| {"id":"def:proper-ker","label":"proper-ker","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:kernel"],"nlp_text":"\\(\\pi \\) is proper iff \\(\\pi (A\\cap B\\mid x)=\\pi (A\\mid x)\\cdot \\mathbf{1}_B(x)\\) for all \\(A\\in \\mathcal X\\), \\(B\\in \\mathcal B\\) and \\(x\\in X\\).","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/ProbabilityTheory.Kernel.IsProper","latex_link":"sect0002.html#def:proper-ker","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/ProbabilityTheory.Kernel.IsProper","lean_decl_name":"ProbabilityTheory.Kernel.IsProper","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"/-- A specification is proper if all its boundary condition kernels are. -/\ndef IsProper (γ : Specification S E) : Prop := ∀ Λ : Finset S, (γ Λ).IsProper\n","line":92,"end_line":93,"proof_status":"formalized"} |
| {"id":"lem:cond-exp-proper-ker-lintegral","label":"cond-exp-proper-ker-lintegral","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:proper-ker","def:cond-exp-ker","lem:lebesgue-cond-exp-char"],"nlp_text":"If \\(\\pi \\in \\text{Ker}_{\\mathcal B, \\mathcal X}\\) is a conditional expectation kernel for \\(\\mu \\), then \\[ \\mu [f\\mid \\mathcal B] = \\int f(x)\\ \\pi (\\partial x\\mid \\cdot )\\ \\mu \\text{-a.e.} \\]","doc_url":null,"latex_link":"sect0002.html#lem:cond-exp-proper-ker-lintegral","lean_link":null,"lean_decl_name":"cond-exp-proper-ker-lintegral","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"def:proper-spec","label":"proper-spec","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:proper-ker","def:spec"],"nlp_text":"A specification \\(\\gamma \\) is proper iff the kernel \\(\\gamma _\\Lambda \\) is proper for every \\(\\Lambda \\in \\operatorname{Finset}(S)\\).","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsProper","latex_link":"sect0002.html#def:proper-spec","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsProper","lean_decl_name":"Specification.IsProper","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"/-- A specification is proper if all its boundary condition kernels are. -/\ndef IsProper (γ : Specification S E) : Prop := ∀ Λ : Finset S, (γ Λ).IsProper\n","line":92,"end_line":93,"proof_status":"formalized"} |
| {"id":"lem:cond-exp-proper-ker-integral","label":"cond-exp-proper-ker-integral","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:proper-ker","def:cond-exp-ker","lem:lebesgue-cond-exp-char"],"nlp_text":"If \\(\\pi \\in \\text{Ker}_{\\mathcal B, \\mathcal X}\\) is a conditional expectation kernel for \\(\\mu \\), then \\[ \\mu (f\\mid \\mathcal B) = \\int f(x)\\ \\pi (\\partial x\\mid \\cdot )\\ \\mu \\text{-a.e.} \\]","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/ProbabilityTheory.Kernel.condExp_ae_eq_integral_kernel","latex_link":"sect0002.html#lem:cond-exp-proper-ker-integral","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/ProbabilityTheory.Kernel.condExp_ae_eq_integral_kernel","lean_decl_name":"ProbabilityTheory.Kernel.condExp_ae_eq_integral_kernel","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Prereqs/Kernel/CondExp.lean","lean4_code":"lemma condExp_ae_eq_integral_kernel (f : X → ℝ) :\n condExp 𝓑 μ f =ᵐ[μ] (fun x₀ ↦ ∫ x, f x ∂(π x₀)) := sorry\n","line":91,"end_line":92,"proof_status":"statement_formalized"} |
| {"id":"lem:proper-ker-integral","label":"proper-ker-integral","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:proper-ker"],"nlp_text":"If \\(\\pi \\) is a proper Markov kernel and \\(g\\) is a bounded function, then \\[ \\int f(x) g(x)\\ \\pi (dx\\mid x_0) = g(x_0)\\int f(x)\\ \\pi (dx\\mid x_0) \\]","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/ProbabilityTheory.Kernel.IsProper.integral_bdd_mul","latex_link":"sect0002.html#lem:proper-ker-integral","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/ProbabilityTheory.Kernel.IsProper.integral_bdd_mul","lean_decl_name":"ProbabilityTheory.Kernel.IsProper.integral_bdd_mul","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/james18lpc/GibbsMeasure/4275e8f4c089b3f14f067e521aed77af94624974/GibbsMeasure/Mathlib/Probability/Kernel/Proper.lean","lean4_code":"lemma IsProper.integral_bdd_mul (h𝓑𝓧 : 𝓑 ≤ 𝓧) (hπ : IsProper π) (hf : Integrable[𝓧] f (π x₀))\n (hg : StronglyMeasurable[𝓑] g) (hgbdd : ∃ C > 0, ∀ x, ‖g x‖ ≤ C) :\n ∫ x, g x * f x ∂(π x₀) = g x₀ * ∫ x, f x ∂(π x₀) := by\n obtain ⟨C, hpC, hC⟩ := hgbdd\n induction f, hf using Integrable.induction' with\n | indicator c s hs _ =>\n simp [← smul_indicator_one_apply, mul_left_comm, integral_const_mul,\n hπ.integral_bdd_mul_indicator h𝓑𝓧 hs hg ⟨C, hC⟩]\n | add f₁ f₂ hf₁ hf₂ _ hgf₁ hgf₂ =>\n have : Integrable (fun x ↦ g x * f₁ x) (π x₀) :=\n hf₁.bdd_mul (hg.mono h𝓑𝓧).aestronglyMeasurable ⟨C, hC⟩\n have : Integrable (fun x ↦ g x * f₂ x) (π x₀) :=\n hf₂.bdd_mul (hg.mono h𝓑𝓧).aestronglyMeasurable ⟨C, hC⟩\n simp [mul_add, integral_add, *]\n | isClosed =>\n refine isClosed_eq ?_ <| by fun_prop\n refine Metric.continuous_iff.mpr fun f2 ε hε ↦ ⟨ε / C, div_pos hε hpC, fun a ha ↦ ?