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import OAI.Computability.PerfectCompleteness.Machines.SignedRuntimeBounds
import OAI.Computability.PerfectCompleteness.Machines.SignedTupleBodyMachine
import OAI.Computability.PerfectCompleteness.Machines.SourceOccurrenceProducerLemmas
import OAI.Computability.PerfectCompleteness.Reduction.CompletedVisitOrder
import OAI.Computability.UniqueGames.Machines.MachineTupleOdometer
namespace OAI
namespace PerfectCompleteness.ExactTargetOdometer
open UniqueGamesTheorem.Foundations.Complexity
open scoped Classical
noncomputable section
variable {width radix : Nat}
def setCoordinate (digits : Fin width → Fin radix) (depth : Nat) (value : Fin radix) :
Fin width → Fin radix :=
fun i => if i.val = depth then value else digits i
@[simp] theorem setCoordinate_self (digits : Fin width → Fin radix)
(i : Fin width) (value : Fin radix) :
setCoordinate digits i.val value i = value := by
simp [setCoordinate]
theorem setCoordinate_eq_update (digits : Fin width → Fin radix)
(coordinate : Fin width) (value : Fin radix) :
setCoordinate digits coordinate.val value = Function.update digits coordinate value := by
funext i
by_cases same : i = coordinate
· subst i
simp [setCoordinate]
· have different : i.val ≠ coordinate.val := fun h => same (Fin.ext h)
simp [setCoordinate, same, different]
theorem setCoordinate_zero (digits : Fin width → Fin radix) (depth : Nat)
(positive : 0 < radix) (zero : ∀ i, i.val = depth → (digits i).val = 0) :
setCoordinate digits depth ⟨0, positive⟩ = digits := by
funext i
by_cases same : i.val = depth
· simp only [setCoordinate, ite_eq_left same]
apply Fin.ext
exact (zero i same).symm
· simp [setCoordinate, same]
def fillCoordinates (digits : Fin width → Fin radix) {depth : Nat}
(tuple : Fin depth → Fin radix) : Fin width → Fin radix :=
fun i => if h : i.val < depth then tuple ⟨i.val, h⟩ else digits i
@[simp] theorem fillCoordinates_zero (digits : Fin width → Fin radix)
(tuple : Fin 0 → Fin radix) : fillCoordinates digits tuple = digits := by
funext i
simp [fillCoordinates]
@[simp] theorem fillCoordinates_all (digits : Fin width → Fin radix)
(tuple : Fin width → Fin radix) : fillCoordinates digits tuple = tuple := by
funext i
simp [fillCoordinates, i.isLt]
theorem fillCoordinates_snoc (digits : Fin width → Fin radix) {depth : Nat}
(tuple : Fin depth → Fin radix) (value : Fin radix) :
fillCoordinates digits (Fin.snoc tuple value) =
fillCoordinates (setCoordinate digits depth value) tuple := by
funext i
by_cases hi : i.val < depth
· have hi' : i.val < depth + 1 := by omega
simp [fillCoordinates, hi, hi', Fin.snoc]
· by_cases same : i.val = depth
· simp [fillCoordinates, Fin.snoc, setCoordinate, same]
· have hi' : ¬ i.val < depth + 1 := by omega
simp [fillCoordinates, hi, hi', setCoordinate, same]
def visits : Nat → (Fin width → Fin radix) → List (Fin width → Fin radix)
| 0, digits => [digits]
| depth + 1, digits => (List.ofFn (fun value : Fin radix => value)).flatMap
(fun value => visits depth (setCoordinate digits depth value))
theorem visits_eq_tupleOrder (depth : Nat) (digits : Fin width → Fin radix) :
visits depth digits = (MachineTupleOdometer.tupleOrder radix depth).map
(fillCoordinates digits) := by
induction depth generalizing digits with
| zero =>
simp only [visits, MachineTupleOdometer.tupleOrder_dimension_zero,
List.map_cons, List.map_nil, fillCoordinates_zero]
| succ depth ih =>
rw [visits, MachineTupleOdometer.tupleOrder_snoc, List.map_flatMap]
apply List.flatMap_congr
intro value _
rw [ih, List.map_map]
apply List.map_congr_left
intro tuple _
exact (fillCoordinates_snoc digits tuple value).symm
@[simp] theorem visits_all (digits : Fin width → Fin radix) :
visits width digits = MachineTupleOdometer.tupleOrder radix width := by
have identity : fillCoordinates digits =
(id : (Fin width → Fin radix) → (Fin width → Fin radix)) := by
funext tuple
exact fillCoordinates_all digits tuple
rw [visits_eq_tupleOrder, identity, List.map_id]
def foldVisits {P : Type*} (step : (Fin width → Fin radix) → P → P)
(depth : Nat) (digits : Fin width → Fin radix) (initial : P) : P :=
(visits depth digits).foldl (fun data tuple => step tuple data) initial
@[simp] theorem foldVisits_zero {P : Type*} (step : (Fin width → Fin radix) → P → P)
(digits : Fin width → Fin radix) (initial : P) :
foldVisits step 0 digits initial = step digits initial := rfl
def blockPrefix {P : Type*} (step : (Fin width → Fin radix) → P → P)
(depth : Nat) (digits : Fin width → Fin radix) (count : Nat) (initial : P) : P :=
((List.ofFn (fun value : Fin radix => value)).take count).foldl
(fun data value => foldVisits step depth (setCoordinate digits depth value) data) initial
@[simp] theorem blockPrefix_zero {P : Type*} (step : (Fin width → Fin radix) → P → P)
(depth : Nat) (digits : Fin width → Fin radix) (initial : P) :
blockPrefix step depth digits 0 initial = initial := rfl
theorem blockPrefix_succ {P : Type*} (step : (Fin width → Fin radix) → P → P)
(depth : Nat) (digits : Fin width → Fin radix) (value : Fin radix) (initial : P) :
blockPrefix step depth digits (value.val + 1) initial =
foldVisits step depth (setCoordinate digits depth value)
(blockPrefix step depth digits value.val initial) := by
have inside : value.val < (List.ofFn (fun value : Fin radix => value)).length := by
simpa only [List.length_ofFn] using value.isLt
simp only [blockPrefix, List.take_succ_eq_append_getElem inside,
List.foldl_append, List.foldl_cons, List.foldl_nil, List.getElem_ofFn]
@[simp] theorem blockPrefix_all {P : Type*} (step : (Fin width → Fin radix) → P → P)
(depth : Nat) (digits : Fin width → Fin radix) (initial : P) :
blockPrefix step depth digits radix initial = foldVisits step (depth + 1) digits initial := by
have complete : (List.ofFn (fun value : Fin radix => value)).take radix =
List.ofFn (fun value : Fin radix => value) :=
List.take_of_length_le (by simp)
simp only [blockPrefix, complete, foldVisits, visits, List.foldl_flatMap]
variable {branch : Nat → Nat} {n t v m : Nat}
theorem visits_sourceOrder
(digits : Fin (TreeCanonical.locationCount branch n t) → Fin m) :
(visits (TreeCanonical.locationCount branch n t) digits).map
SourceQuestionOrder.tupleToQuestion = SourceVisitOrder.visitOrder branch n t m := by
rw [visits_all, SourceVisitOrder.visitOrder_eq_tupleOrder]
variable (clauses : Fin m → SourceClause.NormalizedClause v) (rows repeats : Nat → Nat)
