openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Sampling /SourceChildKernelLaw.lean
| 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 | |