openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Decoding /FixedStoppedNativeProjection.lean
| import OAI.Computability.PerfectCompleteness.Decoding.CleanPhysicalDecoderLaw | |
| import OAI.Computability.PerfectCompleteness.Decoding.CutProjectionGeometry | |
| import OAI.Computability.PerfectCompleteness.Foundations.StoppedProjectedArrays | |
| import OAI.Computability.PerfectCompleteness.Sampling.FixedStoppedPhysicalLawLemmas | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.FixedStoppedNativeProjection | |
| noncomputable section | |
| open scoped Classical | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| open FixedStoppedPhysicalLaw (clauses designated) | |
| section PathTransport | |
| variable {branch : Nat → Nat} {root h t v m : Nat} [NeZero m] | |
| (rows repeats : Nat → Nat) | |
| (outside : Slots branch root → Fin t → MixedSupport.Slot) | |
| (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (designated : Fin (branch h) → Slots branch h) | |
| {p q : Path branch root (h + 1)} | |
| omit [NeZero m] in | |
| theorem leftSlots_sourceSampleEquiv (hp : p = q) | |
| (x : SourceChildKernel.Sample (C := WholeCutCalls.Index rows repeats p) (t := t) | |
| rows clauses designated) : | |
| CleanPhysicalCanonicalQuery.leftSlots rows repeats p outside clauses designated x = | |
| CleanPhysicalCanonicalQuery.leftSlots rows repeats q outside clauses designated | |
| (FixedStoppedNativeLaw.sourceSampleEquiv rows repeats clauses designated hp x) := by | |
| cases hp | |
| rfl | |
| omit [NeZero m] in | |
| theorem rightSlots_sourceSampleEquiv (hp : p = q) | |
| (x : SourceChildKernel.Sample (C := WholeCutCalls.Index rows repeats p) (t := t) | |
| rows clauses designated) : | |
| CleanPhysicalCanonicalQuery.rightSlots rows repeats p outside clauses designated x = | |
| CleanPhysicalCanonicalQuery.rightSlots rows repeats q outside clauses designated | |
| (FixedStoppedNativeLaw.sourceSampleEquiv rows repeats clauses designated hp x) := by | |
| cases hp | |
| rfl | |
| omit [NeZero m] in | |
| theorem projection_sourceSampleEquiv_heq (hp : p = q) | |
| (x : SourceChildKernel.Sample (C := WholeCutCalls.Index rows repeats p) (t := t) | |
| rows clauses designated) (leaf : Slots branch root) (coordinate : Fin t) : | |
| HEq (CleanPhysicalCanonicalQuery.projection rows repeats p outside clauses designated x | |
| leaf coordinate) | |
| (CleanPhysicalCanonicalQuery.projection rows repeats q outside clauses designated | |
| (FixedStoppedNativeLaw.sourceSampleEquiv rows repeats clauses designated hp x) | |
| leaf coordinate) := by | |
| cases hp | |
| rfl | |
| end PathTransport | |
| theorem fillProjection_outsidePresentation_heq | |
| {branch : Nat → Nat} {root h t : Nat} (p : Path branch root h) | |
| (outside otherOutside : Slots branch root → Fin t → MixedSupport.Slot) | |
| (left right : Slots branch h → Fin t → MixedSupport.Slot) | |
| (q : ∀ s k, MixedSupport.Projection (left s k) (right s k)) | |
| (hfillLeft : CutSlotAssembly.fill p outside left = | |
| CutSlotAssembly.fill p otherOutside left) (leaf : Slots branch root) (a : Fin t) : | |
| HEq (CutProjectionAssembly.fillProjection p outside left right q leaf a) | |
| (CutProjectionAssembly.fillProjection p otherOutside left right q leaf a) := by | |
| by_cases hin : leaf ∈ Set.range p.slotEmbedding | |
| · obtain ⟨s, rfl⟩ := hin | |
| exact (CutProjectionAssembly.fillProjection_at_cut p outside left right q s a).trans | |