[NeZero m] (hn : 0 < n) (hbranch : ∀ k < n, 0 < branch k)
(hrows : ∀ k, 0 < rows (k + 1)) (δ : ℚ)
theorem visits_completedOrder
(digits : Fin (TreeCanonical.locationCount branch n t) → Fin m) :
(visits (TreeCanonical.locationCount branch n t) digits).flatMap
(fun tuple => CompletedVisitOrder.localOrder clauses rows repeats hn hbranch hrows δ
(SourceQuestionOrder.tupleToQuestion tuple)) =
CompletedVisitOrder.completedOrder (t := t) clauses rows repeats hn hbranch hrows δ := by
rw [CompletedVisitOrder.completedOrder, ← visits_sourceOrder digits, List.flatMap_map]
end
noncomputable section
variable {width : Nat} {Extra : Type}
def baseTapes (input : NormalizedSourceInput.Input)
(indices : Fin width → Fin (NormalizedSourceInput.clauseCount input))
(extra : Extra → List Bool) : SignedTupleLayout.Tape width Extra → List Bool
| .source .table => NormalizedSourceInput.bits input
| .current i => encodeWord (indices i).val
| .remaining i => encodeWord (NormalizedSourceInput.clauseCount input - 1 - (indices i).val)
| .extra x => extra x
| _ => []
theorem baseTapes_input (input : NormalizedSourceInput.Input)
(indices : Fin width → Fin (NormalizedSourceInput.clauseCount input))
(extra : Extra → List Bool) :
SignedTupleLayout.Input input.formula indices (baseTapes input indices extra) where
toClear := {
rawDigits := fun _ => rfl
rawVariables := fun _ _ => rfl
sourceWork := fun _ => rfl
encodedVariables := fun _ _ => rfl
scratch := rfl
}
table := rfl
current := fun _ => rfl
@[simp] theorem baseTapes_edgeWork (input : NormalizedSourceInput.Input)
(indices : Fin width → Fin (NormalizedSourceInput.clauseCount input))
(extra : Extra → List Bool) (j : Fin 10) :
baseTapes input indices extra (.edgeWork j) = [] := rfl
theorem baseTapes_emptyWork (input : NormalizedSourceInput.Input)
(indices : Fin width → Fin (NormalizedSourceInput.clauseCount input))
(extra : Extra → List Bool) :
∀ j : Fin 10, j ≠ 1 → j ≠ 7 → j ≠ 8 →
baseTapes input indices extra (.edgeWork j) = [] := by
intro j _ _ _
rfl
variable {branch : Nat → Nat} {n t : Nat} [DecidableEq Extra]
local notation "W" => TreeCanonical.locationCount branch n t
def nativeTapes (input : NormalizedSourceInput.Input)
(indices : Fin W → Fin (NormalizedSourceInput.clauseCount input))
(data : SignedTupleBodyMachine.Data W) (extra : Extra → List Bool) :
SignedTupleLayout.Tape W Extra → List Bool :=
SignedTupleBodyMachine.tapes (branch := branch) (n := n) (t := t)
SignedTupleLayout.Tape.edgeWork SignedTupleLayout.Tape.edgeCount
(baseTapes input indices extra) data
@[simp] theorem nativeTapes_table (input : NormalizedSourceInput.Input)
(indices : Fin W → Fin (NormalizedSourceInput.clauseCount input))
(data : SignedTupleBodyMachine.Data W) (extra : Extra → List Bool) :
nativeTapes input indices data extra (.source .table) = NormalizedSourceInput.bits input := by
simp [nativeTapes, SignedTupleBodyMachine.tapes, baseTapes]
@[simp] theorem nativeTapes_current (input : NormalizedSourceInput.Input)
(indices : Fin W → Fin (NormalizedSourceInput.clauseCount input))
(data : SignedTupleBodyMachine.Data W) (extra : Extra → List Bool) (i : Fin W) :
nativeTapes input indices data extra (.current i) = encodeWord (indices i).val := by
simp [nativeTapes, SignedTupleBodyMachine.tapes, baseTapes]
@[simp] theorem nativeTapes_remaining (input : NormalizedSourceInput.Input)
(indices : Fin W → Fin (NormalizedSourceInput.clauseCount input))
(data : SignedTupleBodyMachine.Data W) (extra : Extra → List Bool) (i : Fin W) :
nativeTapes input indices data extra (.remaining i) =
encodeWord (NormalizedSourceInput.clauseCount input - 1 - (indices i).val) := by
simp [nativeTapes, SignedTupleBodyMachine.tapes, baseTapes]
@[simp] theorem nativeTapes_output (input : NormalizedSourceInput.Input)
(indices : Fin W → Fin (NormalizedSourceInput.clauseCount input))
(data : SignedTupleBodyMachine.Data W) (extra : Extra → List Bool) :
nativeTapes input indices data extra (.edgeWork 7) = data.reversedOutput := by
simp [nativeTapes, SignedTupleBodyMachine.tapes]
@[simp] theorem nativeTapes_count (input : NormalizedSourceInput.Input)
(indices : Fin W → Fin (NormalizedSourceInput.clauseCount input))
(data : SignedTupleBodyMachine.Data W) (extra : Extra → List Bool) :
nativeTapes input indices data extra .edgeCount = encodeWord data.count := by
simp [nativeTapes, SignedTupleBodyMachine.tapes]
@[simp] theorem nativeTapes_extra (input : NormalizedSourceInput.Input)
(indices : Fin W → Fin (NormalizedSourceInput.clauseCount input))
(data : SignedTupleBodyMachine.Data W) (extra : Extra → List Bool) (x : Extra) :
nativeTapes input indices data extra (.extra x) = extra x := by
simp [nativeTapes, SignedTupleBodyMachine.tapes, baseTapes]
theorem nativeTapes_input (input : NormalizedSourceInput.Input)
(indices : Fin W → Fin (NormalizedSourceInput.clauseCount input))
(data : SignedTupleBodyMachine.Data W) (extra : Extra → List Bool) :
SignedTupleLayout.Input input.formula indices (nativeTapes input indices data extra) := by
exact SignedTupleBodyMachine.tapes_input (branch := branch) (n := n) (t := t)
input.formula indices (baseTapes input indices extra)
(baseTapes_input input indices extra) data
theorem nativeTapes_emptyWork (input : NormalizedSourceInput.Input)
(indices : Fin W → Fin (NormalizedSourceInput.clauseCount input))
(data : SignedTupleBodyMachine.Data W) (extra : Extra → List Bool)
(j : Fin 10) (h1 : j ≠ 1) (h7 : j ≠ 7) (h8 : j ≠ 8) :
nativeTapes input indices data extra (.edgeWork j) = [] := by
simp [nativeTapes, SignedTupleBodyMachine.tapes, baseTapes, h1, h7, h8]
theorem digitTapes_nativeTapes (input : NormalizedSourceInput.Input)
(indices : Fin W → Fin (NormalizedSourceInput.clauseCount input))
(data : SignedTupleBodyMachine.Data W) (extra : Extra → List Bool)
(coordinate : Fin W) (value : Fin (NormalizedSourceInput.clauseCount input)) :
MachineTupleOdometer.digitTapes
(SignedTupleLayout.Tape.current coordinate)
(SignedTupleLayout.Tape.remaining coordinate)
(nativeTapes input indices data extra)
value.val (NormalizedSourceInput.clauseCount input - 1 - value.val) [] [] =
nativeTapes input (Function.update indices coordinate value) data extra := by
funext tape
cases tape with
| source source =>
cases source <;>
simp [MachineTupleOdometer.digitTapes, MachineUnaryAddAt.unaryTapes,
UniqueGamesTheorem.Reduction.MachineTransfer.tapesAt,
