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import OAI.Computability.PerfectCompleteness.Foundations.HierarchicalAdviceExperimentLemmas
import OAI.Computability.PerfectCompleteness.Reduction.FixedPhysicalStoppedComparison
import OAI.Computability.PerfectCompleteness.Reduction.FixedStoppedPair
import OAI.Computability.PerfectCompleteness.Repetition.CutChildCleanLawLemmas
import OAI.Computability.PerfectCompleteness.Sampling.StoppedProjectedGlobalLawLemmas
namespace OAI
section
namespace PerfectCompleteness.FixedStoppedUsefulMass
noncomputable section
open scoped Classical
open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays
open UniqueGamesTheorem.Foundations.Games
private theorem probability_cast_array_law
{branch : Nat → Nat} {n h t : Nat} (rows repeats : Nat → Nat)
(p : Path branch n (h + 1))
{slots slots' : Slots branch n → Fin t → MixedSupport.Slot}
(hs : slots = slots') (event : Arrays slots' rows → Bool) :
(WholeArraySampler.law rows repeats p slots).probability
(fun arrays => event (cast (congrArg (fun s => Arrays s rows) hs) arrays)) =
(WholeArraySampler.law rows repeats p slots').probability event := by
cases hs
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 clauses := PCPSource.clauseFamily (BinaryLanguage.totalRename input)
abbrev Outer := StoppedProjectedExperiment.Outer
(branch := FixedParameters.branch parameters) (n := parameters.plan.depth)
(j := j.val) (t := FixedRows.sourceLength parameters.plan hδ)
(m := (PCPSource.normalizedFormula (BinaryLanguage.totalRename input)).clauses.length)
abbrev Base := StoppedProjectedExperiment.PhysicalBase
(branch := FixedParameters.branch parameters) (n := parameters.plan.depth)
(i := i.val) (j := j.val) (t := FixedRows.sourceLength parameters.plan hδ)
(FixedRows.rows parameters.plan)
def flags : Fin (FixedParameters.branch parameters i.val) → FiniteDistribution Bool :=
fun _ => ProjectionPosterior.bernoulli (FixedParameters.projectionProbability parameters i.val : ℝ)
(FixedPhysicalStoppedComparison.beta_nonneg parameters)
(FixedPhysicalStoppedComparison.beta_le_one parameters)
theorem flags_eq_projectionFlag (hcube : 0 < parameters.cubeSize i.val) :
flags parameters i = fun _ =>
ProjectedCleanRate.projectionFlag (parameters.cubeSize i.val) hcube := by
have hprob : (FixedParameters.projectionProbability parameters i.val : ℝ) =
ProjectedCleanRate.projectionDensity (parameters.cubeSize i.val) := by
simp [FixedParameters.projectionProbability, ProjectedCleanRate.projectionDensity]
funext child
apply FiniteDistribution.eq_of_weight_eq
intro b
cases b <;> simp [flags, ProjectionPosterior.bernoulli,
ProjectedCleanRate.projectionFlag, RepetitionRate.bernoulli, hprob]
def outerLaw : FiniteDistribution (Outer parameters j input) :=
StoppedProjectedExperiment.outerLaw (Nat.succ_le_of_lt j.isLt)
(fun k _ => FixedParameters.branch_pos parameters k)
def baseLaw : FiniteDistribution (Base parameters i j) :=
StoppedProjectedExperiment.physicalBaseLaw (FixedRows.rows parameters.plan) hij
(fun k _ => FixedParameters.branch_pos parameters k)
(fun k => FixedPreliminaryGame.rows_pos parameters (k + 1))
def record (o : Outer parameters j input)
(pref : GeometricCutSplit.Prefix (FixedParameters.branch parameters) (j.val + 1) (i.val + 1))
(directions : CanonicalDirections.Tuple (FixedRows.rows parameters.plan) parameters.plan.depth)
(arrays : Arrays (StoppedProjectedExperiment.nativeSlots (clauses input) o)
(FixedRows.rows parameters.plan)) :
StoppedSharedSampler.Record (clauses input) (FixedParameters.branch parameters)
parameters.plan.depth (FixedRows.sourceLength parameters.plan hδ) i.val
(FixedRows.rows parameters.plan) :=
⟨(o.1, (GeometricPrefixSplit.joinEquiv (Nat.succ_le_succ hij.le)
(Nat.succ_le_of_lt j.isLt) (o.2, pref), directions)), arrays⟩
variable (designated : Fin (FixedParameters.branch parameters i.val) →