_⟩\n have hInt1 : Integrable (fun x ↦ g x * a x) (π x₀) :=\n (L1.integrable_coeFn a).bdd_mul (hg.mono h𝓑𝓧).aestronglyMeasurable ⟨C, hC⟩\n have hInt2 : Integrable (fun x ↦ g x * f2 x) (π x₀) :=\n (L1.integrable_coeFn f2).bdd_mul (hg.mono h𝓑𝓧).aestronglyMeasurable ⟨C, hC⟩\n have hsub :\n ‖∫ x, g x * a x ∂π x₀ - ∫ x, g x * f2 x ∂π x₀‖ = ‖∫ x, g x * a x - g x * f2 x ∂π x₀‖ := by\n simp [integral_sub hInt1 hInt2]\n have hdiff_ae : (fun x ↦ g x * a x - g x * f2 x) =ᵐ[π x₀] (fun x ↦ g x * (a - f2) x) := by\n filter_upwards [Lp.coeFn_sub a f2] with x hx\n have hmul : g x * (a - f2) x = g x * (a x - f2 x) := by\n simpa [hx] using congrArg (fun t ↦ g x * t) hx\n calc\n g x * a x - g x * f2 x\n = g x * (a x - f2 x) := by simp [mul_sub]\n _ = g x * (a - f2) x := by simp [hmul.symm]\n have hIntDiff : Integrable (fun x ↦ g x * (a - f2) x) (π x₀) :=\n (L1.integrable_coeFn (a - f2)).bdd_mul ((hg.mono h𝓑𝓧).aestronglyMeasurable) ⟨C, hC⟩\n have hdInt : Integrable (fun x ↦ C * ‖(a - f2) x‖) (π x₀) :=\n ((L1.integrable_coeFn (a - f2)).norm.smul C)\n have hle_ae : (fun x ↦ ‖g x * (a - f2) x‖) ≤ᵐ[π x₀] (fun x ↦ C * ‖(a - f2) x‖) :=\n .of_forall fun x ↦ by simpa using mul_le_mul_of_nonneg_right (hC x) (norm_nonneg ((a - f2) x))\n have hle_int : ‖∫ x, g x * (a - f2) x ∂π x₀‖ ≤ C * ‖a - f2‖ := by\n calc\n ‖∫ x, g x * (a - f2) x ∂π x₀‖\n ≤ ∫ x, ‖g x * (a - f2) x‖ ∂π x₀ := norm_integral_le_integral_norm _\n _ ≤ ∫ x, C * ‖(a - f2) x‖ ∂π x₀ := integral_mono_ae hIntDiff.norm hdInt hle_ae\n _ = C * ∫ x, ‖(a - f2) x‖ ∂π x₀ := by simp [integral_const_mul]\n _ = C * ‖a - f2‖ := by rw [(L1.norm_eq_integral_norm (a - f2) (μ := π x₀)).symm]\n have hle : ‖∫ x, g x * a x ∂π x₀ - ∫ x, g x * f2 x ∂π x₀‖ ≤ C * ‖a - f2‖ := by\n have hdiffint : ‖∫ x, g x * a x ∂π x₀ - ∫ x, g x * f2 x ∂π x₀‖\n = ‖∫ x, g x * (a - f2) x ∂π x₀‖ := by\n simpa [integral_congr_ae hdiff_ae,integral_sub hInt1 hInt2] using hsub\n simpa [hdiffint] using hle_int\n have hlt : C * ‖a - f2‖ < ε := by\n have hdist : ‖a - f2‖ < ε / C := by simpa [dist_eq_norm] using ha\n simpa [mul_div,mul_div_cancel_left₀ ε (ne_of_gt hpC)] using (mul_lt_mul_of_pos_left hdist hpC)\n exact lt_of_le_of_lt hle hlt\n | ae_congr f₁ f₂ hf₁ hf hgf₁ =>\n have : (fun x ↦ g x * f₁ x) =ᵐ[π x₀] (fun x ↦ g x * f₂ x) := by\n filter_upwards [hf] with x hx; simp [hx]\n simpa [integral_congr_ae this, integral_congr_ae hf] using hgf₁","line":83,"end_line":140,"proof_status":"formalized"} |
| {"id":"lem:proper-ker-lintegral","label":"proper-ker-lintegral","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:proper-ker"],"nlp_text":"If \\(\\pi \\) is proper, then \\[ \\int f(x) g(x)\\ \\pi (dx\\mid x_0) = g(x_0)\\int f(x)\\ \\pi (dx\\mid x_0) \\]","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/ProbabilityTheory.Kernel.IsProper.lintegral_mul","latex_link":"sect0002.html#lem:proper-ker-lintegral","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/ProbabilityTheory.Kernel.IsProper.lintegral_mul","lean_decl_name":"ProbabilityTheory.Kernel.IsProper.lintegral_mul","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/leanprover-community/mathlib4/029db123ddaa7f8fd0d18cea3b1b33bf84dacd1e/Mathlib/Probability/Kernel/Proper.lean","lean4_code":"lemma IsProper.lintegral_mul (hπ : IsProper π) (h𝓑𝓧 : 𝓑 ≤ 𝓧) (hf : Measurable[𝓧] f)\n (hg : Measurable[𝓑] g) (x₀ : X) :\n ∫⁻ x, g x * f x ∂(π x₀) = g x₀ * ∫⁻ x, f x ∂(π x₀) := by\n refine hg.ennreal_induction ?_ ?_ ?_\n · rintro c A hA\n simp_rw [← smul_indicator_one_apply, smul_mul_assoc, smul_eq_mul]\n rw [lintegral_const_mul, hπ.lintegral_indicator_mul h𝓑𝓧 hf hA]\n · measurability\n · rintro g₁ g₂ - _ hg₂_meas hg₁ hg₂\n simp only [Pi.add_apply, add_mul]\n rw [lintegral_add_right, hg₁, hg₂]\n · exact (hg₂_meas.mono h𝓑𝓧 le_rfl).mul hf\n · rintro g' hg'_meas hg'_mono hg'\n simp_rw [ENNReal.iSup_mul]\n rw [lintegral_iSup (fun n ↦ ((hg'_meas _).mono h𝓑𝓧 le_rfl).mul hf)\n (hg'_mono.mul_const (zero_le _))]\n simp_rw [hg']","line":121,"end_line":137,"proof_status":"formalized"} |
| {"id":"lem:gibbs-meas-tfae","label":"gibbs-meas-tfae","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["lem:cond-exp-proper-ker-lintegral","def:proper-spec","def:gibbs-meas"],"nlp_text":"Let \\(\\gamma \\) be a proper specification with parameter set \\(S\\) and state space \\((E, \\mathcal{E})\\), and let \\(\\nu \\in \\mathfrak {P}(E^S, \\mathcal{E}^S)\\). TFAE: \\(\\nu \\in \\mathcal{G}(\\gamma )\\). \\(\\gamma _\\Lambda \\circ _m\\nu = \\nu \\) for all \\(\\Lambda \\in \\operatorname{Finset}(S)\\). \\(\\gamma _\\Lambda \\circ _m\\nu = \\nu \\) frequently as \\(\\Lambda \\to S\\).","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.isGibbsMeasure_iff_forall_bind_eq","latex_link":"sect0002.html#lem:gibbs-meas-tfae","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.isGibbsMeasure_iff_forall_bind_eq","lean_decl_name":"Specification.isGibbsMeasure_iff_forall_bind_eq","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"lemma isGibbsMeasure_iff_forall_bind_eq (hγ : γ.IsProper) [IsFiniteMeasure μ] :\n γ.IsGibbsMeasure μ ↔ ∀ Λ, μ.bind (γ Λ) = μ :=\n forall_congr' fun _Λ ↦ Kernel.isCondExp_iff_bind_eq_left (hγ _) cylinderEvents_le_pi\n","line":142,"end_line":144,"proof_status":"formalized"} |