nativeTapes, SignedTupleBodyMachine.tapes, baseTapes]
| current position =>
by_cases same : position = coordinate
· subst position
simp [MachineTupleOdometer.digitTapes, MachineUnaryAddAt.unaryTapes,
UniqueGamesTheorem.Reduction.MachineTransfer.tapesAt,
nativeTapes, SignedTupleBodyMachine.tapes, baseTapes]
· simp [MachineTupleOdometer.digitTapes, MachineUnaryAddAt.unaryTapes,
UniqueGamesTheorem.Reduction.MachineTransfer.tapesAt,
nativeTapes, SignedTupleBodyMachine.tapes, baseTapes, same]
| remaining position =>
by_cases same : position = coordinate
· subst position
simp [MachineTupleOdometer.digitTapes, MachineUnaryAddAt.unaryTapes,
UniqueGamesTheorem.Reduction.MachineTransfer.tapesAt,
nativeTapes, SignedTupleBodyMachine.tapes, baseTapes]
· simp [MachineTupleOdometer.digitTapes, MachineUnaryAddAt.unaryTapes,
UniqueGamesTheorem.Reduction.MachineTransfer.tapesAt,
nativeTapes, SignedTupleBodyMachine.tapes, baseTapes, same]
| encodedVariable position slot =>
simp [MachineTupleOdometer.digitTapes, MachineUnaryAddAt.unaryTapes,
UniqueGamesTheorem.Reduction.MachineTransfer.tapesAt,
nativeTapes, SignedTupleBodyMachine.tapes, baseTapes]
| adapterScratch =>
simp [MachineTupleOdometer.digitTapes, MachineUnaryAddAt.unaryTapes,
UniqueGamesTheorem.Reduction.MachineTransfer.tapesAt,
nativeTapes, SignedTupleBodyMachine.tapes, baseTapes]
| edgeWork j =>
simp [MachineTupleOdometer.digitTapes, MachineUnaryAddAt.unaryTapes,
UniqueGamesTheorem.Reduction.MachineTransfer.tapesAt,
nativeTapes, SignedTupleBodyMachine.tapes, baseTapes, Function.update_apply]
| edgeCount =>
simp [MachineTupleOdometer.digitTapes, MachineUnaryAddAt.unaryTapes,
UniqueGamesTheorem.Reduction.MachineTransfer.tapesAt,
nativeTapes, SignedTupleBodyMachine.tapes]
| extra x =>
simp [MachineTupleOdometer.digitTapes, MachineUnaryAddAt.unaryTapes,
UniqueGamesTheorem.Reduction.MachineTransfer.tapesAt,
nativeTapes, SignedTupleBodyMachine.tapes, baseTapes]
end
inductive Label (locations stages : Nat) (Descriptor Extra : Type)
| body (label : SignedTupleBodyMachine.BodyLabel locations stages Descriptor Extra)
| check (coordinate : Fin locations)
| reset (coordinate : Fin locations)
| exit
deriving DecidableEq, Fintype
noncomputable section Native
variable {branch : Nat → Nat} {n t : Nat}
(rows repeats : Nat → Nat) (hn : 0 < n) (hbranch : ∀ k < n, 0 < branch k)
(hrows : ∀ k, 0 < rows (k + 1)) (δ : ℚ)
local notation "width" => TreeCanonical.locationCount branch n t
local notation "q" => FinitePreliminaryCompletion.alphabet branch n t δ
local notation "hq" => CanonicalLocalCompletion.alphabet_positive width δ
local notation "large" => CanonicalLocalCompletion.partitionWidth_le_alphabet width δ
local notation "D" => SignedMultiplicity.denominator branch n t rows repeats hn hbranch hrows q
local notation "Desc" => SignedCompletionSchedule.Descriptor (rows := rows) (repeats := repeats) hq large
abbrev Control := SignedTupleLayout.State width Unit Desc
abbrev Ambient := (SourceTupleMachine.State width Unit × Option Desc) × (Bool × Option Bool)
abbrev NativeLabel (Extra : Type) := Label width D Desc Extra
variable {Extra : Type} [DecidableEq Extra]
local notation "Arena" => SignedTupleLayout.Tape width Extra
local notation "Labels" => NativeLabel (branch := branch) (n := n) (t := t)
rows repeats hn hbranch hrows δ Extra
local notation "Registers" => Control (branch := branch) (n := n) (t := t) rows repeats δ
def checkAt (depth : Nat) : Labels :=
if h : depth < width then .check ⟨depth, h⟩ else .exit
def resetAt (depth : Nat) : Labels :=
if h : depth < width then .reset ⟨depth, h⟩ else .exit
def currentAt (depth : Nat) : Arena :=
if h : depth < width then .current ⟨depth, h⟩ else .adapterScratch
def remainingAt (depth : Nat) : Arena :=
if h : depth < width then .remaining ⟨depth, h⟩ else .edgeCount
def body : Labels := .body .start
def program : Labels → Turing.TM2.Stmt (fun _ : Arena => Bool) Labels Registers
| .body label => SignedTupleBodyMachine.bodyInstruction (branch := branch) (n := n) (t := t)
(rows := rows) (repeats := repeats) hq large hn hbranch hrows
Label.body (some (checkAt (branch := branch) (n := n) (t := t)
(Extra := Extra) rows repeats hn hbranch hrows δ 0)) label
| .check i => MachineTupleOdometer.increment (.current i) (.remaining i)
(body (branch := branch) (n := n) (t := t)
(Extra := Extra) rows repeats hn hbranch hrows δ) (.reset i)
| .reset i => MachineTupleOdometer.reset (.current i) (.remaining i) (.reset i)
(checkAt (branch := branch) (n := n) (t := t)
(Extra := Extra) rows repeats hn hbranch hrows δ (i.val + 1))
| .exit => .halt
def tupleStep (input : NormalizedSourceInput.Input)
(indices : Fin width → Fin (NormalizedSourceInput.clauseCount input))
(data : SignedTupleBodyMachine.Data width) : SignedTupleBodyMachine.Data width :=
SignedTupleBodyMachine.result (branch := branch) (n := n) (t := t)
(rows := rows) (repeats := repeats) hq large hn hbranch hrows
(MetadataFreeSampler.sourceSigns (NormalizedSourceInput.clauses input)
(SourceQuestionOrder.tupleToQuestion indices))
(fun i => (indices i).val)
(fun i slot => ((NormalizedSourceInput.clauses input) (indices i)).variable slot |>.val)
(List.ofFn (fun i : Fin D => i)) data
@[simp] theorem tupleStep_count (input : NormalizedSourceInput.Input)
(indices : Fin width → Fin (NormalizedSourceInput.clauseCount input))
(data : SignedTupleBodyMachine.Data width) :
(tupleStep rows repeats hn hbranch hrows δ input indices data).count = data.count + D := by
simp only [tupleStep, SignedTupleBodyMachine.result_count, List.length_ofFn]
theorem tupleStep_eq (input : NormalizedSourceInput.Input)
(indices : Fin width → Fin (NormalizedSourceInput.clauseCount input))
(data : SignedTupleBodyMachine.Data width) :
tupleStep rows repeats hn hbranch hrows δ input indices data =
SignedTupleBodyMachine.tupleResult (branch := branch) (n := n) (t := t)
(rows := rows) (repeats := repeats)
hq large hn hbranch hrows input.formula indices data := rfl
local notation "runProgram" => program (branch := branch) (n := n) (t := t)
(Extra := Extra) rows repeats hn hbranch hrows δ
local notation "entry" => body (branch := branch) (n := n) (t := t)
(Extra := Extra) rows repeats hn hbranch hrows δ
local notation "atCheck" => checkAt (branch := branch) (n := n) (t := t)
(Extra := Extra) rows repeats hn hbranch hrows δ
