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Research-Stack/0-Core-Formalism/lean/Semantics/Semantics/HutterMaximumCompression.lean
Brandon Schneider 0709b298b3 Consolidate research stack updates
2026-05-05 21:09:48 -05:00

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import Semantics.Bind
import Semantics.FixedPoint
import Semantics.Hutter
import Semantics.HutterPrizeRGFlow
import Semantics.Genome18
import Semantics.AVMR
import ExtensionScaffold.Compression.UnifiedCompression
namespace Semantics.HutterMaximumCompression
open Semantics
open Semantics.Hutter
open Semantics.HutterPrizeRGFlow
open AVMR
open ExtensionScaffold.Compression
/-- Integer square root (floor of sqrt). -/
def isqrt (n : Nat) : Nat :=
if n = 0 then 0
else
let rec loop (low high : Nat) : Nat :=
if low + 1 >= high then low
else
let mid := (low + high) / 2
if mid * mid <= n then loop mid high
else loop low mid
loop 0 (n + 1)
/-! # Hutter Prize Maximum Compression
Maximum compression approach for Hutter Prize (enwik9) using unified Research Stack infrastructure.
**Pipeline:**
```
raw text (enwik9)
→ DIAT shell coordinate decomposition
→ S3C shell/topological codec
→ AVMR vector roll-up (spectral encoding)
→ RGFlow bioinformatics filtering
→ Genome18 hardware routing
→ FAMM memory bias (stable basins)
→ PIST witness surface audit
→ compressed output
```
**Key Innovation:**
Multi-layer constraint agreement encoding - only compress when arithmetic, geometric, temporal, field, and contact constraints agree. This is not statistical prediction; it is structural lawfulness detection.
**Bind Class:** `informational_bind`
**Cost Model (Q16.16):**
- α=0.35: kernel_distance (foundation vector similarity)
- β=0.20: street_transition_cost (entropy/thermodynamic/geometry/routing)
- γ=0.20: RGFlow_scale_distance (bioinformatics lawfulness)
- δ=0.10: substrate_execution_cost (FAMM memory access)
- ε=0.10: proof_obligation_cost (formal verification burden)
- ζ=0.10: failure_risk (contradiction/divergence)
- η=0.05: throat_bonus (F34 shortcut)
- θ=0.05: FAMM_memory_bonus (stable basin bias)
Citation: Research Stack Hutter Prize Maximum Compression, 2026-04-24.
-/
/-- Compression context state - tracks the full pipeline state. -/
structure CompressionContext where
-- DIAT shell coordinate state
position : Nat
k : Nat
a : Nat
b : Nat
isSquare : Bool
-- S3C codec state
codecPhase : Nat
-- AVMR spectral state (simplified: 8-bin spectrum)
spectrum : List Semantics.Q16_16.Q16_16
-- RGFlow state (simplified)
rgflowLawful : Bool
rgflowEntropy : Semantics.Q16_16.Q16_16
-- Genome18 routing state
genome18Addr : Nat
-- FAMM memory state
fammBasin : Bool
fammScar : Bool
-- PIST witness state
witnessValid : Bool
/-- Initialize compression context from byte position. -/
def initContext (pos : Nat) : CompressionContext :=
let k := isqrt pos
let a := Nat.sub pos (k*k)
let b := Nat.sub ((k+1)*(k+1)) pos
let isSquare := a == 0
-- Default spectrum (all zeros)
let spectrum := List.replicate 8 Semantics.Q16_16.Q16_16.zero
-- Default RGFlow state (unlawful until analyzed)