| {"id":"lem:isssd-proper-spec","label":"isssd-proper-spec","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:proper-spec","def:isssd-spec","def:isssd"],"nlp_text":"\\(\\operatorname{ISSSD}(\\nu )\\) is a proper specification.","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsProper.isssd","latex_link":"sect0002.html#lem:isssd-proper-spec","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsProper.isssd","lean_decl_name":"Specification.IsProper.isssd","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"/-- The **Independent Specification with Single Spin Distribution**.\n\nThis is the specification corresponding to the product measure. -/\n@[simps]\ndef isssd : Specification S E where\n toFun := isssdFun ν\n isConsistent' Λ₁ Λ₂ hΛ := by\n classical\n rw [isssdFun_comp_isssdFun]\n ext a s _\n simp only [Kernel.comap_apply, id_eq, isssdFun_apply, Finset.coe_sort_coe]\n rw [Finset.union_eq_right.2 hΛ]\n","line":198,"end_line":209,"proof_status":"formalized"} |
| {"id":"lem:modif-proper","label":"modif-proper","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:proper-spec","lem:proper-ker-lintegral","def:modif"],"nlp_text":"If \\(\\gamma \\) is a specification and \\(\\rho \\) a modifier of \\(\\gamma \\), then \\(\\rho \\gamma \\) is a proper specification.","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsProper.modification","latex_link":"sect0002.html#lem:modif-proper","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsProper.modification","lean_decl_name":"Specification.IsProper.modification","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"/-- Modification specification.\n\nModifying the specification `γ` by a family indexed by finsets `Λ : Finset S` of densities\n`ρ Λ : (S → E) → ℝ≥0∞` results in a family of kernels `γ.modificationKer ρ _ Λ` whose density is\nthat of `γ Λ` multiplied by `ρ Λ`.\n\nWhen the family of densities `ρ` is a modifier (`Specification.IsModifier`), modifying a\nspecification results in a specification `γ.modification ρ _`. -/\nnoncomputable def modification (γ : Specification S E) (ρ : Finset S → (S → E) → ℝ≥0∞)\n (hρ : γ.IsModifier ρ) : Specification S E where\n toFun := modificationKer γ ρ hρ.measurable\n isConsistent' := hρ.isConsistent\n","line":296,"end_line":307,"proof_status":"formalized"} |
| {"id":"lem:isssd-gibbs-meas-uniqueness","label":"isssd-gibbs-meas-uniqueness","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["lem:isssd-proper-spec","def:gibbs-meas"],"nlp_text":"There is at most one Gibbs measure specified by \\(\\operatorname{ISSSD}(\\nu )\\).","doc_url":null,"latex_link":"sect0002.html#lem:isssd-gibbs-meas-uniqueness","lean_link":null,"lean_decl_name":"isssd-gibbs-meas-uniqueness","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"def:spec","label":"spec","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:cylinder-event","def:kernel"],"nlp_text":"A specification is a family of kernels \\(\\gamma : \\operatorname{Finset}S \\to \\operatorname{Ker}_{\\mathcal{F}_{S\\setminus \\Lambda }, \\mathcal{E}^S}\\) which is consistent, in the sense that \\[ \\forall \\Lambda _1, \\Lambda _2 \\in \\operatorname{Finset}(S), \\Lambda _1 \\subseteq \\Lambda _2 \\implies \\gamma _{\\Lambda _1} \\circ _k \\gamma _{\\Lambda _2} = \\gamma _{\\Lambda _2} \\]","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification","latex_link":"sect0002.html#def:spec","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification","lean_decl_name":"Specification","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"./GibbsMeasure/Specification.lean","lean4_code":"structure Specification [MeasurableSpace E] where\n /-- The boundary condition kernels of a specification.\n\n DO NOT USE. Instead use the coercion to function `⇑γ`. Lean should insert it automatically in most\n cases. -/\n toFun (Λ : Finset S) : Kernel[cylinderEvents Λᶜ] (S → E) (S → E)\n /-- The boundary condition kernels of a specification are consistent.\n\n DO NOT USE. Instead use `Specification.isConsistent`. -/\n isConsistent' : IsConsistent toFun\n\nnamespace Specification\n\n","line":37,"end_line":49,"proof_status":"formalized"} |
| {"id":"def:indep-spec","label":"indep-spec","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:spec"],"nlp_text":"A specification \\(\\gamma \\) is independent iff \\[ \\forall \\Lambda _1, \\Lambda _2 \\in \\operatorname{Finset}(S), \\gamma _{\\Lambda _1} \\circ _k \\gamma _{\\Lambda _2} = \\gamma _{\\Lambda _1\\cup \\Lambda _2} \\]","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsIndep","latex_link":"sect0002.html#def:indep-spec","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsIndep","lean_decl_name":"Specification.IsIndep","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"/-- An independent specification is a specification `γ` where `γ Λ₁ ∘ₖ γ Λ₂ = γ (Λ₁ ∪ Λ₂)` for all\n`Λ₁ Λ₂`. -/\ndef IsIndep (γ : Specification S E) : Prop :=\n ∀ ⦃Λ₁ Λ₂⦄ [DecidableEq S] , (γ Λ₁).comap id cylinderEvents_le_pi ∘ₖ γ Λ₂ = (γ (Λ₁ ∪ Λ₂)).comap id\n (measurable_id'' <| by gcongr; exact Finset.subset_union_right)\n","line":70,"end_line":74,"proof_status":"formalized"} |
| {"id":"def:markov-spec","label":"markov-spec","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:spec"],"nlp_text":"A specification \\(\\gamma \\) is a Markov specification iff \\(\\gamma _\\Lambda \\) is a probability kernel for every \\(\\Lambda \\in \\operatorname{Finset}(S)\\).","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsMarkov","latex_link":"sect0002.html#def:markov-spec","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsMarkov","lean_decl_name":"Specification.IsMarkov","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"/-- A Markov specification is a specification whose boundary condition kernels are all Markov\nkernels. -/\nclass IsMarkov (γ : Specification S E) : Prop where\n isMarkovKernel : ∀ Λ, IsMarkovKernel (γ Λ)\n","line":80,"end_line":83,"proof_status":"formalized"} |
