openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Decoding /CutGroupedProjectionLemmas.lean
| import OAI.Computability.PerfectCompleteness.Decoding.ChildAssemblyProjection | |
| import OAI.Computability.PerfectCompleteness.Decoding.CutProjectionGeometry | |
| import OAI.Computability.PerfectCompleteness.Foundations.OriginalTerminalSplitLemmas | |
| import OAI.Computability.PerfectCompleteness.Sampling.CutSamplerReplayGroupingTransport | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.CutGroupedProjection | |
| open PointwiseSpaces RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| open RecursiveSampler TerminalCalls | |
| open scoped BigOperators Classical | |
| abbrev F2 := ZMod 2 | |
| noncomputable section | |
| variable {branch : Nat → Nat} {n m t : Nat} | |
| def cutProjection (p : Path branch n (m + 1)) | |
| {left right : Slots branch n → Fin t → MixedSupport.Slot} | |
| (q : ∀ s k, MixedSupport.Projection (left s k) (right s k)) : | |
| ∀ s k, MixedSupport.Projection | |
| (WholeCutGrouping.cutSlots p left s k) (WholeCutGrouping.cutSlots p right s k) := | |
| fun s k => q (p.slotEmbedding s) k | |
| def terminalPullback (repeats : Nat → Nat) (p : Path branch n (m + 1)) | |
| {left right : Slots branch n → Fin t → MixedSupport.Slot} | |
| (q : ∀ s k, MixedSupport.Projection (left s k) (right s k)) | |
| (terminal : CutTerminalSplit.TerminalTape F2 repeats p (LeafDomain right)) : | |
| CutTerminalSplit.TerminalTape F2 repeats p (LeafDomain left) := | |
| fun call child => ChildBlockProjection.squarePullback | |
| (ChildAssemblyProjection.childProjection (cutProjection p q) child) (terminal call child) | |
| def factorPullback (repeats : Nat → Nat) : | |
| {n m : Nat} → (p : Path branch n (m + 1)) → | |
| (left right : Slots branch n → Fin t → MixedSupport.Slot) → | |
| (∀ s k, MixedSupport.Projection (left s k) (right s k)) → | |
| (j : DrawIndex repeats p) → | |
| CutSamplerRefinement.RefinedSpace F2 repeats p (LeafDomain right) j → | |
| CutSamplerRefinement.RefinedSpace F2 repeats p (LeafDomain left) j | |
| | _ + 1, _, .refl _, _, _, q, _, x => | |
| fun child => ChildBlockProjection.squarePullback | |
| (ChildAssemblyProjection.childProjection q child) (x child) | |
| | _ + 1, _, .step i p, left, right, q, j, x => | |
| match j with | |
| | Sum.inl j => | |
| ChildBlockProjection.squarePullback (ChildAssemblyProjection.childProjection q j.val) x | |
| | Sum.inr j => | |
| factorPullback repeats p (childSlots left i) (childSlots right i) | |
| (ChildAssemblyProjection.childProjection q i) j.2 x | |
| def tapePullback (repeats : Nat → Nat) (p : Path branch n (m + 1)) | |
| (left right : Slots branch n → Fin t → MixedSupport.Slot) | |
| (q : ∀ s k, MixedSupport.Projection (left s k) (right s k)) | |
| (tape : CutSamplerRefinement.Tape F2 repeats p (LeafDomain right)) : | |
| CutSamplerRefinement.Tape F2 repeats p (LeafDomain left) := | |
| fun j => factorPullback repeats p left right q j (tape j) | |
| theorem evaluate_tapePullback (repeats : Nat → Nat) (p : Path branch n (m + 1)) : | |
| ∀ (left right : Slots branch n → Fin t → MixedSupport.Slot) | |
| (q : ∀ s k, MixedSupport.Projection (left s k) (right s k)) | |
| (tape : CutSamplerRefinement.Tape F2 repeats p (LeafDomain right)) (x : Domain left), | |
| (CutSamplerRefinement.evaluate F2 repeats p (LeafDomain left) | |
