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import OAI.Computability.PerfectCompleteness.Decoding.ChildAssemblyProjection
import OAI.Computability.PerfectCompleteness.Sampling.RationalFiniteLawLemmas
import OAI.Computability.PerfectCompleteness.Sampling.SourceChildKernelLaw
namespace OAI
section
namespace PerfectCompleteness.SourceChildKernelMarginal
open scoped BigOperators Classical
open UniqueGamesTheorem.Foundations.Games
open RecursiveSpaces SourceQuestionReconstruction
noncomputable section
theorem nestedKernel_sources {E R T : Type*} [Fintype E] [Fintype R] [Fintype T]
(μ : FiniteDistribution E) (ν : FiniteDistribution R)
(kernel : E → R → FiniteDistribution T) :
(CleanConditioning.kernelJoint μ (fun e => CleanConditioning.kernelJoint ν (kernel e))).pushforward
(fun x => (x.1, x.2.1)) = μ.product ν := by
apply FiniteDistribution.eq_of_weight_eq
rintro ⟨e, r⟩
simp only [FiniteDistribution.pushforward, CleanConditioning.kernelJoint,
FiniteDistribution.product, Fintype.sum_prod_type, Prod.mk.injEq]
rw [Finset.sum_eq_single e]
· rw [Finset.sum_eq_single r]
· simp [← Finset.mul_sum, FiniteDistribution.normalized]
· intro r' _ hne
simp [hne]
· simp
· intro e' _ hne
simp [hne]
· simp
variable {branch : Nat → Nat} {n t v m : Nat} {C : Type*} [Fintype C]
def sources (rows : Nat → Nat) (clauses : Fin m → SourceClause.NormalizedClause v)
(designated : Fin (branch n) → Slots branch n)
(sample : SourceChildKernel.Sample (C := C) (t := t) rows clauses designated) :
SourceChildKernel.Sources (m := m) (t := t) designated :=
fun i => ((sample i).1, (sample i).2.1)
theorem childrenLaw_sources (rows : Nat → Nat)
(clauses : Fin m → SourceClause.NormalizedClause v)
(designated : Fin (branch n) → Slots branch n)
(μ : Fin (branch n) → FiniteDistribution (SourceTuple m t))
(ν : (i : Fin (branch n)) →
FiniteDistribution (SourceChildKernel.Rest (m := m) (t := t) designated i))
(flag : Fin (branch n) → FiniteDistribution Bool) :
(SourceChildKernel.childrenLaw (C := C) rows clauses designated μ ν flag).pushforward
(sources rows clauses designated) =
FiniteProduct.law (fun i => (μ i).product (ν i)) := by
let P (i : Fin (branch n)) : FiniteDistribution
(SourceTuple m t × (SourceChildKernel.Rest (m := m) (t := t) designated i ×
SourceChildKernel.Raw (C := C) (t := t) rows clauses designated i)) :=
CleanConditioning.kernelJoint (μ i) (fun e => CleanConditioning.kernelJoint (ν i)
(fun r => SourceChildKernel.kernel rows clauses designated flag i (e, r)))
let f (i : Fin (branch n))
(x : SourceTuple m t × (SourceChildKernel.Rest (m := m) (t := t) designated i ×
SourceChildKernel.Raw (C := C) (t := t) rows clauses designated i)) :
SourceChildKernel.Source (m := m) (t := t) designated i := (x.1, x.2.1)
change (FiniteProduct.law P).pushforward (fun sample i => f i (sample i)) = _
refine (CleanConditioning.pushforward_law P f).trans ?_
apply congrArg (fun laws : (i : Fin (branch n)) →
FiniteDistribution (SourceChildKernel.Source (m := m) (t := t) designated i) =>
FiniteProduct.law laws)
funext i
exact nestedKernel_sources (μ i) (ν i)
(fun e r => SourceChildKernel.kernel rows clauses designated flag i (e, r))
theorem originalLaw_sources [NeZero m] (rows : Nat → Nat)
(clauses : Fin m → SourceClause.NormalizedClause v)
(designated : Fin (branch n) → Slots branch n)
(flag : Fin (branch n) → FiniteDistribution Bool) :
(SourceChildKernel.originalLaw (C := C) (t := t) rows clauses designated flag).pushforward