local notation "atReset" => resetAt (branch := branch) (n := n) (t := t)
(Extra := Extra) rows repeats hn hbranch hrows δ
local notation "atCurrent" => currentAt (branch := branch) (n := n) (t := t) (Extra := Extra)
local notation "atRemaining" => remainingAt (branch := branch) (n := n) (t := t) (Extra := Extra)
def configuration (label : Labels) (input : NormalizedSourceInput.Input)
(indices : Fin width → Fin (NormalizedSourceInput.clauseCount input))
(data : SignedTupleBodyMachine.Data width) (extra : Extra → List Bool) :
Turing.TM2.Cfg (fun _ : Arena => Bool) Labels Registers :=
⟨some label, SignedTupleLayout.clean (), nativeTapes input indices data extra⟩
local notation "cfg" => configuration rows repeats hn hbranch hrows δ
def leafBudget (input : NormalizedSourceInput.Input)
(indices : Fin width → Fin (NormalizedSourceInput.clauseCount input))
(data : SignedTupleBodyMachine.Data width) (extra : Extra → List Bool) : Nat :=
SignedTupleBodyMachine.bodyBudget (branch := branch) (n := n) (t := t)
(rows := rows) (repeats := repeats)
hq large hn hbranch hrows
input.formula indices (baseTapes input indices extra) data
def leafInTime (input : NormalizedSourceInput.Input)
(indices : Fin width → Fin (NormalizedSourceInput.clauseCount input))
(data : SignedTupleBodyMachine.Data width) (extra : Extra → List Bool) :
StateTransition.EvalsToInTime (Turing.TM2.step runProgram)
(cfg entry input indices data extra)
(some (cfg (atCheck 0) input indices
(tupleStep rows repeats hn hbranch hrows δ input indices data) extra))
(leafBudget rows repeats hn hbranch hrows δ input indices data extra) := by
have run := SignedTupleBodyMachine.bodyInTime (branch := branch) (n := n) (t := t)
(rows := rows) (repeats := repeats) hq large hn hbranch hrows
input.formula indices (baseTapes input indices extra) (baseTapes_input input indices extra)
() data Label.body (some (atCheck 0)) runProgram (fun _ => rfl)
(baseTapes_emptyWork input indices extra)
simpa only [configuration, body, SignedTupleBodyMachine.bodyMain, nativeTapes,
tupleStep_eq, leafBudget] using run
def maxLeafBudget (input : NormalizedSourceInput.Input) (extra : Extra → List Bool) :
Nat → (Fin width → Fin (NormalizedSourceInput.clauseCount input)) →
SignedTupleBodyMachine.Data width → Nat
| 0, indices, data => leafBudget rows repeats hn hbranch hrows δ input indices data extra
| depth + 1, indices, data => Finset.univ.sup fun value =>
maxLeafBudget input extra depth (setCoordinate indices depth value)
(blockPrefix (tupleStep rows repeats hn hbranch hrows δ input) depth indices value.val data)
def CostsAtMost (input : NormalizedSourceInput.Input) (extra : Extra → List Bool) (B : Nat) :
Nat → (Fin width → Fin (NormalizedSourceInput.clauseCount input)) →
SignedTupleBodyMachine.Data width → Prop
| 0, indices, data => leafBudget rows repeats hn hbranch hrows δ input indices data extra ≤ B
| depth + 1, indices, data => ∀ value,
CostsAtMost input extra B depth (setCoordinate indices depth value)
(blockPrefix (tupleStep rows repeats hn hbranch hrows δ input) depth indices value.val data)
theorem costsAtMost_of_max (input : NormalizedSourceInput.Input) (extra : Extra → List Bool)
(depth : Nat) (indices : Fin width → Fin (NormalizedSourceInput.clauseCount input))
(data : SignedTupleBodyMachine.Data width) (B : Nat)
(bounded : maxLeafBudget rows repeats hn hbranch hrows δ input extra depth indices data ≤ B) :
CostsAtMost rows repeats hn hbranch hrows δ input extra B depth indices data := by
induction depth generalizing indices data with
| zero => exact bounded
| succ depth ih =>
intro value
apply ih
exact (Finset.le_sup (Finset.mem_univ value)).trans bounded
private theorem digitTapes_at (input : NormalizedSourceInput.Input)
(indices : Fin width → Fin (NormalizedSourceInput.clauseCount input))
(data : SignedTupleBodyMachine.Data width) (extra : Extra → List Bool)
(depth remainder : Nat) (inside : depth < width)
(remainingBound : remainder ≤ NormalizedSourceInput.clauseCount input - 1) :
MachineTupleOdometer.digitTapes (atCurrent depth) (atRemaining depth)
(nativeTapes input indices data extra)
(NormalizedSourceInput.clauseCount input - 1 - remainder) remainder [] [] =
nativeTapes input (setCoordinate indices depth
⟨NormalizedSourceInput.clauseCount input - 1 - remainder,
by have := NormalizedSourceInput.clauseCount_positive input; omega⟩) data extra := by
have positive := NormalizedSourceInput.clauseCount_positive input
have valueBound : NormalizedSourceInput.clauseCount input - 1 - remainder <
NormalizedSourceInput.clauseCount input := by omega
have complement : NormalizedSourceInput.clauseCount input - 1 -
(NormalizedSourceInput.clauseCount input - 1 - remainder) = remainder := by omega
have h := digitTapes_nativeTapes input indices data extra ⟨depth, inside⟩
⟨NormalizedSourceInput.clauseCount input - 1 - remainder, valueBound⟩
simpa only [currentAt, remainingAt, dite_eq_left inside, complement,
← setCoordinate_eq_update] using h
theorem existsTree (input : NormalizedSourceInput.Input) (extra : Extra → List Bool)
(B depth : Nat) (inside : depth ≤ width)
(indices : Fin width → Fin (NormalizedSourceInput.clauseCount input))
(zero : ∀ i, i.val < depth → (indices i).val = 0)
(data : SignedTupleBodyMachine.Data width)
(costs : CostsAtMost rows repeats hn hbranch hrows δ input extra B depth indices data) :
∃ tree : MachineTupleOdometer.NestedCycle (NormalizedSourceInput.clauseCount input)
atCurrent atRemaining entry atCheck atReset runProgram depth (atCheck depth)
(cfg entry input indices data extra)
(cfg (atCheck depth) input indices
(foldVisits (tupleStep rows repeats hn hbranch hrows δ input) depth indices data) extra),
tree.LeafBound B := by
let m := NormalizedSourceInput.clauseCount input
have positive : 0 < m := NormalizedSourceInput.clauseCount_positive input
let step := tupleStep (branch := branch) (n := n) (t := t) rows repeats hn hbranch hrows δ input
induction depth generalizing indices data with
| zero =>
let run := leafInTime rows repeats hn hbranch hrows δ input indices data extra
refine ⟨.leaf (atCheck 0) (cfg entry input indices data extra)
(cfg (atCheck 0) input indices (foldVisits step 0 indices data) extra)
run.steps rfl rfl run.evals_in_steps, ?_⟩
unfold MachineTupleOdometer.NestedCycle.LeafBound
exact run.steps_le_m.trans costs
| succ depth ih =>
have hlt : depth < width := by omega
let accumulated := blockPrefix step depth indices