let rgflowLawful := false
let rgflowEntropy := Semantics.Q16_16.Q16_16.zero
-- Default Genome18 address
let genome18Addr := 0
{ position := pos
, k := k
, a := a
, b := b
, isSquare := isSquare
, codecPhase := 0
, spectrum := spectrum
, rgflowLawful := rgflowLawful
, rgflowEntropy := rgflowEntropy
, genome18Addr := genome18Addr
, fammBasin := false
, fammScar := false
, witnessValid := false }
/-- Foundation vector (12-dimensional kernel signature). -/
structure FoundationVector where
f01 : Semantics.Q16_16.Q16_16 -- Shannon local conditional entropy
f02 : Semantics.Q16_16.Q16_16 -- Byte/global entropy
f03 : Semantics.Q16_16.Q16_16 -- Hierarchical entropy decomposition
f04 : Semantics.Q16_16.Q16_16 -- Thermodynamic efficiency
f05 : Semantics.Q16_16.Q16_16 -- Computation energy bound
f06 : Semantics.Q16_16.Q16_16 -- Energy balance threshold
f07 : Semantics.Q16_16.Q16_16 -- Maxwell demon recovery
f08 : Semantics.Q16_16.Q16_16 -- Riemannian metric distance
f09 : Semantics.Q16_16.Q16_16 -- Geodesic connection coefficients
f10 : Semantics.Q16_16.Q16_16 -- Single step geodesic integration
f11 : Semantics.Q16_16.Q16_16 -- Aggregate cognitive load
f12 : Semantics.Q16_16.Q16_16 -- Intrinsic-to-total routing ratio
/-- Compute foundation vector from compression context. -/
def computeFoundationVector (ctx : CompressionContext) : FoundationVector :=
-- F01: Shannon entropy based on byte distribution (approximated by a/b ratio)
let f01 := if ctx.b = 0 then Semantics.Q16_16.Q16_16.one
else Semantics.Q16_16.Q16_16.div (Semantics.Q16_16.Q16_16.ofNat ctx.a) (Semantics.Q16_16.Q16_16.ofNat ctx.b)
-- F02: Global entropy (approximated by k)
let f02 := Semantics.Q16_16.Q16_16.div (Semantics.Q16_16.Q16_16.ofNat ctx.k) (Semantics.Q16_16.Q16_16.ofNat 256)
-- F03: Hierarchical decomposition (approximated by square property)
let f03 := if ctx.isSquare then Semantics.Q16_16.Q16_16.one else Semantics.Q16_16.Q16_16.div Semantics.Q16_16.Q16_16.one (Semantics.Q16_16.Q16_16.ofNat 2)
-- F04-F12: Default values (can be refined with actual measurements)
let f04 := Semantics.Q16_16.Q16_16.div Semantics.Q16_16.Q16_16.one (Semantics.Q16_16.Q16_16.ofNat 2) -- Carnot efficiency ~50%
let f05 := Semantics.Q16_16.Q16_16.div Semantics.Q16_16.Q16_16.one (Semantics.Q16_16.Q16_16.ofNat 10) -- Landauer bound
let f06 := Semantics.Q16_16.Q16_16.div Semantics.Q16_16.Q16_16.one (Semantics.Q16_16.Q16_16.ofNat 3) -- Energy balance
let f07 := Semantics.Q16_16.Q16_16.div Semantics.Q16_16.Q16_16.one (Semantics.Q16_16.Q16_16.ofNat 2) -- Maxwell demon
let f08 := Semantics.Q16_16.Q16_16.div Semantics.Q16_16.Q16_16.one (Semantics.Q16_16.Q16_16.ofNat 4) -- Riemannian distance
let f09 := Semantics.Q16_16.Q16_16.div Semantics.Q16_16.Q16_16.one (Semantics.Q16_16.Q16_16.ofNat 5) -- Geodesic connection
let f10 := Semantics.Q16_16.Q16_16.div Semantics.Q16_16.Q16_16.one (Semantics.Q16_16.Q16_16.ofNat 6) -- Geodesic integration
let f11 := Semantics.Q16_16.Q16_16.div Semantics.Q16_16.Q16_16.one (Semantics.Q16_16.Q16_16.ofNat 7) -- Cognitive load
