x = CofiniteTopology.of ↑default}.Infinite"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 570,
"column": 6
} | {
"line": 570,
"column": 64
} | {
"line": 572,
"column": 0
} | [
{
"pp": "case coe\nX✝ : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X✝\ns : Set (OnePoint X✝)\nX : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : DiscreteTopology X\ny : OnePoint X\nx✝¹ : y ∈ univ\nval : X\nx✝ : ↑val ∈ univ\nhxy : ↑val ≠ y\n⊢ IsOpen[inst✝¹] (some ⁻¹' {↑val})",
"ppTerm": "?coe",
"a... | [] | exacts [isOpen_discrete _, (Option.some_ne_none val).symm] | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_exacts_1 | Batteries.Tactic.exacts |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 587,
"column": 28
} | {
"line": 587,
"column": 39
} | {
"line": 587,
"column": 40
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ns : Set (OnePoint X)\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\ninst✝ : CompactSpace Y\ny : Y\nf : X → Y\nhf : IsEmbedding f\nhy : range f = {y}ᶜ\n_i : T2Space X\nN : Set Y\nhN : N ∈ 𝓝 y\nU : Set Y\nhU₁ : U ⊆ N\nhU₂ : IsOpen[inst✝²] U\nhU... | [
"X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ns : Set (OnePoint X)\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\ninst✝ : CompactSpace Y\ny : Y\nf : X → Y\nhf : IsEmbedding f\nhy : range f = {y}ᶜ\n_i : T2Space X\nN : Set Y\nhN : N ∈ 𝓝 y\nU : Set Y\nhU₁ : U ⊆ N\nhU₂ : IsOpen[inst✝²] U\nhU₃ : y ∈ U\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 597,
"column": 34
} | {
"line": 597,
"column": 50
} | {
"line": 597,
"column": 51
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ns : Set (OnePoint X)\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\ninst✝ : CompactSpace Y\ny : Y\nf : X → Y\nhf : IsEmbedding f\nhy : range f = {y}ᶜ\n_i : T2Space X\nthis : Tendsto f (coclosedCompact X) (𝓝 y)\np : X\n⊢ f p ≠ y",
"ppTerm"... | [
"X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ns : Set (OnePoint X)\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\ninst✝ : CompactSpace Y\ny : Y\nf : X → Y\nhf : IsEmbedding f\nhy : range f = {y}ᶜ\n_i : T2Space X\nthis : Tendsto f (coclosedCompact X) (𝓝 y)\np : X\n⊢ ¬f p = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 598,
"column": 10
} | {
"line": 598,
"column": 26
} | {
"line": 598,
"column": 27
} | [
{
"pp": "case coe\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ns : Set (OnePoint X)\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\ninst✝ : CompactSpace Y\ny : Y\nf : X → Y\nhf : IsEmbedding f\nhy : range f = {y}ᶜ\n_i : T2Space X\nthis : Tendsto f (coclosedCompact X) (𝓝 y)\np : X\nhp : f p ≠ y\n... | [
"case coe\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ns : Set (OnePoint X)\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\ninst✝ : CompactSpace Y\ny : Y\nf : X → Y\nhf : IsEmbedding f\nhy : range f = {y}ᶜ\n_i : T2Space X\nthis : Tendsto f (coclosedCompact X) (𝓝 y)\np : X\nhp : f p ≠ y\n⊢ ⋯.choose =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 603,
"column": 10
} | {
"line": 603,
"column": 26
} | {
"line": 603,
"column": 27
} | [
{
"pp": "case inr\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ns : Set (OnePoint X)\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\ninst✝ : CompactSpace Y\ny : Y\nf : X → Y\nhf : IsEmbedding f\nhy : range f = {y}ᶜ\n_i : T2Space X\nthis : Tendsto f (coclosedCompact X) (𝓝 y)\nq : Y\nhq : q ≠ y\nhq... | [
"case inr\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ns : Set (OnePoint X)\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\ninst✝ : CompactSpace Y\ny : Y\nf : X → Y\nhf : IsEmbedding f\nhy : range f = {y}ᶜ\n_i : T2Space X\nthis : Tendsto f (coclosedCompact X) (𝓝 y)\nq : Y\nhq : q ≠ y\nhq' : q ∈ rang... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 599,
"column": 27
} | {
"line": 603,
"column": 42
} | {
"line": 603,
"column": 43
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ns : Set (OnePoint X)\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\ninst✝ : CompactSpace Y\ny : Y\nf : X → Y\nhf : IsEmbedding f\nhy : range f = {y}ᶜ\n_i : T2Space X\nthis : Tendsto f (coclosedCompact X) (𝓝 y)\nq : Y\n⊢ (fun p ↦ p.elim y f) (... | [] | by
rcases eq_or_ne q y with rfl | hq
· simp
· have hq' : q ∈ range f := by simpa [hy]
simpa [hq] using hq'.choose_spec | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.UpperHalfPlane.ProperAction | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 51
} | {
"line": 84,
"column": 52
} | [
{
"pp": "case inr\nK : Set ℍ\nhK : IsCompact K\nhKne : K.Nonempty\nδ : ℝ\nhδ : δ > 0\ng : SL(2, ℝ)\nhg : g • I ∈ K\nhδK : Complex.normSq (denom ((Matrix.SpecialLinearGroup.mapGL ℝ) g) ↑I) ≤ 1 / δ\n⊢ ↑g 1 0 ^ 2 + ↑g 1 1 ^ 2 ≤ 1 / δ",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mp... | [
"case inr\nK : Set ℍ\nhK : IsCompact K\nhKne : K.Nonempty\nδ : ℝ\nhδ : δ > 0\ng : SL(2, ℝ)\nhg : g • I ∈ K\nhδK : Complex.normSq (denom ((Matrix.SpecialLinearGroup.mapGL ℝ) g) ↑I) ≤ 1 / δ\n⊢ ↑g 1 0 * ↑g 1 0 + ↑g 1 1 * ↑g 1 1 ≤ δ⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.CharacteristicFunction | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 15
} | {
"line": 138,
"column": 16
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf₁ f₂ : ℂ → E\nr : ℝ\nh₁f₁ : Meromorphic f₁\nh₁f₂ : Meromorphic f₂\nhr : 1 ≤ r\n⊢ ∀ a ∈ Finset.univ, Meromorphic (![f₁, f₂] a)",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Meromorphic",... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf₁ f₂ : ℂ → E\nr : ℝ\nh₁f₁ : Meromorphic f₁\nh₁f₂ : Meromorphic f₂\nhr : 1 ≤ r\n⊢ Meromorphic f₁ ∧ Meromorphic f₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.CharacteristicFunction | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 13
} | {
"line": 139,
"column": 14
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf₁ f₂ : ℂ → E\nr : ℝ\nh₁f₁ : Meromorphic f₁\nh₁f₂ : Meromorphic f₂\nhr : 1 ≤ r\nh_meromorphic : ∀ a ∈ Finset.univ, Meromorphic (![f₁, f₂] a)\n⊢ characteristic (f₁ + f₂) ⊤ r ≤ characteristic f₁ ⊤ r + characteristic f₂ ⊤ r + log 2",
... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf₁ f₂ : ℂ → E\nr : ℝ\nh₁f₁ : Meromorphic f₁\nh₁f₂ : Meromorphic f₂\nhr : 1 ≤ r\nh_meromorphic : ∀ a ∈ Finset.univ, Meromorphic (![f₁, f₂] a)\n⊢ characteristic (f₁ + f₂) ⊤ r ≤ characteristic f₁ ⊤ r + characteristic f₂ ⊤ r + log 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.ProperAction | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 30
} | {
"line": 100,
"column": 2
} | [
{
"pp": "K : Set ℍ\nhK : IsCompact K\n⊢ IsCompact ((fun g ↦ g • I) ⁻¹' K)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Real.instLE",
"Real",
"instHSMul",
"Matrix.SpecialLinearGroup",
"UpperHalfPlane.SLAction",
"Matrix",
"Matrix.SpecialLinearGro... | [
"K : Set ℍ\nhK : IsCompact K\nA : ℝ\nhA : ∀ (g : SL(2, ℝ)), g • I ∈ K → ↑g 0 0 ^ 2 + ↑g 0 1 ^ 2 ≤ A\n⊢ IsCompact ((fun g ↦ g • I) ⁻¹' K)"
] | obtain ⟨A, hA⟩ := absq_le hK | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Complex.ValueDistribution.Proximity.Basic | {
"line": 149,
"column": 4
} | {
"line": 149,
"column": 30
} | {
"line": 149,
"column": 31
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℂ → E\na : WithTop E\nh : a = ⊤\nr : ℝ\n⊢ 0 r ≤ proximity f a r",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"dite_cond_eq_true",
"Norm.norm",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"R... | [
"case pos\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℂ → E\na : WithTop E\nh : a = ⊤\nr : ℝ\n⊢ 0 ≤ circleAverage (fun x ↦ log⁺ ‖f x‖) 0 r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.Proximity.Basic | {
"line": 149,
"column": 4
} | {
"line": 149,
"column": 30
} | {
"line": 149,
"column": 31
} | [
{
"pp": "case neg\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℂ → E\na : WithTop E\nh : ¬a = ⊤\nr : ℝ\n⊢ 0 r ≤ proximity f a r",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real.instLE",
"Real",... | [
"case neg\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℂ → E\na : WithTop E\nh : ¬a = ⊤\nr : ℝ\n⊢ 0 ≤ circleAverage (fun x ↦ log⁺ ‖f x - a.untop₀‖⁻¹) 0 r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.Proximity.Basic | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 13
} | {
"line": 202,
"column": 14
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf₁ f₂ : ℂ → E\nh₁f₁ : Meromorphic f₁\nh₁f₂ : Meromorphic f₂\n⊢ proximity (f₁ + f₂) ⊤ ≤ proximity f₁ ⊤ + proximity f₂ ⊤ + fun x ↦ log 2",
"ppTerm": "?m.44",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf₁ f₂ : ℂ → E\nh₁f₁ : Meromorphic f₁\nh₁f₂ : Meromorphic f₂\n⊢ proximity (f₁ + f₂) ⊤ ≤ proximity f₁ ⊤ + proximity f₂ ⊤ + fun x ↦ log 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.FirstMainTheorem | {
"line": 112,
"column": 16
} | {
"line": 112,
"column": 27
} | {
"line": 112,
"column": 28
} | [
{
"pp": "f : ℂ → ℂ\nh : Meromorphic f\nR : ℝ\n⊢ ‖(characteristic f ⊤ - characteristic f⁻¹ ⊤) R‖ ≤ max |log ‖f 0‖| |log ‖meromorphicTrailingCoeffAt f 0‖| * ‖1 R‖",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
... | [
"f : ℂ → ℂ\nh : Meromorphic f\nR : ℝ\n⊢ |characteristic f ⊤ R - characteristic f⁻¹ ⊤ R| ≤ |log ‖f 0‖| ∨\n |characteristic f ⊤ R - characteristic f⁻¹ ⊤ R| ≤ |log ‖meromorphicTrailingCoeffAt f 0‖|"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.Proximity.IntegralPresentation | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 60
} | {
"line": 56,
"column": 61
} | [
{
"pp": "f : ℂ → ℂ\nR β : ℝ\n⊢ IntervalIntegrable (fun x ↦ cartanKernel f R x β) volume 0 (2 * π)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Real",
"Real.pi",
"HMul.hMul",
"Real.instZero",
"IntervalIntegrable",
"ValueDistribution.Cartan.cartanKerne... | [
"f : ℂ → ℂ\nR β : ℝ\n⊢ IntervalIntegrable (fun x ↦ log ‖f (circleMap 0 R β) - circleMap 0 1 x‖) volume 0 (2 * π)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.FirstMainTheorem | {
"line": 146,
"column": 6
} | {
"line": 147,
"column": 13
} | {
"line": 147,
"column": 14
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na₀ : E\nf : ℂ → E\nr : ℝ\nh✝ : Meromorphic f\nh₁f : CircleIntegrable (fun x ↦ log⁺ ‖f x‖) 0 r\nh₂f : CircleIntegrable (fun x ↦ log⁺ ‖f x - a₀‖) 0 r\nθ : ℂ\nhθ : θ ∈ sphere 0 |r|\nh : 0 ≤ log⁺ ‖f θ‖ - log⁺ ‖f θ - a₀‖\n⊢ |log... | [
"case pos\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na₀ : E\nf : ℂ → E\nr : ℝ\nh✝ : Meromorphic f\nh₁f : CircleIntegrable (fun x ↦ log⁺ ‖f x‖) 0 r\nh₂f : CircleIntegrable (fun x ↦ log⁺ ‖f x - a₀‖) 0 r\nθ : ℂ\nhθ : θ ∈ sphere 0 |r|\nh : 0 ≤ log⁺ ‖f θ‖ - log⁺ ‖f θ - a₀‖\n⊢ log⁺ ‖f θ‖ ≤ log... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.FirstMainTheorem | {
"line": 163,
"column": 16
} | {
"line": 163,
"column": 27
} | {
"line": 163,
"column": 28
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na₀ : E\nf : ℂ → E\nh : Meromorphic f\nR : ℝ\n⊢ ‖(characteristic f ⊤ - characteristic (fun x ↦ f x - a₀) ⊤) R‖ ≤ (log⁺ ‖a₀‖ + log 2) * ‖1 R‖",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Norm.norm",
... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na₀ : E\nf : ℂ → E\nh : Meromorphic f\nR : ℝ\n⊢ |characteristic f ⊤ R - characteristic (fun x ↦ f x - a₀) ⊤ R| ≤ log⁺ ‖a₀‖ + log 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.Proximity.IntegralPresentation | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 40
} | {
"line": 131,
"column": 41
} | [
{
"pp": "f : ℂ → ℂ\nR : ℝ\nh : Meromorphic f\nthis : Measurable f\n⊢ Integrable (fun p ↦ cartanKernel f R p.1 p.2)\n ((volume.restrict (uIoc 0 (2 * π))).prod (volume.restrict (uIoc 0 (2 * π))))",
"ppTerm": "?m.88",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"Norme... | [
"f : ℂ → ℂ\nR : ℝ\nh : Meromorphic f\nthis : Measurable f\n⊢ Integrable (fun p ↦ cartanKernel f R p.1 p.2)\n ((volume.restrict (Ioc 0 (2 * π))).prod (volume.restrict (Ioc 0 (2 * π))))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.Proximity.IntegralPresentation | {
"line": 144,
"column": 2
} | {
"line": 145,
