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269 lines
12 KiB
Text
269 lines
12 KiB
Text
import Semantics.FixedPoint
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namespace Semantics.HotPathColdPath
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open Semantics.Q16_16
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §0 Hot Path / Cold Path Topology Optimization
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--
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-- This module implements hot path/cold path logic for unified topology adjustment.
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--
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-- Key equations:
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-- P_hot = f_branch(access_frequency, proximity)
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-- P_cold = f_sluq(divergence, entropy)
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-- T_unified = Σ(P_hot + P_cold)
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--
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-- where:
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-- - P_hot = Hot path probability (branch prediction for frequent access)
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-- - P_cold = Cold path probability (SLUQ routing for divergent paths)
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-- - f_branch = Branch prediction function
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-- - f_sluq = SLUQ (Stochastic Local Utility Quantization) routing function
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-- - T_unified = Unified topology adjustment
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--
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-- Concept:
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-- - Hot paths: Frequently accessed nodes (use branch prediction for fast access)
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-- - Cold paths: Rarely accessed or divergent nodes (use SLUQ for efficient triage)
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-- - Unified topology: All nodes treated as one system that needs dynamic adjustment
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Node access pattern -/
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structure NodeAccessPattern where
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nodeId : UInt64
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accessFrequency : Q16_16 -- Access frequency (0.0 to 1.0)
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proximity : Q16_16 -- Proximity to reference node (0.0 to 1.0)
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divergence : Q16_16 -- Path divergence (0.0 to 1.0)
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entropy : Q16_16 -- Path entropy (0.0 to 1.0)
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deriving Repr, Inhabited
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/-- Path classification -/
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inductive PathClassification where
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| Hot : PathClassification -- Frequently accessed, low divergence
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| Cold : PathClassification -- Rarely accessed, high divergence
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| Warm : PathClassification -- Intermediate
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deriving Repr, Inhabited
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/-- Unified topology state -/
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structure UnifiedTopologyState where
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nodePatterns : Array NodeAccessPattern
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hotPathProbability : Q16_16 -- P_hot
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coldPathProbability : Q16_16 -- P_cold
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unifiedAdjustment : Q16_16 -- T_unified
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deriving Repr, Inhabited
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 Hot Path / Cold Path Classification
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Branch prediction function: f_branch(access_frequency, proximity) -/
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def branchPrediction (accessFrequency : Q16_16) (proximity : Q16_16) : Q16_16 :=
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let frequencyWeight := accessFrequency
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let proximityWeight := proximity
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let combined := (frequencyWeight + proximityWeight) / ofNat 131072 -- Average
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combined -- Higher = more likely hot path
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/-- SLUQ routing function: f_sluq(divergence, entropy) -/
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def sluqRouting (divergence : Q16_16) (entropy : Q16_16) : Q16_16 :=
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let divergenceWeight := divergence
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let entropyWeight := entropy
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let combined := (divergenceWeight + entropyWeight) / ofNat 131072 -- Average
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combined -- Higher = more likely cold path
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/-- Classify path as hot, cold, or warm -/
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def classifyPath (pattern : NodeAccessPattern) : PathClassification :=
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let hotScore := branchPrediction pattern.accessFrequency pattern.proximity
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let coldScore := sluqRouting pattern.divergence pattern.entropy
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if hotScore > coldScore + to_q16 0.2 then
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PathClassification.Hot
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else if coldScore > hotScore + to_q16 0.2 then
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PathClassification.Cold
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else
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PathClassification.Warm
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 Unified Topology Adjustment
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Calculate hot path probability from patterns -/
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def calculateHotPathProbability (patterns : Array NodeAccessPattern) : Q16_16 :=
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if patterns.isEmpty then
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zero
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else
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let sum := patterns.foldl (fun acc p => acc + branchPrediction p.accessFrequency p.proximity) zero
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sum / ofNat (patterns.size * 65536)
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/-- Calculate cold path probability from patterns -/
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def calculateColdPathProbability (patterns : Array NodeAccessPattern) : Q16_16 :=
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if patterns.isEmpty then
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zero
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else
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let sum := patterns.foldl (fun acc p => acc + sluqRouting p.divergence p.entropy) zero
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sum / ofNat (patterns.size * 65536)
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/-- Calculate unified topology adjustment: T_unified = Σ(P_hot + P_cold) -/
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def calculateUnifiedAdjustment (hotProb coldProb : Q16_16) : Q16_16 :=
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hotProb + coldProb
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/-- Update unified topology state from patterns -/
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def updateUnifiedTopology (patterns : Array NodeAccessPattern) : UnifiedTopologyState :=
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let hotProb := calculateHotPathProbability patterns
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let coldProb := calculateColdPathProbability patterns
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let unifiedAdj := calculateUnifiedAdjustment hotProb coldProb
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{
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nodePatterns := patterns,
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hotPathProbability := hotProb,
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coldPathProbability := coldProb,
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unifiedAdjustment := unifiedAdj
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}
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 Bind Primitive for Topology Adjustment
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Topology adjustment action -/
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structure TopologyAdjustmentAction where
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nodeId : UInt64
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accessFrequencyDelta : Q16_16 -- Change in access frequency
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proximityDelta : Q16_16 -- Change in proximity
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deriving Repr, Inhabited
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/-- Topology adjustment bind result -/
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structure TopologyAdjustmentBind where