| (CutProjectionAssembly.fillProjection_at_cut p otherOutside left right q s a).symm | |
| · have hkeep (s u : MixedSupport.Slot) (hsu : s = u) : | |
| HEq (MixedSupport.Projection.keep s) (MixedSupport.Projection.keep u) := by | |
| cases hsu | |
| rfl | |
| exact (CutProjectionGeometry.fillProjection_outside p outside left right q leaf hin a).trans | |
| ((hkeep _ _ (congrFun (congrFun hfillLeft leaf) a)).trans | |
| (CutProjectionGeometry.fillProjection_outside p otherOutside left right q leaf hin a).symm) | |
| private theorem fullProjection_sources_heq | |
| {branch : Nat → Nat} {root h t v m : Nat} [NeZero m] | |
| (p : Path branch root (h + 1)) | |
| (outside : Slots branch root → Fin t → MixedSupport.Slot) | |
| (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (designated : Fin (branch h) → Slots branch h) | |
| {sources other : SourceChildKernel.Sources (m := m) (t := t) designated} | |
| (hsources : sources = other) (flags : Fin (branch h) → Bool) | |
| (leaf : Slots branch root) (a : Fin t) : | |
| HEq (SourcePhysicalTapeLaw.fullProjection p outside clauses designated (sources, flags) leaf a) | |
| (SourcePhysicalTapeLaw.fullProjection p outside clauses designated (other, flags) leaf a) := by | |
| cases hsources | |
| rfl | |
| private theorem projection_function_heq | |
| {branch : Nat → Nat} {n t : Nat} | |
| {left left' right right' : Slots branch n → Fin t → MixedSupport.Slot} | |
| (hl : left = left') (hr : right = right') | |
| (p : ∀ leaf a, MixedSupport.Projection (left leaf a) (right leaf a)) | |
| (q : ∀ leaf a, MixedSupport.Projection (left' leaf a) (right' leaf a)) | |
| (hpq : ∀ leaf a, HEq (p leaf a) (q leaf a)) : HEq p q := by | |
| cases hl | |
| cases hr | |
| exact heq_of_eq (funext (fun leaf => funext (fun a => eq_of_heq (hpq leaf a)))) | |
| private theorem projectedSlots_physicalRead_eq | |
| {branch : Nat → Nat} {n i j t v m : Nat} [NeZero m] | |
| (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (rows repeats : Nat → Nat) (hupper : j + 1 ≤ n) (hij : i < j) | |
| (designated : Fin (branch i) → Slots branch i) | |
| (o : StoppedProjectedExperiment.Outer | |
| (branch := branch) (n := n) (j := j) (t := t) (m := m)) | |
| (sample : StoppedProjectedExperiment.PhysicalSample | |
| clauses rows repeats hupper hij designated o) : | |
| StoppedProjectedExperiment.projectedSlots clauses rows hupper hij designated o | |
| (StoppedProjectedExperiment.physicalRead clauses rows repeats hupper hij designated o sample).1 = | |
| SourcePhysicalTapeLaw.projectedSlots | |
| (StoppedProjectedExperiment.stoppedPath rows hupper hij o | |
| (StoppedProjectedExperiment.baseTag rows sample.1)) | |
| (StoppedProjectedExperiment.nativeSlots clauses o) clauses designated | |
| (StoppedProjectedExperiment.physicalSources rows hupper hij designated o sample.1, | |
| (StoppedProjectedExperiment.physicalRead | |
| clauses rows repeats hupper hij designated o sample).1.2.1.2) := rfl | |
| private theorem projectedSlots_tag_eq_rightSlots | |
| {branch : Nat → Nat} {root h t v m : Nat} [NeZero m] | |
| (rows repeats : Nat → Nat) (p : Path branch root (h + 1)) | |
| (outside : Slots branch root → Fin t → MixedSupport.Slot) | |
| (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (designated : Fin (branch h) → Slots branch h) | |
| (sample : SourceChildKernel.Sample (C := WholeCutCalls.Index rows repeats p) (t := t) | |
| rows clauses designated) : | |
| SourcePhysicalTapeLaw.projectedSlots p outside clauses designated | |