let schedule : MachineTupleOdometer.CycleSchedule Arena (Ambient (branch := branch) (n := n) (t := t) rows repeats δ) :=
{ currentSuffix := []
remainingSuffix := []
ambient := fun _ => (SignedTupleLayout.clean («width» := width) («Desc» := Desc) ()).1
register := fun _ => none
base := fun remainder => nativeTapes input indices (accumulated (m - remainder) data) extra }
have children : ∀ r : Fin m,
∃ tree : MachineTupleOdometer.NestedCycle m atCurrent atRemaining entry atCheck
atReset runProgram depth (atCheck depth)
(MachineTupleOdometer.bodyConfiguration (atCurrent depth) (atRemaining depth)
entry (m - 1) r.val schedule.currentSuffix schedule.remainingSuffix
schedule.ambient schedule.base)
(MachineTupleOdometer.checkConfiguration (atCurrent depth) (atRemaining depth)
(atCheck depth) (m - 1) r.val schedule.currentSuffix schedule.remainingSuffix
schedule.ambient schedule.register schedule.base), tree.LeafBound B := by
intro r
have hr : r.val ≤ m - 1 := by omega
let value : Fin m := ⟨m - 1 - r.val, by omega⟩
have before : m - (r.val + 1) = value.val := by dsimp [value]; omega
have after : m - r.val = value.val + 1 := by dsimp [value]; omega
have childZero : ∀ i, i.val < depth →
(setCoordinate indices depth value i).val = 0 := by
intro i hi
simp only [setCoordinate, ite_eq_right (by omega : i.val ≠ depth)]
exact zero i (by omega)
have childCosts : CostsAtMost rows repeats hn hbranch hrows δ input extra B depth
(setCoordinate indices depth value) (accumulated (m - (r.val + 1)) data) := by
rw [before]
exact costs value
obtain ⟨child, bounded⟩ := ih (by omega) (setCoordinate indices depth value)
childZero (accumulated (m - (r.val + 1)) data) childCosts
have outputData : foldVisits step depth (setCoordinate indices depth value)
(accumulated (m - (r.val + 1)) data) = accumulated (m - r.val) data := by
rw [before, after]
exact (blockPrefix_succ step depth indices value data).symm
have childExists :
∃ child : MachineTupleOdometer.NestedCycle m atCurrent atRemaining entry atCheck
atReset runProgram depth (atCheck depth)
(cfg entry input (setCoordinate indices depth value)
(accumulated (m - (r.val + 1)) data) extra)
(cfg (atCheck depth) input (setCoordinate indices depth value)
(foldVisits step depth (setCoordinate indices depth value)
(accumulated (m - (r.val + 1)) data)) extra), child.LeafBound B :=
⟨child, bounded⟩
rw [outputData] at childExists
have childStart :
cfg entry input (setCoordinate indices depth value)
(accumulated (m - (r.val + 1)) data) extra =
MachineTupleOdometer.bodyConfiguration (atCurrent depth) (atRemaining depth)
entry (m - 1) r.val schedule.currentSuffix schedule.remainingSuffix
schedule.ambient schedule.base := by
dsimp only [MachineTupleOdometer.bodyConfiguration, schedule, configuration]
rw [digitTapes_at input indices _ extra depth r.val hlt hr]
rfl
have childFinish :
cfg (atCheck depth) input (setCoordinate indices depth value)
(accumulated (m - r.val) data) extra =
MachineTupleOdometer.checkConfiguration (atCurrent depth) (atRemaining depth)
(atCheck depth) (m - 1) r.val schedule.currentSuffix schedule.remainingSuffix
schedule.ambient schedule.register schedule.base := by
dsimp only [MachineTupleOdometer.checkConfiguration, schedule, configuration]
rw [digitTapes_at input indices _ extra depth r.val hlt hr]
rfl
rw [childStart, childFinish] at childExists
exact childExists
let chosen := fun r => Classical.choose (children r)
have chosenBound : ∀ r, (chosen r).LeafBound B :=
fun r => Classical.choose_spec (children r)
have distinct : atCurrent depth ≠ atRemaining depth := by
simp [currentAt, remainingAt, hlt]
have checkInstruction : runProgram (atCheck depth) =
MachineTupleOdometer.increment (atCurrent depth) (atRemaining depth) entry (atReset depth) := by
simp [program, checkAt, currentAt, remainingAt, resetAt, hlt]
have resetInstruction : runProgram (atReset depth) =
MachineTupleOdometer.reset (atCurrent depth) (atRemaining depth)
(atReset depth) (atCheck (depth + 1)) := by
simp [program, resetAt, currentAt, remainingAt, hlt]
let tree := MachineTupleOdometer.NestedCycle.node depth (atCheck (depth + 1))
positive distinct checkInstruction resetInstruction schedule chosen
have treeBound : tree.LeafBound B := by
simpa only [tree, MachineTupleOdometer.NestedCycle.LeafBound] using chosenBound
have unchanged : setCoordinate indices depth ⟨0, positive⟩ = indices :=
setCoordinate_zero indices depth positive (fun i hi => zero i (by omega))
have startTapes : MachineTupleOdometer.digitTapes (atCurrent depth) (atRemaining depth)
(nativeTapes input indices (accumulated (m - ((m - 1) + 1)) data) extra)
(m - 1 - (m - 1)) (m - 1) [] [] = nativeTapes input indices data extra := by
rw [digitTapes_at input indices _ extra depth (m - 1) hlt (Nat.le_refl _)]
have full : m - 1 + 1 = m := by omega
dsimp only [m] at full unchanged ⊢
simp only [Nat.sub_self, unchanged, full, accumulated, blockPrefix_zero]
have finishTapes : MachineTupleOdometer.digitTapes (atCurrent depth) (atRemaining depth)
(nativeTapes input indices (accumulated (m - 0) data) extra) 0 (m - 1) [] [] =
nativeTapes input indices (foldVisits step (depth + 1) indices data) extra := by
have h := digitTapes_at input indices (accumulated m data) extra depth (m - 1)
hlt (Nat.le_refl _)
dsimp only [m] at h unchanged ⊢
simpa only [Nat.sub_self, Nat.sub_zero, unchanged, accumulated, blockPrefix_all] using h
have nodeExists :
∃ node : MachineTupleOdometer.NestedCycle m atCurrent atRemaining entry atCheck
atReset runProgram (depth + 1) (atCheck (depth + 1))
(MachineTupleOdometer.bodyConfiguration (atCurrent depth) (atRemaining depth)
entry (m - 1) (m - 1) schedule.currentSuffix schedule.remainingSuffix
schedule.ambient schedule.base)
(MachineTupleOdometer.cycleExit (atCurrent depth) (atRemaining depth)
(atCheck (depth + 1)) (m - 1) schedule.currentSuffix schedule.remainingSuffix
schedule.ambient schedule.base), node.LeafBound B := ⟨tree, treeBound⟩
have nodeStart :
MachineTupleOdometer.bodyConfiguration (atCurrent depth) (atRemaining depth)
entry (m - 1) (m - 1) schedule.currentSuffix schedule.remainingSuffix
schedule.ambient schedule.base = cfg entry input indices data extra := by
dsimp only [MachineTupleOdometer.bodyConfiguration, schedule, configuration]
rw [startTapes]
rfl
have nodeFinish :
MachineTupleOdometer.cycleExit (atCurrent depth) (atRemaining depth)
