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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 46 47 48 49 50 51 52 | import OAI.Probability.SATComputability.HierarchyTable
import OAI.Probability.SATComputability.FiniteTrialEvaluation
namespace OAI
namespace FixedClauseThreshold.Computability
open DilutedSpinGlass Nat.Partrec FiniteArithmetic
attribute [local irreducible] normalizedBranches branchWeights
noncomputable def leafFields (L : ℕ) (cs : List Code) : Fin L → ℝ :=
fun i => decodedRational (cs[i.val]?.getD .zero)
theorem leafFields_ofFn {L : ℕ} (cs : Fin L → Code) :
leafFields L (List.ofFn cs) = fun i => (decodedRational (cs i) : ℝ) := by
funext i
simp [leafFields, i.isLt]
theorem listLogMean_eq_finite {L : ℕ} (ms : List ℚ) (g : (Fin L → ℝ) → ℝ)
(cs : Fin L → Code) :
listLogMean ms (fun ds => g (leafFields L ds)) (List.ofFn cs) =
finiteLogMean ms.length g (fun i => (ms.get i : ℝ))
(fun i => decodeTree ms.length (cs i)) := by
induction ms generalizing cs with
| nil =>
change g (leafFields L (List.ofFn cs)) = g (fun i => (decodedRational (cs i) : ℝ))
rw [leafFields_ofFn]
| cons m ms ih =>
rw [listLogMean, chooseChildren_expect_ofFn]
simp_rw [ih]
simp only [List.length_cons, finiteLogMean, decodeTree, List.get_eq_getElem,
List.getElem_cons_zero, List.getElem_cons_succ, Fin.val_zero, Fin.val_succ]
rw [show (instDecidableEqFin L : DecidableEq (Fin L)) =
(fun a b => Classical.propDecidable (a = b)) from Subsingleton.elim _ _]
theorem listTrialLog_eq_finite {L : ℕ} (ms : List ℚ) (g : (Fin L → ℝ) → ℝ)
(root : Code) :
branchSum (chooseChildren (List.replicate L root))
(listLogMean ms (fun ds => g (leafFields L ds))) =
finiteTrialLog ms.length (decodeTree (ms.length+1) root)
(fun i => (ms.get i : ℝ)) g := by
rw [← List.ofFn_const L root, chooseChildren_expect_ofFn]
simp_rw [listLogMean_eq_finite]
simp only [finiteTrialLog, decodeTree]
rw [show (instDecidableEqFin L : DecidableEq (Fin L)) =
(fun a b => Classical.propDecidable (a = b)) from Subsingleton.elim _ _]
rfl
end FixedClauseThreshold.Computability
end OAI
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