File size: 3,555 Bytes
2f887db | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 | import Mathlib.Algebra.Order.BigOperators.Group.Finset
import OAI.Computability.PerfectCompleteness.Construction.TreeSourceSpaces
import OAI.Computability.PerfectCompleteness.Foundations.CanonicalKeys
import OAI.Computability.PerfectCompleteness.Reduction.SourceGame
namespace OAI
section
namespace PerfectCompleteness.CanonicalKeys
open ClauseSupport MixedSupport
open scoped BigOperators
variable {n : Nat} {Y : Type*}
abbrev KeyLabel (k : Key n) := ↥k.partition
instance KeyLabel_finite (k : Key n) : Finite (KeyLabel k) := by
let : Finite ReducedValue := ReducedDomains.reducedValue_finite
exact Finite.of_injective (fun P : KeyLabel k => P.val) Subtype.val_injective
instance keyLabel_nonempty (side : Side) (slots : Fin n → Slot)
(f : Assignment slots → Y) : Nonempty (KeyLabel (key side slots f)) := by
change Nonempty (Label slots f)
infer_instance
theorem label_finite (slots : Fin n → Slot) (f : Assignment slots → Y) :
Finite (Label slots f) := by
infer_instance
theorem label_nonempty (slots : Fin n → Slot) (f : Assignment slots → Y) :
Nonempty (Label slots f) := by
infer_instance
theorem card_label_le_output [Finite Y] (slots : Fin n → Slot)
(f : Assignment slots → Y) : Nat.card (Label slots f) ≤ Nat.card Y :=
Nat.card_le_card_of_injective (restore slots f) (restore_injective slots f)
theorem card_label_le_assignment (slots : Fin n → Slot) (f : Assignment slots → Y) :
Nat.card (Label slots f) ≤ Nat.card (Assignment slots) :=
ReducedPartition.card_parts_le (reduction slots f) f
def answerEquivFinSeven (signs : SourceClause.Triple) : Answer signs ≃ Fin 7 where
toFun u := SourceGame.encodeLabel signs u.val
((SourceClause.mem_satisfyingTriples signs u.val).mp u.property)
invFun i := ⟨SourceGame.decodeLabel signs i, SourceGame.decodeLabel_mem signs i⟩
left_inv u := by
apply Subtype.ext
exact SourceGame.decodeLabel_encodeLabel signs u.val _
right_inv i := by
apply SourceGame.decodeLabel_injective signs
exact SourceGame.decodeLabel_encodeLabel signs (SourceGame.decodeLabel signs i) _
theorem card_answer (signs : SourceClause.Triple) : Nat.card (Answer signs) = 7 := by
simpa only [Nat.card_fin] using Nat.card_congr (answerEquivFinSeven signs)
theorem card_slotDomain_le_seven (s : Slot) : Nat.card s.Domain ≤ 7 := by
cases s with
| clause occurrence variableIDs signs => exact (card_answer signs).le
| bit variableID =>
change Nat.card Bool ≤ 7
simp
theorem card_assignment_le_seven_pow (slots : Fin n → Slot) :
Nat.card (Assignment slots) ≤ 7 ^ n := by
change Nat.card (∀ i : Fin n, (slots i).Domain) ≤ _
rw [Nat.card_pi]
calc
(∏ i : Fin n, Nat.card (slots i).Domain) ≤ ∏ _i : Fin n, (7 : Nat) :=
Finset.prod_le_prod (fun i _ => card_slotDomain_le_seven (slots i))
_ = 7 ^ n := by simp
theorem card_label_le_seven_pow (slots : Fin n → Slot) (f : Assignment slots → Y) :
Nat.card (Label slots f) ≤ 7 ^ n :=
(card_label_le_assignment slots f).trans (card_assignment_le_seven_pow slots)
theorem card_keyLabel_le_output [Finite Y] (side : Side) (slots : Fin n → Slot)
(f : Assignment slots → Y) : Nat.card (KeyLabel (key side slots f)) ≤ Nat.card Y :=
card_label_le_output slots f
theorem card_keyLabel_le_seven_pow (side : Side) (slots : Fin n → Slot)
(f : Assignment slots → Y) : Nat.card (KeyLabel (key side slots f)) ≤ 7 ^ n :=
card_label_le_seven_pow slots f
end PerfectCompleteness.CanonicalKeys
end
end OAI
|