"column": 9
} | {
"line": 145,
"column": 10
} | [
{
"pp": "f : ℂ → ℂ\nR : ℝ\nh : Meromorphic f\nh_int :\n Integrable (fun p ↦ cartanKernel f R p.1 p.2)\n ((volume.restrict (Ioc 0 (2 * π))).prod (volume.restrict (Ioc 0 (2 * π))))\n⊢ IntegrableOn (fun x ↦ ∫ (α : ℝ) in 0..2 * π, cartanKernel f R α x) (Ioc 0 (2 * π)) volume",
"ppTerm": "?m.91",
"assign... | [
"f : ℂ → ℂ\nR : ℝ\nh : Meromorphic f\nh_int :\n Integrable (fun p ↦ cartanKernel f R p.1 p.2)\n ((volume.restrict (Ioc 0 (2 * π))).prod (volume.restrict (Ioc 0 (2 * π))))\n⊢ Integrable (fun x ↦ ∫ (α : ℝ) in Ioc 0 (2 * π), log ‖f (circleMap 0 R x) - circleMap 0 1 α‖ ∂volume)\n (volume.restrict (Ioc 0 (2 * π))... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.Proximity.IntegralPresentation | {
"line": 156,
"column": 2
} | {
"line": 157,
"column": 9
} | {
"line": 157,
"column": 10
} | [
{
"pp": "f : ℂ → ℂ\nR : ℝ\nh : Meromorphic f\nh_int :\n Integrable (fun p ↦ cartanKernel f R p.1 p.2)\n ((volume.restrict (Ioc 0 (2 * π))).prod (volume.restrict (Ioc 0 (2 * π))))\n⊢ IntegrableOn (fun x ↦ ∫ (β : ℝ) in 0..2 * π, cartanKernel f R x β) (Ioc 0 (2 * π)) volume",
"ppTerm": "?m.91",
"assign... | [
"f : ℂ → ℂ\nR : ℝ\nh : Meromorphic f\nh_int :\n Integrable (fun p ↦ cartanKernel f R p.1 p.2)\n ((volume.restrict (Ioc 0 (2 * π))).prod (volume.restrict (Ioc 0 (2 * π))))\n⊢ Integrable (fun x ↦ ∫ (β : ℝ) in Ioc 0 (2 * π), log ‖f (circleMap 0 R β) - circleMap 0 1 x‖ ∂volume)\n (volume.restrict (Ioc 0 (2 * π))... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.Cartan | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 30
} | {
"line": 87,
"column": 31
} | [
{
"pp": "case neg.inr.inl\nf : ℂ → ℂ\nh : ¬¬MeromorphicAt f 0\nhzero : meromorphicOrderAt f 0 = 0\n⊢ CircleIntegrable (fun x ↦ log ‖meromorphicTrailingCoeffAt f 0 - x‖) 0 1",
"ppTerm": "?neg.inr.inl✝",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"... | [
"case neg.inr.inl\nf : ℂ → ℂ\nh : ¬¬MeromorphicAt f 0\nhzero : meromorphicOrderAt f 0 = 0\n⊢ CircleIntegrable (fun x ↦ log ‖x - meromorphicTrailingCoeffAt f 0‖) 0 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ConstantSpeed | {
"line": 67,
"column": 2
} | {
"line": 69,
"column": 53
} | {
"line": 71,
"column": 0
} | [
{
"pp": "E : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns : Set ℝ\nhs : s.Subsingleton\nl : ℝ≥0\n⊢ HasConstantSpeedOnWith f s l",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"sub_self",
"ENNReal.ofReal",
"congrArg"... | [] | rintro x hx y hy; cases hs hx hy
rw [eVariationOn.subsingleton f (fun y hy z hz => hs hy.1 hz.1 : (s ∩ Icc x x).Subsingleton)]
simp only [sub_self, mul_zero, ENNReal.ofReal_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.ConstantSpeed | {
"line": 67,
"column": 2
} | {
"line": 69,
"column": 53
} | {
"line": 71,
"column": 0
} | [
{
"pp": "E : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns : Set ℝ\nhs : s.Subsingleton\nl : ℝ≥0\n⊢ HasConstantSpeedOnWith f s l",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"sub_self",
"ENNReal.ofReal",
"congrArg"... | [] | rintro x hx y hy; cases hs hx hy
rw [eVariationOn.subsingleton f (fun y hy z hz => hs hy.1 hz.1 : (s ∩ Icc x x).Subsingleton)]
simp only [sub_self, mul_zero, ENNReal.ofReal_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.ValueDistribution.Cartan | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 13
} | {
"line": 155,
"column": 14
} | [
{
"pp": "f : ℂ → ℂ\nR : ℝ\nh : Meromorphic f\nhR : ¬R = 0\nx✝ : ℂ\n⊢ logCounting f (↑x✝) R =\n (((fun a ↦ circleAverage (fun x ↦ log ‖f x - a‖) 0 ?neg.convert_1✝) + fun x ↦ logCounting f ⊤ R) - fun a ↦\n log ‖meromorphicTrailingCoeffAt (fun x ↦ ?neg.convert_2✝ x - a) 0‖)\n x✝",
"ppTerm": "?m.... | [
"f : ℂ → ℂ\nR : ℝ\nh : Meromorphic f\nhR : ¬R = 0\nx✝ : ℂ\n⊢ logCounting f (↑x✝) R =\n circleAverage (fun x ↦ log ‖f x - x✝‖) 0 ?neg.convert_1✝ + logCounting f ⊤ R -\n log ‖meromorphicTrailingCoeffAt (fun x ↦ ?neg.convert_2✝ x - x✝) 0‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 101,
"column": 4
} | {
"line": 117,
"column": 10
} | {
"line": 119,
"column": 0
} | [
{
"pp": "E✝ : Type u_1\ninst✝² : NormedAddCommGroup E✝\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nD₁ D₂ : locallyFinsupp E ℤ\n⊢ (fun r ↦ ∑ᶠ (z : E), ↑(((toClosedBall r) (D₁ + D₂)) z) * log (r * ‖z‖⁻¹) + ↑((D₁ + D₂) 0) * log r) =\n (fun r ↦ ∑ᶠ (z : E), ↑(((toClosedBall r) D₁) z) * lo... | [] | simp only [map_add, coe_add, Pi.add_apply, Int.cast_add]
ext r
have {A B C D : ℝ} : A + B + (C + D) = A + C + (B + D) := by ring
rw [Pi.add_apply, this]
congr 1
· have h₁s : ((D₁.toClosedBall r).support ∪ (D₂.toClosedBall r).support).Finite := by
apply Set.finite_union.2
constructor
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 101,
"column": 4
} | {
"line": 117,
"column": 10
} | {
"line": 119,
"column": 0
} | [
{
"pp": "E✝ : Type u_1\ninst✝² : NormedAddCommGroup E✝\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nD₁ D₂ : locallyFinsupp E ℤ\n⊢ (fun r ↦ ∑ᶠ (z : E), ↑(((toClosedBall r) (D₁ + D₂)) z) * log (r * ‖z‖⁻¹) + ↑((D₁ + D₂) 0) * log r) =\n (fun r ↦ ∑ᶠ (z : E), ↑(((toClosedBall r) D₁) z) * lo... | [] | simp only [map_add, coe_add, Pi.add_apply, Int.cast_add]
ext r
have {A B C D : ℝ} : A + B + (C + D) = A + C + (B + D) := by ring
rw [Pi.add_apply, this]
congr 1
· have h₁s : ((D₁.toClosedBall r).support ∪ (D₂.toClosedBall r).support).Finite := by
apply Set.finite_union.2
constructor
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Asymptotic | {
"line": 81,
"column": 49
} | {
"line": 81,
"column": 66
} | {
"line": 81,
"column": 67
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nD : locallyFinsuppWithin univ ℤ\nh : 0 ≤ D\nh₁ : ¬D = 0\ne : E\nhe : single e 1 ≤ D\na : ℝ\nha : a > 0\nb c : ℝ\nhc : ∀ (b : ℝ), c ≤ b → ‖1 b‖ ≤ a * ‖logCounting (single e 1) b‖\nℓ : ℝ := 1 + max ‖e‖ (max |b| |c|)\nh₁ℓ : c ≤ ℓ\nh₂ℓ : 1... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nD : locallyFinsuppWithin univ ℤ\nh : 0 ≤ D\nh₁ : ¬D = 0\ne : E\nhe : single e 1 ≤ D\na : ℝ\nha : a > 0\nb c : ℝ\nhc : ∀ (b : ℝ), c ≤ b → ‖1 b‖ ≤ a * ‖logCounting (single e 1) b‖\nℓ : ℝ := 1 + max ‖e‖ (max |b| |c|)\nh₁ℓ : c ≤ ℓ\nh₂ℓ : 1 ≤ ℓ\n⊢ 1 ≤ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ConstantSpeed | {
