The probability of h₀ over a concrete database

Lax794877.ExampleDatabaseProbability · concepts/Lax794877/ExampleDatabaseProbability.lean · lax-794877

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Natural Language Statement

Theorem

Over a probabilistic database whose uncertain facts have positive total weights, the probability of $h_0$ in its encoding is its count divided by its total, two numbers computed by enumerating its worlds.

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thm✓Lax794877.ExampleDatabaseProbability

⊢Lax794877Proofs.Bridge.probDb_worldProb_eq

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1	import Lax366625.CountingClasses
2	import Lax366625.CountingProblems
3	import Lax366625.QuantitativeLogic
4	import Lax366625.WitnessCounting
5	import Lax799700.Common
6	import Lax859101.OneCallReductions
7	import Lax904597.Machines
8	import Lax904597.SecondOrder
9	import Mathlib.Algebra.BigOperators.Field
10	import Mathlib.Algebra.BigOperators.Fin
11	import Mathlib.Algebra.BigOperators.Finprod
12	import Mathlib.Algebra.BigOperators.Group.Finset.Basic
13	import Mathlib.Algebra.BigOperators.Pi
14	import Mathlib.Algebra.BigOperators.Ring.Finset
15	import Mathlib.Algebra.Group.Action.Defs
16	import Mathlib.Algebra.Order.BigOperators.Group.Finset
17	import Mathlib.Algebra.Order.BigOperators.Ring.Finset
18	import Mathlib.Algebra.Order.Field.Basic
19	import Mathlib.Algebra.Order.Ring.Rat
20	import Mathlib.Data.Finite.Sigma
21	import Mathlib.Data.Finset.Max
22	import Mathlib.Data.Fintype.BigOperators
23	import Mathlib.Data.Fintype.Card
24	import Mathlib.Data.Fintype.EquivFin
25	import Mathlib.Data.Fintype.Lattice
26	import Mathlib.Data.Fintype.Pi
27	import Mathlib.Data.Fintype.Pigeonhole
28	import Mathlib.Data.Fintype.Sort
29	import Mathlib.Data.Nat.Bitwise
30	import Mathlib.Data.Prod.Lex
31	import Mathlib.Data.Set.Card
32	import Mathlib.Data.Set.Finite.Lemmas
33	import Mathlib.Dynamics.FixedPoints.Basic
34	import Mathlib.Logic.Equiv.Fin.Basic
35	import Mathlib.Logic.Equiv.Prod
36	import Mathlib.ModelTheory.Complexity
37	import Mathlib.ModelTheory.Graph
38	import Mathlib.ModelTheory.Order
39	import Mathlib.ModelTheory.Semantics
40	import Mathlib.ModelTheory.Syntax
41	import Mathlib.Order.Hom.Set
42	import Mathlib.Order.Lattice.Nat
43	import Mathlib.Order.PiLex
44	import Mathlib.SetTheory.Cardinal.Finite
45	import Mathlib.Tactic.FieldSimp
46	import Mathlib.Tactic.FinCases
47	import Mathlib.Tactic.Linarith
48	import Mathlib.Tactic.Ring
49	import Lax794877.PossibleWorlds
50	import Lax794877.WeightedWorlds
51	import Lax794877.Queries
52	import Lax794877.ExampleDatabase
53
54	/-!
55	---
56	title: The probability of h₀ over a concrete database
57	type: theorem
58	---
59	Over a probabilistic database whose uncertain facts have positive total
60	weights, the probability of $h_0$ in its encoding is its count divided by
61	its total, two numbers computed by enumerating its worlds.
62	-/
63
64	namespace Lax794877.ExampleDatabaseProbability
65
66	open Lax904597.SecondOrder
67	open FirstOrder
68	open Language Structure
69	open Lax794877.PossibleWorlds Lax799700.Common Lax904597.SecondOrder
70	open FirstOrder.Language
71	open Lax366625.CountingProblems Lax366625.CountingClasses Lax366625.WitnessCounting
72	Lax366625.QuantitativeLogic
73	open Lax859101.OneCallReductions Lax904597.Machines
74	open Lax794877.PossibleWorlds Lax794877.WeightedWorlds Lax794877.Queries Lax794877.ExampleDatabase
75
76	/-- The probability of h₀ over a database is its count divided by its total. -/
77	axiom probDb_worldProb_eq :
78	∀ (i : ProbDb) (hpos : ∀ x : {q : @Fact rst (Fin i.n) // @IsOpen _ _ (probDbStructure i) q},
79	0 < @presWeight _ _ (probDbStructure i) x + @absWeight _ _ (probDbStructure i) x),
80	@worldProb _ _ _ _ (probDbStructure i) _ h0 hpos = (i.count : ℚ) / (i.total : ℚ)
81
82	end Lax794877.ExampleDatabaseProbability
83

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The probability of h₀ over a concrete database

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