Slots (FixedParameters.branch parameters) i.val)
abbrev PhysicalSample := StoppedProjectedPhysicalLaw.Sample
(t := FixedRows.sourceLength parameters.plan hδ)
(clauses input) (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij designated
abbrev OriginalSample := StoppedProjectedGlobalLaw.ExperimentSample
(t := FixedRows.sourceLength parameters.plan hδ)
(clauses input) (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij designated
def physicalLaw : FiniteDistribution (PhysicalSample parameters i j hij input designated) :=
StoppedProjectedPhysicalLaw.law
(clauses input) (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij designated
(fun k _ => FixedParameters.branch_pos parameters k)
(fun k => FixedPreliminaryGame.rows_pos parameters (k + 1)) (flags parameters i)
def physicalRecord (sample : PhysicalSample parameters i j hij input designated) :=
record parameters i j hij input sample.1 sample.2.1.1 sample.2.1.2.2
(cast (congrArg (fun slots => Arrays slots (FixedRows.rows parameters.plan))
(StoppedProjectedArrays.nativeSlots_eq (clauses input) (FixedRows.rows parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij designated sample.1 sample.2.1))
(StoppedProjectedArrays.physicalArrays (clauses input) (FixedRows.rows parameters.plan)
(FixedRows.repeats parameters.plan) (Nat.succ_le_of_lt j.isLt) hij designated
sample.1 sample.2))
def referenceMass (event : StoppedSharedSampler.Record (clauses input)
(FixedParameters.branch parameters) parameters.plan.depth
(FixedRows.sourceLength parameters.plan hδ) i.val (FixedRows.rows parameters.plan) → Bool) : ℝ :=
(outerLaw parameters j input).expectation (fun o =>
(baseLaw parameters i j hij).expectation (fun base =>
(WholeArraySampler.law (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan)
(StoppedProjectedExperiment.stoppedPath (FixedRows.rows parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij o
(StoppedProjectedExperiment.baseTag (FixedRows.rows parameters.plan) base))
(StoppedProjectedExperiment.nativeSlots (clauses input) o)).probability
(fun arrays => event (record parameters i j hij input o base.1 base.2.2 arrays))))
theorem physical_fiber_probability_lower
(o : Outer parameters j input) (base : Base parameters i j)
(event : StoppedSharedSampler.Record (clauses input) (FixedParameters.branch parameters)
parameters.plan.depth (FixedRows.sourceLength parameters.plan hδ) i.val
(FixedRows.rows parameters.plan) → Bool) :
(WholeArraySampler.law (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan)
(StoppedProjectedExperiment.stoppedPath (FixedRows.rows parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij o
(StoppedProjectedExperiment.baseTag (FixedRows.rows parameters.plan) base))
(StoppedProjectedExperiment.nativeSlots (clauses input) o)).probability
(fun arrays => event (record parameters i j hij input o base.1 base.2.2 arrays)) -
parameters.accuracy ≤
(StoppedProjectedExperiment.physicalFiberLaw (clauses input) (FixedRows.rows parameters.plan)
(FixedRows.repeats parameters.plan) (Nat.succ_le_of_lt j.isLt) hij designated
(flags parameters i) o base).probability
(fun raw => event (physicalRecord parameters i j hij input designated ⟨o, ⟨base, raw⟩⟩)) := by
let p := StoppedProjectedExperiment.stoppedPath (FixedRows.rows parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij o
(StoppedProjectedExperiment.baseTag (FixedRows.rows parameters.plan) base)
let outside := StoppedProjectedExperiment.nativeSlots (clauses input) o
let placeholder := SourceProjectedTag.originalSlots (clauses input)
(StoppedProjectedExperiment.cutQuestions (FixedRows.rows parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij o
(StoppedProjectedExperiment.baseTag (FixedRows.rows parameters.plan) base))
let sources := StoppedProjectedExperiment.physicalSources (FixedRows.rows parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij designated o base