"line": 112,
"column": 4
} | {
"line": 117,
"column": 89
} | {
"line": 118,
"column": 4
} | [
{
"pp": "case inl.inr\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns : Set ℝ\nl : ℝ≥0\nt : Set ℝ\nhfs : ∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → x ≤ y → eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (↑l * (y - x))\nhft : ∀ ⦃x : ℝ⦄, x ∈ t → ∀ ⦃y : ℝ⦄, y ∈ t → x ≤ y → eVariationOn f (t ∩ Icc x y) = ENNRea... | [
"case inl.inr\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns : Set ℝ\nl : ℝ≥0\nt : Set ℝ\nhfs : ∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → x ≤ y → eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (↑l * (y - x))\nhft : ∀ ⦃x : ℝ⦄, x ∈ t → ∀ ⦃y : ℝ⦄, y ∈ t → x ≤ y → eVariationOn f (t ∩ Icc x y) = ENNReal.ofReal (↑l... | have : (s ∪ t) ∩ Icc z y = s ∩ Icc z x ∪ t ∩ Icc x y := by
ext w; constructor
· rintro ⟨ws | wt, zw, wy⟩
exacts [Or.inl ⟨ws, zw, hs.2 ws⟩, Or.inr ⟨wt, ht.2 wt, wy⟩]
· rintro (⟨ws, zw, wx⟩ | ⟨wt, xw, wy⟩)
exacts [⟨Or.inl ws, zw, wx.trans (ht.2 yt)⟩, ⟨Or.inr wt, (hs.2 zs).trans xw, wy⟩] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Semicontinuity.Lindelof | {
"line": 83,
"column": 4
} | {
"line": 83,
"column": 70
} | {
"line": 83,
"column": 71
} | [
{
"pp": "X : Type u_1\nE : Type u_2\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : HereditarilyLindelofSpace X\ninst✝⁴ : LinearOrder E\ninst✝³ : TopologicalSpace E\ninst✝² : OrderClosedTopology E\ninst✝¹ : DenselyOrdered E\ninst✝ : SeparableSpace E\ns : X → E\n𝓕 : Set (X → E)\nh𝓕_cont : ∀ f ∈ 𝓕, UpperSemicontinuous ... | [
"X : Type u_1\nE : Type u_2\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : HereditarilyLindelofSpace X\ninst✝⁴ : LinearOrder E\ninst✝³ : TopologicalSpace E\ninst✝² : OrderClosedTopology E\ninst✝¹ : DenselyOrdered E\ninst✝ : SeparableSpace E\ns : X → E\n𝓕 : Set (X → E)\nh𝓕_cont : ∀ f ∈ 𝓕, UpperSemicontinuous f\nh𝓕 : ∀ (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ConstantSpeed | {
"line": 166,
"column": 4
} | {
"line": 166,
"column": 19
} | {
"line": 166,
"column": 20
} | [
{
"pp": "case mpr\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns : Set ℝ\nh : eVariationOn f s = 0\nx : ℝ\nx✝¹ : x ∈ s\ny : ℝ\nx✝ : y ∈ s\n⊢ eVariationOn f (s ∩ Icc x y) = 0",
"ppTerm": "?mpr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case mpr\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns : Set ℝ\nh : eVariationOn f s = 0\nx : ℝ\nx✝¹ : x ∈ s\ny : ℝ\nx✝ : y ∈ s\n⊢ eVariationOn f (s ∩ Icc x y) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 212,
"column": 6
} | {
"line": 212,
"column": 27
} | {
"line": 212,
"column": 28
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : DecidableEq E\ninst✝ : ProperSpace E\nD : locallyFinsupp E ℤ\ne : E\nhD : single e 1 ≤ D\na : ℝ\nha : a ∈ Ioi ‖e‖\nb : ℝ\nhb : b ∈ Ioi ‖e‖\nhab : a ≤ b\n⊢ a ∈ Ioi 0",
"ppTerm": "?m.178",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : DecidableEq E\ninst✝ : ProperSpace E\nD : locallyFinsupp E ℤ\ne : E\nhD : single e 1 ≤ D\na : ℝ\nha : a ∈ Ioi ‖e‖\nb : ℝ\nhb : b ∈ Ioi ‖e‖\nhab : a ≤ b\n⊢ 0 < a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 209,
"column": 2
} | {
"line": 212,
"column": 55
} | {
"line": 214,
"column": 0
} | [
{
"pp": "case hg\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : DecidableEq E\ninst✝ : ProperSpace E\nD : locallyFinsupp E ℤ\ne : E\nhD : single e 1 ≤ D\n⊢ MonotoneOn (logCounting (D - single e 1)) (Ioi ‖e‖)",
"ppTerm": "?hg",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup"... | [] | · intro a ha b hb hab
apply logCounting_mono _ _ ((norm_nonneg e).trans_lt hb) hab
· simp [hD]
· simpa [mem_Ioi] using (norm_nonneg e).trans_lt ha | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 222,
"column": 54
} | {
"line": 222,
"column": 65
} | {
"line": 222,
"column": 66
} | [
{
"pp": "E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nf : locallyFinsupp E ℤ\nr : ℝ\nh : 0 ≤ f\nhr : 1 ≤ r\nh₃r : 0 < r\nthis : ∀ (z : E), 0 ≤ ↑(((toClosedBall r) f) z) * log (r * ‖z‖⁻¹)\n⊢ 0 ≤ ↑(f 0)",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"Int.cast_no... | [
"E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nf : locallyFinsupp E ℤ\nr : ℝ\nh : 0 ≤ f\nhr : 1 ≤ r\nh₃r : 0 < r\nthis : ∀ (z : E), 0 ≤ ↑(((toClosedBall r) f) z) * log (r * ‖z‖⁻¹)\n⊢ 0 ≤ f 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 228,
"column": 6
} | {
"line": 228,
"column": 23
} | {
"line": 228,
"column": 24
} | [
{
"pp": "case pos.refine_1\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nf : locallyFinsupp E ℤ\nr : ℝ\nh : 0 ≤ f\nhr : 1 ≤ r\nh₃r : 0 < r\na : E\nh₁a : ¬a = 0\nh₂a : a ∈ closedBall 0 |r|\n⊢ 0 ≤ ↑(((toClosedBall r) f) a)",
"ppTerm": "?pos.refine_1✝",
"assigned": true,
"usedCon... | [
"case pos.refine_1\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nf : locallyFinsupp E ℤ\nr : ℝ\nh : 0 ≤ f\nhr : 1 ≤ r\nh₃r : 0 < r\na : E\nh₁a : ¬a = 0\nh₂a : a ∈ closedBall 0 |r|\n⊢ 0 ≤ f a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 229,
"column": 6
} | {
"line": 229,
"column": 86
} | {
"line": 229,
"column": 87
} | [
{
"pp": "case pos.refine_2\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nf : locallyFinsupp E ℤ\nr : ℝ\nh : 0 ≤ f\nhr : 1 ≤ r\nh₃r : 0 < r\na : E\nh₁a : ¬a = 0\nh₂a : a ∈ closedBall 0 |r|\n⊢ 1 ≤ r * ‖a‖⁻¹",
"ppTerm": "?pos.refine_2✝",
"assigned": true,
"usedConstants": [
... | [
"case pos.refine_2\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nf : locallyFinsupp E ℤ\nr : ℝ\nh : 0 ≤ f\nhr : 1 ≤ r\nh₃r : 0 < r\na : E\nh₁a : ¬a = 0\nh₂a : a ∈ closedBall 0 |r|\n⊢ ‖a‖ ≤ r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 239,
"column": 2
} | {
"line": 239,
"column": 13
} | {
"line": 239,