| {"id":"def:modif","label":"modif","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:spec"],"nlp_text":"A modifier of \\(\\gamma \\) is a family \\[ \\rho : \\operatorname{Finset}(S) \\to \\Omega \\to [0, \\infty [ \\]","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsModifier","latex_link":"sect0002.html#def:modif","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsModifier","lean_decl_name":"Specification.IsModifier","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"/-- A modifier of a specification `γ` is a family indexed by finsets `Λ : Finset S` of densities\n`ρ Λ : (S → E) → ℝ≥0∞` such that:\n* Each `ρ Λ` is measurable.\n* `γ.modificationKer ρ` (informally, `ρ * γ`) is consistent. -/\n@[mk_iff]\nstructure IsModifier (γ : Specification S E) (ρ : Finset S → (S → E) → ℝ≥0∞) : Prop where\n measurable Λ : Measurable (ρ Λ)\n isConsistent : IsConsistent (modificationKer γ ρ measurable)\n","line":264,"end_line":271,"proof_status":"formalized"} |
| {"id":"def:isssd-spec","label":"isssd-spec","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:spec"],"nlp_text":"\\(\\operatorname{ISSSD}(\\nu )\\) is a specification.","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.isssd","latex_link":"sect0002.html#def:isssd-spec","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.isssd","lean_decl_name":"Specification.isssd","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"/-- The **Independent Specification with Single Spin Distribution**.\n\nThis is the specification corresponding to the product measure. -/\n@[simps]\ndef isssd : Specification S E where\n toFun := isssdFun ν\n isConsistent' Λ₁ Λ₂ hΛ := by\n classical\n rw [isssdFun_comp_isssdFun]\n ext a s _\n simp only [Kernel.comap_apply, id_eq, isssdFun_apply, Finset.coe_sort_coe]\n rw [Finset.union_eq_right.2 hΛ]\n","line":198,"end_line":209,"proof_status":"formalized"} |
| {"id":"def:gibbs-meas","label":"gibbs-meas","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:spec","def:cond-exp-ker"],"nlp_text":"Given a specification \\(\\gamma \\), a Gibbs measures specified by \\(\\gamma \\) is a measure \\(\\nu \\in \\mathfrak {M}(E^S, \\mathcal{E}^S)\\) such that \\(\\gamma _\\Lambda (A|\\cdot )\\) is a conditional expectation kernel for \\(\\nu \\) for all \\(A \\in \\mathcal E^S\\) and \\(\\Lambda \\in \\operatorname{Finset}(S)\\).","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsGibbsMeasure","latex_link":"sect0002.html#def:gibbs-meas","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsGibbsMeasure","lean_decl_name":"Specification.IsGibbsMeasure","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"/-- For a specification `γ`, a Gibbs measure is a measure whose conditional expectation kernels\nconditionally on configurations exterior to finite sets agree with the boundary condition kernels\nof the specification `γ`. -/\ndef IsGibbsMeasure (γ : Specification S E) (μ : Measure (S → E)) : Prop := ∀ Λ, (γ Λ).IsCondExp μ\n","line":136,"end_line":139,"proof_status":"formalized"} |
| {"id":"prop:modif-indep-char","label":"modif-indep-char","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:indep-spec","def:modif"],"nlp_text":"If \\(\\rho \\) is a family of measurable densities and \\(\\gamma \\) is a proper independent specification, then TFAE \\(\\rho \\) is a modifier of \\(\\gamma \\) For all \\(\\Lambda _1, \\Lambda _2\\) with \\(\\Lambda _1 \\subseteq \\Lambda _2\\), \\(\\eta : S \\to E\\) and \\(\\gamma _{\\Lambda _2 \\setminus \\Lambda _1}(\\cdot |\\alpha )\\)-almost all \\(\\eta _2 : S \\to E\\), we have \\[ \\rho _{\\Lambda _2}(\\zeta _1)\\rho _{\\Lambda _1}(\\zeta _2) = \\rho _{\\Lambda _2}(\\zeta _2) \\rho _{\\Lambda _1}(\\zeta _1) \\]","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.isModifier_iff_ae_comm","latex_link":"sect0002.html#prop:modif-indep-char","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.isModifier_iff_ae_comm","lean_decl_name":"Specification.isModifier_iff_ae_comm","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"lemma isModifier_iff_ae_comm [DecidableEq S] :\n γ.IsModifier ρ ↔ (∀ Λ, Measurable (ρ Λ)) ∧\n ∀ ⦃Λ₁ Λ₂⦄, Λ₁ ⊆ Λ₂ → ∀ η₁, ∀ᵐ η₂ ∂γ (Λ₂ \\ Λ₁) η₁, ∀ᵐ ζ ∂(γ Λ₁ η₂).prod (γ Λ₂ η₂),\n ρ Λ₂ ζ.1 * ρ Λ₁ ζ.2 = ρ Λ₂ ζ.2 * ρ Λ₁ ζ.1 := by\n -- simp only [isModifier_iff_ae_eq, and_congr_right_iff]\n -- refine fun hρ ↦ forall₄_congr fun Λ₁ Λ₂ hΛ η ↦ ?_\n sorry\n","line":288,"end_line":294,"proof_status":"formalized"} |
| {"id":"lem:isssd-isIndep","label":"isssd-isIndep","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:indep-spec","def:isssd"],"nlp_text":"\\(\\operatorname{ISSSD}(\\nu )\\) is independent.","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.isssdFun_comp_isssdFun","latex_link":"sect0002.html#lem:isssd-isIndep","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.isssdFun_comp_isssdFun","lean_decl_name":"Specification.isssdFun_comp_isssdFun","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"/-- The ISSSD of a measure is strongly consistent. -/\nlemma isssdFun_comp_isssdFun [DecidableEq S] (Λ₁ Λ₂ : Finset S) :\n (isssdFun ν Λ₁).comap id cylinderEvents_le_pi ∘ₖ isssdFun ν Λ₂ =\n (isssdFun ν (Λ₁ ∪ Λ₂)).comap id\n (measurable_id'' <| by gcongr; exact Finset.subset_union_right) :=\n sorry\n","line":191,"end_line":196,"proof_status":"formalized"} |
