File size: 5,262 Bytes
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 | import OAI.Computability.PerfectCompleteness.Construction.SourceChildKernelLemmas
import OAI.Computability.PerfectCompleteness.Sampling.RationalFiniteLawLemmas
namespace OAI
section
namespace PerfectCompleteness.SourceChildKernelLaw
open UniqueGamesTheorem.Foundations.Games
open RecursiveSpaces TreeSourceSpaces SourceChildKernel
open scoped Classical
noncomputable section
variable {branch : Nat → Nat} {n t v m : Nat} {C : Type*} [Fintype C]
variable (rows : Nat → Nat) (clauses : Fin m → SourceClause.NormalizedClause v)
(designated : Fin (branch n) → Slots branch n)
abbrev Physical (i : Fin (branch n)) (s : Source (m := m) (t := t) designated i) :=
Σ b : Bool, Block C rows (mixedSlots clauses designated i s b)
def physical (i : Fin (branch n)) (s : Source (m := m) (t := t) designated i)
(raw : Raw (C := C) (t := t) rows clauses designated i) :
Physical (C := C) (t := t) rows clauses designated i s :=
⟨rawProjected rows clauses designated i raw,
decodeRight rows clauses designated i s raw⟩
@[simp] theorem physical_inl (i : Fin (branch n))
(s : Source (m := m) (t := t) designated i)
(raw : D (C := C) (t := t) rows clauses designated i) :
physical rows clauses designated i s (.inl raw) =
(⟨false, decodeNative rows clauses designated i s raw⟩ :
Physical (C := C) (t := t) rows clauses designated i s) := rfl
@[simp] theorem physical_inr (i : Fin (branch n))
(s : Source (m := m) (t := t) designated i)
(raw : UniformLatent.Raw (Factors (C := C) (t := t) rows clauses designated i)) :
physical rows clauses designated i s (.inr raw) =
(⟨true, decodeProjected rows clauses designated i s raw⟩ :
Physical (C := C) (t := t) rows clauses designated i s) := rfl
def physicalLaw (flag : Fin (branch n) → FiniteDistribution Bool)
(i : Fin (branch n)) (s : Source (m := m) (t := t) designated i) :
FiniteDistribution (Physical (C := C) (t := t) rows clauses designated i s) :=
(flag i).mixture (fun b =>
(FiniteDistribution.uniform (Block C rows (mixedSlots clauses designated i s b))).pushforward
(fun block => (⟨b, block⟩ : Physical (C := C) (t := t) rows clauses designated i s)))
theorem kernel_physical (flag : Fin (branch n) → FiniteDistribution Bool)
(i : Fin (branch n)) (s : Source (m := m) (t := t) designated i) :
(kernel (C := C) (t := t) rows clauses designated flag i s).pushforward
(physical rows clauses designated i s) =
physicalLaw (C := C) (t := t) rows clauses designated flag i s := by
unfold kernel UniformLatent.projectedKernel physicalLaw
rw [FiniteDistribution.pushforward_mixture]
apply congrArg (fun laws : Bool →
FiniteDistribution (Physical (C := C) (t := t) rows clauses designated i s) =>
(flag i).mixture laws)
funext b
cases b with
| false =>
change ((nonprojected (C := C) (t := t) rows clauses designated i s).pushforward
Sum.inl).pushforward (physical rows clauses designated i s) =
(FiniteDistribution.uniform (Native (C := C) (t := t) rows clauses designated i s)).pushforward
(fun block => (⟨false, block⟩ : Physical (C := C) (t := t) rows clauses designated i s))
rw [FiniteDistribution.pushforward_comp]
calc
_ = ((nonprojected (C := C) (t := t) rows clauses designated i s).pushforward
(decodeNative rows clauses designated i s)).pushforward
(fun block => (⟨false, block⟩ :
Physical (C := C) (t := t) rows clauses designated i s)) :=
(FiniteDistribution.pushforward_comp _ _ _).symm
_ = _ := by rw [decodeNative_law]
| true =>
change ((UniformLatent.latentKernel
(Factors (C := C) (t := t) rows clauses designated i) s).pushforward
Sum.inr).pushforward (physical rows clauses designated i s) =
(FiniteDistribution.uniform
(Projected (C := C) (t := t) rows clauses designated i s)).pushforward
(fun block => (⟨true, block⟩ : Physical (C := C) (t := t) rows clauses designated i s))
rw [FiniteDistribution.pushforward_comp]
calc
_ = ((UniformLatent.latentKernel
(Factors (C := C) (t := t) rows clauses designated i) s).pushforward
(decodeProjected rows clauses designated i s)).pushforward
(fun block => (⟨true, block⟩ :
Physical (C := C) (t := t) rows clauses designated i s)) :=
(FiniteDistribution.pushforward_comp _ _ _).symm
_ = _ := by rw [decodeProjected_law]
theorem kernels_physical (flag : Fin (branch n) → FiniteDistribution Bool)
(sources : Sources (m := m) (t := t) designated) :
(FiniteProduct.law (fun i =>
kernel (C := C) (t := t) rows clauses designated flag i (sources i))).pushforward
(fun raw i => physical rows clauses designated i (sources i) (raw i)) =
FiniteProduct.law (fun i =>
physicalLaw (C := C) (t := t) rows clauses designated flag i (sources i)) := by
rw [FiniteProduct.pushforward_map]
exact congrArg FiniteProduct.law
(funext fun i => kernel_physical rows clauses designated flag i (sources i))
end
end PerfectCompleteness.SourceChildKernelLaw
end
end OAI
|