"column": 14
} | [
{
"pp": "E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nf₁ f₂ : locallyFinsupp E ℤ\nr : ℝ\nh : 0 ≤ f₂ - f₁\nhr : 1 ≤ r\n⊢ 0 ≤ logCounting f₂ r - logCounting f₁ r",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
... | [
"E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nf₁ f₂ : locallyFinsupp E ℤ\nr : ℝ\nh : 0 ≤ f₂ - f₁\nhr : 1 ≤ r\n⊢ logCounting f₁ r ≤ logCounting f₂ r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 274,
"column": 4
} | {
"line": 274,
"column": 39
} | {
"line": 275,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : ProperSpace 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nU : Set 𝕜\nf g : 𝕜 → E\na : WithTop E\na₀ : E\nh : a = ⊤\n⊢ ℝ → ℝ",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"... | [] | exact (divisor f univ)⁻.logCounting | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 274,
"column": 4
} | {
"line": 274,
"column": 39
} | {
"line": 275,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : ProperSpace 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nU : Set 𝕜\nf g : 𝕜 → E\na : WithTop E\na₀ : E\nh : a = ⊤\n⊢ ℝ → ℝ",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"... | [] | exact (divisor f univ)⁻.logCounting | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 274,
"column": 4
} | {
"line": 274,
"column": 39
} | {
"line": 275,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : ProperSpace 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nU : Set 𝕜\nf g : 𝕜 → E\na : WithTop E\na₀ : E\nh : a = ⊤\n⊢ ℝ → ℝ",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"... | [] | exact (divisor f univ)⁻.logCounting | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 358,
"column": 4
} | {
"line": 358,
"column": 32
} | {
"line": 358,
"column": 33
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : ProperSpace 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\ne : WithTop E\nh : e = ⊤\n⊢ MonotoneOn (logCounting f e) (Ioi 0)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": ... | [
"case pos\n𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : ProperSpace 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\ne : WithTop E\nh : e = ⊤\n⊢ MonotoneOn (locallyFinsuppWithin.logCounting (divisor f univ)⁻) (Ioi 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 358,
"column": 4
} | {
"line": 358,
"column": 32
} | {
"line": 358,
"column": 33
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : ProperSpace 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\ne : WithTop E\nh : ¬e = ⊤\n⊢ MonotoneOn (logCounting f e) (Ioi 0)",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants":... | [
"case neg\n𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : ProperSpace 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\ne : WithTop E\nh : ¬e = ⊤\n⊢ MonotoneOn (locallyFinsuppWithin.logCounting (divisor (fun x ↦ f x - e.untop₀) univ)⁺) (Ioi 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 366,
"column": 4
} | {
"line": 367,
"column": 43
} | {
"line": 368,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : ProperSpace 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nr : ℝ\nf : 𝕜 → E\ne : WithTop E\nhr : 1 ≤ r\nh : e = ⊤\n⊢ 0 ≤ logCounting f e r",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstant... | [] | simp [logCounting, h, locallyFinsuppWithin.logCounting_nonneg
(negPart_nonneg (divisor f univ)) hr] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 366,
"column": 4
} | {
"line": 367,
"column": 43
} | {
"line": 368,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : ProperSpace 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nr : ℝ\nf : 𝕜 → E\ne : WithTop E\nhr : 1 ≤ r\nh : e = ⊤\n⊢ 0 ≤ logCounting f e r",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstant... | [] | simp [logCounting, h, locallyFinsuppWithin.logCounting_nonneg
(negPart_nonneg (divisor f univ)) hr] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 366,
"column": 4
} | {
"line": 367,
"column": 43
} | {
"line": 368,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : ProperSpace 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nr : ℝ\nf : 𝕜 → E\ne : WithTop E\nhr : 1 ≤ r\nh : e = ⊤\n⊢ 0 ≤ logCounting f e r",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstant... | [] | simp [logCounting, h, locallyFinsuppWithin.logCounting_nonneg
(negPart_nonneg (divisor f univ)) hr] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 598,
"column": 4
} | {
"line": 598,
"column": 19
} | {
"line": 600,
"column": 0
} | [
{
"pp": "case e_a.hf\nR : ℝ\nf : ℂ → ℂ\nh : Meromorphic f\nhR : R ≠ 0\nh₁f : MeromorphicOn f (closedBall 0 |R|)\n⊢ MeromorphicOn f (closedBall 0 |R|)",
"ppTerm": "?e_a.hf✝",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"h₁f"
],
"usedGoals": []
},
{
"pp": "case e_a.... | [] | all_goals aesop | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.Combinatorics.Hall.Finite | {
"line": 62,
"column": 33
} | {
"line": 62,
"column": 50
} | {
"line": 62,
"column": 51
} | [
{
"pp": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nx : ι\na : α\ns' : Finset ↑{x' | x' ≠ x}\nthis : DecidableEq ι\nha : s'.Nonempty → image (fun z ↦ ↑z) s' ≠ univ → #s' < #((image (fun z ↦ ↑z) s').biUnion t)\nhe : s'.Nonempty\nh : image (fun z ↦ ↑z) s' = univ\n⊢ False"... | [
"ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nx : ι\na : α\ns' : Finset ↑{x' | x' ≠ x}\nthis : DecidableEq ι\nha : s'.Nonempty → image (fun z ↦ ↑z) s' ≠ univ → #s' < #((image (fun z ↦ ↑z) s').biUnion t)\nhe : s'.Nonempty\nh : image (fun z ↦ ↑z) s' = univ\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Approximation | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 51
} | {
"line": 74,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ns : Set E\nφ : E → ℝ\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : IsScalarTower ℝ 𝕜 E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nx : E\na... | [