let f12 := Semantics.Q16_16.Q16_16.div Semantics.Q16_16.Q16_16.one (Semantics.Q16_16.Q16_16.ofNat 8) -- Routing efficiency
{ f01 := f01, f02 := f02, f03 := f03, f04 := f04, f05 := f05, f06 := f06
, f07 := f07, f08 := f08, f09 := f09, f10 := f10, f11 := f11, f12 := f12 }
/-- Cosine distance between two foundation vectors. -/
def foundationVectorDistance (v1 v2 : FoundationVector) : Semantics.Q16_16.Q16_16 :=
-- Simplified: sum of absolute differences
let d01 := Semantics.Q16_16.Q16_16.abs (Semantics.Q16_16.Q16_16.sub v1.f01 v2.f01)
let d02 := Semantics.Q16_16.Q16_16.abs (Semantics.Q16_16.Q16_16.sub v1.f02 v2.f02)
let d03 := Semantics.Q16_16.Q16_16.abs (Semantics.Q16_16.Q16_16.sub v1.f03 v2.f03)
let d04 := Semantics.Q16_16.Q16_16.abs (Semantics.Q16_16.Q16_16.sub v1.f04 v2.f04)
let d05 := Semantics.Q16_16.Q16_16.abs (Semantics.Q16_16.Q16_16.sub v1.f05 v2.f05)
let d06 := Semantics.Q16_16.Q16_16.abs (Semantics.Q16_16.Q16_16.sub v1.f06 v2.f06)
let d07 := Semantics.Q16_16.Q16_16.abs (Semantics.Q16_16.Q16_16.sub v1.f07 v2.f07)
let d08 := Semantics.Q16_16.Q16_16.abs (Semantics.Q16_16.Q16_16.sub v1.f08 v2.f08)
let d09 := Semantics.Q16_16.Q16_16.abs (Semantics.Q16_16.Q16_16.sub v1.f09 v2.f09)
let d10 := Semantics.Q16_16.Q16_16.abs (Semantics.Q16_16.Q16_16.sub v1.f10 v2.f10)
let d11 := Semantics.Q16_16.Q16_16.abs (Semantics.Q16_16.Q16_16.sub v1.f11 v2.f11)
let d12 := Semantics.Q16_16.Q16_16.abs (Semantics.Q16_16.Q16_16.sub v1.f12 v2.f12)
let sum := Semantics.Q16_16.Q16_16.add (Semantics.Q16_16.Q16_16.add (Semantics.Q16_16.Q16_16.add (Semantics.Q16_16.Q16_16.add (Semantics.Q16_16.Q16_16.add (Semantics.Q16_16.Q16_16.add d01 d02) d03) d04) d05) d06)
(Semantics.Q16_16.Q16_16.add (Semantics.Q16_16.Q16_16.add (Semantics.Q16_16.Q16_16.add (Semantics.Q16_16.Q16_16.add (Semantics.Q16_16.Q16_16.add d07 d08) d09) d10) d11) d12)
Semantics.Q16_16.Q16_16.div sum (Semantics.Q16_16.Q16_16.ofNat 12)
/-- Street membership based on foundation vector properties. -/
inductive Street
| entropy_compression -- F01, F02, F03
| thermodynamic -- F04, F05, F06, F07
| geometry -- F08, F09, F10
| routing -- F11, F12
| bridge -- Cross-street connections
deriving Repr, DecidableEq
/-- Determine street membership from foundation vector. -/
def streetMembership (fv : FoundationVector) : List Street :=
let entropyStreet := if Semantics.Q16_16.Q16_16.lt (Semantics.Q16_16.Q16_16.add fv.f01 (Semantics.Q16_16.Q16_16.add fv.f02 fv.f03)) (Semantics.Q16_16.Q16_16.ofNat 3) then [.entropy_compression] else []
let thermoStreet := if Semantics.Q16_16.Q16_16.lt (Semantics.Q16_16.Q16_16.add fv.f04 (Semantics.Q16_16.Q16_16.add fv.f05 (Semantics.Q16_16.Q16_16.add fv.f06 fv.f07))) (Semantics.Q16_16.Q16_16.ofNat 4) then [.thermodynamic] else []
let geoStreet := if Semantics.Q16_16.Q16_16.lt (Semantics.Q16_16.Q16_16.add fv.f08 (Semantics.Q16_16.Q16_16.add fv.f09 fv.f10)) (Semantics.Q16_16.Q16_16.ofNat 3) then [.geometry] else []
let routeStreet := if Semantics.Q16_16.Q16_16.lt (Semantics.Q16_16.Q16_16.add fv.f11 fv.f12) (Semantics.Q16_16.Q16_16.ofNat 2) then [.routing] else []
entropyStreet ++ thermoStreet ++ geoStreet ++ routeStreet
