openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Construction /CutChildGrouping.lean
| import OAI.Computability.PerfectCompleteness.Repetition.CleanTreeFactors | |
| import OAI.Computability.PerfectCompleteness.Sampling.UniformChildSum | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.CutChildGrouping | |
| open scoped BigOperators Classical | |
| open TreeSourceSpaces HierarchicalArrays PointwiseSpaces | |
| open UniqueGamesTheorem.Foundations.Games | |
| abbrev F2 := ZMod 2 | |
| noncomputable section | |
| variable {branch : Nat → Nat} {n t : Nat} {C : Type*} [Fintype C] | |
| (slots : RecursiveSpaces.Slots branch (n + 1) → Fin t → MixedSupport.Slot) | |
| (rows : Nat → Nat) | |
| abbrev Child (i : Fin (branch n)) := | |
| (C → squareSpace (H (childSlots slots i))) × Arrays (childSlots slots i) rows | |
| instance childFintype (i : Fin (branch n)) : Fintype (Child (C := C) slots rows i) := | |
| Fintype.ofFinite _ | |
| abbrev Raw := (i : Fin (branch n)) → Child (C := C) slots rows i | |
| abbrev Assembled := | |
| (C → H slots) × ((i : Fin (branch n)) → Arrays (childSlots slots i) rows) | |
| instance assembledFintype : Fintype (Assembled (C := C) slots rows) := Fintype.ofFinite _ | |
| def rawLaw : FiniteDistribution (Raw (C := C) slots rows) := | |
| FiniteProduct.law (fun i => FiniteDistribution.uniform (Child (C := C) slots rows i)) | |
| def assemble : Raw (C := C) slots rows →ₗ[F2] Assembled (C := C) slots rows where | |
| toFun x := | |
| (fun q => UniformChildSum.recursiveSum (TreeSourceSpaces.LeafDomain slots) | |
| (fun i => (x i).1 q), fun i => (x i).2) | |
| map_add' x y := by | |
| apply Prod.ext | |
| · funext q | |
| exact map_add (UniformChildSum.recursiveSum (TreeSourceSpaces.LeafDomain slots)) | |
| (fun i => (x i).1 q) (fun i => (y i).1 q) | |
| · rfl | |
| map_smul' c x := by | |
| apply Prod.ext | |
| · funext q | |
| exact map_smul (UniformChildSum.recursiveSum (TreeSourceSpaces.LeafDomain slots)) c | |
| (fun i => (x i).1 q) | |
| · rfl | |
| omit [Fintype C] in | |
| @[simp] theorem assemble_calls (x : Raw (C := C) slots rows) (q : C) : | |
| (assemble slots rows x).1 q = | |
| UniformChildSum.recursiveSum (TreeSourceSpaces.LeafDomain slots) | |
| (fun i => (x i).1 q) := rfl | |
| omit [Fintype C] in | |
| @[simp] theorem assemble_arrays (x : Raw (C := C) slots rows) (i : Fin (branch n)) : | |
| (assemble slots rows x).2 i = (x i).2 := rfl | |
| omit [Fintype C] in | |
| theorem assemble_surjective : Function.Surjective (assemble (C := C) slots rows) := by | |
| rintro ⟨calls, arrays⟩ | |
| have hex : ∀ q : C, ∃ u : (i : Fin (branch n)) → squareSpace (H (childSlots slots i)), | |
| UniformChildSum.recursiveSum (TreeSourceSpaces.LeafDomain slots) u = calls q := by | |
| intro q | |
| exact UniformChildSum.recursiveSum_surjective (TreeSourceSpaces.LeafDomain slots) (calls q) | |
| choose u hu using hex | |
| refine ⟨fun i => (fun q => u q i, arrays i), ?_⟩ | |
| apply Prod.ext | |
| · funext q | |
| exact hu q | |
| · rfl | |
| theorem assemble_law : | |
| (rawLaw (C := C) slots rows).pushforward (assemble slots rows) = | |
| FiniteDistribution.uniform (Assembled (C := C) slots rows) := by | |
| unfold rawLaw | |
| rw [UniformLinearImage.law_uniform] | |
| exact UniformLinearImage.uniform_pushforward_linearMap | |
| (assemble slots rows) (assemble_surjective slots rows) | |
| theorem projected_child_type (calls : Nat) | |
| (ids : RecursiveSpaces.Slots branch (n + 1) → Fin t → Nat) | |
| (i : Fin (branch n)) : | |
| Child (C := Fin calls) (ProjectedCardinality.projectedSlots ids) rows i = | |
| CleanTreeFactors.Raw calls rows (fun s k => ids (i, s) k) := rfl | |
| end | |
| end PerfectCompleteness.CutChildGrouping | |
| end | |
| end OAI | |