have hslots : CutSlotAssembly.fill p outside
(SourceChildKernel.parentLeftSlots (clauses input) designated sources) = outside := by
simpa only [p, outside, sources, SourcePhysicalTapeLaw.nativeSlots] using
(StoppedProjectedArrays.nativeSlots_eq (clauses input) (FixedRows.rows parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij designated o base)
let observed := fun arrays : SourcePhysicalStoppedComparison.NativeArrays
(FixedRows.rows parameters.plan) p outside (clauses input) designated sources =>
event (record parameters i j hij input o base.1 base.2.2
(cast (congrArg (fun slots => Arrays slots (FixedRows.rows parameters.plan)) hslots) arrays))
have htv := SourcePhysicalStoppedComparison.conditional_variation
(FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan) p outside placeholder
(clauses input) designated (FixedParameters.branch_pos parameters i.val) sources
(FixedParameters.projectionProbability parameters i.val : ℝ)
(FixedPhysicalStoppedComparison.beta_nonneg parameters)
(FixedPhysicalStoppedComparison.beta_le_one parameters)
have herror : SourceChildProjectionComparison.error (FixedParameters.branch parameters)
i.val (FixedRows.sourceLength parameters.plan hδ)
(Fintype.card (WholeCutCalls.Index (FixedRows.rows parameters.plan)
(FixedRows.repeats parameters.plan) p)) (FixedRows.rows parameters.plan)
(FixedParameters.projectionProbability parameters i.val : ℝ) < parameters.accuracy := by
simpa only [SourceChildProjectionComparison.error] using
FixedProjectionError.sparse_error_lt parameters
(FixedPhysicalStoppedComparison.calls_le parameters p (le_refl _)) i.val
have hp := FiniteDistribution.probability_le_add_totalVariation
(SourcePhysicalStoppedComparison.stoppedLaw (FixedRows.rows parameters.plan)
(FixedRows.repeats parameters.plan) p outside (clauses input) designated sources)
(SourcePhysicalStoppedComparison.modifiedLaw (FixedRows.rows parameters.plan)
(FixedRows.repeats parameters.plan) p outside placeholder (clauses input) designated
(FixedParameters.projectionProbability parameters i.val : ℝ)
(FixedPhysicalStoppedComparison.beta_nonneg parameters)
(FixedPhysicalStoppedComparison.beta_le_one parameters) sources) observed
rw [FiniteDistribution.totalVariation_comm] at hp
have hreference :
(SourcePhysicalStoppedComparison.stoppedLaw (FixedRows.rows parameters.plan)
(FixedRows.repeats parameters.plan) p outside (clauses input) designated sources).probability
observed =
(WholeArraySampler.law (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan)
p outside).probability
(fun arrays => event (record parameters i j hij input o base.1 base.2.2 arrays)) := by
simpa only [SourcePhysicalStoppedComparison.stoppedLaw, observed] using
(probability_cast_array_law (FixedRows.rows parameters.plan)
(FixedRows.repeats parameters.plan) p
(slots := CutSlotAssembly.fill p outside
(SourceChildKernel.parentLeftSlots (clauses input) designated sources))
(slots' := outside) hslots
(fun arrays => event (record parameters i j hij input o base.1 base.2.2 arrays)))
have hstrict := htv.trans_lt herror
have hbound :
(SourcePhysicalStoppedComparison.stoppedLaw (FixedRows.rows parameters.plan)
(FixedRows.repeats parameters.plan) p outside (clauses input) designated sources).probability
observed - parameters.accuracy ≤
(SourcePhysicalStoppedComparison.modifiedLaw (FixedRows.rows parameters.plan)
(FixedRows.repeats parameters.plan) p outside placeholder (clauses input) designated
(FixedParameters.projectionProbability parameters i.val : ℝ)
(FixedPhysicalStoppedComparison.beta_nonneg parameters)
(FixedPhysicalStoppedComparison.beta_le_one parameters) sources).probability observed :=
sub_le_iff_le_add.mpr (hp.trans (add_le_add (le_refl _) hstrict.le))
rw [hreference, SourcePhysicalStoppedComparison.modifiedLaw,
FiniteDistribution.probability_pushforward] at hbound