(atCheck (depth + 1)) (m - 1) schedule.currentSuffix schedule.remainingSuffix
schedule.ambient schedule.base =
cfg (atCheck (depth + 1)) input indices (foldVisits step (depth + 1) indices data) extra := by
dsimp only [MachineTupleOdometer.cycleExit, schedule, configuration]
rw [finishTapes]
rfl
rw [nodeStart, nodeFinish] at nodeExists
exact nodeExists
def result (input : NormalizedSourceInput.Input) (data : SignedTupleBodyMachine.Data width) :
SignedTupleBodyMachine.Data width :=
foldVisits (tupleStep rows repeats hn hbranch hrows δ input) width
(fun _ => ⟨0, NormalizedSourceInput.clauseCount_positive input⟩) data
theorem result_count (input : NormalizedSourceInput.Input)
(data : SignedTupleBodyMachine.Data width) :
(result rows repeats hn hbranch hrows δ input data).count =
data.count + NormalizedSourceInput.clauseCount input ^ width * D := by
have countFold (xs : List (Fin width → Fin (NormalizedSourceInput.clauseCount input)))
(initial : SignedTupleBodyMachine.Data width) :
(xs.foldl (fun current indices => tupleStep rows repeats hn hbranch hrows δ input indices current)
initial).count = initial.count + xs.length * D := by
induction xs generalizing initial with
| nil => simp
| cons indices xs ih =>
simp only [List.foldl_cons, ih, tupleStep_count, List.length_cons, Nat.add_mul, Nat.one_mul]
omega
unfold result foldVisits
rw [countFold, visits_all, MachineTupleOdometer.tupleOrder_length]
def enumerateInTime (input : NormalizedSourceInput.Input)
(data : SignedTupleBodyMachine.Data width) (extra : Extra → List Bool) :
StateTransition.EvalsToInTime (Turing.TM2.step runProgram)
(cfg entry input (fun _ => ⟨0, NormalizedSourceInput.clauseCount_positive input⟩) data extra)
(some (cfg .exit input (fun _ => ⟨0, NormalizedSourceInput.clauseCount_positive input⟩)
(result rows repeats hn hbranch hrows δ input data) extra))
((maxLeafBudget rows repeats hn hbranch hrows δ input extra width
(fun _ => ⟨0, NormalizedSourceInput.clauseCount_positive input⟩) data + 2 * width) *
NormalizedSourceInput.clauseCount input ^ width) := by
let indices : Fin width → Fin (NormalizedSourceInput.clauseCount input) :=
fun _ => ⟨0, NormalizedSourceInput.clauseCount_positive input⟩
let B := maxLeafBudget rows repeats hn hbranch hrows δ input extra width indices data
let witness := existsTree rows repeats hn hbranch hrows δ input extra B width (Nat.le_refl _)
indices (fun _ _ => rfl) data
(costsAtMost_of_max rows repeats hn hbranch hrows δ input extra width indices data B (Nat.le_refl _))
let tree := Classical.choose witness
have bounded := Classical.choose_spec witness
refine { steps := tree.steps, evals_in_steps := ?_, steps_le_m := ?_ }
· change (MachineComposition.advance (Turing.TM2.step runProgram))^[tree.steps]
(some (cfg entry input indices data extra)) = _
simpa only [checkAt, Nat.lt_irrefl, ↓reduceDIte, result] using tree.trace
· exact tree.steps_le_power (NormalizedSourceInput.clauseCount_positive input) B bounded
def enumerateInTimeOfBound (input : NormalizedSourceInput.Input)
(data : SignedTupleBodyMachine.Data width) (extra : Extra → List Bool) (B : Nat)
(costs : CostsAtMost rows repeats hn hbranch hrows δ input extra B width
(fun _ => ⟨0, NormalizedSourceInput.clauseCount_positive input⟩) data) :
StateTransition.EvalsToInTime (Turing.TM2.step runProgram)
(cfg entry input (fun _ => ⟨0, NormalizedSourceInput.clauseCount_positive input⟩) data extra)
(some (cfg .exit input (fun _ => ⟨0, NormalizedSourceInput.clauseCount_positive input⟩)
(result rows repeats hn hbranch hrows δ input data) extra))
((B + 2 * width) * NormalizedSourceInput.clauseCount input ^ width) := by
let witness := existsTree rows repeats hn hbranch hrows δ input extra B width (Nat.le_refl _)
(fun _ => ⟨0, NormalizedSourceInput.clauseCount_positive input⟩) (fun _ _ => rfl) data costs
let tree := Classical.choose witness
have bounded := Classical.choose_spec witness
refine { steps := tree.steps, evals_in_steps := ?_, steps_le_m := ?_ }
· change (MachineComposition.advance (Turing.TM2.step runProgram))^[tree.steps]
(some (cfg entry input (fun _ => ⟨0, NormalizedSourceInput.clauseCount_positive input⟩) data extra)) = _
simpa only [checkAt, Nat.lt_irrefl, ↓reduceDIte, result] using tree.trace
· exact tree.steps_le_power (NormalizedSourceInput.clauseCount_positive input) B bounded
end Native
noncomputable section Semantics
theorem foldl_chunks_eq_prefix {α β S : Type*}
(order : List β) (chunk : α → List β) (step : α → S → S) (pack : List β → S)
(hstep : ∀ (a : α) (earlier suffix : List β),
order = earlier ++ chunk a ++ suffix →
step a (pack earlier) = pack (earlier ++ chunk a))
(xs : List α) (earlier suffix : List β)
(split : order = earlier ++ xs.flatMap chunk ++ suffix) :
xs.foldl (fun state a => step a state) (pack earlier) =
pack (earlier ++ xs.flatMap chunk) := by
induction xs generalizing earlier with
| nil =>
simp only [List.foldl_nil, List.flatMap_nil, List.append_nil]
| cons a xs ih =>
have currentSplit : order = earlier ++ chunk a ++ (xs.flatMap chunk ++ suffix) := by
simpa only [List.flatMap_cons, List.append_assoc] using split
rw [List.foldl_cons, hstep a earlier (xs.flatMap chunk ++ suffix) currentSplit]
have restSplit : order = (earlier ++ chunk a) ++ xs.flatMap chunk ++ suffix := by
simpa only [List.flatMap_cons, List.append_assoc] using split
simpa only [List.flatMap_cons, List.append_assoc] using
ih (earlier ++ chunk a) restSplit
variable {branch : Nat → Nat} {n t : Nat}
(rows repeats : Nat → Nat) (hn : 0 < n) (hbranch : ∀ k < n, 0 < branch k)
(hrows : ∀ k, 0 < rows (k + 1)) (δ : ℚ)
(input : NormalizedSourceInput.Input)
local notation "width" => TreeCanonical.locationCount branch n t
local notation "C" => CompletedSignedLaw.Completed (t := t) (NormalizedSourceInput.clauses input)
rows repeats hn hbranch hrows δ
local notation "order" => CompletedVisitOrder.completedOrder
(t := t) (NormalizedSourceInput.clauses input) rows repeats hn hbranch hrows δ
local notation "localOrder" => CompletedVisitOrder.localOrder
(t := t) (NormalizedSourceInput.clauses input) rows repeats hn hbranch hrows δ
local notation "pack" => SignedTupleSemantics.dataOfPrefix
(t := t) (NormalizedSourceInput.clauses input) rows repeats hn hbranch hrows δ
theorem tupleStep_dataOfPrefix
(indices : Fin width → Fin (NormalizedSourceInput.clauseCount input))
(earlier suffix : List C) (tail : List Bool)