| (tapePullback repeats p left right q tape)).val x = | |
| (CutSamplerRefinement.evaluate F2 repeats p (LeafDomain right) tape).val | |
| (sourceProjection q x) := by | |
| induction n generalizing m with | |
| | zero => | |
| have h := p.height_le | |
| omega | |
| | succ n ih => | |
| cases p with | |
| | refl => | |
| intro left right q tape x | |
| exact congrArg (fun f : H left => f.val x) | |
| (ChildAssemblyProjection.recursiveSum_pullback q (tape ())) | |
| | step i p => | |
| intro left right q tape x | |
| simp only [CutSamplerLocality.evaluate_step, RecursiveSampler.combine, | |
| RecursiveSampler.combineFunction, Finset.sum_apply, Pi.add_apply, Pi.mul_apply, | |
| PointwiseSpaces.pullback_apply] | |
| change | |
| (∑ j : OffPath i, | |
| (ChildBlockProjection.squarePullback (ChildAssemblyProjection.childProjection q j.val) | |
| (tape (.inl j))).val (restrictChild left j.val x)) + | |
| (∑ h : Fin (repeats (n + 1)), | |
| (CutSamplerRefinement.evaluate F2 repeats p (LeafDomain (childSlots left i)) | |
| (tapePullback repeats p (childSlots left i) (childSlots right i) | |
| (ChildAssemblyProjection.childProjection q i) | |
| (CutSamplerLocality.subTape F2 repeats i p (LeafDomain right) tape h false))).val | |
| (restrictChild left i x) * | |
| (CutSamplerRefinement.evaluate F2 repeats p (LeafDomain (childSlots left i)) | |
| (tapePullback repeats p (childSlots left i) (childSlots right i) | |
| (ChildAssemblyProjection.childProjection q i) | |
| (CutSamplerLocality.subTape F2 repeats i p (LeafDomain right) tape h true))).val | |
| (restrictChild left i x)) = | |
| (∑ j : OffPath i, (tape (.inl j)).val (restrictChild right j.val (sourceProjection q x))) + | |
| (∑ h : Fin (repeats (n + 1)), | |
| (CutSamplerRefinement.evaluate F2 repeats p (LeafDomain (childSlots right i)) | |
| (CutSamplerLocality.subTape F2 repeats i p (LeafDomain right) tape h false)).val | |
| (restrictChild right i (sourceProjection q x)) * | |
| (CutSamplerRefinement.evaluate F2 repeats p (LeafDomain (childSlots right i)) | |
| (CutSamplerLocality.subTape F2 repeats i p (LeafDomain right) tape h true)).val | |
| (restrictChild right i (sourceProjection q x))) | |
| apply congrArg₂ (fun a b : F2 => a + b) | |
| · exact Finset.sum_congr rfl (fun _ _ => rfl) | |
| · apply Finset.sum_congr rfl | |
| intro h _ | |
| apply congrArg₂ (fun a b : F2 => a * b) | |
| · exact ih p (childSlots left i) (childSlots right i) | |
| (ChildAssemblyProjection.childProjection q i) | |
| (CutSamplerLocality.subTape F2 repeats i p (LeafDomain right) tape h false) | |
| (restrictChild left i x) | |
| · exact ih p (childSlots left i) (childSlots right i) | |
| (ChildAssemblyProjection.childProjection q i) | |
| (CutSamplerLocality.subTape F2 repeats i p (LeafDomain right) tape h true) | |
| (restrictChild left i x) | |
| theorem terminalFactor_pullback (repeats : Nat → Nat) (p : Path branch n (m + 1)) : | |
| ∀ (left right : Slots branch n → Fin t → MixedSupport.Slot) | |
| (q : ∀ s k, MixedSupport.Projection (left s k) (right s k)) | |
| (call : TerminalIndex repeats p) | |
| (x : CutSamplerRefinement.RefinedSpace F2 repeats p (LeafDomain right) | |
| (drawIndex repeats p call)), | |
| CutTerminalSplit.terminalFactorEquiv F2 repeats p (LeafDomain left) call | |
| (factorPullback repeats p left right q (drawIndex repeats p call) x) = | |