| {"id":"lem:modif-modif","label":"modif-modif","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:modif"],"nlp_text":"Modifying a specification \\(\\gamma \\) by \\(\\rho _1\\) then \\(\\rho _2\\) is the same as modifying it by their product.","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.modification_modification","latex_link":"sect0002.html#lem:modif-modif","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.modification_modification","lean_decl_name":"Specification.modification_modification","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"@[simp] lemma modification_modification (γ : Specification S E) (ρ₁ ρ₂ : Finset S → (S → E) → ℝ≥0∞)\n (hρ₁ : γ.IsModifier ρ₁) (hρ₂ : (γ.modification ρ₁ hρ₁).IsModifier ρ₂) :\n (γ.modification ρ₁ hρ₁).modification ρ₂ hρ₂ = γ.modification (ρ₁ * ρ₂) (hρ₁.mul hρ₂) := by\n ext Λ σ s hs\n simp only [modification_apply, Pi.mul_apply]\n rw [withDensity_apply _ hs, withDensity_apply _ hs,\n setLIntegral_withDensity_eq_setLIntegral_mul _ (hρ₁.measurable Λ) (hρ₂.1 Λ) hs]\n","line":329,"end_line":335,"proof_status":"formalized"} |
| {"id":"lem:exists-modif-isssd","label":"exists-modif-isssd","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:modif","def:isssd"],"nlp_text":"If \\(E\\) is countable, \\(\\nu \\) is the counting measure and \\(\\gamma \\) is any specification, then \\[ \\rho _\\Lambda (\\eta ) = \\gamma _\\Lambda (\\{ \\sigma _\\Lambda = \\eta _\\Lambda \\} | \\eta ) \\]","doc_url":null,"latex_link":"sect0002.html#lem:exists-modif-isssd","lean_link":null,"lean_decl_name":"exists-modif-isssd","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"prop:modif-char","label":"modif-char","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:modif"],"nlp_text":"If \\(\\rho \\) is a family of measurable densities and \\(\\gamma \\) is a proper specification, then TFAE \\(\\rho \\) is a modifier of \\(\\gamma \\) For all \\(\\Lambda _1, \\Lambda _2\\) with \\(\\Lambda _1 \\subseteq \\Lambda _2\\) and all \\(\\eta : S \\to E\\), we have \\[ \\rho _{\\Lambda _2} = \\rho _{\\Lambda _1}\\cdot (\\gamma _{\\Lambda _1} \\rho _{\\Lambda _2}) \\quad \\gamma _{\\Lambda _2}(\\cdot |\\eta )\\text{-a.e.} \\]","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.isModifier_iff_ae_eq","latex_link":"sect0002.html#prop:modif-char","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.isModifier_iff_ae_eq","lean_decl_name":"Specification.isModifier_iff_ae_eq","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"lemma isModifier_iff_ae_eq (hγ : γ.IsProper) :\n γ.IsModifier ρ ↔ (∀ Λ, Measurable (ρ Λ)) ∧ ∀ ⦃Λ₁ Λ₂⦄, Λ₁ ⊆ Λ₂ → ∀ η,\n ρ Λ₂ =ᵐ[γ Λ₂ η] fun η ↦ ∫⁻ ζ, ρ Λ₂ ζ ∂(γ Λ₁ η).withDensity (ρ Λ₁) := by\n simp only [isModifier_iff, IsConsistent, modificationKer, Kernel.ext_iff, Kernel.comp_apply,\n Kernel.coe_mk, Kernel.coe_comap, CompTriple.comp_eq, Measure.ext_iff, exists_prop,\n and_congr_right_iff]\n refine fun hρ ↦ forall₄_congr fun Λ₁ Λ₂ hΛ η ↦ ?_\n sorry\n","line":279,"end_line":286,"proof_status":"statement_formalized"} |
| {"id":"lem:isssd-gibbs-meas-existence","label":"isssd-gibbs-meas-existence","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:isssd-spec","def:gibbs-meas","def:product-probability-measure"],"nlp_text":"The product measure \\(\\nu ^S\\) is a Gibbs measure specified by \\(\\operatorname{ISSSD}(\\nu )\\).","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.isGibbsMeasure_isssd_productMeasure","latex_link":"sect0002.html#lem:isssd-gibbs-meas-existence","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.isGibbsMeasure_isssd_productMeasure","lean_decl_name":"Specification.isGibbsMeasure_isssd_productMeasure","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"/-- The product measure `ν ^ S` is a `isssd μ`-Gibbs measure. -/\nlemma isGibbsMeasure_isssd_productMeasure (ν : Measure E) [IsProbabilityMeasure ν] :\n (isssd ν).IsGibbsMeasure (.infinitePi fun _ : S ↦ ν) := by\n rintro Λ\n sorry\n","line":229,"end_line":233,"proof_status":"statement_formalized"} |
| {"id":"lem:premodif-modif","label":"premodif-modif","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["prop:modif-indep-char","def:premodif"],"nlp_text":"If \\(h\\) is a premodifier and \\(\\nu \\) is such that \\(0 {\\lt} \\nu _\\Lambda h_\\Lambda {\\lt} \\infty \\) for all \\(\\Lambda \\), then \\[ \\rho _\\Lambda := \\frac{h_\\Lambda }{\\operatorname{ISSSD}(\\nu )_\\Lambda h_\\Lambda } \\]","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsPremodifier.isModifier_div","latex_link":"sect0002.html#lem:premodif-modif","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsPremodifier.isModifier_div","lean_decl_name":"Specification.IsPremodifier.isModifier_div","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/james18lpc/GibbsMeasure/4275e8f4c089b3f14f067e521aed77af94624974/GibbsMeasure/Specification.lean","lean4_code":"lemma IsPremodifier.isModifier_div (hρ : IsPremodifier ρ) (ν : Measure E) [IsProbabilityMeasure ν] :\n (isssd ν).IsModifier fun Λ σ ↦ ρ Λ σ / ∫⁻ x, ρ Λ x ∂(isssd ν Λ σ) where\n measurable Λ :=\n (hρ.measurable Λ).div ((hρ.measurable Λ).lintegral_kernel.mono cylinderEvents_le_pi le_rfl)\n isConsistent Λ₁ Λ₂ hΛ := by\n sorry","line":353,"end_line":358,"proof_status":"statement_formalized"} |
