openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Decoding /CutProjectionGeometry.lean
| import OAI.Computability.PerfectCompleteness.Decoding.ChildAssemblyProjection | |
| import OAI.Computability.PerfectCompleteness.Decoding.CutProjectionAssembly | |
| import OAI.Computability.PerfectCompleteness.Foundations.CutSlotOutside | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.CutProjectionGeometry | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays PointwiseSpaces | |
| open scoped Classical | |
| noncomputable section | |
| variable {branch : Nat → Nat} {n t : Nat} | |
| theorem sourceProjection_eq_cast | |
| (slots projected : Slots branch n → Fin t → MixedSupport.Slot) | |
| (q : ∀ s k, MixedSupport.Projection (slots s k) (projected s k)) | |
| (hs : slots = projected) | |
| (hk : ∀ s k, HEq (q s k) (MixedSupport.Projection.keep (slots s k))) | |
| (x : Domain slots) : | |
| sourceProjection q x = cast (congrArg Domain hs) x := by | |
| cases hs | |
| have hq : q = fun s k => MixedSupport.Projection.keep (slots s k) := by | |
| funext s k | |
| exact eq_of_heq (hk s k) | |
| subst q | |
| rfl | |
| theorem HPullback_eq_cast | |
| (slots projected : Slots branch n → Fin t → MixedSupport.Slot) | |
| (q : ∀ s k, MixedSupport.Projection (slots s k) (projected s k)) | |
| (hs : slots = projected) | |
| (hk : ∀ s k, HEq (q s k) (MixedSupport.Projection.keep (slots s k))) | |
| (f : H projected) : | |
| HPullback q f = cast (congrArg | |
| (fun ss : Slots branch n → Fin t → MixedSupport.Slot => (H ss : Type)) hs.symm) f := by | |
| cases hs | |
| have hq : q = fun s k => MixedSupport.Projection.keep (slots s k) := by | |
| funext s k | |
| exact eq_of_heq (hk s k) | |
| subst q | |
| apply Subtype.ext | |
| rfl | |
| theorem squarePullback_eq_cast | |
| (slots projected : Slots branch n → Fin t → MixedSupport.Slot) | |
| (q : ∀ s k, MixedSupport.Projection (slots s k) (projected s k)) | |
| (hs : slots = projected) | |
| (hk : ∀ s k, HEq (q s k) (MixedSupport.Projection.keep (slots s k))) | |
| (f : squareSpace (H projected)) : | |
| ChildBlockProjection.squarePullback q f = cast (congrArg | |
| (fun ss : Slots branch n → Fin t → MixedSupport.Slot => | |
| (squareSpace (H ss) : Type)) hs.symm) f := by | |
| cases hs | |
| have hq : q = fun s k => MixedSupport.Projection.keep (slots s k) := by | |
| funext s k | |
| exact eq_of_heq (hk s k) | |
| subst q | |
| apply Subtype.ext | |
| rfl | |
| theorem arraysPullback_eq_cast (rows : Nat → Nat) | |
| (slots projected : Slots branch n → Fin t → MixedSupport.Slot) | |
| (q : ∀ s k, MixedSupport.Projection (slots s k) (projected s k)) | |
| (hs : slots = projected) | |
| (hk : ∀ s k, HEq (q s k) (MixedSupport.Projection.keep (slots s k))) | |
| (arrays : Arrays projected rows) : | |
| ChildBlockProjection.arraysPullback rows q arrays = cast (congrArg | |
| (fun ss : Slots branch n → Fin t → MixedSupport.Slot => Arrays ss rows) hs.symm) arrays := by | |
| cases hs | |
| have hq : q = fun s k => MixedSupport.Projection.keep (slots s k) := by | |
| funext s k | |
| exact eq_of_heq (hk s k) | |
| subst q | |
| funext node row | |
| apply Subtype.ext | |
| rfl | |
| theorem rawPullback_eq_cast (calls : Nat) (rows : Nat → Nat) | |
| (slots projected : Slots branch n → Fin t → MixedSupport.Slot) | |
| (q : ∀ s k, MixedSupport.Projection (slots s k) (projected s k)) | |
| (hs : slots = projected) | |