| fun child => ChildBlockProjection.squarePullback | |
| (ChildAssemblyProjection.childProjection (cutProjection p q) child) | |
| (CutTerminalSplit.terminalFactorEquiv F2 repeats p (LeafDomain right) call x child) := by | |
| induction n generalizing m with | |
| | zero => | |
| have h := p.height_le | |
| omega | |
| | succ n ih => | |
| cases p with | |
| | refl => | |
| intro left right q call x | |
| cases call | |
| rfl | |
| | step i p => | |
| intro left right q call x | |
| exact ih p (childSlots left i) (childSlots right i) | |
| (ChildAssemblyProjection.childProjection q i) call.2 x | |
| theorem exteriorFactor_pullback (repeats : Nat → Nat) (p : Path branch n (m + 1)) : | |
| ∀ (left right : Slots branch n → Fin t → MixedSupport.Slot) | |
| (q : ∀ s k, MixedSupport.Projection (left s k) (right s k)) | |
| (hs : ∀ s, WholeCutExteriorTransport.Outside p s → left s = right s) | |
| (_hk : ∀ s (_hout : WholeCutExteriorTransport.Outside p s) k, | |
| HEq (q s k) (MixedSupport.Projection.keep (left s k))) | |
| (j : DrawIndex repeats p) (hj : ¬ IsTerminal repeats p j) | |
| (x : CutSamplerRefinement.RefinedSpace F2 repeats p (LeafDomain right) j), | |
| factorPullback repeats p left right q j x = | |
| cast (WholeCutExteriorTransport.factor_eq repeats p right left | |
| (fun s h => (hs s h).symm) j hj) x := by | |
| induction n generalizing m with | |
| | zero => | |
| have h := p.height_le | |
| omega | |
| | succ n ih => | |
| cases p with | |
| | refl => | |
| intro left right q hs hk j hj x | |
| exact False.elim (hj True.intro) | |
| | step i p => | |
| intro left right q hs hk j hj x | |
| cases j with | |
| | inl j => | |
| exact CutProjectionGeometry.squarePullback_eq_cast | |
| (childSlots left j.val) (childSlots right j.val) | |
| (ChildAssemblyProjection.childProjection q j.val) | |
| (WholeCutExteriorTransport.ordinarySlots_eq i p left right hs j) | |
| (fun s k => hk (j.val, s) (WholeCutExteriorTransport.outside_ordinary i p j s) k) x | |
| | inr j => | |
| exact ih p (childSlots left i) (childSlots right i) | |
| (ChildAssemblyProjection.childProjection q i) | |
| (fun s h => hs (i, s) (WholeCutExteriorTransport.outside_selected i p s h)) | |
| (fun s h k => hk (i, s) (WholeCutExteriorTransport.outside_selected i p s h) k) | |
| j.2 hj x | |
| theorem splitTape_tapePullback (repeats : Nat → Nat) (p : Path branch n (m + 1)) | |
| (left right : Slots branch n → Fin t → MixedSupport.Slot) | |
| (q : ∀ s k, MixedSupport.Projection (left s k) (right s k)) | |
| (hs : ∀ s, WholeCutExteriorTransport.Outside p s → left s = right s) | |
| (hk : ∀ s (_hout : WholeCutExteriorTransport.Outside p s) k, | |
| HEq (q s k) (MixedSupport.Projection.keep (left s k))) | |
| (tape : CutSamplerRefinement.Tape F2 repeats p (LeafDomain right)) : | |
| CutTerminalSplit.splitTape F2 repeats p (LeafDomain left) | |
| (tapePullback repeats p left right q tape) = | |
| (terminalPullback repeats p q (CutTerminalSplit.splitTape F2 repeats p (LeafDomain right) tape).1, | |
| cast (WholeCutExteriorTransport.scalarExterior_eq repeats p right left | |
| (fun s h => (hs s h).symm)) | |
| (CutTerminalSplit.splitTape F2 repeats p (LeafDomain right) tape).2) := by | |
| apply Prod.ext | |
| · funext call | |
| exact terminalFactor_pullback repeats p left right q call (tape (drawIndex repeats p call)) | |
