openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Sampling /CutSamplerReplayGroupingTransport.lean
| import OAI.Computability.PerfectCompleteness.Foundations.WholeCutExteriorTransportLemmas | |
| import OAI.Computability.PerfectCompleteness.Foundations.WholeCutReplayKeys | |
| import OAI.Computability.PerfectCompleteness.Sampling.CutSamplerKeyLocalityLemmas | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.CutSamplerReplayGroupingTransport | |
| open PointwiseSpaces RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| open RecursiveSampler TerminalCalls CutSamplerKeyLocality | |
| open scoped Classical | |
| abbrev F2 := ZMod 2 | |
| noncomputable section | |
| universe u v | |
| theorem cast_pi_apply {I : Type u} {A B : I → Type v} | |
| (h : ((i : I) → A i) = ((i : I) → B i)) (hA : ∀ i, A i = B i) | |
| (f : (i : I) → A i) (i : I) : | |
| (cast h f) i = cast (hA i) (f i) := by | |
| have hAB : A = B := funext hA | |
| cases hAB | |
| rfl | |
| theorem cast_prod_fst {A A' : Type u} {B B' : Type v} | |
| (h : (A × B) = (A' × B')) (hA : A = A') (hB : B = B') (x : A × B) : | |
| (cast h x).1 = cast hA x.1 := by | |
| cases hA | |
| cases hB | |
| rfl | |
| theorem cast_prod_snd {A A' : Type u} {B B' : Type v} | |
| (h : (A × B) = (A' × B')) (hA : A = A') (hB : B = B') (x : A × B) : | |
| (cast h x).2 = cast hB x.2 := by | |
| cases hA | |
| cases hB | |
| rfl | |
| variable {branch : Nat → Nat} {n m t : Nat} | |
| theorem outside_kept (p : Path branch n (m + 1)) (clean : Fin (branch m) → Prop) : | |
| ∀ s, WholeCutExteriorTransport.Outside p s → keptLeaf p clean s := by | |
| induction n generalizing m with | |
| | zero => | |
| have h := p.height_le | |
| omega | |
| | succ n ih => | |
| cases p with | |
| | refl => | |
| intro s hs | |
| exact False.elim (hs ⟨s, rfl⟩) | |
| | step i p => | |
| rintro ⟨j, s⟩ hs | |
| by_cases hji : j = i | |
| · subst j | |
| apply Or.inr | |
| apply ih p clean s | |
| rintro ⟨z, hz⟩ | |
| exact hs ⟨z, congrArg (fun a => (i, a)) hz⟩ | |
| · exact Or.inl hji | |
| theorem outside_eq (p : Path branch n (m + 1)) | |
| (slots target : Slots branch n → Fin t → MixedSupport.Slot) | |
| (clean : Fin (branch m) → Prop) | |
| (hs : ∀ s, keptLeaf p clean s → slots s = target s) : | |
| ∀ s, WholeCutExteriorTransport.Outside p s → slots s = target s := | |
| fun s hout => hs s (outside_kept p clean s hout) | |
| theorem retainedChildSlots_eq (p : Path branch n (m + 1)) | |
| (slots target : Slots branch n → Fin t → MixedSupport.Slot) | |
| (clean : Fin (branch m) → Prop) | |
| (hs : ∀ s, keptLeaf p clean s → slots s = target s) | |
| (i : {i : Fin (branch m) // ¬ clean i}) : | |
| childSlots (WholeCutGrouping.cutSlots p slots) i.val = | |
| childSlots (WholeCutGrouping.cutSlots p target) i.val := by | |
| funext s | |
| exact hs (p.slotEmbedding (i.val, s)) ((keptLeaf_at_cut p clean i.val s).mpr i.property) | |
| theorem squareEquiv_eq_cast | |
| (slots target : Slots branch n → Fin t → MixedSupport.Slot) (hs : slots = target) | |
| (f : squareSpace (H slots)) : | |
| (RecursiveSpaceEquiv.squareEquiv (𝕜 := F2) | |
| (fun s => Equiv.cast (congrArg MixedSupport.Assignment (congrFun hs s)))).symm f = | |
| cast (congrArg | |
| (fun ss : Slots branch n → Fin t → MixedSupport.Slot => (squareSpace (H ss) : Type)) | |
| hs) f := by | |