| (hk : ∀ s k, HEq (q s k) (MixedSupport.Projection.keep (slots s k))) | |
| (block : ChildBlockCardinality.Raw calls rows projected) : | |
| ChildBlockProjection.rawPullback calls rows q block = cast (congrArg | |
| (fun ss : Slots branch n → Fin t → MixedSupport.Slot => | |
| ChildBlockCardinality.Raw calls rows ss) hs.symm) block := by | |
| cases hs | |
| have hq : q = fun s k => MixedSupport.Projection.keep (slots s k) := by | |
| funext s k | |
| exact eq_of_heq (hk s k) | |
| subst q | |
| apply Prod.ext | |
| · funext call | |
| apply Subtype.ext | |
| rfl | |
| · funext node row | |
| apply Subtype.ext | |
| rfl | |
| theorem childPullback_heq {C : Type*} [Fintype C] (rows : Nat → Nat) | |
| {left right left' right' : Slots branch (n + 1) → Fin t → MixedSupport.Slot} | |
| (hL : left = left') (hR : right = right') | |
| (q : ∀ s k, MixedSupport.Projection (left s k) (right s k)) | |
| (q' : ∀ s k, MixedSupport.Projection (left' s k) (right' s k)) | |
| (hq : ∀ s k, HEq (q s k) (q' s k)) (i : Fin (branch n)) | |
| (block : CutChildGrouping.Child (C := C) right rows i) | |
| (block' : CutChildGrouping.Child (C := C) right' rows i) | |
| (hb : HEq block block') : | |
| HEq (ChildAssemblyProjection.childPullback rows q i block) | |
| (ChildAssemblyProjection.childPullback rows q' i block') := by | |
| cases hL | |
| cases hR | |
| have hqq' : q = q' := by | |
| funext s k | |
| exact eq_of_heq (hq s k) | |
| cases hqq' | |
| have hbb' : block = block' := eq_of_heq hb | |
| cases hbb' | |
| rfl | |
| private theorem keep_heq {a b : MixedSupport.Slot} (h : a = b) : | |
| HEq (MixedSupport.Projection.keep a) (MixedSupport.Projection.keep b) := by | |
| cases h | |
| rfl | |
| theorem fillProjection_outside {m : Nat} (p : Path branch n m) | |
| (outside : Slots branch n → Fin t → MixedSupport.Slot) | |
| (left right : Slots branch m → Fin t → MixedSupport.Slot) | |
| (q : ∀ s k, MixedSupport.Projection (left s k) (right s k)) | |
| (s : Slots branch n) (hout : s ∉ Set.range p.slotEmbedding) (k : Fin t) : | |
| HEq (CutProjectionAssembly.fillProjection p outside left right q s k) | |
| (MixedSupport.Projection.keep (CutSlotAssembly.fill p outside left s k)) := by | |
| revert outside left right q s hout k | |
| induction p with | |
| | refl _ => | |
| intro outside left right q s hout k | |
| exact False.elim (hout ⟨s, rfl⟩) | |
| | step i p ih => | |
| intro outside left right q s hout k | |
| obtain ⟨j, s⟩ := s | |
| by_cases hji : j = i | |
| · subst j | |
| have htail : s ∉ Set.range p.slotEmbedding := by | |
| rintro ⟨a, ha⟩ | |
| exact hout ⟨a, congrArg (fun b => (i, b)) ha⟩ | |
| have hfill : CutSlotAssembly.fill (.step i p) outside left (i, s) = | |
| CutSlotAssembly.fill p (fun b => outside (i, b)) left s := | |
| ite_eq_left (show i = i from rfl) | |
| simp only [CutProjectionAssembly.fillProjection] | |
| exact (CutProjectionAssembly.castProjection_heq _ _ _).trans | |
| ((ih (fun b => outside (i, b)) left right q s htail k).trans | |
| (keep_heq (congrFun hfill k).symm)) | |
| · have hfill : CutSlotAssembly.fill (.step i p) outside left (j, s) = outside (j, s) := | |
| ite_eq_right hji | |
| simp only [CutProjectionAssembly.fillProjection, dite_eq_right hji] | |
| exact (CutProjectionAssembly.castProjection_heq _ _ _).trans | |
| (keep_heq (congrFun hfill k).symm) | |
| end | |
| end PerfectCompleteness.CutProjectionGeometry | |
| end | |
| end OAI | |