"𝕜 : Type u_1\nE : Type u_2\ns : Set E\nφ : E → ℝ\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : IsScalarTower ℝ 𝕜 E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nx : E\na : ℝ\nhx : x... | let A := { p : E × 𝕜 | p.1 ∈ s ∧ φ p.1 ≤ re p.2 } | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Combinatorics.Hall.Finite | {
"line": 113,
"column": 6
} | {
"line": 113,
"column": 27
} | {
"line": 113,
"column": 28
} | [
{
"pp": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = n + 1\nht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), #s' ≤ #(s'.biUnio... | [
"ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = n + 1\nht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), #s' ≤ #(s'.biUnion t')) → ∃ f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Approximation | {
"line": 84,
"column": 6
} | {
"line": 85,
"column": 29
} | {
"line": 85,
"column": 30
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ns : Set E\nφ : E → ℝ\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : IsScalarTower ℝ 𝕜 E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nx : E\na... | [
"𝕜 : Type u_1\nE : Type u_2\ns : Set E\nφ : E → ℝ\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : IsScalarTower ℝ 𝕜 E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nx : E\na : ℝ\nhx : x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Approximation | {
"line": 87,
"column": 4
} | {
"line": 88,
"column": 11
} | {
"line": 88,
"column": 12
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nE : Type u_2\ns : Set E\nφ : E → ℝ\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : IsScalarTower ℝ 𝕜 E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpac... | [
"case refine_1\n𝕜 : Type u_1\nE : Type u_2\ns : Set E\nφ : E → ℝ\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : IsScalarTower ℝ 𝕜 E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nx : E... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.BetweenList | {
"line": 137,
"column": 33
} | {
"line": 137,
"column": 44
} | {
"line": 137,
"column": 45
} | [
{
"pp": "case cons.refine_2.refine_1.cons.inl.cons\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : IsOrderedRing R\nhead head3 : P\ntail : List P\nx✝ :\n (Pairwise (Sbtw R head) (head :: head3 :: ... | [
"case cons.refine_2.refine_1.cons.inl.cons\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : IsOrderedRing R\nhead head3 : P\ntail : List P\nx✝ :\n (Pairwise (Sbtw R head) (head :: head3 :: tail) ∧ Trip... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.BetweenList | {
"line": 138,
"column": 12
} | {
"line": 138,
"column": 23
} | {
"line": 138,
"column": 24
} | [
{
"pp": "case cons.refine_2.refine_1.cons.inr\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : IsOrderedRing R\nhead head2 : P\ntail : List P\nx✝ :\n (Pairwise (Sbtw R head) (head2 :: tail) ∧ Tripl... | [
"case cons.refine_2.refine_1.cons.inr\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : IsOrderedRing R\nhead head2 : P\ntail : List P\nx✝ :\n (Pairwise (Sbtw R head) (head2 :: tail) ∧ Triplewise (Sbtw ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Approximation | {
"line": 96,
"column": 32
} | {
"line": 96,
"column": 48
} | {
"line": 96,
"column": 49
} | [
{
"pp": "s : Set ℝ\nf : ℝ → ℝ\nx a : ℝ\nhx : x ∈ s\nhax : a < f x\nhsc : IsClosed s\nhfc : LowerSemicontinuousOn f s\nhf : ConvexOn ℝ s f\nl : ℝ →L[ℝ] ℝ\nc' : ℝ\nhlc'_le : s.restrict (⇑re ∘ ⇑l) + const (↑s) c' ≤ s.restrict f\nhlc'_eq : re (l x) + c' = a\nh1 : ∀ (y : ℝ), l 1 * y = l y\ny : ℝ\nhy : y ∈ s\n⊢ l 1 *... | [
"s : Set ℝ\nf : ℝ → ℝ\nx a : ℝ\nhx : x ∈ s\nhax : a < f x\nhsc : IsClosed s\nhfc : LowerSemicontinuousOn f s\nhf : ConvexOn ℝ s f\nl : ℝ →L[ℝ] ℝ\nc' : ℝ\nhlc'_le : s.restrict (⇑re ∘ ⇑l) + const (↑s) c' ≤ s.restrict f\nhlc'_eq : re (l x) + c' = a\nh1 : ∀ (y : ℝ), l 1 * y = l y\ny : ℝ\nhy : y ∈ s\n⊢ l y + c' ≤ f y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Approximation | {
"line": 96,
"column": 69
} | {
"line": 96,
"column": 85
} | {
"line": 96,
"column": 86
} | [
{
"pp": "s : Set ℝ\nf : ℝ → ℝ\nx a : ℝ\nhx : x ∈ s\nhax : a < f x\nhsc : IsClosed s\nhfc : LowerSemicontinuousOn f s\nhf : ConvexOn ℝ s f\nl : ℝ →L[ℝ] ℝ\nc' : ℝ\nhlc'_le : s.restrict (⇑re ∘ ⇑l) + const (↑s) c' ≤ s.restrict f\nhlc'_eq : re (l x) + c' = a\nh1 : ∀ (y : ℝ), l 1 * y = l y\n⊢ l 1 * x + c' = a",
"... | [
"s : Set ℝ\nf : ℝ → ℝ\nx a : ℝ\nhx : x ∈ s\nhax : a < f x\nhsc : IsClosed s\nhfc : LowerSemicontinuousOn f s\nhf : ConvexOn ℝ s f\nl : ℝ →L[ℝ] ℝ\nc' : ℝ\nhlc'_le : s.restrict (⇑re ∘ ⇑l) + const (↑s) c' ≤ s.restrict f\nhlc'_eq : re (l x) + c' = a\nh1 : ∀ (y : ℝ), l 1 * y = l y\n⊢ l x + c' = a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Approximation | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 51
} | {
"line": 115,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ns : Set E\nφ : E → ℝ\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : IsScalarTower ℝ 𝕜 E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nhsc : Is... | [
"𝕜 : Type u_1\nE : Type u_2\ns : Set E\nφ : E → ℝ\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : IsScalarTower ℝ 𝕜 E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nhsc : IsClosed s\nhφ... | let A := { p : E × 𝕜 | p.1 ∈ s ∧ φ p.1 ≤ re p.2 } | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.Convex.Between | {
"line": 384,
"column": 4
} | {
"line": 384,
"column": 37
} | {
"line": 384,
"column": 38
} | [
{
"pp": "case refine_1\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : IsOrderedRing R\nx y : P\nh : Wbtw R x y x\n⊢ y = x",
"ppTerm": "?refine_1",
"assigned": false,
"usedConstants": [... | [
"case refine_1\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : IsOrderedRing R\nx y : P\nh : Wbtw R x y x\n⊢ y = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Hall.Basic | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 37
} | {
"line": 69,
"column": 38
} | [
{