| · funext j | |
| exact (exteriorFactor_pullback repeats p left right q hs hk j.val j.property (tape j.val)).trans | |
| (CutSamplerReplayGroupingTransport.cast_pi_apply | |
| (WholeCutExteriorTransport.scalarExterior_eq repeats p right left | |
| (fun s h => (hs s h).symm)) | |
| (fun j => WholeCutExteriorTransport.factor_eq repeats p right left | |
| (fun s h => (hs s h).symm) j.val j.property) | |
| (CutTerminalSplit.splitTape F2 repeats p (LeafDomain right) tape).2 j).symm | |
| theorem evaluate_rebuild (repeats : Nat → Nat) (p : Path branch n (m + 1)) | |
| (left right : Slots branch n → Fin t → MixedSupport.Slot) | |
| (q : ∀ s k, MixedSupport.Projection (left s k) (right s k)) | |
| (hs : ∀ s, WholeCutExteriorTransport.Outside p s → left s = right s) | |
| (hk : ∀ s (_hout : WholeCutExteriorTransport.Outside p s) k, | |
| HEq (q s k) (MixedSupport.Projection.keep (left s k))) | |
| (terminal : CutTerminalSplit.TerminalTape F2 repeats p (LeafDomain right)) | |
| (exterior : CutTerminalSplit.ExteriorTape F2 repeats p (LeafDomain right)) (x : Domain left) : | |
| (CutSamplerRefinement.evaluate F2 repeats p (LeafDomain left) | |
| ((CutTerminalSplit.splitTape F2 repeats p (LeafDomain left)).symm | |
| (terminalPullback repeats p q terminal, | |
| cast (WholeCutExteriorTransport.scalarExterior_eq repeats p right left | |
| (fun s h => (hs s h).symm)) exterior))).val x = | |
| (CutSamplerRefinement.evaluate F2 repeats p (LeafDomain right) | |
| ((CutTerminalSplit.splitTape F2 repeats p (LeafDomain right)).symm (terminal, exterior))).val | |
| (sourceProjection q x) := by | |
| let tape := (CutTerminalSplit.splitTape F2 repeats p (LeafDomain right)).symm (terminal, exterior) | |
| have hsplit : | |
| CutTerminalSplit.splitTape F2 repeats p (LeafDomain left) | |
| (tapePullback repeats p left right q tape) = | |
| (terminalPullback repeats p q terminal, | |
| cast (WholeCutExteriorTransport.scalarExterior_eq repeats p right left | |
| (fun s h => (hs s h).symm)) exterior) := by | |
| simpa only [tape, Equiv.apply_symm_apply] using | |
| splitTape_tapePullback repeats p left right q hs hk tape | |
| have hleft := congrArg (CutTerminalSplit.splitTape F2 repeats p (LeafDomain left)).symm hsplit | |
| simp only [Equiv.symm_apply_apply] at hleft | |
| rw [← hleft] | |
| exact evaluate_tapePullback repeats p left right q tape x | |
| end | |
| end PerfectCompleteness.CutGroupedProjection | |
| end | |
| section | |
| namespace PerfectCompleteness.OriginalScalarProjection | |
| open PointwiseSpaces RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| open RecursiveSampler TerminalCalls | |
| open scoped BigOperators Classical | |
| abbrev F2 := ZMod 2 | |
| noncomputable section | |
| variable {branch : Nat → Nat} {n m t : Nat} | |
| theorem factor_eq (repeats : Nat → Nat) (p : Path branch n m) : | |
| ∀ (left right : Slots branch n → Fin t → MixedSupport.Slot), | |
| (∀ s, WholeCutExteriorTransport.Outside p s → left s = right s) → | |
| ∀ (j : DrawIndex repeats p), ¬ IsTerminal repeats p j → | |
| DrawSpace F2 repeats p (LeafDomain left) j = | |
| DrawSpace F2 repeats p (LeafDomain right) j := by | |
| induction p with | |
| | refl n => | |
| intro left right hs j hj | |
| exact False.elim (hj True.intro) | |
| | @step n m i p ih => | |
| intro left right hs j hj | |
| cases j with | |
