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a2ee19a | 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 | import OAI.Probability.SATComputability.MaskPermutations
import OAI.Probability.SATComputability.GridBlockLaw
namespace OAI
namespace FixedClauseThreshold.Computability
open DilutedSpinGlass _root_.MeasureTheory _root_.OAI.MeasureTheory
open scoped NNReal Classical
theorem maskGrid_permutation {n k M : ℕ} (e : Equiv.Perm (Fin n))
(c : (Fin k → SignedLiteral n) × Fin M → ℕ) :
(fun j => renameMask e (maskGrid clauseMask c j)) =
maskGrid clauseMask (fun p => c ((clausePermutation e k).symm p.1,p.2)) := by
funext j
exact countsMask_permutation e (fun a => c (a,j))
theorem gridMaskLaw_permutation {n k M : ℕ} (e : Equiv.Perm (Fin n)) (r : ℝ≥0)
(f : (Fin M → Finset (DeletionCandidate n)) → ℝ) :
(gridMaskLaw M r (clauseMask (n := n) (k := k))).expect
(fun xs => f (fun j => renameMask e (xs j))) =
(gridMaskLaw M r (clauseMask (n := n) (k := k))).expect f := by
simp only [gridMaskLaw_expect, maskGrid_permutation]
have h := poissonCountLaw_equiv
((clausePermutation e k).prodCongr (Equiv.refl (Fin M))) r
(fun c => f (maskGrid clauseMask c))
exact h
theorem candidatePath_permutation {n : ℕ} [NeZero n] (M k : ℕ)
(e : Equiv.Perm (Fin n)) (r : ℝ≥0)
(f : (Fin M → Finset (DeletionCandidate n)) → ℝ) :
(FiniteLaw.pi (fun _ : Fin M => candidateBlock r k Finset.univ)).expect
(fun xs => f (fun j => renameMask e (xs j))) =
(FiniteLaw.pi (fun _ : Fin M => candidateBlock r k Finset.univ)).expect f := by
have hc : (Fintype.card (Fin k → SignedLiteral n) : ℝ≥0) ≠ 0 := by
exact_mod_cast Fintype.card_ne_zero
have h := gridMaskLaw_permutation (k := k) (M := M) e
(r/Fintype.card (Fin k → SignedLiteral n)) f
rw [gridMaskLaw_clauses, div_mul_cancel₀ _ hc] at h
exact h
end FixedClauseThreshold.Computability
end OAI
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