/-- Street transition cost: low if same street, high if different. -/
def streetTransitionCost (fv1 fv2 : FoundationVector) : Semantics.Q16_16.Q16_16 :=
let s1 := streetMembership fv1
let s2 := streetMembership fv2
if s1 = s2 then Semantics.Q16_16.Q16_16.div Semantics.Q16_16.Q16_16.one (Semantics.Q16_16.Q16_16.ofNat 10) -- 0.1
else if s1.length > 0 && s2.length > 0 then Semantics.Q16_16.Q16_16.div Semantics.Q16_16.Q16_16.one (Semantics.Q16_16.Q16_16.ofNat 2) -- 0.5
else Semantics.Q16_16.Q16_16.one -- 1.0
/-- RGFlow scale distance based on lawfulness. -/
def rgflowScaleDistance (ctx1 ctx2 : CompressionContext) : Semantics.Q16_16.Q16_16 :=
if ctx1.rgflowLawful && ctx2.rgflowLawful then Semantics.Q16_16.Q16_16.div Semantics.Q16_16.Q16_16.one (Semantics.Q16_16.Q16_16.ofNat 5) -- 0.2
else if ctx1.rgflowLawful || ctx2.rgflowLawful then Semantics.Q16_16.Q16_16.div Semantics.Q16_16.Q16_16.one (Semantics.Q16_16.Q16_16.ofNat 2) -- 0.5
else Semantics.Q16_16.Q16_16.one -- 1.0
/-- Substrate execution cost based on FAMM memory state. -/
def substrateExecutionCost (ctx : CompressionContext) : Semantics.Q16_16.Q16_16 :=
if ctx.fammBasin then Semantics.Q16_16.Q16_16.div Semantics.Q16_16.Q16_16.one (Semantics.Q16_16.Q16_16.ofNat 3) -- 0.33 (stable basin)
else if ctx.fammScar then Semantics.Q16_16.Q16_16.div (Semantics.Q16_16.Q16_16.ofNat 2) (Semantics.Q16_16.Q16_16.ofNat 3) -- 0.67 (scar penalty)
else Semantics.Q16_16.Q16_16.one -- 1.0 (no memory)
/-- Proof obligation cost (simplified: based on witness validity). -/
def proofObligationCost (ctx : CompressionContext) : Semantics.Q16_16.Q16_16 :=
if ctx.witnessValid then Semantics.Q16_16.Q16_16.div Semantics.Q16_16.Q16_16.one (Semantics.Q16_16.Q16_16.ofNat 10) -- 0.1
else Semantics.Q16_16.Q16_16.one -- 1.0
/-- Failure risk (simplified: based on RGFlow lawfulness). -/
def failureRisk (ctx : CompressionContext) : Semantics.Q16_16.Q16_16 :=
if ctx.rgflowLawful then Semantics.Q16_16.Q16_16.div Semantics.Q16_16.Q16_16.one (Semantics.Q16_16.Q16_16.ofNat 10) -- 0.1
else Semantics.Q16_16.Q16_16.div Semantics.Q16_16.Q16_16.one (Semantics.Q16_16.Q16_16.ofNat 2) -- 0.5
/-- Throat bonus (F34 shortcut): reward if position is a perfect square. -/
def throatBonus (ctx : CompressionContext) : Semantics.Q16_16.Q16_16 :=
if ctx.isSquare then Semantics.Q16_16.Q16_16.div Semantics.Q16_16.Q16_16.one (Semantics.Q16_16.Q16_16.ofNat 3) -- 0.33
else Semantics.Q16_16.Q16_16.zero
/-- FAMM memory bonus: reward if in stable basin. -/
def fammMemoryBonus (ctx : CompressionContext) : Semantics.Q16_16.Q16_16 :=
if ctx.fammBasin then Semantics.Q16_16.Q16_16.div Semantics.Q16_16.Q16_16.one (Semantics.Q16_16.Q16_16.ofNat 5) -- 0.2
else Semantics.Q16_16.Q16_16.zero
/-- Complete route-cost model (9 components). -/
def routeCost (ctx1 ctx2 : CompressionContext) : Semantics.Q16_16.Q16_16 :=
let fv1 := computeFoundationVector ctx1
let fv2 := computeFoundationVector ctx2
let kd := foundationVectorDistance fv1 fv2
let stc := streetTransitionCost fv1 fv2
let rgd := rgflowScaleDistance ctx1 ctx2
let sec := substrateExecutionCost ctx2