(split : order = earlier ++
localOrder (SourceQuestionOrder.tupleToQuestion (branch := branch) (n := n) (t := t) indices) ++
suffix) :
ExactTargetOdometer.tupleStep rows repeats hn hbranch hrows δ input indices
(pack earlier tail) =
pack (earlier ++ localOrder
(SourceQuestionOrder.tupleToQuestion (branch := branch) (n := n) (t := t) indices)) tail := by
simpa only [ExactTargetOdometer.tupleStep, List.finRange,
SignedTupleSemantics.sourceIDs_tuple, SignedTupleSemantics.sourceVariables_tuple] using
SignedTupleSemantics.full_result_eq (t := t)
(NormalizedSourceInput.clauses input) rows repeats hn hbranch hrows δ
(SourceQuestionOrder.tupleToQuestion (branch := branch) (n := n) (t := t) indices)
earlier suffix tail split
theorem fold_tupleSteps_dataOfPrefix
(xs : List (Fin width → Fin (NormalizedSourceInput.clauseCount input)))
(earlier suffix : List C) (tail : List Bool)
(split : order = earlier ++ xs.flatMap
(fun indices => localOrder
(SourceQuestionOrder.tupleToQuestion (branch := branch) (n := n) (t := t) indices)) ++ suffix) :
xs.foldl
(fun data indices =>
ExactTargetOdometer.tupleStep rows repeats hn hbranch hrows δ input indices data)
(pack earlier tail) =
pack (earlier ++ xs.flatMap
(fun indices => localOrder
(SourceQuestionOrder.tupleToQuestion (branch := branch) (n := n) (t := t) indices))) tail := by
exact foldl_chunks_eq_prefix order
(fun indices : Fin width → Fin (NormalizedSourceInput.clauseCount input) =>
localOrder (SourceQuestionOrder.tupleToQuestion (branch := branch) (n := n) (t := t) indices))
(ExactTargetOdometer.tupleStep rows repeats hn hbranch hrows δ input)
(fun before => pack before tail)
(fun indices before after hsplit =>
tupleStep_dataOfPrefix rows repeats hn hbranch hrows δ input indices before after tail hsplit)
xs earlier suffix split
theorem result_dataOfPrefix (tail : List Bool) :
ExactTargetOdometer.result rows repeats hn hbranch hrows δ input (pack [] tail) =
pack order tail := by
let digits : Fin width → Fin (NormalizedSourceInput.clauseCount input) :=
fun _ => ⟨0, NormalizedSourceInput.clauseCount_positive input⟩
let xs := ExactTargetOdometer.visits width digits
have chronology : xs.flatMap
(fun indices => localOrder
(SourceQuestionOrder.tupleToQuestion (branch := branch) (n := n) (t := t) indices)) = order :=
ExactTargetOdometer.visits_completedOrder (t := t)
(NormalizedSourceInput.clauses input) rows repeats hn hbranch hrows δ digits
have split : order = [] ++ xs.flatMap
(fun indices => localOrder
(SourceQuestionOrder.tupleToQuestion (branch := branch) (n := n) (t := t) indices)) ++ [] := by
simpa only [List.nil_append, List.append_nil] using chronology.symm
have folded := fold_tupleSteps_dataOfPrefix rows repeats hn hbranch hrows δ input
xs [] [] tail split
change xs.foldl
(fun data indices =>
ExactTargetOdometer.tupleStep rows repeats hn hbranch hrows δ input indices data)
(pack [] tail) = pack order tail
simpa only [List.nil_append, chronology] using folded
def targetInTime {Extra : Type} [DecidableEq Extra] (tail : List Bool)
(extra : Extra → List Bool) :
StateTransition.EvalsToInTime
(Turing.TM2.step (program (branch := branch) (n := n) (t := t)
(Extra := Extra) rows repeats hn hbranch hrows δ))
(configuration rows repeats hn hbranch hrows δ
(body (branch := branch) (n := n) (t := t) (Extra := Extra) rows repeats hn hbranch hrows δ)
input (fun _ => ⟨0, NormalizedSourceInput.clauseCount_positive input⟩) (pack [] tail) extra)
(some (configuration rows repeats hn hbranch hrows δ .exit input
(fun _ => ⟨0, NormalizedSourceInput.clauseCount_positive input⟩) (pack order tail) extra))
((maxLeafBudget rows repeats hn hbranch hrows δ input extra width
(fun _ => ⟨0, NormalizedSourceInput.clauseCount_positive input⟩) (pack [] tail) + 2 * width) *
NormalizedSourceInput.clauseCount input ^ width) := by
simpa only [result_dataOfPrefix] using
enumerateInTime rows repeats hn hbranch hrows δ input (pack [] tail) extra
end Semantics
noncomputable section ChronologicalCosts
theorem visits_split_at {width radix : Nat} (depth : Nat)
(indices : Fin width → Fin radix) (value : Fin radix) :
ExactTargetOdometer.visits (depth + 1) indices =
((List.ofFn (fun a : Fin radix => a)).take value.val).flatMap
(fun a => ExactTargetOdometer.visits depth
(ExactTargetOdometer.setCoordinate indices depth a)) ++
ExactTargetOdometer.visits depth
(ExactTargetOdometer.setCoordinate indices depth value) ++
((List.ofFn (fun a : Fin radix => a)).drop (value.val + 1)).flatMap
(fun a => ExactTargetOdometer.visits depth
(ExactTargetOdometer.setCoordinate indices depth a)) := by
let choices : List (Fin radix) := List.ofFn (fun a : Fin radix => a)
let lower : Fin radix → List (Fin width → Fin radix) :=
fun a => ExactTargetOdometer.visits depth (ExactTargetOdometer.setCoordinate indices depth a)
have inside : value.val < choices.length := by
simpa only [choices, List.length_ofFn] using value.isLt
have selected : choices[value.val]'inside = value := by
simp only [choices, List.getElem_ofFn, Fin.eta]
have split : choices = choices.take value.val ++ choices.drop value.val :=
(List.take_append_drop value.val choices).symm
rw [List.drop_eq_getElem_cons inside, selected] at split
change choices.flatMap lower =
(choices.take value.val).flatMap lower ++ lower value ++
(choices.drop (value.val + 1)).flatMap lower
simpa only [List.flatMap_append, List.flatMap_cons, List.append_assoc] using
congrArg (fun xs : List (Fin radix) => xs.flatMap lower) split
variable {branch : Nat → Nat} {n t : Nat}
(rows repeats : Nat → Nat) (hn : 0 < n) (hbranch : ∀ k < n, 0 < branch k)
(hrows : ∀ k, 0 < rows (k + 1)) (δ : ℚ)
(input : NormalizedSourceInput.Input)
local notation "width" => TreeCanonical.locationCount branch n t
local notation "Digits" => Fin width → Fin (NormalizedSourceInput.clauseCount input)
local notation "C" => CompletedSignedLaw.Completed (t := t) (NormalizedSourceInput.clauses input)
rows repeats hn hbranch hrows δ
local notation "order" => CompletedVisitOrder.completedOrder
(t := t) (NormalizedSourceInput.clauses input) rows repeats hn hbranch hrows δ
local notation "localOrder" => CompletedVisitOrder.localOrder
(t := t) (NormalizedSourceInput.clauses input) rows repeats hn hbranch hrows δ
local notation "pack" => SignedTupleSemantics.dataOfPrefix
(t := t) (NormalizedSourceInput.clauses input) rows repeats hn hbranch hrows δ