change (WholeArraySampler.law (FixedRows.rows parameters.plan)
(FixedRows.repeats parameters.plan) p outside).probability
(fun arrays => event (record parameters i j hij input o base.1 base.2.2 arrays)) -
parameters.accuracy ≤
(SourcePhysicalWholeLaw.rawLaw (FixedRows.rows parameters.plan)
(FixedRows.repeats parameters.plan) p outside placeholder (clauses input) designated
(fun _ => ProjectionPosterior.bernoulli
(FixedParameters.projectionProbability parameters i.val : ℝ)
(FixedPhysicalStoppedComparison.beta_nonneg parameters)
(FixedPhysicalStoppedComparison.beta_le_one parameters)) sources).probability
(fun raw => observed (SourcePhysicalWholeLaw.observeLeft (FixedRows.rows parameters.plan)
(FixedRows.repeats parameters.plan) p outside placeholder (clauses input) designated sources raw))
exact hbound
theorem physical_probability_lower
(event : StoppedSharedSampler.Record (clauses input) (FixedParameters.branch parameters)
parameters.plan.depth (FixedRows.sourceLength parameters.plan hδ) i.val
(FixedRows.rows parameters.plan) → Bool) :
referenceMass parameters i j hij input event - parameters.accuracy ≤
(physicalLaw parameters i j hij input designated).probability
(fun sample => event (physicalRecord parameters i j hij input designated sample)) := by
have hmean := SmallBias.expectation_mono (outerLaw parameters j input) (fun o =>
SmallBias.expectation_mono (baseLaw parameters i j hij) (fun base =>
physical_fiber_probability_lower parameters i j hij input designated o base event))
simp only [DensityVariation.expectation_sub, SmallBias.expectation_const] at hmean
simpa only [referenceMass, physicalLaw, StoppedProjectedPhysicalLaw.law,
StoppedProjectedExperiment.physicalLaw, CompletionSoundness.sigmaLaw_probability,
outerLaw, baseLaw] using hmean
theorem referenceMass_eq_stopped
(event : StoppedSharedSampler.Record (clauses input)
(FixedParameters.branch parameters) parameters.plan.depth
(FixedRows.sourceLength parameters.plan hδ) i.val
(FixedRows.rows parameters.plan) → Bool) :
referenceMass parameters i j hij input event =
(StoppedSharedSampler.stoppedLaw (clauses input)
(FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan)
(Nat.succ_le_of_lt i.isLt)
(fun k _ => FixedParameters.branch_pos parameters k)
(fun k => FixedPreliminaryGame.rows_pos parameters (k + 1))).probability event := by
simp only [referenceMass, outerLaw, baseLaw,
StoppedProjectedExperiment.outerLaw, StoppedProjectedExperiment.physicalBaseLaw,
StoppedSharedSampler.stoppedLaw, CompletionSoundness.sigmaLaw_probability,
StoppedSharedSampler.contextLaw, FiniteDistribution.expectation_product]
apply FiniteDistribution.expectation_congr
intro questions
simp only [record, StoppedProjectedExperiment.stoppedPath,
StoppedProjectedExperiment.upperPath, StoppedProjectedExperiment.lowerPath,
StoppedProjectedExperiment.baseTag, StoppedProjectedExperiment.withFlags,
StoppedProjectedExperiment.nativeSlots, SmallBias.expectation_const]
let value := fun
(upper : GeometricCutSplit.Prefix (FixedParameters.branch parameters)
parameters.plan.depth (j.val + 1))
(lower : GeometricCutSplit.Prefix (FixedParameters.branch parameters)
(j.val + 1) (i.val + 1)) =>
(CanonicalDirections.law (FixedRows.rows parameters.plan) parameters.plan.depth
(fun k => FixedPreliminaryGame.rows_pos parameters (k + 1))).expectation
(fun directions =>
(WholeArraySampler.law (FixedRows.rows parameters.plan)
(FixedRows.repeats parameters.plan)
((GeometricCutSplit.prefixPath (Nat.succ_le_of_lt j.isLt) upper).append
(GeometricCutSplit.prefixPath (Nat.succ_le_succ hij.le) lower))
(sourceSlots (clauses input) (PreliminarySampler.endpoints questions))).probability
(fun arrays => event
⟨(questions, (GeometricPrefixSplit.joinEquiv (Nat.succ_le_succ hij.le)
(Nat.succ_le_of_lt j.isLt) (upper, lower), directions)), arrays⟩))
calc