| {"id":"lem:modif-premodif","label":"modif-premodif","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["prop:modif-char","def:premodif"],"nlp_text":"If \\(\\rho \\) is a modifier of \\(\\operatorname{ISSSD}(\\nu ^S)\\), then it is a premodifier if any of the following conditions hold: \\(E\\) is countable and \\(\\nu \\) is equivalent to the counting measure. \\(E\\) is a second countable Borel space. \\(\\nu \\) is everywhere dense. For all \\(\\Lambda _1 \\subseteq \\Lambda _2\\) and all \\(\\eta : S \\to E\\), \\(\\zeta \\mapsto \\rho _{\\Lambda _1}(\\zeta \\eta _{\\Lambda _1^c})\\) is continuous on \\(E^{\\Lambda _1}\\).","doc_url":null,"latex_link":"sect0002.html#lem:modif-premodif","lean_link":null,"lean_decl_name":"modif-premodif","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"lem:cond-exp-proper-ker-bind","label":"cond-exp-proper-ker-bind","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:cond-exp-ker","lem:lebesgue-cond-exp-char"],"nlp_text":"Let \\(\\mu \\in \\mathfrak {M}(X,\\mathcal X)\\) be a finite measure and let \\(\\pi \\in \\text{Ker}_{\\mathcal B,\\mathcal X}\\) be a proper kernel. Then, \\[ \\pi \\text{ is a conditional expectation kernel for }\\mu \\iff \\mu \\pi = \\mu \\]","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/ProbabilityTheory.Kernel.isCondExp_iff_bind_eq_left","latex_link":"sect0002.html#lem:cond-exp-proper-ker-bind","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/ProbabilityTheory.Kernel.isCondExp_iff_bind_eq_left","lean_decl_name":"ProbabilityTheory.Kernel.isCondExp_iff_bind_eq_left","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Prereqs/Kernel/CondExp.lean","lean4_code":"lemma isCondExp_iff_bind_eq_left (hπ : π.IsProper) (h𝓑𝓧 : 𝓑 ≤ 𝓧) [SigmaFinite (μ.trim h𝓑𝓧)] :\n IsCondExp π μ ↔ μ.bind π = μ := by\n simp_rw [isCondExp_iff, Filter.eventuallyEq_comm,\n toReal_ae_eq_indicator_condExp_iff_forall_meas_inter_eq h𝓑𝓧, Measure.ext_iff]\n refine ⟨fun h A hA ↦ ?_, fun h A hA B hB ↦ ?_⟩\n · rw [eq_comm, Measure.bind_apply hA (π.measurable.mono h𝓑𝓧 le_rfl).aemeasurable]\n simpa using h hA _ .univ\n · rw [hπ.setLIntegral_eq_comp h𝓑𝓧 hA hB, eq_comm]\n exact h _ (by measurability)\n","line":15,"end_line":23,"proof_status":"formalized"} |
| {"id":"def:cond-exp-ker","label":"cond-exp-ker","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:kernel"],"nlp_text":"Let \\(\\mu \\in \\mathfrak {M}(X,\\mathcal X)\\). Then, \\(\\pi \\in \\text{Ker}_{\\mathcal B,\\mathcal X}\\) is a conditional expectation kernel for \\(\\mu \\) if \\(\\mu (A\\mid \\mathcal B)=\\pi (A\\mid \\cdot )\\) \\(\\mu \\)-a.e.","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/ProbabilityTheory.condExpKernel","latex_link":"sect0002.html#def:cond-exp-ker","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/ProbabilityTheory.condExpKernel","lean_decl_name":"ProbabilityTheory.condExpKernel","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Probability/Kernel/Condexp.lean","lean4_code":"noncomputable irreducible_def condExpKernel (μ : Measure Ω) [IsFiniteMeasure μ]\n (m : MeasurableSpace Ω) : @Kernel Ω Ω m mΩ :=\n if _h : Nonempty Ω then\n Kernel.comap (@condDistrib Ω Ω Ω mΩ _ _ mΩ (m ⊓ mΩ) id id μ _) id\n (measurable_id'' (inf_le_left : m ⊓ mΩ ≤ m))\n else 0\n\n","line":71,"end_line":77,"proof_status":"formalized"} |
| {"id":"def:premodif","label":"premodif","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A family of measurable functions \\(h_\\Lambda : (S \\to E) \\to [0, \\infty [\\) is a premodifier if \\[ h_{\\Lambda _2}(\\zeta )h_{\\Lambda _1}(\\eta ) = h_{\\Lambda _1}(\\zeta )h_{\\Lambda _2}(\\eta ) \\]","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsPremodifier","latex_link":"sect0002.html#def:premodif","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsPremodifier","lean_decl_name":"Specification.IsPremodifier","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"/-- A premodifier is a family indexed by finsets `Λ : Finset S` of densities\n`ρ Λ : (S → E) → ℝ≥0∞` such that:\n* each `ρ Λ` is measurable,\n* `ρ Λ₂ ζ * ρ Λ₁ η = ρ Λ₁ ζ * ρ Λ₂ η` for all `Λ₁ Λ₂ : Finset S` and `ζ η : S → E` such that\n `Λ₁ ⊆ Λ₂` and `∀ (s : Λ₁ᶜ), ζ s = η s`. -/\nstructure IsPremodifier [MeasurableSpace E] (ρ : Finset S → (S → E) → ℝ≥0∞) : Prop where\n measurable Λ : Measurable (ρ Λ)\n comm_of_subset ⦃Λ₁ Λ₂ : Finset S⦄ ⦃ζ η : S → E⦄ (hΛ : Λ₁ ⊆ Λ₂)\n (hrestrict : ∀ s ∉ Λ₁, ζ s = η s) : ρ Λ₂ ζ * ρ Λ₁ η = ρ Λ₁ ζ * ρ Λ₂ η\n","line":343,"end_line":351,"proof_status":"formalized"} |