local notation "step" => ExactTargetOdometer.tupleStep rows repeats hn hbranch hrows δ input
variable {Extra : Type} [DecidableEq Extra]
theorem costsAtMost_of_chronological (extra : Extra → List Bool) (B : Nat) (tail : List Bool)
(bounded : ∀ (indices : Digits) (before after : List C),
order = before ++ localOrder
(SourceQuestionOrder.tupleToQuestion (branch := branch) (n := n) (t := t) indices) ++ after →
ExactTargetOdometer.leafBudget rows repeats hn hbranch hrows δ input indices
(pack before tail) extra ≤ B)
(depth : Nat) (indices : Digits) (earlier suffix : List C)
(split : order = earlier ++ (ExactTargetOdometer.visits depth indices).flatMap
(fun tuple => localOrder
(SourceQuestionOrder.tupleToQuestion (branch := branch) (n := n) (t := t) tuple)) ++ suffix) :
ExactTargetOdometer.CostsAtMost rows repeats hn hbranch hrows δ input extra B
depth indices (pack earlier tail) := by
let chunk : Digits → List C := fun tuple => localOrder
(SourceQuestionOrder.tupleToQuestion (branch := branch) (n := n) (t := t) tuple)
change order = earlier ++ (ExactTargetOdometer.visits depth indices).flatMap chunk ++ suffix at split
induction depth generalizing indices earlier suffix with
| zero =>
change ExactTargetOdometer.leafBudget rows repeats hn hbranch hrows δ input indices
(pack earlier tail) extra ≤ B
apply bounded indices earlier suffix
simpa only [ExactTargetOdometer.visits, List.flatMap_singleton, chunk] using split
| succ depth ih =>
intro value
let beforeTuples : List Digits :=
((List.ofFn (fun a : Fin (NormalizedSourceInput.clauseCount input) => a)).take value.val).flatMap
(fun a => ExactTargetOdometer.visits depth
(ExactTargetOdometer.setCoordinate indices depth a))
let afterTuples : List Digits :=
((List.ofFn (fun a : Fin (NormalizedSourceInput.clauseCount input) => a)).drop
(value.val + 1)).flatMap
(fun a => ExactTargetOdometer.visits depth
(ExactTargetOdometer.setCoordinate indices depth a))
have decomposition : ExactTargetOdometer.visits (depth + 1) indices =
beforeTuples ++ ExactTargetOdometer.visits depth
(ExactTargetOdometer.setCoordinate indices depth value) ++ afterTuples :=
visits_split_at depth indices value
have beforeSplit : order = earlier ++ beforeTuples.flatMap chunk ++
((ExactTargetOdometer.visits depth
(ExactTargetOdometer.setCoordinate indices depth value)).flatMap chunk ++
afterTuples.flatMap chunk ++ suffix) := by
simpa only [decomposition, List.flatMap_append, List.append_assoc] using split
have prefixState :
ExactTargetOdometer.blockPrefix step depth indices value.val (pack earlier tail) =
pack (earlier ++ beforeTuples.flatMap chunk) tail := by
have folded := ExactTargetOdometer.fold_tupleSteps_dataOfPrefix
rows repeats hn hbranch hrows δ input beforeTuples earlier
((ExactTargetOdometer.visits depth
(ExactTargetOdometer.setCoordinate indices depth value)).flatMap chunk ++
afterTuples.flatMap chunk ++ suffix) tail beforeSplit
simpa only [beforeTuples, ExactTargetOdometer.blockPrefix,
ExactTargetOdometer.foldVisits, List.foldl_flatMap] using folded
rw [prefixState]
apply ih (ExactTargetOdometer.setCoordinate indices depth value)
(earlier ++ beforeTuples.flatMap chunk) (afterTuples.flatMap chunk ++ suffix)
simpa only [decomposition, List.flatMap_append, List.append_assoc] using split
theorem costsAtMost_full (extra : Extra → List Bool) (B : Nat) (tail : List Bool)
(bounded : ∀ (indices : Digits) (before after : List C),
order = before ++ localOrder
(SourceQuestionOrder.tupleToQuestion (branch := branch) (n := n) (t := t) indices) ++ after →
ExactTargetOdometer.leafBudget rows repeats hn hbranch hrows δ input indices
(pack before tail) extra ≤ B) :
ExactTargetOdometer.CostsAtMost rows repeats hn hbranch hrows δ input extra B width
(fun _ => ⟨0, NormalizedSourceInput.clauseCount_positive input⟩) (pack [] tail) := by
apply costsAtMost_of_chronological rows repeats hn hbranch hrows δ input extra B tail bounded
width (fun _ => ⟨0, NormalizedSourceInput.clauseCount_positive input⟩) [] []
simpa only [List.nil_append, List.append_nil] using
(ExactTargetOdometer.visits_completedOrder (t := t)
(NormalizedSourceInput.clauses input) rows repeats hn hbranch hrows δ
(fun _ => ⟨0, NormalizedSourceInput.clauseCount_positive input⟩)).symm
local notation "q" => FinitePreliminaryCompletion.alphabet branch n t δ
local notation "D" => SignedMultiplicity.denominator branch n t rows repeats hn hbranch hrows q
def timeBound (size : Nat) : Nat :=
(SignedCostEnvelope.bodyCoeff width D + 2 * width) *
(size + 1) ^ (SignedCostEnvelope.degree width + width)
def targetInTimePolynomial (tail : List Bool) (extra : Extra → List Bool) :
StateTransition.EvalsToInTime
(Turing.TM2.step (program (branch := branch) (n := n) (t := t)
(Extra := Extra) rows repeats hn hbranch hrows δ))
(configuration rows repeats hn hbranch hrows δ
(body (branch := branch) (n := n) (t := t) (Extra := Extra) rows repeats hn hbranch hrows δ)
input (fun _ => ⟨0, NormalizedSourceInput.clauseCount_positive input⟩) (pack [] tail) extra)
(some (configuration rows repeats hn hbranch hrows δ .exit input
(fun _ => ⟨0, NormalizedSourceInput.clauseCount_positive input⟩) (pack order tail) extra))
(timeBound (branch := branch) (n := n) (t := t)
rows repeats hn hbranch hrows δ (NormalizedSourceInput.bits input).length) := by
let B := SignedCostEnvelope.bodyCoeff width D *
((NormalizedSourceInput.bits input).length + 1) ^ SignedCostEnvelope.degree width
have bounded : CostsAtMost rows repeats hn hbranch hrows δ input extra B width
(fun _ => ⟨0, NormalizedSourceInput.clauseCount_positive input⟩) (pack [] tail) := by
apply costsAtMost_full rows repeats hn hbranch hrows δ input extra B tail
intro indices before after split
simpa only [leafBudget] using SignedRuntimeBounds.bodyBudget_le_polynomial (t := t)
input rows repeats hn hbranch hrows δ indices (baseTapes input indices extra)
(baseTapes_input input indices extra) before after tail split
let run := enumerateInTimeOfBound rows repeats hn hbranch hrows δ input
(pack [] tail) extra B bounded
refine { steps := run.steps, evals_in_steps := ?_, steps_le_m := ?_ }
· simpa only [result_dataOfPrefix] using run.evals_in_steps
· exact run.steps_le_m.trans
(SignedRuntimeBounds.outerBudget_le_polynomial (t := t) input rows repeats hn hbranch hrows δ)
end ChronologicalCosts
end PerfectCompleteness.ExactTargetOdometer
end OAI
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