"pp": "ι : Type u\nα : Type v\nt : ι → Finset α\nι' ι'' : Finset ι\nh : ι' ⊆ ι''\nf : ↑(hallMatchingsOn t ι'')\nhinj : Injective ↑f\nhc : ∀ (x : ↥ι''), ↑f x ∈ t ↑x\ni : ι\nhi : i ∈ ι'\nj : ι\nhj : j ∈ ι'\nhh : (fun i ↦ ↑f ⟨↑i, ⋯⟩) ⟨i, hi⟩ = (fun i ↦ ↑f ⟨↑i, ⋯⟩) ⟨j, hj⟩\n⊢ ⟨i, hi⟩ = ⟨j, hj⟩",
"ppTerm": "?m... | [
"ι : Type u\nα : Type v\nt : ι → Finset α\nι' ι'' : Finset ι\nh : ι' ⊆ ι''\nf : ↑(hallMatchingsOn t ι'')\nhinj : Injective ↑f\nhc : ∀ (x : ↥ι''), ↑f x ∈ t ↑x\ni : ι\nhi : i ∈ ι'\nj : ι\nhj : j ∈ ι'\nhh : (fun i ↦ ↑f ⟨↑i, ⋯⟩) ⟨i, hi⟩ = (fun i ↦ ↑f ⟨↑i, ⋯⟩) ⟨j, hj⟩\n⊢ i = j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Hall.Basic | {
"line": 104,
"column": 4
} | {
"line": 104,
"column": 19
} | {
"line": 104,
"column": 20
} | [
{
"pp": "ι : Type u\nα : Type v\nt : ι → Finset α\nι' : Finset ι\ng : ↑(hallMatchingsOn t ι') → ↥ι' → ↥(ι'.biUnion t) := fun f i ↦ ⟨↑f i, ⋯⟩\nf f' : ↑(hallMatchingsOn t ι')\nh : ∀ (x : ↥ι'), g f x = g f' x\na : ↥ι'\n⊢ ↑f a = ↑f' a",
"ppTerm": "?m.79",
"assigned": false,
"usedConstants": [],
"use... | [
"ι : Type u\nα : Type v\nt : ι → Finset α\nι' : Finset ι\ng : ↑(hallMatchingsOn t ι') → ↥ι' → ↥(ι'.biUnion t) := fun f i ↦ ⟨↑f i, ⋯⟩\nf f' : ↑(hallMatchingsOn t ι')\nh : ∀ (x : ↥ι'), g f x = g f' x\na : ↥ι'\n⊢ ↑f a = ↑f' a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Approximation | {
"line": 227,
"column": 2
} | {
"line": 227,
"column": 13
} | {
"line": 227,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nφ : E → ℝ\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : IsScalarTower ℝ 𝕜 E\ninst✝³ : IsTopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : LocallyConvexSpace ℝ E\ninst✝ : Hereditari... | [
"𝕜 : Type u_1\nE : Type u_2\nφ : E → ℝ\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : IsScalarTower ℝ 𝕜 E\ninst✝³ : IsTopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : LocallyConvexSpace ℝ E\ninst✝ : HereditarilyLindelofSp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.CofilteredSystem | {
"line": 282,
"column": 2
} | {
"line": 282,
"column": 13
} | {
"line": 282,
"column": 14
} | [
{
"pp": "J : Type u\ninst✝¹ : Category.{v_1, u} J\nF : J ⥤ Type v\ninst✝ : IsCofilteredOrEmpty J\nFsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective ⇑(ConcreteCategory.hom (F.map f))\ni j : J\nf g : i ⟶ j\nk : J\nφ : k ⟶ i\nhφ : φ ≫ f = φ ≫ g\nthis :\n (fun x ↦ (ConcreteCategory.hom (F.map f)) ((ConcreteCateg... | [
"J : Type u\ninst✝¹ : Category.{v_1, u} J\nF : J ⥤ Type v\ninst✝ : IsCofilteredOrEmpty J\nFsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective ⇑(ConcreteCategory.hom (F.map f))\ni j : J\nf g : i ⟶ j\nk : J\nφ : k ⟶ i\nhφ : φ ≫ f = φ ≫ g\nthis :\n (fun x ↦ (ConcreteCategory.hom (F.map f)) ((ConcreteCategory.hom (F.m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Permutation | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 95
} | {
"line": 118,
"column": 6
} | [
{
"pp": "n : Type u_1\ninst✝³ : DecidableEq n\nσ : Perm n\ninst✝² : Fintype n\n𝕜 : Type u_3\ninst✝¹ : RCLike 𝕜\ninst✝ : Nonempty n\ninhabited_h : Inhabited n\n⊢ 1 ≤ ‖Perm.permMatrix 𝕜 σ‖",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : Type u_1\ninst✝³ : DecidableEq n\nσ : Perm n\ninst✝² : Fintype n\n𝕜 : Type u_3\ninst✝¹ : RCLike 𝕜\ninst✝ : Nonempty n\ninhabited_h : Inhabited n\n⊢ 1 ≤ ‖Perm.permMatrix 𝕜 σ‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Between | {
"line": 735,
"column": 2
} | {
"line": 735,
"column": 13
} | {
"line": 735,
"column": 14
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y z : P\nh : Wbtw R x y z\n⊢ SameRay R (y -ᵥ x) (z -ᵥ y)",
"ppTerm": "?m.25",
"assigned": false,
"us... | [
"R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y z : P\nh : Wbtw R x y z\n⊢ SameRay R (y -ᵥ x) (z -ᵥ y)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Stochastic | {
"line": 51,
"column": 4
} | {
"line": 51,
"column": 36
} | {
"line": 52,
"column": 2
} | [
{
"pp": "R✝ : Type u_1\nn✝ : Type u_2\ninst✝⁹ : Fintype n✝\ninst✝⁸ : DecidableEq n✝\ninst✝⁷ : Semiring R✝\ninst✝⁶ : PartialOrder R✝\ninst✝⁵ : IsOrderedRing R✝\nM✝ : Matrix n✝ n✝ R✝\nx : n✝ → R✝\nR : Type u_3\nn : Type u_4\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Semiring R\ninst✝¹ : PartialOrder R\... | [] | rw [← mulVec_mulVec, hN.2, hM.2] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Matrix.Stochastic | {
"line": 213,
"column": 6
} | {
"line": 213,
"column": 31
} | {
"line": 213,
"column": 31
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Semiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\nσ : Equiv.Perm n\n⊢ Equiv.Perm.permMatrix R σ ∈ colStochastic R n",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Matrix.colStochas... | [
"R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Semiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\nσ : Equiv.Perm n\n⊢ (∀ (i j : n), 0 ≤ Equiv.Perm.permMatrix R σ i j) ∧ ∀ (j : n), ∑ i, Equiv.Perm.permMatrix R σ i j = 1"
] | mem_colStochastic_iff_sum | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Stochastic | {
"line": 238,
"column": 16
} | {
"line": 238,
"column": 27
} | {
"line": 238,
"column": 28
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝⁶ : Fintype n\ninst✝⁵ : DecidableEq n\ninst✝⁴ : Semiring R\ninst✝³ : PartialOrder R\ninst✝² : IsOrderedRing R\nm : Type u_3\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\nM : Matrix n n R\ne₁ e₂ : n ≃ m\nhM : M ∈ rowStochastic R n\nx✝¹ x✝ : m\n⊢ 0 ≤ (reindex e₁ e₂) M x✝¹ x... | [
"R : Type u_1\nn : Type u_2\ninst✝⁶ : Fintype n\ninst✝⁵ : DecidableEq n\ninst✝⁴ : Semiring R\ninst✝³ : PartialOrder R\ninst✝² : IsOrderedRing R\nm : Type u_3\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\nM : Matrix n n R\ne₁ e₂ : n ≃ m\nhM : M ∈ rowStochastic R n\nx✝¹ x✝ : m\n⊢ 0 ≤ M (e₁.symm x✝¹) (e₂.symm x✝)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.DoublyStochasticMatrix | {
"line": 102,
"column": 88
} | {
"line": 105,
"column": 78
} | {
"line": 107,
"column": 0