| {"id":"lem:lebesgue-cond-exp-char","label":"lebesgue-cond-exp-char","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:lebesgue-cond-exp"],"nlp_text":"If \\(f : X \\to [0, \\infty ]\\) is a \\(\\mathcal X\\)-measurable function, then \\(\\mu [f | \\mathcal B]\\) is the \\(\\mu \\)-ae unique \\(\\mathcal B\\)-measurable function \\(X \\to [0, \\infty ]\\) such that \\[ \\int _B \\mu [f | \\mathcal B]\\ \\partial \\mu = \\int _B f\\ \\partial \\mu \\]","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/MeasureTheory.ae_eq_lcondExp_of_forall_setLIntegral_eq","latex_link":"sect0001.html#lem:lebesgue-cond-exp-char","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/MeasureTheory.ae_eq_lcondExp_of_forall_setLIntegral_eq","lean_decl_name":"MeasureTheory.ae_eq_lcondExp_of_forall_setLIntegral_eq","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Prereqs/LebesgueCondExp.lean","lean4_code":" rw [lintegral_lcondExp, lintegral_indicator hA, setLIntegral_const, one_mul]\n\n/-- **Uniqueness of the conditional expectation**\n\nIf a function is a.e. `m`-measurable, verifies an integrability condition and has same lintegral\nas `f` on all `m`-measurable sets, then it is a.e. equal to `μ⁻[f|hm]`. -/\nlemma ae_eq_lcondExp_of_forall_setLIntegral_eq (hm : m ≤ m₀) [SigmaFinite (μ.trim hm)]\n {f g : α → ℝ≥0∞}\n (hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫⁻ x in s, g x ∂μ = ∫⁻ x in s, f x ∂μ)\n (hgm : AEStronglyMeasurable[m] g μ) : g =ᵐ[μ] μ⁻[f|m] := by\n refine ae_eq_of_forall_setLIntegral_eq_of_sigmaFinite' hm (fun s hs hμs => ?_) hgm sorry\n","line":113,"end_line":123,"proof_status":"statement_formalized"} |
| {"id":"def:lebesgue-cond-exp","label":"lebesgue-cond-exp","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The conditional expectation of a \\(\\mathcal X\\)-measurable function \\(f : X \\to [0, \\infty ]\\) is \\[ \\mu [f | \\mathcal B] = ?? \\]","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/MeasureTheory.lcondExp","latex_link":"sect0001.html#def:lebesgue-cond-exp","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/MeasureTheory.lcondExp","lean_decl_name":"MeasureTheory.lcondExp","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Prereqs/LebesgueCondExp.lean","lean4_code":"/-- Lebesgue conditional expectation of an `ℝ≥0∞`-valued function. It is defined as `0` if any of\nthe following conditions holds:\n* `m` is not a sub-σ-algebra of `m₀`,\n* `μ` is not σ-finite with respect to `m`,\n* `f` is not `μ`-integrable. -/\nnoncomputable def lcondExp : α → ℝ≥0∞ :=\n open scoped Classical in\n if hm : m ≤ m₀ then\n if _h : SigmaFinite (μ.trim hm) then\n if Measurable[m] f then f\n else if hf : Measurable[m₀] f then\n ENNReal.ofReal ∘\n ⨆ n, μ[ENNReal.toReal ∘ (hf.stronglyMeasurable.finStronglyMeasurable μ).approx n | m]\n else 0\n else 0\n","line":12,"end_line":26,"proof_status":"formalized"} |
| {"id":"def:cylinder-event","label":"cylinder-event","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\((E,\\mathcal{E})\\) be a measurable space, and let \\(S\\) be a set. Then, \\begin{align} \\mathcal{F}:\\mathcal{P}(S)& \\to \\left\\{ \\text{sigma algebras on } E^S \\right\\} \\\\ \\Delta & \\mapsto \\sigma (\\{ \\text{proj}_\\delta :E^S\\to E\\mid \\delta \\in \\Delta \\} ) \\end{align}","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/MeasureTheory.cylinderEvents","latex_link":"sect0002.html#def:cylinder-event","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/MeasureTheory.cylinderEvents","lean_decl_name":"MeasureTheory.cylinderEvents","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/MeasureTheory/Constructions/Cylinders.lean","lean4_code":"def cylinderEvents (Δ : Set ι) : MeasurableSpace (∀ i, X i) := ⨆ i ∈ Δ, (m i).comap fun σ ↦ σ i\n\n","line":394,"end_line":395,"proof_status":"formalized"} |
| {"id":"def:product-probability-measure","label":"product-probability-measure","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(I\\) be a set. Suppose for each \\(i\\in I\\) that \\((\\Omega _i,\\mathcal B_i,P_i)\\) is a probability space. Then, \\(P:=\\bigotimes _{i\\in I}P_i\\) is a well-defined product probability measure on \\(\\prod _{i\\in I}\\Omega _i\\).","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/MeasureTheory.productMeasure","latex_link":"sect0002.html#def:product-probability-measure","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/MeasureTheory.productMeasure","lean_decl_name":"MeasureTheory.productMeasure","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/james18lpc/GibbsMeasure/4275e8f4c089b3f14f067e521aed77af94624974/GibbsMeasure/KolmogorovExtension4/ProductMeasure.lean","lean4_code":"/-- The product measure of an arbitrary family of probability measures. It is defined as the unique\nextension of the function which gives to cylinders the measure given by the associated product\nmeasure. -/\nnoncomputable def productMeasure : Measure ((i : ι) → X i) := (sorry : (∀ i, Measure (X i)) → _) μ","line":106,"end_line":109,"proof_status":"formalized"} |
| {"id":"def:isssd","label":"isssd","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:juxtaposition"],"nlp_text":"The Independent Specification with Single Spin Distribution \\(\\nu \\) is \\begin{align} \\operatorname{ISSSD}:\\mathfrak {P}(E, \\mathcal{E})& \\to \\operatorname{Finset}(S)\\to \\mathcal{E}^S\\times E^S\\to \\overline{\\mathbb {R}_{\\geq 0}}\\\\ \\nu & \\mapsto \\Lambda \\mapsto (A\\mid \\omega )\\mapsto \\left(\\nu ^\\Lambda \\left(\\operatorname{Juxt}_{\\omega }^{-1}(A)\\right)\\right) \\end{align}","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.isssdFun","latex_link":"sect0002.html#def:isssd","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.isssdFun","lean_decl_name":"Specification.isssdFun","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"/-- Auxiliary definition for `Specification.isssd`. -/\n@[simps -fullyApplied]\ndef isssdFun (ν : Measure E) (Λ : Finset S) : Kernel[cylinderEvents Λᶜ] (S → E) (S → E) :=\n @Kernel.mk _ _ (_) _\n (fun η ↦ Measure.map (juxt Λ η) (Measure.pi fun _ : Λ ↦ ν))\n (measurable_isssdFun ν Λ)\n","line":184,"end_line":189,"proof_status":"statement_formalized"} |
