File size: 3,584 Bytes
f0f05de | 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 | import OAI.Combinatorics.BalancedRyser.Local.LocalSpecificationEncoding
import OAI.Combinatorics.BalancedRyser.Probability.FiniteDependencyRank
namespace OAI
namespace Balanced.Local.ConditionSpec
noncomputable section
open scoped Classical
variable {b κ n : ℕ}
theorem exists_goodSpecification_of_acyclic_fixed
(u : Position b → Position b) (I T : Finset (Position b))
(hTI : T ⊆ I) (hcard : T.card = n) (hfix : ∀ v ∈ T, u v = v)
(code : Position b → Code b κ)
(hwf : WellFounded (fun x y : {v // v ∈ T} =>
y.val ∈ underlyingAnchors u (code x.val)))
(E : Position b → Code b κ → Prop) (hE : ∀ v ∈ T, E v (code v)) :
∃ σ : Fin n → Position b × Code b κ,
GoodSpecification u I σ ∧
(∀ k, E (σ k).1 (σ k).2) ∧
Finset.univ.image (fun k => (σ k).1) = T := by
obtain ⟨rank, hrank⟩ := exists_nat_rank_on_finset T
(fun v w => w ∈ underlyingAnchors u (code v)) hwf
apply exists_goodSpecification_of_finset u I T hTI hcard
(fun v hv w hw h => by simpa only [hfix v hv, hfix w hw] using h)
code rank _ E hE
intro v hv w hw hdep
rw [hfix v hv, hfix w hw]
exact hrank v hv w hw (by simpa only [underlyingAnchors, hfix w hw] using hdep)
variable {F : Type*} [Field F] [Fintype F] [DecidableEq F]
theorem badCover_zero_reuse_exists_goodSpecification
(P : EqualityPattern b 0) (a : DirectionAssignment F b 0)
(bits : Bits b) (p : Point F) (M : MergedPairs F)
(hb : 3 ≤ b) (ha : Eligible P a)
(hadm : Admissible P a bits p M) (hbad : BadCover P a bits p M) :
∃ σ : Fin 3 → Position b × Code b 0,
GoodSpecification id Finset.univ σ ∧
∀ k, Holds P a bits p M (σ k).1 (σ k).2 := by
obtain ⟨T, code, hcard, hT, hwf⟩ :=
badCover_zero_reuse_exists_acyclic_spec P a bits p M hb ha hadm hbad
obtain ⟨σ, hσ, hE, _⟩ := exists_goodSpecification_of_acyclic_fixed
id Finset.univ T (Finset.subset_univ _) hcard (fun _ _ => rfl) code
(by simpa only [underlyingAnchors, Finset.image_id] using hwf)
(Holds P a bits p M) (fun v hv => (hT v hv).1)
exact ⟨σ, hσ, hE⟩
theorem badCover_one_reuse_exists_goodSpecification
(P : EqualityPattern b 1) (a : DirectionAssignment F b 1)
(bits : Bits b) (p : Point F) (M : MergedPairs F)
(hb : 3 ≤ b) (ha : Eligible P a)
(hadm : Admissible P a bits p M) (hbad : BadCover P a bits p M)
(root child : Fin b) (r s : Fin 4)
(hshared : P.classOf root r = P.classOf child s) :
let u := ParentMap.underlying root child r s (bits root r) (bits child s)
∃ σ : Fin 2 → Position b × Code b 1,
GoodSpecification u (Finset.univ \ ParentMap.dependent child s) σ ∧
∀ k, Holds P a bits p M (σ k).1 (σ k).2 := by
dsimp only
let u := ParentMap.underlying root child r s (bits root r) (bits child s)
obtain ⟨T, code, hcard, hT, hwf⟩ :=
badCover_one_reuse_exists_acyclic_spec P a bits p M hb ha hadm hbad
root child r s hshared
have hTI : T ⊆ Finset.univ \ ParentMap.dependent child s := by
intro v hv
exact Finset.mem_sdiff.mpr ⟨Finset.mem_univ _, (hT v hv).1⟩
have hfix : ∀ v ∈ T, u v = v := by
intro v hv
exact ParentMap.underlying_fixed root child r s (bits root r) (bits child s)
(hT v hv).1
obtain ⟨σ, hσ, hE, _⟩ := exists_goodSpecification_of_acyclic_fixed u
(Finset.univ \ ParentMap.dependent child s) T hTI hcard hfix code hwf
(Holds P a bits p M) (fun v hv => (hT v hv).2.1)
exact ⟨σ, hσ, hE⟩
end
end Balanced.Local.ConditionSpec
end OAI
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