| cases hs | |
| apply Subtype.ext | |
| funext x | |
| rfl | |
| def castTerminal (repeats : Nat → Nat) (p : Path branch n (m + 1)) | |
| (slots target : Slots branch n → Fin t → MixedSupport.Slot) | |
| (clean : Fin (branch m) → Prop) | |
| (hs : ∀ s, keptLeaf p clean s → slots s = target s) | |
| (terminal : CutSamplerReplayGrouping.RetainedTerminal F2 repeats p (LeafDomain slots) clean) : | |
| CutSamplerReplayGrouping.RetainedTerminal F2 repeats p (LeafDomain target) clean := | |
| fun call child => cast (congrArg | |
| (fun ss : Slots branch m → Fin t → MixedSupport.Slot => (squareSpace (H ss) : Type)) | |
| (retainedChildSlots_eq p slots target clean hs child)) (terminal call child) | |
| theorem subExterior_cast (repeats : Nat → Nat) (i : Fin (branch n)) | |
| (p : Path branch n (m + 1)) | |
| (slots target : Slots branch (n + 1) → Fin t → MixedSupport.Slot) | |
| (clean : Fin (branch m) → Prop) | |
| (hs : ∀ s, keptLeaf (.step i p) clean s → slots s = target s) | |
| (call : Fin (repeats (n + 1)) × Bool) | |
| (exterior : CutTerminalSplit.ExteriorTape F2 repeats (.step i p) (LeafDomain slots)) : | |
| CutSamplerReplayGrouping.subExterior F2 repeats i p (LeafDomain target) call | |
| (cast (WholeCutExteriorTransport.scalarExterior_eq repeats (.step i p) slots target | |
| (outside_eq (.step i p) slots target clean hs)) exterior) = | |
| cast (WholeCutExteriorTransport.scalarExterior_eq repeats p | |
| (childSlots slots i) (childSlots target i) | |
| (outside_eq p (childSlots slots i) (childSlots target i) clean | |
| (fun s h => hs (i, s) (Or.inr h)))) | |
| (CutSamplerReplayGrouping.subExterior F2 repeats i p (LeafDomain slots) call exterior) := by | |
| funext j | |
| let hext := WholeCutExteriorTransport.scalarExterior_eq repeats (.step i p) slots target | |
| (outside_eq (.step i p) slots target clean hs) | |
| let hsub := WholeCutExteriorTransport.scalarExterior_eq repeats p | |
| (childSlots slots i) (childSlots target i) | |
| (outside_eq p (childSlots slots i) (childSlots target i) clean | |
| (fun s h => hs (i, s) (Or.inr h))) | |
| change (cast hext exterior) (CutSamplerReplayGrouping.exteriorStepIndex repeats i p call j) = | |
| (cast hsub | |
| (CutSamplerReplayGrouping.subExterior F2 repeats i p (LeafDomain slots) call exterior)) j | |
| rw [cast_pi_apply hext | |
| (fun j => WholeCutExteriorTransport.factor_eq repeats (.step i p) slots target | |
| (outside_eq (.step i p) slots target clean hs) j.val j.property)] | |
| rw [cast_pi_apply hsub | |
| (fun j => WholeCutExteriorTransport.factor_eq repeats p (childSlots slots i) | |
| (childSlots target i) | |
| (outside_eq p (childSlots slots i) (childSlots target i) clean | |
| (fun s h => hs (i, s) (Or.inr h))) j.val j.property)] | |
| rfl | |
| theorem transport_assemble (repeats : Nat → Nat) (p : Path branch n (m + 1)) : | |
| ∀ (slots target : Slots branch n → Fin t → MixedSupport.Slot) | |
| (clean : Fin (branch m) → Prop) | |
| (hs : ∀ s, keptLeaf p clean s → slots s = target s) | |
| (exterior : CutTerminalSplit.ExteriorTape F2 repeats p (LeafDomain slots)) | |
| (terminal : CutSamplerReplayGrouping.RetainedTerminal F2 repeats p (LeafDomain slots) clean), | |
| CutSamplerReplayTransport.transport F2 repeats p (LeafDomain slots) (LeafDomain target) | |