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Semiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\n⊢ Convex R ↑(doublyStochastic R n)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Analysis.Convex... | [] | by
intro x hx y hy a b ha hb h
simp only [SetLike.mem_coe, mem_doublyStochastic_iff_sum] at hx hy ⊢
simp [add_nonneg, ha, hb, mul_nonneg, hx, hy, sum_add_distrib, ← mul_sum, h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Between | {
"line": 813,
"column": 2
} | {
"line": 813,
"column": 20
} | {
"line": 814,
"column": 2
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : IsTorsionFree R V\nt : Affine.Triangle R P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\nh3 ... | [
"R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : IsTorsionFree R V\nt : Affine.Triangle R P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\nh3 : ∀ (i : Fin... | refine ⟨hs i₁, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Convex.Caratheodory | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 15
} | {
"line": 71,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u\ninst✝⁵ : Field 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : DecidableEq E\nt : Finset E\nf : E → 𝕜\nfpos : ∀ y ∈ t, 0 ≤ f y\nfsum : ∑ y ∈ t, f y = 1\ng : E → 𝕜\ngcombo : ∑ e ∈ t, g e • e = 0\ngsum : ∑ e... | [] | exact mem.2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Convex.Birkhoff | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 22
} | {
"line": 73,
"column": 23
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Semifield R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nM : Matrix n n R\ns : R\nhs : 0 < s\nhM : (∀ (i j : n), 0 ≤ M i j) ∧ (∀ (i : n), ∑ j, M i j = s) ∧ ∀ (j : n), ∑ i, M i j = s\nf : n → Finset n := fun i ↦ ... | [
"R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Semifield R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nM : Matrix n n R\ns : R\nhs : 0 < s\nhM : (∀ (i j : n), 0 ≤ M i j) ∧ (∀ (i : n), ∑ j, M i j = s) ∧ ∀ (j : n), ∑ i, M i j = s\nf : n → Finset n := fun i ↦ {j | M i j ≠... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Birkhoff | {
"line": 109,
"column": 4
} | {
"line": 109,
"column": 38
} | {
"line": 109,
"column": 39
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Field R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nh✝ : Nonempty n\nd : ℕ\nih :\n ∀ m < d,\n ∀ (M : Matrix n n R) (s : R),\n 0 ≤ s →\n (∃ M' ∈ doublyStochastic R n, M = s • M') →\n #{i... | [
"R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Field R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nh✝ : Nonempty n\nd : ℕ\nih :\n ∀ m < d,\n ∀ (M : Matrix n n R) (s : R),\n 0 ≤ s →\n (∃ M' ∈ doublyStochastic R n, M = s • M') →\n #{i | M i.1 i.2... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Between | {
"line": 819,
"column": 4
} | {
"line": 819,
"column": 15
} | {
"line": 819,
"column": 16
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : IsTorsionFree R V\nt : Affine.Triangle R P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\nh3 ... | [
"R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : IsTorsionFree R V\nt : Affine.Triangle R P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\nh3 : ∀ (i : Fin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Caratheodory | {
"line": 93,
"column": 12
} | {
"line": 93,
"column": 67
} | {
"line": 93,
"column": 68
} | [
{
"pp": "case hb\n𝕜 : Type u_1\nE : Type u\ninst✝⁵ : Field 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : DecidableEq E\nt : Finset E\nf : E → 𝕜\nfpos : ∀ y ∈ t, 0 ≤ f y\nfsum : ∑ y ∈ t, f y = 1\ng : E → 𝕜\ngcombo : ∑ e ∈ t, g e • e = 0\ng... | [
"case hb\n𝕜 : Type u_1\nE : Type u\ninst✝⁵ : Field 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : DecidableEq E\nt : Finset E\nf : E → 𝕜\nfpos : ∀ y ∈ t, 0 ≤ f y\nfsum : ∑ y ∈ t, f y = 1\ng : E → 𝕜\ngcombo : ∑ e ∈ t, g e • e = 0\ngsum : ∑ e ∈ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Between | {
"line": 820,
"column": 4
} | {
"line": 820,
"column": 24
} | {
"line": 820,
"column": 25
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : IsTorsionFree R V\nt : Affine.Triangle R P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\nh3 ... | [
"R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : IsTorsionFree R V\nt : Affine.Triangle R P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\nh3 : ∀ (i : Fin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Birkhoff | {
"line": 131,
"column": 8
} | {
"line": 131,
"column": 42
} | {
"line": 131,
"column": 43
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Field R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nh✝ : Nonempty n\nd : ℕ\nih :\n ∀ m < d,\n ∀ (M : Matrix n n R) (s : R),\n 0 ≤ s →\n (∃ M' ∈ doublyStochastic R n, M = s • M') →\n #{i... | [
"R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Field R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nh✝ : Nonempty n\nd : ℕ\nih :\n ∀ m < d,\n ∀ (M : Matrix n n R) (s : R),\n 0 ≤ s →\n (∃ M' ∈ doublyStochastic R n, M = s • M') →\n #{i | M i.1 i.2... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Cone.Basic | {
"line": 192,
"column": 2
} | {
"line": 192,
"column": 45
} | {
"line": 192,
"column": 46
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\ninst✝⁹ : AddCommMonoid E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Semifield 𝕜\ninst✝⁶ : LinearOrder 𝕜\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : TopologicalSpace 𝕜\ninst✝³ : OrderTopology 𝕜\ninst✝² : DenselyOrdered 𝕜\ninst✝¹ : NoMaxOrder 𝕜\ninst✝ : ContinuousSMul 𝕜 E\nC : ConvexC... | [
"𝕜 : Type u_1\nE : Type u_3\ninst✝⁹ : AddCommMonoid E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Semifield 𝕜\ninst✝⁶ : LinearOrder 𝕜\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : TopologicalSpace 𝕜\ninst✝³ : OrderTopology 𝕜\ninst✝² : DenselyOrdered 𝕜\ninst✝¹ : NoMaxOrder 𝕜\ninst✝ : ContinuousSMul 𝕜 E\nC : ConvexCone 𝕜 E\nhS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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