| | inl j => | |
| exact congrArg | |
| (fun ss : Slots branch n → Fin t → MixedSupport.Slot => | |
| (squareSpace (space F2 branch n (LeafDomain ss)) : Type)) | |
| (WholeCutExteriorTransport.ordinarySlots_eq i p left right hs j) | |
| | inr j => | |
| exact ih (childSlots left i) (childSlots right i) | |
| (fun s h => hs (i, s) (WholeCutExteriorTransport.outside_selected i p s h)) j.2 hj | |
| theorem scalarExterior_eq (repeats : Nat → Nat) (p : Path branch n (m + 1)) | |
| (left right : Slots branch n → Fin t → MixedSupport.Slot) | |
| (hs : ∀ s, WholeCutExteriorTransport.Outside p s → left s = right s) : | |
| OriginalTerminalSplit.ExteriorTape F2 repeats p (LeafDomain left) = | |
| OriginalTerminalSplit.ExteriorTape F2 repeats p (LeafDomain right) := | |
| congrArg | |
| (fun A : OriginalTerminalSplit.ExteriorIndex repeats p → Type => | |
| (j : OriginalTerminalSplit.ExteriorIndex repeats p) → A j) | |
| (funext (fun j => factor_eq repeats p left right hs j.val j.property)) | |
| def terminalPullback (repeats : Nat → Nat) (p : Path branch n (m + 1)) | |
| {left right : Slots branch n → Fin t → MixedSupport.Slot} | |
| (q : ∀ s k, MixedSupport.Projection (left s k) (right s k)) | |
| (values : OriginalScalarReconstruction.TerminalValues F2 repeats p (LeafDomain right)) : | |
| OriginalScalarReconstruction.TerminalValues F2 repeats p (LeafDomain left) := | |
| fun call => HPullback (CutGroupedProjection.cutProjection p q) (values call) | |
| def factorPullback (repeats : Nat → Nat) : | |
| {n m : Nat} → (p : Path branch n (m + 1)) → | |
| (left right : Slots branch n → Fin t → MixedSupport.Slot) → | |
| (∀ s k, MixedSupport.Projection (left s k) (right s k)) → | |
| (j : DrawIndex repeats p) → | |
| RecursiveSampler.DrawSpace F2 repeats p (LeafDomain right) j → | |
| RecursiveSampler.DrawSpace F2 repeats p (LeafDomain left) j | |
| | _ + 1, _, .refl _, _, _, q, _, x => HPullback q x | |
| | _ + 1, _, .step i p, left, right, q, j, x => | |
| match j with | |
| | Sum.inl j => | |
| ChildBlockProjection.squarePullback (ChildAssemblyProjection.childProjection q j.val) x | |
| | Sum.inr j => | |
| factorPullback repeats p (childSlots left i) (childSlots right i) | |
| (ChildAssemblyProjection.childProjection q i) j.2 x | |
| def tapePullback (repeats : Nat → Nat) (p : Path branch n (m + 1)) | |
| (left right : Slots branch n → Fin t → MixedSupport.Slot) | |
| (q : ∀ s k, MixedSupport.Projection (left s k) (right s k)) | |
| (tape : RecursiveSampler.Tape F2 repeats p (LeafDomain right)) : | |
| RecursiveSampler.Tape F2 repeats p (LeafDomain left) := | |
| fun j => factorPullback repeats p left right q j (tape j) | |
| theorem evaluate_tapePullback (repeats : Nat → Nat) (p : Path branch n (m + 1)) : | |
| ∀ (left right : Slots branch n → Fin t → MixedSupport.Slot) | |
| (q : ∀ s k, MixedSupport.Projection (left s k) (right s k)) | |
| (tape : RecursiveSampler.Tape F2 repeats p (LeafDomain right)) (x : Domain left), | |
| (RecursiveSampler.evaluate F2 repeats p (LeafDomain left) | |
| (tapePullback repeats p left right q tape)).val x = | |
| (RecursiveSampler.evaluate F2 repeats p (LeafDomain right) tape).val | |
| (sourceProjection q x) := by | |
| induction n generalizing m with | |
| | zero => | |
| have h := p.height_le | |
| omega | |
| | succ n ih => | |
| cases p with | |
| | refl => | |