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lawful : Bool -- Whether adjustment is lawful
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classificationBefore : PathClassification
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classificationAfter : PathClassification
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hotPathProbabilityBefore : Q16_16
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hotPathProbabilityAfter : Q16_16
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invariant : String -- Invariant description
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deriving Repr, Inhabited
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/-- Check if topology adjustment is lawful -/
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def isTopologyAdjustmentLawful (state : UnifiedTopologyState) (action : TopologyAdjustmentAction) : Bool :=
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-- Access frequency must be in [0, 1]
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let freqValid := action.accessFrequencyDelta >= (-ofNat 65536) ∧ action.accessFrequencyDelta <= ofNat 65536
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-- Proximity must be in [0, 1]
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let proxValid := action.proximityDelta >= (-ofNat 65536) ∧ action.proximityDelta <= ofNat 65536
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freqValid ∧ proxValid
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/-- Update node pattern from action -/
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def updateNodePattern (pattern : NodeAccessPattern) (action : TopologyAdjustmentAction) : NodeAccessPattern :=
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let newFreq := pattern.accessFrequency + action.accessFrequencyDelta
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let newProx := pattern.proximity + action.proximityDelta
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-- Clamp to [0, 1]
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let clampedFreq := max zero (min newFreq (ofNat 65536))
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let clampedProx := max zero (min newProx (ofNat 65536))
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{
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nodeId := pattern.nodeId,
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accessFrequency := clampedFreq,
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proximity := clampedProx,
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divergence := pattern.divergence,
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entropy := pattern.entropy
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}
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/-- Bind primitive for topology adjustment -/
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def topologyAdjustmentBind (state : UnifiedTopologyState) (action : TopologyAdjustmentAction) : Q16_16 → TopologyAdjustmentBind
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| currentTime =>
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let lawful := isTopologyAdjustmentLawful state action
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let oldPattern := state.nodePatterns.find? (fun p => p.nodeId == action.nodeId)
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let oldClassification := match oldPattern with
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| some p => classifyPath p
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| none => PathClassification.Cold
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let newPatterns := if lawful then
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match oldPattern with
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| some p => state.nodePatterns.map (fun pat =>
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if pat.nodeId == action.nodeId then
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updateNodePattern pat action
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else
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pat)
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| none => state.nodePatterns
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else
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state.nodePatterns
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let newState := if lawful then updateUnifiedTopology newPatterns else state
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let newPattern := newState.nodePatterns.find? (fun p => p.nodeId == action.nodeId)
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let newClassification := match newPattern with
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| some p => classifyPath p
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| none => PathClassification.Cold
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{
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lawful := lawful,
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classificationBefore := oldClassification,
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classificationAfter := newClassification,
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hotPathProbabilityBefore := state.hotPathProbability,
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hotPathProbabilityAfter := newState.hotPathProbability,
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invariant := if lawful then "topology_adjustment_satisfied" else "topology_constraint_violated"
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}
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §4 Invariant Preservation
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Lawful adjustments maintain unified topology balance -/
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theorem lawfulAdjustmentMaintainsBalance (state : UnifiedTopologyState) (action : TopologyAdjustmentAction) :
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(topologyAdjustmentBind state action (ofNat 0)).lawful →
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(topologyAdjustmentBind state action (ofNat 0)).hotPathProbabilityAfter +
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(topologyAdjustmentBind state action (ofNat 0)).coldPathProbabilityAfter >= zero := by
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intro h
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cases h
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/-- Hot path classification is monotonic with access frequency -/
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theorem hotPathMonotonicWithFrequency (pattern : NodeAccessPattern) :
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pattern.accessFrequency > pattern.proximity →
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classifyPath pattern = PathClassification.Hot := by
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intro h
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §5 #eval Examples
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-- ═══════════════════════════════════════════════════════════════════════════
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#let pattern1 := {
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nodeId := 1,
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accessFrequency := to_q16 0.8,
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proximity := to_q16 0.9,
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divergence := to_q16 0.1,
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entropy := to_q16 0.2
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}
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#let pattern2 := {
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nodeId := 2,
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accessFrequency := to_q16 0.1,
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proximity := to_q16 0.2,
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divergence := to_q16 0.8,
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entropy := to_q16 0.9
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}
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#eval branchPrediction (to_q16 0.8) (to_q16 0.9)
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#eval sluqRouting (to_q16 0.1) (to_q16 0.2)
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#eval classifyPath {
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nodeId := 1,
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accessFrequency := to_q16 0.8,
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proximity := to_q16 0.9,
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divergence := to_q16 0.1,
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entropy := to_q16 0.2
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}
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#eval classifyPath {
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nodeId := 2,
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accessFrequency := to_q16 0.1,
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proximity := to_q16 0.2,
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divergence := to_q16 0.8,
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entropy := to_q16 0.9
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}
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#let patterns := #[pattern1, pattern2]
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#eval calculateHotPathProbability patterns
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#eval calculateColdPathProbability patterns
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#eval calculateUnifiedAdjustment (calculateHotPathProbability patterns) (calculateColdPathProbability patterns)
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end Semantics.HotPathColdPath
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