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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 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 | import OAI.Computability.PerfectCompleteness.Foundations.SourceStrategy
namespace OAI
section
namespace PerfectCompleteness.SourceCompleteness
open SourceClause ClauseSupport MixedSupport CanonicalKeys SourceKeys
variable {v m n : Nat} {Y Z : Type*}
structure Presentation (clauses : Fin m → NormalizedClause v) (n : Nat) (Y : Type*) where
endpoints : Fin n → Endpoint v m
joint : Assignment (slot clauses ∘ endpoints) → Y
variable {clauses : Fin m → NormalizedClause v} {side : Side}
noncomputable def presentationKey (side : Side) (P : Presentation clauses n Y) : Key n :=
key side (slot clauses ∘ P.endpoints) P.joint
noncomputable def generatedKeys (side : Side) (clauses : Fin m → NormalizedClause v)
(n : Nat) (Y : Type*) : Set (Key n) :=
Set.range (presentationKey (clauses := clauses) (n := n) (Y := Y) side)
abbrev Vertex (side : Side) (clauses : Fin m → NormalizedClause v) (n : Nat) (Y : Type*) :=
↥(generatedKeys side clauses n Y)
noncomputable def vertexOf (side : Side) (P : Presentation clauses n Y) :
Vertex side clauses n Y :=
⟨presentationKey side P, ⟨P, rfl⟩⟩
abbrev VertexLabel (vertex : Vertex side clauses n Y) :=
{part : Set (Fin n → ReducedValue) // part ∈ vertex.val.partition}
noncomputable def representation (vertex : Vertex side clauses n Y) :
Presentation clauses n Y :=
vertex.property.choose
theorem representation_key (vertex : Vertex side clauses n Y) :
presentationKey side (representation vertex) = vertex.val :=
vertex.property.choose_spec
noncomputable def evaluatedLabel (assignment : Fin v → Bool)
(hsat : ∀ c, (clauses c).clause.eval assignment = true)
(P : Presentation clauses n Y) : Label (slot clauses ∘ P.endpoints) P.joint :=
evaluateLabel (slot clauses ∘ P.endpoints) P.joint
(realizingAssignment clauses assignment hsat P.endpoints)
noncomputable def strategy (assignment : Fin v → Bool)
(hsat : ∀ c, (clauses c).clause.eval assignment = true)
(vertex : Vertex side clauses n Y) : VertexLabel vertex := by
refine ⟨(evaluatedLabel assignment hsat (representation vertex)).val, ?_⟩
rw [← representation_key vertex]
exact (evaluatedLabel assignment hsat (representation vertex)).property
theorem strategy_value_eq (assignment : Fin v → Bool)
(hsat : ∀ c, (clauses c).clause.eval assignment = true)
(vertex : Vertex side clauses n Y) (P : Presentation clauses n Z)
(hkey : presentationKey side P = vertex.val) :
(strategy assignment hsat vertex).val = (evaluatedLabel assignment hsat P).val := by
change (evaluatedLabel assignment hsat (representation vertex)).val =
(evaluatedLabel assignment hsat P).val
exact evaluated_label_eq_of_key_eq clauses assignment hsat side side
(representation vertex).endpoints P.endpoints
(representation vertex).joint P.joint
((representation_key vertex).trans hkey.symm)
def labelInPresentation (vertex : Vertex side clauses n Y)
(P : Presentation clauses n Z) (hkey : presentationKey side P = vertex.val)
(label : VertexLabel vertex) : Label (slot clauses ∘ P.endpoints) P.joint :=
⟨label.val, by
change label.val ∈ (presentationKey side P).partition
rw [hkey]
exact label.property⟩
theorem strategy_in_presentation (assignment : Fin v → Bool)
(hsat : ∀ c, (clauses c).clause.eval assignment = true)
(vertex : Vertex side clauses n Y) (P : Presentation clauses n Z)
(hkey : presentationKey side P = vertex.val) :
labelInPresentation vertex P hkey (strategy assignment hsat vertex) =
evaluatedLabel assignment hsat P := by
apply Subtype.ext
exact strategy_value_eq assignment hsat vertex P hkey
theorem restore_strategy_in_presentation (assignment : Fin v → Bool)
(hsat : ∀ c, (clauses c).clause.eval assignment = true)
(vertex : Vertex side clauses n Y) (P : Presentation clauses n Z)
(hkey : presentationKey side P = vertex.val) :
restore (slot clauses ∘ P.endpoints) P.joint
(labelInPresentation vertex P hkey (strategy assignment hsat vertex)) =
P.joint (realizingAssignment clauses assignment hsat P.endpoints) := by
rw [strategy_in_presentation]
exact restore_evaluateLabel _ _ _
def Presentation.postcompose (P : Presentation clauses n Y) (p : Y → Z) :
Presentation clauses n Z where
endpoints := P.endpoints
joint := p ∘ P.joint
noncomputable def coarseningEdge (sourceSide targetSide : Side)
(P : Presentation clauses n Y) (p : Y → Z)
(label : VertexLabel (vertexOf sourceSide P)) :
VertexLabel (vertexOf targetSide (P.postcompose p)) := by
let original := labelInPresentation (vertexOf sourceSide P) P rfl label
let coarse := CanonicalEdges.coarsen (slot clauses ∘ P.endpoints) P.joint p original
exact ⟨coarse.val, coarse.property⟩
theorem coarseningEdge_strategy (assignment : Fin v → Bool)
(hsat : ∀ c, (clauses c).clause.eval assignment = true)
(sourceSide targetSide : Side) (P : Presentation clauses n Y) (p : Y → Z) :
coarseningEdge sourceSide targetSide P p
(strategy assignment hsat (vertexOf sourceSide P)) =
strategy assignment hsat (vertexOf targetSide (P.postcompose p)) := by
have hsource : presentationKey sourceSide P = (vertexOf sourceSide P).val := rfl
have htarget : presentationKey targetSide (P.postcompose p) =
(vertexOf targetSide (P.postcompose p)).val := rfl
have hlabel : labelInPresentation (vertexOf sourceSide P) P hsource
(strategy assignment hsat (vertexOf sourceSide P)) = evaluatedLabel assignment hsat P :=
strategy_in_presentation assignment hsat (vertexOf sourceSide P) P hsource
apply Subtype.ext
calc
_ = (CanonicalEdges.coarsen (slot clauses ∘ P.endpoints) P.joint p
(evaluatedLabel assignment hsat P)).val :=
congrArg (fun L : Label (slot clauses ∘ P.endpoints) P.joint =>
(CanonicalEdges.coarsen (slot clauses ∘ P.endpoints) P.joint p L).val) hlabel
_ = (evaluatedLabel assignment hsat (P.postcompose p)).val :=
congrArg (fun L : Label (slot clauses ∘ P.endpoints) (p ∘ P.joint) => L.val)
(CanonicalEdges.coarsen_evaluateLabel (slot clauses ∘ P.endpoints) P.joint p
(realizingAssignment clauses assignment hsat P.endpoints))
_ = _ := (strategy_value_eq assignment hsat (vertexOf targetSide (P.postcompose p))
(P.postcompose p) htarget).symm
end PerfectCompleteness.SourceCompleteness
end
end OAI
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