| (SourcePhysicalTapeLaw.tag rows repeats p clauses designated sample) = | |
| CleanPhysicalCanonicalQuery.rightSlots rows repeats p outside clauses designated sample := rfl | |
| variable {δ : ℚ} {hδ : 0 < δ} (parameters : FixedParameters.Parameters δ hδ) | |
| (i j : Fin parameters.plan.depth) (hij : i < j) (input : List Bool) | |
| abbrev physicalSample (sample : FixedStoppedNativeLaw.Sample parameters i j hij input) := | |
| FixedStoppedNativeLaw.read parameters i j hij input sample | |
| abbrev innerTag (sample : FixedStoppedNativeLaw.Sample parameters i j hij input) := | |
| (StoppedProjectedExperiment.physicalRead (clauses input) | |
| (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan) | |
| (Nat.succ_le_of_lt j.isLt) hij (designated parameters i) | |
| (physicalSample parameters i j hij input sample).1 | |
| (physicalSample parameters i j hij input sample).2.1).1 | |
| abbrev originalQuestions (sample : FixedStoppedNativeLaw.Sample parameters i j hij input) := | |
| (physicalSample parameters i j hij input sample).1.1 | |
| abbrev path (sample : FixedStoppedNativeLaw.Sample parameters i j hij input) := | |
| FixedStoppedPhysicalLaw.path parameters i j hij sample.1.1 | |
| abbrev childSample (sample : FixedStoppedNativeLaw.Sample parameters i j hij input) := | |
| FixedStoppedNativeLaw.childRead parameters i j hij input sample.1 sample.2 | |
| abbrev sources (sample : FixedStoppedNativeLaw.Sample parameters i j hij input) := | |
| SourcePhysicalTapeLaw.sourceTag (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan) | |
| (path parameters i j hij input sample) (clauses input) (designated parameters i) | |
| (childSample parameters i j hij input sample) | |
| abbrev flags (sample : FixedStoppedNativeLaw.Sample parameters i j hij input) := | |
| fun child => SourceChildKernel.rawProjected (FixedRows.rows parameters.plan) | |
| (clauses input) (designated parameters i) child | |
| ((childSample parameters i j hij input sample) child).2.2 | |
| theorem physicalSources_eq (sample : FixedStoppedNativeLaw.Sample parameters i j hij input) : | |
| StoppedProjectedExperiment.physicalSources (FixedRows.rows parameters.plan) | |
| (Nat.succ_le_of_lt j.isLt) hij (designated parameters i) | |
| (physicalSample parameters i j hij input sample).1 | |
| (physicalSample parameters i j hij input sample).2.1.1 = | |
| sources parameters i j hij input sample := by | |
| change SourceQuestionKernelJoint.sources (designated parameters i) | |
| (SourceQuestionCutSplit.restrict (path parameters i j hij input sample) | |
| (SourceQuestionCutSplit.join (path parameters i j hij input sample) | |
| (SourceQuestionRawSwap.insideQuestions (FixedRows.rows parameters.plan) | |
| (clauses input) (designated parameters i) (childSample parameters i j hij input sample)) | |
| sample.1.2.1)) | |
| (SourceQuestionRawSwap.positions (FixedRows.rows parameters.plan) | |
| (clauses input) (designated parameters i) (childSample parameters i j hij input sample)) = _ | |
| have hrestrict := SourceQuestionRawSwap.restrict_join (path parameters i j hij input sample) | |
| (SourceQuestionRawSwap.insideQuestions (FixedRows.rows parameters.plan) | |
| (clauses input) (designated parameters i) (childSample parameters i j hij input sample)) | |
| sample.1.2.1 | |
| have hsources := congrArg (fun questions => SourceQuestionKernelJoint.sources | |
| (designated parameters i) questions | |
| (SourceQuestionRawSwap.positions (FixedRows.rows parameters.plan) | |