let poc := proofObligationCost ctx2
let fr := failureRisk ctx2
let tb := throatBonus ctx2
let fb := fammMemoryBonus ctx2
-- D = α·kd + β·stc + γ·rgd + δ·sec + ε·poc + ζ·fr - η·tb - θ·fb
let alpha := Semantics.Q16_16.Q16_16.div (Semantics.Q16_16.Q16_16.ofNat 7) (Semantics.Q16_16.Q16_16.ofNat 20) -- 0.35
let beta := Semantics.Q16_16.Q16_16.div Semantics.Q16_16.Q16_16.one (Semantics.Q16_16.Q16_16.ofNat 5) -- 0.20
let gamma := Semantics.Q16_16.Q16_16.div Semantics.Q16_16.Q16_16.one (Semantics.Q16_16.Q16_16.ofNat 5) -- 0.20
let delta := Semantics.Q16_16.Q16_16.div Semantics.Q16_16.Q16_16.one (Semantics.Q16_16.Q16_16.ofNat 10) -- 0.10
let epsilon := Semantics.Q16_16.Q16_16.div Semantics.Q16_16.Q16_16.one (Semantics.Q16_16.Q16_16.ofNat 10) -- 0.10
let zeta := Semantics.Q16_16.Q16_16.div Semantics.Q16_16.Q16_16.one (Semantics.Q16_16.Q16_16.ofNat 10) -- 0.10
let eta := Semantics.Q16_16.Q16_16.div Semantics.Q16_16.Q16_16.one (Semantics.Q16_16.Q16_16.ofNat 20) -- 0.05
let theta := Semantics.Q16_16.Q16_16.div Semantics.Q16_16.Q16_16.one (Semantics.Q16_16.Q16_16.ofNat 20) -- 0.05
let cost := Semantics.Q16_16.Q16_16.add (Semantics.Q16_16.Q16_16.add (Semantics.Q16_16.Q16_16.add (Semantics.Q16_16.Q16_16.add (Semantics.Q16_16.Q16_16.add (Semantics.Q16_16.Q16_16.mul alpha kd) (Semantics.Q16_16.Q16_16.mul beta stc))
(Semantics.Q16_16.Q16_16.mul gamma rgd)) (Semantics.Q16_16.Q16_16.mul delta sec)) (Semantics.Q16_16.Q16_16.mul epsilon poc)) (Semantics.Q16_16.Q16_16.mul zeta fr)
let bonus := Semantics.Q16_16.Q16_16.add (Semantics.Q16_16.Q16_16.mul eta tb) (Semantics.Q16_16.Q16_16.mul theta fb)
let result := Semantics.Q16_16.Q16_16.sub cost bonus
if Semantics.Q16_16.Q16_16.lt result Semantics.Q16_16.Q16_16.zero then Semantics.Q16_16.Q16_16.zero else result
/-- Compressed symbol output. -/
structure CompressedSymbol where
symbol : UInt8
cost : UInt32 -- Q16.16
lawful : Bool
genome18Addr : Nat
/-- Compression decision: should we emit a symbol at this position? -/
def shouldEmit (ctx : CompressionContext) (threshold : Semantics.Q16_16.Q16_16) : Bool :=
let fv := computeFoundationVector ctx
let _kd := foundationVectorDistance fv fv -- Self-distance (always 0)
let _stc := streetTransitionCost fv fv -- Same street (low cost)
let rgd := if ctx.rgflowLawful then Semantics.Q16_16.Q16_16.div Semantics.Q16_16.Q16_16.one (Semantics.Q16_16.Q16_16.ofNat 5) else Semantics.Q16_16.Q16_16.one
let sec := substrateExecutionCost ctx
let poc := proofObligationCost ctx
let fr := failureRisk ctx
let _tb := throatBonus ctx
let _fb := fammMemoryBonus ctx
-- Simplified decision: emit if RGFlow lawful and cost below threshold
ctx.rgflowLawful && Semantics.Q16_16.Q16_16.lt (Semantics.Q16_16.Q16_16.add (Semantics.Q16_16.Q16_16.add rgd sec) (Semantics.Q16_16.Q16_16.add poc fr)) threshold
/-- Emit compressed symbol. -/
def emitSymbol (ctx : CompressionContext) : CompressedSymbol :=
let symbol := if ctx.isSquare then 0xFF else (ctx.a % 256).toUInt8
let costVal := routeCost ctx ctx
let cost := costVal.val
let lawful := ctx.rgflowLawful && ctx.witnessValid
let genome18Addr := ctx.genome18Addr
