openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Construction /ChildCallNumbering.lean
| import OAI.Computability.PerfectCompleteness.Algebra.UniformLinearImage | |
| import OAI.Computability.PerfectCompleteness.Decoding.ChildBlockProjection | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.ChildCallNumbering | |
| open RecursiveSpaces TreeSourceSpaces HierarchicalArrays PointwiseSpaces | |
| open UniqueGamesTheorem.Foundations.Games | |
| open scoped Classical | |
| noncomputable section | |
| variable {C : Type*} [Fintype C] {branch : Nat → Nat} {n t : Nat} | |
| abbrev Raw (rows : Nat → Nat) (slots : Slots branch n → Fin t → MixedSupport.Slot) := | |
| (C → squareSpace (H slots)) × Arrays slots rows | |
| instance rawFintype (rows : Nat → Nat) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) : | |
| Fintype (Raw (C := C) rows slots) := Fintype.ofFinite _ | |
| instance rawNonempty (rows : Nat → Nat) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) : | |
| Nonempty (Raw (C := C) rows slots) := ⟨(fun _ => 0, fun _ _ => 0)⟩ | |
| def equiv (rows : Nat → Nat) (slots : Slots branch n → Fin t → MixedSupport.Slot) : | |
| Raw (C := C) rows slots ≃ₗ[F2] | |
| ChildBlockCardinality.Raw (Fintype.card C) rows slots where | |
| toFun x := (fun k => x.1 ((Fintype.equivFin C).symm k), x.2) | |
| invFun x := (fun c => x.1 (Fintype.equivFin C c), x.2) | |
| left_inv x := by | |
| apply Prod.ext | |
| · funext c | |
| simp only [Equiv.symm_apply_apply] | |
| · rfl | |
| right_inv x := by | |
| apply Prod.ext | |
| · funext k | |
| simp only [Equiv.apply_symm_apply] | |
| · rfl | |
| map_add' x y := rfl | |
| map_smul' c x := rfl | |
| @[simp] theorem equiv_calls (rows : Nat → Nat) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) | |
| (x : Raw (C := C) rows slots) (k : Fin (Fintype.card C)) : | |
| (equiv rows slots x).1 k = x.1 ((Fintype.equivFin C).symm k) := rfl | |
| @[simp] theorem equiv_arrays (rows : Nat → Nat) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) | |
| (x : Raw (C := C) rows slots) : (equiv rows slots x).2 = x.2 := rfl | |
| @[simp] theorem equiv_symm_calls (rows : Nat → Nat) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) | |
| (x : ChildBlockCardinality.Raw (Fintype.card C) rows slots) (c : C) : | |
| ((equiv (C := C) rows slots).symm x).1 c = x.1 (Fintype.equivFin C c) := rfl | |
| @[simp] theorem equiv_symm_arrays (rows : Nat → Nat) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) | |
| (x : ChildBlockCardinality.Raw (Fintype.card C) rows slots) : | |
| ((equiv (C := C) rows slots).symm x).2 = x.2 := rfl | |
| def pullback (rows : Nat → Nat) | |
| {slots projected : Slots branch n → Fin t → MixedSupport.Slot} | |
| (p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k)) : | |
| Raw (C := C) rows projected →ₗ[F2] Raw (C := C) rows slots where | |
| toFun x := (fun c => ChildBlockProjection.squarePullback p (x.1 c), | |
| ChildBlockProjection.arraysPullback rows p x.2) | |
| map_add' x y := by | |
| apply Prod.ext | |
| · funext c | |
| exact map_add (ChildBlockProjection.squarePullback p) (x.1 c) (y.1 c) | |
| · exact map_add (ChildBlockProjection.arraysPullback rows p) x.2 y.2 | |
| map_smul' a x := by | |
| apply Prod.ext | |
| · funext c | |
| exact map_smul (ChildBlockProjection.squarePullback p) a (x.1 c) | |
| · exact map_smul (ChildBlockProjection.arraysPullback rows p) a x.2 | |
| omit [Fintype C] in | |
| @[simp] theorem pullback_calls (rows : Nat → Nat) | |
| {slots projected : Slots branch n → Fin t → MixedSupport.Slot} | |
| (p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k)) | |
| (x : Raw (C := C) rows projected) (c : C) : | |
| (pullback rows p x).1 c = ChildBlockProjection.squarePullback p (x.1 c) := rfl | |
| omit [Fintype C] in | |
| @[simp] theorem pullback_arrays (rows : Nat → Nat) | |
| {slots projected : Slots branch n → Fin t → MixedSupport.Slot} | |
| (p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k)) | |
| (x : Raw (C := C) rows projected) : | |
| (pullback rows p x).2 = ChildBlockProjection.arraysPullback rows p x.2 := rfl | |
| theorem equiv_pullback (rows : Nat → Nat) | |
| {slots projected : Slots branch n → Fin t → MixedSupport.Slot} | |
| (p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k)) | |
| (x : Raw (C := C) rows projected) : | |
| equiv rows slots (pullback rows p x) = | |
| ChildBlockProjection.rawPullback (Fintype.card C) rows p | |
| (equiv rows projected x) := rfl | |
| omit [Fintype C] in | |
| theorem pullback_cast (rows : Nat → Nat) | |
| {slots target projected : Slots branch n → Fin t → MixedSupport.Slot} | |
| (h : slots = target) | |
| (p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k)) | |
| (x : Raw (C := C) rows projected) : | |
| cast (congrArg (Raw (C := C) rows) h) (pullback rows p x) = | |
| pullback rows | |
| (cast (congrArg | |
| (fun ss : Slots branch n → Fin t → MixedSupport.Slot => | |
| ∀ s k, MixedSupport.Projection (ss s k) (projected s k)) h) p) x := by | |
| cases h | |
| rfl | |
| theorem uniform_equiv (rows : Nat → Nat) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) : | |
| (FiniteDistribution.uniform (Raw (C := C) rows slots)).pushforward | |
| (equiv (C := C) rows slots) = | |
| FiniteDistribution.uniform (ChildBlockCardinality.Raw (Fintype.card C) rows slots) := | |
| UniformLinearImage.uniform_pushforward_linearMap | |
| (equiv (C := C) rows slots).toLinearMap (equiv (C := C) rows slots).surjective | |
| theorem card_le (rows : Nat → Nat) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) : | |
| Fintype.card (Raw (C := C) rows slots) ≤ | |
| ChildBlockCardinality.bound branch n t (Fintype.card C) rows := by | |
| calc | |
| Fintype.card (Raw (C := C) rows slots) = | |
| Fintype.card (ChildBlockCardinality.Raw (Fintype.card C) rows slots) := | |
| Fintype.card_congr (equiv (C := C) rows slots).toEquiv | |
| _ ≤ _ := by | |
| simpa only [Nat.card_eq_fintype_card] using | |
| ChildBlockCardinality.card_le (Fintype.card C) rows slots | |
| end | |
| end PerfectCompleteness.ChildCallNumbering | |
| end | |
| end OAI | |