| (clauses input) (designated parameters i) (childSample parameters i j hij input sample))) hrestrict | |
| exact hsources.trans | |
| (SourceQuestionRawSwap.sources_insideQuestions_positions (FixedRows.rows parameters.plan) | |
| (clauses input) (designated parameters i) (childSample parameters i j hij input sample)) | |
| @[simp] theorem flags_eq (sample : FixedStoppedNativeLaw.Sample parameters i j hij input) : | |
| (innerTag parameters i j hij input sample).2.1.2 = flags parameters i j hij input sample := rfl | |
| theorem outside_eq (sample : FixedStoppedNativeLaw.Sample parameters i j hij input) : | |
| (FixedStoppedNativeLaw.setup parameters i j hij input sample.1).outside = | |
| SourcePhysicalExteriorSwap.canonicalSlots (clauses input) (path parameters i j hij input sample) | |
| (SourceQuestionCutSplit.outside (path parameters i j hij input sample) | |
| (originalQuestions parameters i j hij input sample)) := by | |
| change SourcePhysicalExteriorSwap.canonicalSlots (clauses input) | |
| (path parameters i j hij input sample) sample.1.2.1 = | |
| SourcePhysicalExteriorSwap.canonicalSlots (clauses input) (path parameters i j hij input sample) | |
| (SourceQuestionCutSplit.outside (path parameters i j hij input sample) | |
| (SourceQuestionCutSplit.join (path parameters i j hij input sample) _ sample.1.2.1)) | |
| exact (congrArg (SourcePhysicalExteriorSwap.canonicalSlots (clauses input) | |
| (path parameters i j hij input sample)) | |
| (SourcePhysicalSourceSwap.outside_join (path parameters i j hij input sample) | |
| (SourceQuestionRawSwap.insideQuestions (FixedRows.rows parameters.plan) | |
| (clauses input) (designated parameters i) (childSample parameters i j hij input sample)) | |
| sample.1.2.1)).symm | |
| theorem fill_outside_eq (sample : FixedStoppedNativeLaw.Sample parameters i j hij input) | |
| (inside : Slots (FixedParameters.branch parameters) (i.val + 1) → | |
| Fin (FixedRows.sourceLength parameters.plan hδ) → MixedSupport.Slot) : | |
| CutSlotAssembly.fill (path parameters i j hij input sample) | |
| (FixedStoppedNativeLaw.setup parameters i j hij input sample.1).outside inside = | |
| CutSlotAssembly.fill (path parameters i j hij input sample) | |
| (StoppedProjectedExperiment.nativeSlots (clauses input) | |
| (physicalSample parameters i j hij input sample).1) inside := by | |
| rw [outside_eq] | |
| exact (SourcePhysicalExteriorSwap.fill_eq (clauses input) | |
| (path parameters i j hij input sample) (originalQuestions parameters i j hij input sample) | |
| inside).symm | |
| theorem leftSlots_eq (sample : FixedStoppedNativeLaw.Sample parameters i j hij input) : | |
| CleanNativeReplay.leftSlots (FixedStoppedNativeLaw.setup parameters i j hij input sample.1) | |
| sample.2 = | |
| StoppedProjectedExperiment.nativeSlots (clauses input) | |
| (physicalSample parameters i j hij input sample).1 := by | |
| have hp := leftSlots_sourceSampleEquiv (FixedRows.rows parameters.plan) | |
| (FixedRows.repeats parameters.plan) | |
| (FixedStoppedNativeLaw.setup parameters i j hij input sample.1).outside | |
| (clauses input) (designated parameters i) | |
| (FixedStoppedNativeLaw.path_eq parameters i j hij input sample.1) sample.2 | |
| refine hp.trans ((fill_outside_eq parameters i j hij input sample _).trans ?_) | |
| exact (congrArg (SourcePhysicalTapeLaw.nativeSlots (path parameters i j hij input sample) | |
| (StoppedProjectedExperiment.nativeSlots (clauses input) | |