(sources rows clauses designated) =
FiniteProduct.law (fun i : Fin (branch n) =>
(SourceOddLists.tupleLaw m t).product
(FiniteProduct.law (fun _ : {leaf : Slots branch n // leaf ≠ designated i} =>
SourceOddLists.tupleLaw m t))) :=
childrenLaw_sources rows clauses designated _ _ flag
end
end PerfectCompleteness.SourceChildKernelMarginal
end
section
namespace PerfectCompleteness.SourceChildNativeLaw
open UniqueGamesTheorem.Foundations.Games
open RecursiveSpaces TreeSourceSpaces SourceChildKernel
open scoped Classical
noncomputable section
def falseFlag : FiniteDistribution Bool where
weight b := if b then 0 else 1
nonnegative b := by cases b <;> simp
normalized := by simp
variable {branch : Nat → Nat} {n t v m : Nat} {C : Type*} [Fintype C]
(rows : Nat → Nat) (clauses : Fin m → SourceClause.NormalizedClause v)
(designated : Fin (branch n) → Slots branch n)
def pullback (i : Fin (branch n)) (s : Source (m := m) (t := t) designated i) (b : Bool)
(block : Block C rows (mixedSlots clauses designated i s b)) :
Native (C := C) (t := t) rows clauses designated i s :=
(fun call => ChildBlockProjection.squarePullback (projection clauses designated i s b)
(block.1 call),
ChildBlockProjection.arraysPullback rows (projection clauses designated i s b) block.2)
omit [Fintype C] in
@[simp] theorem pullback_false (i : Fin (branch n))
(s : Source (m := m) (t := t) designated i)
(block : Native (C := C) (t := t) rows clauses designated i s) :
pullback rows clauses designated i s false block = block := by
apply Prod.ext
· funext call
apply Subtype.ext
funext x
rfl
· funext node row
apply Subtype.ext
funext x
rfl
def physicalLeft (i : Fin (branch n)) (s : Source (m := m) (t := t) designated i)
(x : SourceChildKernelLaw.Physical (C := C) (t := t) rows clauses designated i s) :
Native (C := C) (t := t) rows clauses designated i s :=
pullback rows clauses designated i s x.1 x.2
def decodeLeft (i : Fin (branch n)) (s : Source (m := m) (t := t) designated i)
(raw : Raw (C := C) (t := t) rows clauses designated i) :
Native (C := C) (t := t) rows clauses designated i s :=
physicalLeft rows clauses designated i s
(SourceChildKernelLaw.physical rows clauses designated i s raw)
theorem decodeLeft_eq_leftChildren (sources : Sources (m := m) (t := t) designated)
(raw : (i : Fin (branch n)) → Raw (C := C) (t := t) rows clauses designated i)
(i : Fin (branch n)) :
decodeLeft rows clauses designated i (sources i) (raw i) =
leftChildren rows clauses designated sources raw i := rfl
def leftLaw (flag : Fin (branch n) → FiniteDistribution Bool)
(i : Fin (branch n)) (s : Source (m := m) (t := t) designated i) :
FiniteDistribution (Native (C := C) (t := t) rows clauses designated i s) :=
(flag i).mixture (fun b => if b then
(FiniteDistribution.uniform (Projected (C := C) (t := t) rows clauses designated i s)).pushforward
(pullback rows clauses designated i s true)
else FiniteDistribution.uniform (Native (C := C) (t := t) rows clauses designated i s))
theorem physicalLaw_left (flag : Fin (branch n) → FiniteDistribution Bool)
(i : Fin (branch n)) (s : Source (m := m) (t := t) designated i) :
(SourceChildKernelLaw.physicalLaw (C := C) (t := t) rows clauses designated flag i s).pushforward
(physicalLeft rows clauses designated i s) =
leftLaw (C := C) (t := t) rows clauses designated flag i s := by
unfold SourceChildKernelLaw.physicalLaw leftLaw
rw [FiniteDistribution.pushforward_mixture]
apply congrArg (fun laws : Bool →
