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760f90e | 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 | import OAI.Computability.UniqueGames.Games.UniformTarget
namespace OAI
section
namespace UniqueGamesTheorem.Explicit.UniformRoundingTarget
open UniqueGamesTheorem.Foundations.Target
open UniqueGamesTheorem.MachineUniformRounding
open UniqueGamesTheorem.MachineUniformCopyOrder
/-- Apply the concrete largest-remainder list of occurrence IDs to the original
list. Equal records at different indices remain separate input occurrences. -/
def rows {α : Type*} (C : Nat) (es : List α) : List α :=
(Rounding.copies (Rounding.denominator es.length 1 C) es.length
(fun _ => 1)).map es.get
theorem rows_eq_occurrenceMajor {α : Type*} (C : Nat) (es : List α)
(h : 0 < es.length) : rows C es = occurrenceMajor C es := by
unfold rows
rw [copies_uniform es.length C h]
simp only [List.map_flatten, List.map_ofFn, Function.comp_def, List.map_replicate]
change (List.ofFn (fun i => List.replicate C (es.get i))).flatten =
(es.map (fun e => List.replicate C e)).flatten
congr 1
calc
List.ofFn (fun i => List.replicate C (es.get i)) =
(List.ofFn es.get).map (fun e => List.replicate C e) := List.map_ofFn.symm
_ = es.map (fun e => List.replicate C e) := by rw [List.ofFn_get]
/-- Exact multiplicities and full permutation records agree with the machine's
copy-major target; only the enumeration order changes. -/
theorem rows_perm_construct {q : Nat} (C : Nat) (hC : 0 < C) (g : Instance q) :
(rows C g.constraints).Perm (UniformTarget.construct C hC g).constraints := by
rw [rows_eq_occurrenceMajor C g.constraints g.constraintCount_positive]
exact occurrenceMajor_perm_copyMajor C g.constraints
theorem selected_count_construct {q : Nat} (C : Nat) (hC : 0 < C) (g : Instance q)
(a : Fin g.vertices → Fin q) :
countSatisfied a (rows C g.constraints) =
countSatisfied a (UniformTarget.construct C hC g).constraints :=
Integration.InstanceEquivalences.countSatisfied_perm a (rows_perm_construct C hC g)
end UniqueGamesTheorem.Explicit.UniformRoundingTarget
end
end OAI
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