An instance has 2^k possible worlds

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

proven

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Natural Language Statement

Theorem

An instance with $k$ uncertain facts has $2^k$ possible worlds: a world is a choice of the uncertain facts it keeps.

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Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A

Evidence

Each proof establishes this claim relative to its assumptions.

no assumptions

thm✓Lax794877.WorldCount

⊢Lax794877Proofs.Bridge.card_isWorld

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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: An instance has 2^k possible worlds
57	type: theorem
58	---
59	An instance with $k$ uncertain facts has $2^k$ possible worlds: a world is a
60	choice of the uncertain facts it keeps.
61	-/
62
63	namespace Lax794877.WorldCount
64
65	open Lax904597.SecondOrder
66	open FirstOrder
67	open Language Structure
68	open Lax794877.PossibleWorlds Lax799700.Common Lax904597.SecondOrder
69	open FirstOrder.Language
70	open Lax366625.CountingProblems Lax366625.CountingClasses Lax366625.WitnessCounting
71	Lax366625.QuantitativeLogic
72	open Lax859101.OneCallReductions Lax904597.Machines
73	open Lax794877.PossibleWorlds Lax794877.WeightedWorlds Lax794877.Queries Lax794877.ExampleDatabase
74
75	/-- An instance with `k` uncertain facts has `2 ^ k` possible worlds. -/
76	axiom card_isWorld :
77	∀ {L : Language.{0, 0}} [Finite (Σ n, L.Relations n)]
78	{A : Type} [(L.sum L).Structure A] [Finite A],
79	Nat.card {ρ : (worldBlock L).Assignment A // IsWorld ρ} =
80	2 ^ Nat.card {q : Σ p : Σ n, L.Relations n, Fin p.1 → A // IsOpenFact q}
81
82	end Lax794877.WorldCount
83

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An instance has 2^k possible worlds

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