FiniteDistribution (Native (C := C) (t := t) rows clauses designated i s) =>
(flag i).mixture laws)
funext b
cases b with
| false =>
rw [FiniteDistribution.pushforward_comp]
calc
_ = (FiniteDistribution.uniform (Native (C := C) (t := t) rows clauses designated i s)).pushforward
(id : Native (C := C) (t := t) rows clauses designated i s → _) := by
apply congrArg (fun f : Native (C := C) (t := t) rows clauses designated i s →
Native (C := C) (t := t) rows clauses designated i s =>
(FiniteDistribution.uniform (Native (C := C) (t := t) rows clauses designated i s)).pushforward f)
funext block
exact pullback_false rows clauses designated i s block
_ = _ := UniqueGamesTheorem.Foundations.Games.FiniteDistribution.pushforward_id
(FiniteDistribution.uniform (Native (C := C) (t := t) rows clauses designated i s))
| true =>
exact FiniteDistribution.pushforward_comp _ _ _
theorem kernel_decodeLeft (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
(decodeLeft rows clauses designated i s) =
leftLaw (C := C) (t := t) rows clauses designated flag i s := by
calc
_ = ((kernel (C := C) (t := t) rows clauses designated flag i s).pushforward
(SourceChildKernelLaw.physical rows clauses designated i s)).pushforward
(physicalLeft rows clauses designated i s) :=
(FiniteDistribution.pushforward_comp _ _ _).symm
_ = _ := by rw [SourceChildKernelLaw.kernel_physical, physicalLaw_left]
theorem leftLaw_false (i : Fin (branch n)) (s : Source (m := m) (t := t) designated i) :
leftLaw (C := C) (t := t) rows clauses designated (fun _ => falseFlag) i s =
FiniteDistribution.uniform (Native (C := C) (t := t) rows clauses designated i s) := by
apply FiniteDistribution.eq_of_weight_eq
intro block
simp [leftLaw, falseFlag, FiniteDistribution.mixture]
theorem kernel_decodeLeft_false (i : Fin (branch n)) (s : Source (m := m) (t := t) designated i) :
(kernel (C := C) (t := t) rows clauses designated (fun _ => falseFlag) i s).pushforward
(decodeLeft rows clauses designated i s) =
FiniteDistribution.uniform (Native (C := C) (t := t) rows clauses designated i s) := by
rw [kernel_decodeLeft, leftLaw_false]
theorem kernels_decodeLeft (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 => decodeLeft rows clauses designated i (sources i) (raw i)) =
FiniteProduct.law (fun i => leftLaw (C := C) (t := t) rows clauses designated flag i (sources i)) := by
rw [FiniteProduct.pushforward_map]
exact congrArg FiniteProduct.law
(funext fun i => kernel_decodeLeft rows clauses designated flag i (sources i))
theorem kernels_leftChildren (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
(leftChildren rows clauses designated sources) =
FiniteProduct.law (fun i => leftLaw (C := C) (t := t) rows clauses designated flag i (sources i)) :=
kernels_decodeLeft rows clauses designated flag sources
theorem kernels_leftChildren_false (sources : Sources (m := m) (t := t) designated) :
(FiniteProduct.law (fun i => kernel (C := C) (t := t) rows clauses designated
(fun _ => falseFlag) i (sources i))).pushforward
(leftChildren rows clauses designated sources) =
FiniteProduct.law (fun i =>
FiniteDistribution.uniform (Native (C := C) (t := t) rows clauses designated i (sources i))) := by
rw [kernels_leftChildren]
exact congrArg FiniteProduct.law
(funext fun i => leftLaw_false rows clauses designated i (sources i))
end
end PerfectCompleteness.SourceChildNativeLaw
end
end OAI
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