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ea76aac | 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 | import Mathlib
import OAI.Computability.MaxCut.Model
namespace OAI
/-!
Finite occurrence presentations used by the v2 rounding/subdivision construction.
An occurrence has its own identifier: its endpoints need not be distinct and
different occurrences may have the same endpoints. The bipartite target records
injectivity of the pair of endpoints, which is precisely the required simplicity.
No hardness or analytic premise enters these structures.
-/
namespace MaxCutGames.Explicit
structure OccurrenceGame (V E A : Type*) where
source : E → V
target : E → V
permutation : E → Equiv.Perm A
structure BipartiteGame (L R E A : Type*) where
left : E → L
right : E → R
permutation : E → Equiv.Perm A
simple : Function.Injective (fun e => (left e, right e))
namespace OccurrenceGame
def Satisfied {V E A : Type*} (G : OccurrenceGame V E A)
(labeling : V → A) (e : E) : Prop :=
labeling (G.target e) = G.permutation e (labeling (G.source e))
instance satisfiedDecidable {V E A : Type*} [DecidableEq A]
(G : OccurrenceGame V E A) (labeling : V → A) (e : E) :
Decidable (G.Satisfied labeling e) := inferInstanceAs (Decidable (_ = _))
def IsTranslation {V E A : Type*} [Add A] (G : OccurrenceGame V E A) : Prop :=
∃ offset : E → A, ∀ e a, G.permutation e a = a + offset e
end OccurrenceGame
namespace BipartiteGame
def Satisfied {L R E A : Type*} (G : BipartiteGame L R E A)
(leftLabeling : L → A) (rightLabeling : R → A) (e : E) : Prop :=
rightLabeling (G.right e) = G.permutation e (leftLabeling (G.left e))
instance satisfiedDecidable {L R E A : Type*} [DecidableEq A]
(G : BipartiteGame L R E A) (a : L → A) (b : R → A) (e : E) :
Decidable (G.Satisfied a b e) := inferInstanceAs (Decidable (_ = _))
def IsTranslation {L R E A : Type*} [Add A] (G : BipartiteGame L R E A) : Prop :=
∃ offset : E → A, ∀ e a, G.permutation e a = a + offset e
end BipartiteGame
end MaxCutGames.Explicit
end OAI
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