| intro left right q tape x | |
| rfl | |
| | step i p => | |
| intro left right q tape x | |
| simp only [RecursiveSampler.evaluate_step, RecursiveSampler.combine, | |
| RecursiveSampler.combineFunction, Finset.sum_apply, Pi.add_apply, Pi.mul_apply, | |
| PointwiseSpaces.pullback_apply] | |
| change | |
| (∑ j : OffPath i, | |
| (ChildBlockProjection.squarePullback (ChildAssemblyProjection.childProjection q j.val) | |
| (tape (.inl j))).val (restrictChild left j.val x)) + | |
| (∑ h : Fin (repeats (n + 1)), | |
| (RecursiveSampler.evaluate F2 repeats p (LeafDomain (childSlots left i)) | |
| (tapePullback repeats p (childSlots left i) (childSlots right i) | |
| (ChildAssemblyProjection.childProjection q i) | |
| (RecursiveSampler.subTape F2 repeats i p (LeafDomain right) tape h false))).val | |
| (restrictChild left i x) * | |
| (RecursiveSampler.evaluate F2 repeats p (LeafDomain (childSlots left i)) | |
| (tapePullback repeats p (childSlots left i) (childSlots right i) | |
| (ChildAssemblyProjection.childProjection q i) | |
| (RecursiveSampler.subTape F2 repeats i p (LeafDomain right) tape h true))).val | |
| (restrictChild left i x)) = | |
| (∑ j : OffPath i, (tape (.inl j)).val (restrictChild right j.val (sourceProjection q x))) + | |
| (∑ h : Fin (repeats (n + 1)), | |
| (RecursiveSampler.evaluate F2 repeats p (LeafDomain (childSlots right i)) | |
| (RecursiveSampler.subTape F2 repeats i p (LeafDomain right) tape h false)).val | |
| (restrictChild right i (sourceProjection q x)) * | |
| (RecursiveSampler.evaluate F2 repeats p (LeafDomain (childSlots right i)) | |
| (RecursiveSampler.subTape F2 repeats i p (LeafDomain right) tape h true)).val | |
| (restrictChild right i (sourceProjection q x))) | |
| apply congrArg₂ (fun a b : F2 => a + b) | |
| · exact Finset.sum_congr rfl (fun _ _ => rfl) | |
| · apply Finset.sum_congr rfl | |
| intro h _ | |
| apply congrArg₂ (fun a b : F2 => a * b) | |
| · exact ih p (childSlots left i) (childSlots right i) | |
| (ChildAssemblyProjection.childProjection q i) | |
| (RecursiveSampler.subTape F2 repeats i p (LeafDomain right) tape h false) | |
| (restrictChild left i x) | |
| · exact ih p (childSlots left i) (childSlots right i) | |
| (ChildAssemblyProjection.childProjection q i) | |
| (RecursiveSampler.subTape F2 repeats i p (LeafDomain right) tape h true) | |
| (restrictChild left i x) | |
| theorem terminalFactor_pullback (repeats : Nat → Nat) (p : Path branch n (m + 1)) : | |
| ∀ (left right : Slots branch n → Fin t → MixedSupport.Slot) | |
| (q : ∀ s k, MixedSupport.Projection (left s k) (right s k)) | |
| (call : TerminalIndex repeats p) | |
| (x : RecursiveSampler.DrawSpace F2 repeats p (LeafDomain right) | |
| (drawIndex repeats p call)), | |
| TerminalCalls.terminalSpaceEquiv F2 repeats p (LeafDomain left) call | |
| (factorPullback repeats p left right q (drawIndex repeats p call) x) = | |
| HPullback (CutGroupedProjection.cutProjection p q) | |
| (TerminalCalls.terminalSpaceEquiv F2 repeats p (LeafDomain right) call x) := by | |
| induction n generalizing m with | |
| | zero => | |
| have h := p.height_le | |
| omega | |
| | succ n ih => | |
| cases p with | |
| | refl => | |
| intro left right q call x | |
| cases call | |
| rfl | |
| | step i p => | |
| intro left right q call x | |
| exact ih p (childSlots left i) (childSlots right i) | |
