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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
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