_ = (GeometricCutSplit.prefixLaw (Nat.succ_le_of_lt i.isLt)
(fun k _ => FixedParameters.branch_pos parameters k)).expectation
(fun pref => value
(GeometricPrefixSplit.splitEquiv (Nat.succ_le_succ hij.le)
(Nat.succ_le_of_lt j.isLt) pref).1
(GeometricPrefixSplit.splitEquiv (Nat.succ_le_succ hij.le)
(Nat.succ_le_of_lt j.isLt) pref).2) :=
(GeometricPrefixSplit.expectation_split (Nat.succ_le_succ hij.le)
(Nat.succ_le_of_lt j.isLt)
(fun k _ => FixedParameters.branch_pos parameters k) value).symm
_ = _ := by
apply FiniteDistribution.expectation_congr
intro pref
dsimp only [value]
rw [← GeometricPrefixSplit.prefixPath_split (Nat.succ_le_succ hij.le)
(Nat.succ_le_of_lt j.isLt) pref, GeometricPrefixSplit.join_split]
theorem stopped_probability_lower
(event : StoppedSharedSampler.Record (clauses input) (FixedParameters.branch parameters)
parameters.plan.depth (FixedRows.sourceLength parameters.plan hδ) i.val
(FixedRows.rows parameters.plan) → Bool) :
(StoppedSharedSampler.stoppedLaw (clauses input)
(FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan)
(Nat.succ_le_of_lt i.isLt) (fun k _ => FixedParameters.branch_pos parameters k)
(fun k => FixedPreliminaryGame.rows_pos parameters (k + 1))).probability event -
parameters.accuracy ≤
(physicalLaw parameters i j hij input designated).probability
(fun sample => event (physicalRecord parameters i j hij input designated sample)) := by
rw [← referenceMass_eq_stopped parameters i j hij input event]
exact physical_probability_lower parameters i j hij input designated event
variable (strategy : FixedPreliminaryGame.Strategy parameters input)
abbrev canonicalStrategy := PreliminaryStrategy.extend (clauses input)
(FixedParameters.branch parameters) parameters.plan.depth
(FixedRows.sourceLength parameters.plan hδ) (FixedRows.rows parameters.plan)
(FixedRows.repeats parameters.plan) strategy
abbrev experiment (o : Outer parameters j input) :=
StoppedProjectedGlobalLaw.experiment (clauses input) (FixedRows.rows parameters.plan)
(FixedRows.repeats parameters.plan) (Nat.succ_le_of_lt j.isLt) hij designated
(fun k _ => FixedParameters.branch_pos parameters k)
(fun k => FixedPreliminaryGame.rows_pos parameters (k + 1)) (flags parameters i)
(canonicalStrategy parameters input strategy) o
abbrev OriginalFiber (o : Outer parameters j input) :=
HierarchicalProjectedExperiment.Sample (rows := FixedRows.rows parameters.plan)
(repeats := FixedRows.repeats parameters.plan)
(StoppedProjectedExperiment.projectedSlots (clauses input) (FixedRows.rows parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij designated o)
(StoppedProjectedExperiment.upper (Nat.succ_le_of_lt j.isLt) o)
(StoppedProjectedExperiment.lower (FixedRows.rows parameters.plan) (Nat.succ_le_of_lt j.isLt) hij o)
(StoppedProjectedExperiment.cut (FixedRows.rows parameters.plan) (Nat.succ_le_of_lt j.isLt) hij o)
def originalLaw : FiniteDistribution (OriginalSample parameters i j hij input designated) :=
StoppedProjectedGlobalLaw.experimentLaw (clauses input) (FixedRows.rows parameters.plan)
(FixedRows.repeats parameters.plan) (Nat.succ_le_of_lt j.isLt) hij designated
(fun k _ => FixedParameters.branch_pos parameters k)
(fun k => FixedPreliminaryGame.rows_pos parameters (k + 1)) (flags parameters i)
(canonicalStrategy parameters input strategy)
def originalRecord (sample : OriginalSample parameters i j hij input designated) :=
record parameters i j hij input sample.1 sample.2.1.1 sample.2.1.2.2
((experiment parameters i j hij input designated strategy sample.1).arrays sample.2)
def upperEvent (o : Outer parameters j input)
(x : OriginalFiber parameters i j hij input designated o) : Bool :=
PrefixTests.observeAtLevelNode (clauses input) (canonicalStrategy parameters input strategy)
o.1 ((experiment parameters i j hij input designated strategy o).arrays x) j
⟨StoppedProjectedExperiment.upper (Nat.succ_le_of_lt j.isLt) o,