| (ChildAssemblyProjection.childProjection q i) call.2 x | |
| theorem exteriorFactor_pullback (repeats : Nat → Nat) (p : Path branch n (m + 1)) : | |
| ∀ (left right : Slots branch n → Fin t → MixedSupport.Slot) | |
| (q : ∀ s k, MixedSupport.Projection (left s k) (right s k)) | |
| (hs : ∀ s, WholeCutExteriorTransport.Outside p s → left s = right s) | |
| (_hk : ∀ s (_hout : WholeCutExteriorTransport.Outside p s) k, | |
| HEq (q s k) (MixedSupport.Projection.keep (left s k))) | |
| (j : DrawIndex repeats p) (hj : ¬ IsTerminal repeats p j) | |
| (x : RecursiveSampler.DrawSpace F2 repeats p (LeafDomain right) j), | |
| factorPullback repeats p left right q j x = | |
| cast (factor_eq repeats p right left | |
| (fun s h => (hs s h).symm) j hj) x := by | |
| induction n generalizing m with | |
| | zero => | |
| have h := p.height_le | |
| omega | |
| | succ n ih => | |
| cases p with | |
| | refl => | |
| intro left right q hs hk j hj x | |
| exact False.elim (hj True.intro) | |
| | step i p => | |
| intro left right q hs hk j hj x | |
| cases j with | |
| | inl j => | |
| exact CutProjectionGeometry.squarePullback_eq_cast | |
| (childSlots left j.val) (childSlots right j.val) | |
| (ChildAssemblyProjection.childProjection q j.val) | |
| (WholeCutExteriorTransport.ordinarySlots_eq i p left right hs j) | |
| (fun s k => hk (j.val, s) (WholeCutExteriorTransport.outside_ordinary i p j s) k) x | |
| | inr j => | |
| exact ih p (childSlots left i) (childSlots right i) | |
| (ChildAssemblyProjection.childProjection q i) | |
| (fun s h => hs (i, s) (WholeCutExteriorTransport.outside_selected i p s h)) | |
| (fun s h k => hk (i, s) (WholeCutExteriorTransport.outside_selected i p s h) k) | |
| j.2 hj x | |
| theorem prefixSplit_tapePullback (repeats : Nat → Nat) (p : Path branch n (m + 1)) | |
| (left right : Slots branch n → Fin t → MixedSupport.Slot) | |
| (q : ∀ s k, MixedSupport.Projection (left s k) (right s k)) | |
| (hs : ∀ s, WholeCutExteriorTransport.Outside p s → left s = right s) | |
| (hk : ∀ s (_hout : WholeCutExteriorTransport.Outside p s) k, | |
| HEq (q s k) (MixedSupport.Projection.keep (left s k))) | |
| (tape : RecursiveSampler.Tape F2 repeats p (LeafDomain right)) : | |
| OriginalScalarReconstruction.prefixSplit F2 repeats p (LeafDomain left) | |
| (tapePullback repeats p left right q tape) = | |
| (terminalPullback repeats p q (OriginalScalarReconstruction.prefixSplit F2 repeats p (LeafDomain right) tape).1, | |
| cast (scalarExterior_eq repeats p right left | |
| (fun s h => (hs s h).symm)) | |
| (OriginalScalarReconstruction.prefixSplit F2 repeats p (LeafDomain right) tape).2) := by | |
| apply Prod.ext | |
| · funext call | |
| exact terminalFactor_pullback repeats p left right q call (tape (drawIndex repeats p call)) | |
| · funext j | |
| exact (exteriorFactor_pullback repeats p left right q hs hk j.val j.property (tape j.val)).trans | |
| (CutSamplerReplayGroupingTransport.cast_pi_apply | |
| (scalarExterior_eq repeats p right left | |
| (fun s h => (hs s h).symm)) | |
| (fun j => factor_eq repeats p right left | |
| (fun s h => (hs s h).symm) j.val j.property) | |
| (OriginalScalarReconstruction.prefixSplit F2 repeats p (LeafDomain right) tape).2 j).symm | |
| theorem reconstruct_pullback_apply (repeats : Nat → Nat) (p : Path branch n (m + 1)) | |
| (left right : Slots branch n → Fin t → MixedSupport.Slot) | |