| {"id":"def:kernel","label":"kernel","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\((X,\\mathcal X)\\) and \\((Y,\\mathcal{Y})\\) be measurable spaces. Then, \\[ \\text{Ker}_{\\mathcal{Y},\\mathcal X}:=\\left\\{ \\pi :\\mathcal X\\times Y\\to [0,\\infty ]~ \\middle \\vert ~ \\forall y\\in Y,\\pi (\\cdot \\mid y)\\in \\mathfrak {M}(X,\\mathcal X);~ \\forall A\\in \\mathcal X,\\pi (A\\mid \\cdot )\\text{ is }\\mathcal{Y}\\text{-measurable}\\right\\} \\]","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/ProbabilityTheory.Kernel","latex_link":"sect0002.html#def:kernel","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/ProbabilityTheory.Kernel","lean_decl_name":"ProbabilityTheory.Kernel","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Probability/Moments/SubGaussian.lean","lean4_code":"structure Kernel.HasSubgaussianMGF (X : Ω → ℝ) (c : ℝ≥0)\n (κ : Kernel Ω' Ω) (ν : Measure Ω' := by volume_tac) : Prop where\n integrable_exp_mul : ∀ t, Integrable (fun ω ↦ exp (t * X ω)) (κ ∘ₘ ν)\n mgf_le : ∀ᵐ ω' ∂ν, ∀ t, mgf X (κ ω') t ≤ exp (c * t ^ 2 / 2)\n```\nAn interesting point is that the integrability condition is not integrability of `exp (t * X)`\nwith respect to `κ ω'` for `ν`-almost all `ω'`, but integrability with respect to `κ ∘ₘ ν`.\nThis is a stronger condition, as the weaker one did not allow to prove interesting results about\nthe sum of two sub-Gaussian random variables.\n\nFor the conditional case, that integrability condition reduces to integrability of `exp (t * X)`\nwith respect to `μ`.\n\n### Definition of `HasCondSubgaussianMGF`\n\nWe define `HasCondSubgaussianMGF` as a special case of `Kernel.HasSubgaussianMGF` with the\nconditional expectation kernel for `m`, `condExpKernel μ m`, and the restriction of `μ` to `m`,\n`μ.trim hm` (where `hm` states that `m` is a sub-sigma-algebra).\nNote that `condExpKernel μ m ∘ₘ μ.trim hm = μ`. The definition is equivalent to the two\nconditions\n* for all `t`, `exp (t * X)` is `μ`-integrable,\n* for `μ.trim hm`-almost all `ω`, for all `t`, the mgf with respect to the the conditional\n distribution `condExpKernel μ m ω` is bounded by `exp (c * t ^ 2 / 2)`.\n\nFor any `t`, we can write the mgf of `X` with respect to the conditional expectation kernel as\na conditional expectation, `(μ.trim hm)`-almost surely:\n`mgf X (condExpKernel μ m ·) t =ᵐ[μ.trim hm] μ[fun ω' ↦ exp (t * X ω') | m]`.\n\n## References\n\n* [R. Vershynin, *High-dimensional probability: An introduction with applications in data\n science*][vershynin2018high]\n\n-/\n\nopen MeasureTheory Real\n\nopen scoped ENNReal NNReal Topology\n\nnamespace ProbabilityTheory\n\nsection Kernel\n\nvariable {Ω Ω' : Type*} {mΩ : MeasurableSpace Ω} {mΩ' : MeasurableSpace Ω'}\n {ν : Measure Ω'} {κ : Kernel Ω' Ω} {X : Ω → ℝ} {c : ℝ≥0}\n\n/-! ### Sub-Gaussian with respect to a kernel and a measure -/\n\n","line":86,"end_line":133,"proof_status":"formalized"} |
| {"id":"def:markov-ker","label":"markov-ker","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:kernel"],"nlp_text":"Let \\((X,\\mathcal X)\\) and \\((Y,\\mathcal{Y})\\) be measurable spaces. We say that \\(\\pi \\in \\text{Ker}_{\\mathcal{Y},\\mathcal X}\\) is a Markov kernel iff \\(\\pi (X\\mid \\cdot )=1\\).","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/ProbabilityTheory.IsMarkovKernel","latex_link":"sect0002.html#def:markov-ker","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/ProbabilityTheory.IsMarkovKernel","lean_decl_name":"ProbabilityTheory.IsMarkovKernel","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"instance IsMarkov.toIsMarkovKernel [γ.IsMarkov] {Λ : Finset S} : IsMarkovKernel (γ Λ) :=\n isMarkovKernel _\n","line":85,"end_line":86,"proof_status":"formalized"} |
| {"id":"def:juxtaposition","label":"juxtaposition","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(E\\) and \\(S\\) be sets. Let \\(\\Delta \\in \\mathcal{P}(S)\\), and let \\(\\omega \\in E^S\\). We define \\begin{align} \\operatorname{Juxt}_{\\omega } : E^\\Delta & \\to E^S\\\\ \\zeta & \\mapsto \\delta \\mapsto \\begin{cases} \\zeta _\\delta & \\delta \\in \\Delta \\\\ \\omega _\\delta & \\delta \\notin \\Delta \\end{cases}\\end{align}","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/juxt","latex_link":"sect0002.html#def:juxtaposition","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/juxt","lean_decl_name":"juxt","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Prereqs/Juxt.lean","lean4_code":"noncomputable def juxt (Λ : Set S) (η : S → E) (ζ : Λ → E) (x : S) : E := by\n classical exact dite (x ∈ Λ) (fun h ↦ ζ ⟨x, h⟩) (fun _ ↦ η x)\n","line":8,"end_line":9,"proof_status":"formalized"} |
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