| (physicalSample parameters i j hij input sample).1) (clauses input) (designated parameters i)) | |
| (physicalSources_eq parameters i j hij input sample)).symm.trans | |
| (StoppedProjectedArrays.nativeSlots_eq (clauses input) (FixedRows.rows parameters.plan) | |
| (Nat.succ_le_of_lt j.isLt) hij (designated parameters i) | |
| (physicalSample parameters i j hij input sample).1 | |
| (physicalSample parameters i j hij input sample).2.1.1) | |
| theorem physicalRightSlots_eq (sample : FixedStoppedNativeLaw.Sample parameters i j hij input) : | |
| StoppedProjectedExperiment.projectedSlots (clauses input) (FixedRows.rows parameters.plan) | |
| (Nat.succ_le_of_lt j.isLt) hij (designated parameters i) | |
| (physicalSample parameters i j hij input sample).1 | |
| (innerTag parameters i j hij input sample) = | |
| CleanPhysicalCanonicalQuery.rightSlots (FixedRows.rows parameters.plan) | |
| (FixedRows.repeats parameters.plan) (path parameters i j hij input sample) | |
| (StoppedProjectedExperiment.nativeSlots (clauses input) | |
| (physicalSample parameters i j hij input sample).1) | |
| (clauses input) (designated parameters i) (childSample parameters i j hij input sample) := by | |
| have hread := projectedSlots_physicalRead_eq (clauses input) (FixedRows.rows parameters.plan) | |
| (FixedRows.repeats parameters.plan) (Nat.succ_le_of_lt j.isLt) hij (designated parameters i) | |
| (physicalSample parameters i j hij input sample).1 | |
| (physicalSample parameters i j hij input sample).2.1 | |
| have htags := congrArg₂ Prod.mk (physicalSources_eq parameters i j hij input sample) | |
| (flags_eq parameters i j hij input sample) | |
| have hslots := congrArg (SourcePhysicalTapeLaw.projectedSlots | |
| (path parameters i j hij input sample) | |
| (StoppedProjectedExperiment.nativeSlots (clauses input) | |
| (physicalSample parameters i j hij input sample).1) (clauses input) (designated parameters i)) htags | |
| have hright := projectedSlots_tag_eq_rightSlots (FixedRows.rows parameters.plan) | |
| (FixedRows.repeats parameters.plan) (path parameters i j hij input sample) | |
| (StoppedProjectedExperiment.nativeSlots (clauses input) | |
| (physicalSample parameters i j hij input sample).1) (clauses input) (designated parameters i) | |
| (childSample parameters i j hij input sample) | |
| exact hread.trans (hslots.trans hright) | |
| theorem rightSlots_eq (sample : FixedStoppedNativeLaw.Sample parameters i j hij input) : | |
| CleanNativeReplay.rightSlots (FixedStoppedNativeLaw.setup parameters i j hij input sample.1) | |
| sample.2 = | |
| StoppedProjectedExperiment.projectedSlots (clauses input) (FixedRows.rows parameters.plan) | |
| (Nat.succ_le_of_lt j.isLt) hij (designated parameters i) | |
| (physicalSample parameters i j hij input sample).1 | |
| (innerTag parameters i j hij input sample) := by | |
| have hp := rightSlots_sourceSampleEquiv (FixedRows.rows parameters.plan) | |
| (FixedRows.repeats parameters.plan) | |
| (FixedStoppedNativeLaw.setup parameters i j hij input sample.1).outside | |
| (clauses input) (designated parameters i) | |
| (FixedStoppedNativeLaw.path_eq parameters i j hij input sample.1) sample.2 | |
| refine hp.trans ((fill_outside_eq parameters i j hij input sample _).trans ?_) | |
| exact (physicalRightSlots_eq parameters i j hij input sample).symm | |
| theorem projection_apply_heq (sample : FixedStoppedNativeLaw.Sample parameters i j hij input) | |
| (leaf : Slots (FixedParameters.branch parameters) parameters.plan.depth) | |
