The probability of ∃x y, R(x) ∧ S(x, y) is easy

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

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Natural Language Statement

Theorem

On the same weighted instances, the numerator of the probability of the hierarchical query $\exists x\, y,\ R(x) \wedge S(x, y)$ is in FP. Given which $R$ -facts are present, the query fails when no present $R(x)$ has a present $S(x, y)$ , a condition on each $S$ -fact separately, so the weight of the worlds in which it fails is a product over $x$ , times the total weight of the $T$ -facts. The weight of the worlds in which it holds is computed the same way, without a subtraction, as a term of quantitative first-order logic.

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Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A

Evidence

Each proof establishes this claim relative to its assumptions.

- lem✓Lax794877.WorldsValues
→
thm✓Lax794877.HierarchicalQueryInFP
⊢Lax794877Proofs.Bridge.weightedWorlds_rs_mem_FP

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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 ∃x y, R(x) ∧ S(x, y) is easy
57	type: theorem
58	---
59	On the same weighted instances, the numerator of the probability of the
60	hierarchical query $\exists x\, y,\ R(x) \wedge S(x, y)$ is in FP. Given
61	which $R$ -facts are present, the query fails when no present $R(x)$ has a
62	present $S(x, y)$ , a condition on each $S$ -fact separately, so the weight of
63	the worlds in which it fails is a product over $x$ , times the total weight
64	of the $T$ -facts. The weight of the worlds in which it holds is computed the
65	same way, without a subtraction, as a term of quantitative first-order
66	logic.
67	-/
68
69	namespace Lax794877.HierarchicalQueryInFP
70
71	open Lax904597.SecondOrder
72	open FirstOrder
73	open Language Structure
74	open Lax794877.PossibleWorlds Lax799700.Common Lax904597.SecondOrder
75	open FirstOrder.Language
76	open Lax366625.CountingProblems Lax366625.CountingClasses Lax366625.WitnessCounting
77	Lax366625.QuantitativeLogic
78	open Lax859101.OneCallReductions Lax904597.Machines
79	open Lax794877.PossibleWorlds Lax794877.WeightedWorlds Lax794877.Queries Lax794877.ExampleDatabase
80
81	/-- The numerator of the probability of the hierarchical query is in FP. -/
82	axiom weightedWorlds_rs_mem_FP :
83	FP.Mem (WeightedWorlds rs)
84
85	end Lax794877.HierarchicalQueryInFP
86

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The probability of ∃x y, R(x) ∧ S(x, y) is easy

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