| (q : ∀ s k, MixedSupport.Projection (left s k) (right s k)) | |
| (hs : ∀ s, WholeCutExteriorTransport.Outside p s → left s = right s) | |
| (hk : ∀ s (_hout : WholeCutExteriorTransport.Outside p s) k, | |
| HEq (q s k) (MixedSupport.Projection.keep (left s k))) | |
| (values : OriginalScalarReconstruction.TerminalValues F2 repeats p (LeafDomain right)) | |
| (exterior : OriginalTerminalSplit.ExteriorTape F2 repeats p (LeafDomain right)) | |
| (x : Domain left) : | |
| (OriginalScalarReconstruction.reconstruct F2 repeats p (LeafDomain left) | |
| (fun call => HPullback (CutGroupedProjection.cutProjection p q) (values call)) | |
| (cast (scalarExterior_eq repeats p right left | |
| (fun s h => (hs s h).symm)) exterior)).val x = | |
| (OriginalScalarReconstruction.reconstruct F2 repeats p (LeafDomain right) | |
| values exterior).val (sourceProjection q x) := by | |
| let tape := (OriginalScalarReconstruction.prefixSplit F2 repeats p (LeafDomain right)).symm | |
| (values, exterior) | |
| have hsplit : | |
| OriginalScalarReconstruction.prefixSplit F2 repeats p (LeafDomain left) | |
| (tapePullback repeats p left right q tape) = | |
| (terminalPullback repeats p q values, | |
| cast (scalarExterior_eq repeats p right left | |
| (fun s h => (hs s h).symm)) exterior) := by | |
| simpa only [tape, Equiv.apply_symm_apply] using | |
| prefixSplit_tapePullback repeats p left right q hs hk tape | |
| have hleft := congrArg | |
| (OriginalScalarReconstruction.prefixSplit F2 repeats p (LeafDomain left)).symm hsplit | |
| simp only [Equiv.symm_apply_apply] at hleft | |
| unfold OriginalScalarReconstruction.reconstruct | |
| change (RecursiveSampler.evaluate F2 repeats p (LeafDomain left) | |
| ((OriginalScalarReconstruction.prefixSplit F2 repeats p (LeafDomain left)).symm | |
| (terminalPullback repeats p q values, | |
| cast (scalarExterior_eq repeats p right left | |
| (fun s h => (hs s h).symm)) exterior))).val x = | |
| (RecursiveSampler.evaluate F2 repeats p (LeafDomain right) tape).val (sourceProjection q x) | |
| rw [← hleft] | |
| exact evaluate_tapePullback repeats p left right q tape x | |
| theorem reconstruct_pullback (repeats : Nat → Nat) (p : Path branch n (m + 1)) | |
| (left right : Slots branch n → Fin t → MixedSupport.Slot) | |
| (q : ∀ s k, MixedSupport.Projection (left s k) (right s k)) | |
| (hs : ∀ s, WholeCutExteriorTransport.Outside p s → left s = right s) | |
| (hk : ∀ s (_hout : WholeCutExteriorTransport.Outside p s) k, | |
| HEq (q s k) (MixedSupport.Projection.keep (left s k))) | |
| (values : OriginalScalarReconstruction.TerminalValues F2 repeats p (LeafDomain right)) | |
| (exterior : OriginalTerminalSplit.ExteriorTape F2 repeats p (LeafDomain right)) : | |
| OriginalScalarReconstruction.reconstruct F2 repeats p (LeafDomain left) | |
| (fun call => HPullback (CutGroupedProjection.cutProjection p q) (values call)) | |
| (cast (scalarExterior_eq repeats p right left | |
| (fun s h => (hs s h).symm)) exterior) = | |
| HPullback q (OriginalScalarReconstruction.reconstruct F2 repeats p (LeafDomain right) | |
| values exterior) := by | |
| apply Subtype.ext | |
| funext x | |
| exact reconstruct_pullback_apply repeats p left right q hs hk values exterior x | |
| end | |
| end PerfectCompleteness.OriginalScalarProjection | |
| end | |
| end OAI | |