| (a : Fin (FixedRows.sourceLength parameters.plan hδ)) : | |
| HEq (CleanPhysicalDecoderLaw.physicalProjection | |
| (FixedStoppedNativeLaw.setup parameters i j hij input sample.1) sample.2 leaf a) | |
| (StoppedProjectedExperiment.projection (clauses input) (FixedRows.rows parameters.plan) | |
| (Nat.succ_le_of_lt j.isLt) hij (designated parameters i) | |
| (physicalSample parameters i j hij input sample).1 | |
| (innerTag parameters i j hij input sample) leaf a) := by | |
| have hp := projection_sourceSampleEquiv_heq (FixedRows.rows parameters.plan) | |
| (FixedRows.repeats parameters.plan) | |
| (FixedStoppedNativeLaw.setup parameters i j hij input sample.1).outside | |
| (clauses input) (designated parameters i) | |
| (FixedStoppedNativeLaw.path_eq parameters i j hij input sample.1) sample.2 leaf a | |
| have hout := fillProjection_outsidePresentation_heq (path parameters i j hij input sample) | |
| (FixedStoppedNativeLaw.setup parameters i j hij input sample.1).outside | |
| (StoppedProjectedExperiment.nativeSlots (clauses input) | |
| (physicalSample parameters i j hij input sample).1) | |
| (CleanPhysicalReplay.leftInside (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan) | |
| (path parameters i j hij input sample) (clauses input) (designated parameters i) | |
| (childSample parameters i j hij input sample)) | |
| (CleanPhysicalReplay.rightInside (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan) | |
| (path parameters i j hij input sample) (clauses input) (designated parameters i) | |
| (childSample parameters i j hij input sample)) | |
| (SourceChildKernel.parentProjection (FixedRows.rows parameters.plan) (clauses input) | |
| (designated parameters i) (sources parameters i j hij input sample) | |
| (fun child => ((childSample parameters i j hij input sample) child).2.2)) | |
| (fill_outside_eq parameters i j hij input sample _) leaf a | |
| have hs := fullProjection_sources_heq (path parameters i j hij input sample) | |
| (StoppedProjectedExperiment.nativeSlots (clauses input) | |
| (physicalSample parameters i j hij input sample).1) (clauses input) (designated parameters i) | |
| (physicalSources_eq parameters i j hij input sample).symm | |
| (flags parameters i j hij input sample) leaf a | |
| have hphysical := StoppedProjectedArrays.projection_heq (clauses input) | |
| (FixedRows.rows parameters.plan) (Nat.succ_le_of_lt j.isLt) hij (designated parameters i) | |
| (physicalSample parameters i j hij input sample).1 | |
| (physicalSample parameters i j hij input sample).2.1.1 | |
| (flags parameters i j hij input sample) leaf a | |
| exact hp.trans (hout.trans (hs.trans hphysical.symm)) | |
| theorem projection_heq (sample : FixedStoppedNativeLaw.Sample parameters i j hij input) : | |
| HEq (CleanPhysicalDecoderLaw.physicalProjection | |
| (FixedStoppedNativeLaw.setup parameters i j hij input sample.1) sample.2) | |
| (StoppedProjectedExperiment.projection (clauses input) (FixedRows.rows parameters.plan) | |
| (Nat.succ_le_of_lt j.isLt) hij (designated parameters i) | |
| (physicalSample parameters i j hij input sample).1 | |
| (innerTag parameters i j hij input sample)) := | |
| projection_function_heq (leftSlots_eq parameters i j hij input sample) | |
| (rightSlots_eq parameters i j hij input sample) _ _ | |
| (projection_apply_heq parameters i j hij input sample) | |
| end | |
| end PerfectCompleteness.FixedStoppedNativeProjection | |
| end | |
| end OAI | |