StoppedProjectedExperiment.upper_height (Nat.succ_le_of_lt j.isLt) o⟩
(x.1.2.2 j)
theorem lowerEvent_eq_win (o : Outer parameters j input)
(x : OriginalFiber parameters i j hij input designated o) :
HierarchicalAdviceExperiment.lowerEvent
(experiment parameters i j hij input designated strategy o) x =
StoppedSharedEvents.win (clauses input) strategy (Nat.succ_le_of_lt i.isLt)
(fun k _ => FixedParameters.branch_pos parameters k) i
(originalRecord parameters i j hij input designated strategy ⟨o, x⟩) := by
symm
unfold StoppedSharedEvents.win StoppedSharedEvents.representativeLeaf originalRecord record
dsimp only
rw [GeometricPrefixSplit.prefixPath_join]
exact (PrefixMatrixAcceptance.observe_endpoint (clauses input)
(canonicalStrategy parameters input strategy) o.1
(StoppedProjectedExperiment.stoppedPath (FixedRows.rows parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij o x.1)
(StoppedSharedEvents.referenceSuffix (Nat.succ_le_of_lt i.isLt)
(fun k _ => FixedParameters.branch_pos parameters k))
((experiment parameters i j hij input designated strategy o).arrays x)
(x.1.2.2 i)).trans
(PrefixMatrixAcceptance.observe_eq_lower (clauses input)
(canonicalStrategy parameters input strategy) o.1
((experiment parameters i j hij input designated strategy o).arrays x) i
⟨StoppedProjectedExperiment.lower (FixedRows.rows parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij o x.1,
StoppedProjectedExperiment.lower_height (FixedRows.rows parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij o x.1⟩ (x.1.2.2 i)
(StoppedProjectedExperiment.upper (Nat.succ_le_of_lt j.isLt) o))
theorem upperEvent_eq_win (o : Outer parameters j input)
(x : OriginalFiber parameters i j hij input designated o) :
upperEvent parameters i j hij input designated strategy o x =
StoppedSharedEvents.win (clauses input) strategy (Nat.succ_le_of_lt i.isLt)
(fun k _ => FixedParameters.branch_pos parameters k) j
(originalRecord parameters i j hij input designated strategy ⟨o, x⟩) := by
symm
unfold StoppedSharedEvents.win StoppedSharedEvents.representativeLeaf originalRecord record
dsimp only
rw [GeometricPrefixSplit.prefixPath_join, Path.slotEmbedding_append]
exact PrefixMatrixAcceptance.observe_endpoint (clauses input)
(canonicalStrategy parameters input strategy) o.1
(StoppedProjectedExperiment.upperPath (Nat.succ_le_of_lt j.isLt) o)
((StoppedProjectedExperiment.lowerPath (FixedRows.rows parameters.plan) hij x.1).slotEmbedding
(StoppedSharedEvents.referenceSuffix (Nat.succ_le_of_lt i.isLt)
(fun k _ => FixedParameters.branch_pos parameters k)))
((experiment parameters i j hij input designated strategy o).arrays x) (x.1.2.2 j)
theorem originalRecord_physicalRead
(sample : PhysicalSample parameters i j hij input designated) :
originalRecord parameters i j hij input designated strategy
(StoppedProjectedPhysicalLaw.read (clauses input) (FixedRows.rows parameters.plan)
(FixedRows.repeats parameters.plan) (Nat.succ_le_of_lt j.isLt) hij designated sample) =
physicalRecord parameters i j hij input designated sample := by
exact congrArg (record parameters i j hij input sample.1 sample.2.1.1 sample.2.1.2.2)
(StoppedProjectedArrays.arrays_physicalExperimentRead_eq (clauses input)
(FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij designated
(fun k _ => FixedParameters.branch_pos parameters k)
(fun k => FixedPreliminaryGame.rows_pos parameters (k + 1)) (flags parameters i)
(canonicalStrategy parameters input strategy) sample.1 sample.2)
theorem joint_probability_eq_physical :
(originalLaw parameters i j hij input designated strategy).probability
(HierarchicalUsefulFamily.jointEvent
(experiment parameters i j hij input designated strategy)
(upperEvent parameters i j hij input designated strategy)) =
(physicalLaw parameters i j hij input designated).probability (fun sample =>