| clean (WholeCutReplayKeys.leafEquiv p slots target clean hs) | |
| (CutSamplerReplayGrouping.assemble F2 repeats p (LeafDomain slots) clean exterior terminal) = | |
| CutSamplerReplayGrouping.assemble F2 repeats p (LeafDomain target) clean | |
| (cast (WholeCutExteriorTransport.scalarExterior_eq repeats p slots target | |
| (outside_eq p slots target clean hs)) exterior) | |
| (castTerminal repeats p slots target clean hs terminal) := by | |
| induction n generalizing m with | |
| | zero => | |
| have h := p.height_le | |
| omega | |
| | succ n ih => | |
| cases p with | |
| | refl => | |
| intro slots target clean hs exterior terminal | |
| funext child | |
| exact squareEquiv_eq_cast (childSlots slots child.val) (childSlots target child.val) | |
| (retainedChildSlots_eq (.refl (n + 1)) slots target clean hs child) (terminal () child) | |
| | step i p => | |
| intro slots target clean hs exterior terminal | |
| apply Prod.ext | |
| · funext j | |
| let hext := WholeCutExteriorTransport.scalarExterior_eq repeats (.step i p) | |
| slots target (outside_eq (.step i p) slots target clean hs) | |
| let hslots := WholeCutExteriorTransport.ordinarySlots_eq i p slots target | |
| (outside_eq (.step i p) slots target clean hs) j | |
| change (RecursiveSpaceEquiv.squareEquiv (𝕜 := F2) | |
| (fun s => WholeCutReplayKeys.leafEquiv (.step i p) slots target clean hs | |
| (j.val, s) (Or.inl j.property))).symm | |
| (exterior ⟨.inl j, not_isTerminal_ordinary repeats i p j⟩) = | |
| (cast hext exterior) ⟨.inl j, not_isTerminal_ordinary repeats i p j⟩ | |
| rw [cast_pi_apply hext | |
| (fun j => WholeCutExteriorTransport.factor_eq repeats (.step i p) slots target | |
| (outside_eq (.step i p) slots target clean hs) j.val j.property)] | |
| exact squareEquiv_eq_cast (childSlots slots j.val) (childSlots target j.val) hslots | |
| (exterior ⟨.inl j, not_isTerminal_ordinary repeats i p j⟩) | |
| · funext call | |
| change CutSamplerReplayTransport.transport F2 repeats p | |
| (LeafDomain (childSlots slots i)) (LeafDomain (childSlots target i)) clean | |
| (WholeCutReplayKeys.leafEquiv p (childSlots slots i) (childSlots target i) clean | |
| (fun s h => hs (i, s) (Or.inr h))) | |
| (CutSamplerReplayGrouping.assemble F2 repeats p (LeafDomain (childSlots slots i)) | |
| clean (CutSamplerReplayGrouping.subExterior F2 repeats i p (LeafDomain slots) | |
| call exterior) (fun location child => terminal (call, location) child)) = | |
| CutSamplerReplayGrouping.assemble F2 repeats p (LeafDomain (childSlots target i)) | |
| clean (CutSamplerReplayGrouping.subExterior F2 repeats i p (LeafDomain target) call | |
| (cast (WholeCutExteriorTransport.scalarExterior_eq repeats (.step i p) slots target | |
| (outside_eq (.step i p) slots target clean hs)) exterior)) | |
| (fun location child => castTerminal repeats (.step i p) slots target clean hs | |
| terminal (call, location) child) | |
| rw [subExterior_cast repeats i p slots target clean hs call exterior] | |
| exact ih p (childSlots slots i) (childSlots target i) clean | |
| (fun s h => hs (i, s) (Or.inr h)) | |
| (CutSamplerReplayGrouping.subExterior F2 repeats i p (LeafDomain slots) call exterior) | |
| (fun location child => terminal (call, location) child) | |
| end | |
| end PerfectCompleteness.CutSamplerReplayGroupingTransport | |
| end | |
| end OAI | |