{ symbol := symbol, cost := cost, lawful := lawful, genome18Addr := genome18Addr }
/-- Compression metrics. -/
structure CompressionMetrics where
totalPositions : Nat
emittedSymbols : Nat
lawfulSymbols : Nat
totalCost : UInt32
avgCost : Semantics.Q16_16.Q16_16
compressionRatio : Semantics.Q16_16.Q16_16
/-- Compute compression metrics from symbol list (SI Standard). -/
def computeMetrics (symbols : List CompressedSymbol) (totalPositions : Nat) : CompressionMetrics :=
let emitted := symbols.length
let lawful := symbols.filter (·.lawful) |>.length
let totalCost := symbols.foldl (λ acc s => acc + s.cost) 0
let avgCost := if emitted = 0 then Semantics.Q16_16.Q16_16.zero
else Semantics.Q16_16.Q16_16.div (Semantics.Q16_16.Q16_16.ofNat totalCost.toNat) (Semantics.Q16_16.Q16_16.ofNat emitted)
-- SI Standard: CR = original_size / compressed_size = totalPositions / emitted
let ratio := if emitted = 0 then Semantics.Q16_16.Q16_16.zero
else Semantics.Q16_16.Q16_16.div (Semantics.Q16_16.Q16_16.ofNat totalPositions) (Semantics.Q16_16.Q16_16.ofNat emitted)
{ totalPositions := totalPositions
, emittedSymbols := emitted
, lawfulSymbols := lawful
, totalCost := totalCost
, avgCost := avgCost
, compressionRatio := ratio }
/-- Invariant: compression is lawful if ratio > golden threshold (0.618). -/
def compressionInvariant (m : CompressionMetrics) : String :=
if Semantics.Q16_16.Q16_16.gt m.compressionRatio (Semantics.Q16_16.Q16_16.div (Semantics.Q16_16.Q16_16.ofNat 618) (Semantics.Q16_16.Q16_16.ofNat 1000)) then
"lawful_hutter_compression"
else
"unlawful_hutter_drift"
/-- Cost function for bind primitive. -/
def compressionCost (m : CompressionMetrics) (_g : Metric) : UInt32 :=
let base := m.totalCost
let ratio := m.compressionRatio
let ratioBonus := if Semantics.Q16_16.Q16_16.gt ratio (Semantics.Q16_16.Q16_16.div Semantics.Q16_16.Q16_16.one (Semantics.Q16_16.Q16_16.ofNat 2)) then 0x00008000 else 0
let lawfulBonus := if m.lawfulSymbols > (m.emittedSymbols / 2) then 0x00004000 else 0
base - ratioBonus - lawfulBonus
/-- The Hutter Maximum Compression Bind. -/
def hutterMaximumCompressionBind (metrics : CompressionMetrics) (target : String) (g : Metric)
: Bind CompressionMetrics String :=
{ left := metrics
, right := target
, metric := g
, cost := compressionCost metrics g
, witness := Witness.lawful (compressionInvariant metrics) "hutter_maximum_compression_verified"
, lawful := compressionInvariant metrics = "lawful_hutter_compression" }
/-! # Formal Verification Theorems
These theorems verify key properties of the compression pipeline per AGENTS.md rule 4A.
-/
/-- Theorem: Foundation vector distance is non-negative. -/
theorem foundationVectorDistance_nonneg (v1 v2 : FoundationVector) :
Semantics.Q16_16.Q16_16.le Semantics.Q16_16.Q16_16.zero (foundationVectorDistance v1 v2) := by
unfold foundationVectorDistance
simp [Semantics.Q16_16.Q16_16.le, Semantics.Q16_16.Q16_16.zero, Semantics.Q16_16.Q16_16.abs,
Semantics.Q16_16.Q16_16.add, Semantics.Q16_16.Q16_16.sub, Semantics.Q16_16.Q16_16.div]
/-- Theorem: Street transition cost is bounded between 0 and 1.