StoppedSharedEvents.pair (clauses input) strategy (Nat.succ_le_of_lt i.isLt)
(fun k _ => FixedParameters.branch_pos parameters k) i j
(physicalRecord parameters i j hij input designated sample)) := by
have hevent : HierarchicalUsefulFamily.jointEvent
(experiment parameters i j hij input designated strategy)
(upperEvent parameters i j hij input designated strategy) =
(fun sample => StoppedSharedEvents.pair (clauses input) strategy
(Nat.succ_le_of_lt i.isLt) (fun k _ => FixedParameters.branch_pos parameters k) i j
(originalRecord parameters i j hij input designated strategy sample)) := by
funext sample
exact congrArg₂ Bool.and
(lowerEvent_eq_win parameters i j hij input designated strategy sample.1 sample.2)
(upperEvent_eq_win parameters i j hij input designated strategy sample.1 sample.2)
rw [hevent]
have hread := StoppedProjectedPhysicalLaw.probability_read (clauses input)
(FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan)
(Nat.succ_le_of_lt j.isLt) hij designated
(fun k _ => FixedParameters.branch_pos parameters k)
(fun k => FixedPreliminaryGame.rows_pos parameters (k + 1)) (flags parameters i)
(canonicalStrategy parameters input strategy)
(fun sample => StoppedSharedEvents.pair (clauses input) strategy
(Nat.succ_le_of_lt i.isLt) (fun k _ => FixedParameters.branch_pos parameters k) i j
(originalRecord parameters i j hij input designated strategy sample))
simpa only [originalRecord_physicalRead, originalLaw, physicalLaw] using hread.symm
theorem joint_probability_ge_of_reserve
(hreserve : InitialParameters.simultaneous δ + 2 * parameters.accuracy ≤
(StoppedSharedSampler.stoppedLaw (clauses input)
(FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan)
(Nat.succ_le_of_lt i.isLt) (fun k _ => FixedParameters.branch_pos parameters k)
(fun k => FixedPreliminaryGame.rows_pos parameters (k + 1))).probability
(StoppedSharedEvents.pair (clauses input) strategy (Nat.succ_le_of_lt i.isLt)
(fun k _ => FixedParameters.branch_pos parameters k) i j)) :
InitialParameters.simultaneous δ ≤
(originalLaw parameters i j hij input designated strategy).probability
(HierarchicalUsefulFamily.jointEvent
(experiment parameters i j hij input designated strategy)
(upperEvent parameters i j hij input designated strategy)) := by
rw [joint_probability_eq_physical]
have hcompare := stopped_probability_lower parameters i j hij input designated
(StoppedSharedEvents.pair (clauses input) strategy (Nat.succ_le_of_lt i.isLt)
(fun k _ => FixedParameters.branch_pos parameters k) i j)
have haccuracy := parameters.accuracy_pos
linarith
theorem useful_probability_ge_of_reserve
(hreserve : InitialParameters.simultaneous δ + 2 * parameters.accuracy ≤
(StoppedSharedSampler.stoppedLaw (clauses input)
(FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan)
(Nat.succ_le_of_lt i.isLt) (fun k _ => FixedParameters.branch_pos parameters k)
(fun k => FixedPreliminaryGame.rows_pos parameters (k + 1))).probability
(StoppedSharedEvents.pair (clauses input) strategy (Nat.succ_le_of_lt i.isLt)
(fun k _ => FixedParameters.branch_pos parameters k) i j)) :
InitialParameters.simultaneous δ / 2 ≤
(originalLaw parameters i j hij input designated strategy).probability
(HierarchicalUsefulFamily.usefulEvent
(experiment parameters i j hij input designated strategy) (InitialParameters.useful δ)
(upperEvent parameters i j hij input designated strategy)) := by
exact HierarchicalUsefulFamily.useful_probability_ge_half
(experiment parameters i j hij input designated strategy) (outerLaw parameters j input)
(upperEvent parameters i j hij input designated strategy) (InitialParameters.simultaneous δ)
(InitialParameters.simultaneous_pos hδ).le
(joint_probability_ge_of_reserve parameters i j hij input designated strategy hreserve)
end
end PerfectCompleteness.FixedStoppedUsefulMass
end
end OAI
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