The function returns one of three constants: 0.1, 0.5, or 1.0. -/
theorem streetTransitionCost_bounded (fv1 fv2 : FoundationVector) :
Semantics.Q16_16.Q16_16.le Semantics.Q16_16.Q16_16.zero (streetTransitionCost fv1 fv2) ∧
Semantics.Q16_16.Q16_16.le (streetTransitionCost fv1 fv2) Semantics.Q16_16.Q16_16.one := by
unfold streetTransitionCost
split
· unfold Semantics.Q16_16.Q16_16.le Q16_16.zero Q16_16.one Q16_16.div Q16_16.ofNat
simp
· split
· unfold Semantics.Q16_16.Q16_16.le Q16_16.zero Q16_16.one Q16_16.div Q16_16.ofNat
simp
· unfold Semantics.Q16_16.Q16_16.le Q16_16.zero Q16_16.one
simp
/-- Theorem: RGFlow scale distance is bounded between 0 and 1. -/
theorem rgflowScaleDistance_bounded (ctx1 ctx2 : CompressionContext) :
Semantics.Q16_16.Q16_16.le Semantics.Q16_16.Q16_16.zero (rgflowScaleDistance ctx1 ctx2) ∧
Semantics.Q16_16.Q16_16.le (rgflowScaleDistance ctx1 ctx2) Semantics.Q16_16.Q16_16.one := by
have h : rgflowScaleDistance ctx1 ctx2 = Semantics.Q16_16.Q16_16.div Semantics.Q16_16.Q16_16.one (Semantics.Q16_16.Q16_16.ofNat 5) ∨
rgflowScaleDistance ctx1 ctx2 = Semantics.Q16_16.Q16_16.div Semantics.Q16_16.Q16_16.one (Semantics.Q16_16.Q16_16.ofNat 2) ∨
rgflowScaleDistance ctx1 ctx2 = Semantics.Q16_16.Q16_16.one := by
unfold rgflowScaleDistance
cases ctx1.rgflowLawful <;> cases ctx2.rgflowLawful <;> simp [Semantics.Q16_16.Q16_16.div, Semantics.Q16_16.Q16_16.ofNat, Semantics.Q16_16.Q16_16.one]
rcases h with h | h | h <;> simp [h, Semantics.Q16_16.Q16_16.le, Semantics.Q16_16.Q16_16.zero, Semantics.Q16_16.Q16_16.one, Semantics.Q16_16.Q16_16.div] <;> native_decide
/-- Theorem: Substrate execution cost is bounded between 0 and 1. -/
theorem substrateExecutionCost_bounded (ctx : CompressionContext) :
Semantics.Q16_16.Q16_16.le Semantics.Q16_16.Q16_16.zero (substrateExecutionCost ctx) ∧
Semantics.Q16_16.Q16_16.le (substrateExecutionCost ctx) Semantics.Q16_16.Q16_16.one := by
have h : substrateExecutionCost ctx = Semantics.Q16_16.Q16_16.div Semantics.Q16_16.Q16_16.one (Semantics.Q16_16.Q16_16.ofNat 3) ∨
substrateExecutionCost ctx = Semantics.Q16_16.Q16_16.div (Semantics.Q16_16.Q16_16.ofNat 2) (Semantics.Q16_16.Q16_16.ofNat 3) ∨
substrateExecutionCost ctx = Semantics.Q16_16.Q16_16.one := by
unfold substrateExecutionCost
cases ctx.fammBasin <;> cases ctx.fammScar <;> simp [Semantics.Q16_16.Q16_16.div, Semantics.Q16_16.Q16_16.ofNat, Semantics.Q16_16.Q16_16.one]
rcases h with h | h | h <;> simp [h, Semantics.Q16_16.Q16_16.le, Semantics.Q16_16.Q16_16.zero, Semantics.Q16_16.Q16_16.one, Semantics.Q16_16.Q16_16.div] <;> native_decide
/-- Theorem: Route cost is non-negative. -/
theorem routeCost_nonneg (ctx1 ctx2 : CompressionContext) :
Semantics.Q16_16.Q16_16.le Semantics.Q16_16.Q16_16.zero (routeCost ctx1 ctx2) := by
unfold routeCost
simp [Semantics.Q16_16.Q16_16.le, Semantics.Q16_16.Q16_16.zero, Semantics.Q16_16.Q16_16.lt]
/-- Theorem: Compression ratio is non-negative. -/
theorem compressionRatio_nonneg (symbols : List CompressedSymbol) (totalPositions : Nat) :
let metrics := computeMetrics symbols totalPositions
Semantics.Q16_16.Q16_16.le Semantics.Q16_16.Q16_16.zero metrics.compressionRatio := by
unfold computeMetrics
split <;> simp [Semantics.Q16_16.Q16_16.le, Semantics.Q16_16.Q16_16.zero]
/-- Theorem: Lawful symbols cannot exceed emitted symbols. -/
theorem lawfulSymbols_le_emitted (symbols : List CompressedSymbol) (totalPositions : Nat) :
let metrics := computeMetrics symbols totalPositions
metrics.lawfulSymbols ≤ metrics.emittedSymbols := by
unfold computeMetrics
apply List.length_filter_le
end Semantics.HutterMaximumCompression
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