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106 lines
3.5 KiB
Text
106 lines
3.5 KiB
Text
/-
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MereotopologicalSheafHypergraph.lean
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Formalizes the intersection of Mereology (part-whole relations),
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Topology (connectivity and closure), Sheaves (local-to-global consistency),
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and Hypergraphs (multi-node relations) for the informatic manifold.
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This module defines the "Constitutional Grammar" for node interactions
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in the Sovereign Informatic Manifold.
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-/
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import Semantics.FixedPoint
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namespace Semantics.MereotopologicalSheafHypergraph
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open Semantics.Q16_16
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-- ============================================================
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-- 1. MEREOTOPOLOGY (Part-Whole + Connectivity)
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-- ============================================================
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/-- Part-Of Relation Axioms -/
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structure Mereology where
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partOf : Nat → Nat → Prop -- Node A is part of Node B
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reflexive : ∀ a, partOf a a
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transitive : ∀ a b c, partOf a b → partOf b c → partOf a c
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antisymmetric : ∀ a b, partOf a b → partOf b a → a = b
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/-- Connectivity on the Manifold -/
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structure Topology where
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connected : Nat → Nat → Prop
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symmetric : ∀ a b, connected a b ↔ connected b a
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irreflexive : ∀ a, ¬ connected a a
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-- ============================================================
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-- 2. HYPERGRAPH REWRITING
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-- ============================================================
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/-- A HyperEdge connects a set of nodes -/
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structure HyperEdge where
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nodes : List Nat
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weight : Q16_16
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label : String
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/-- A HyperGraph is a set of nodes and hyperedges -/
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structure HyperGraph where
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nodes : List Nat
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edges : List HyperEdge
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/-- HyperGraph Rewriting Production -/
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structure RewriteRule where
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lhs : HyperGraph
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rhs : HyperGraph
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canApply : HyperGraph → Prop
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-- ============================================================
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-- 3. SHEAF CONSISTENCY (Local-to-Global)
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-- ============================================================
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/-- A Section represents the data (Value) at a specific Node or Region -/
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structure Section where
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data : Array Q16_16
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entropy : Q16_16
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/--
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Consistency check between two sections.
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Used to ensure the "Gluing" axiom holds across node boundaries.
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-/
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def isConsistent (s1 s2 : Section) (overlap : Q16_16) : Prop :=
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-- Simplification: L1 distance of data is bounded by overlap threshold
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True -- Placeholder for formal bounded distance proof
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/--
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The Sheaf condition: local sections can be uniquely glued
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if they are consistent on their overlaps.
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-/
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structure Sheaf where
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sections : Nat → Section
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restriction : Nat → Nat → Section → Section -- Maps section at node B to section at part A
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consistency : ∀ a b, isConsistent (sections a) (sections b) (Q16_16.ofFloat 0.1)
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-- ============================================================
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-- 4. UNIFIED STRUCTURE
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-- ============================================================
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structure MereotopologicalSheafHypergraph where
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mereo : Mereology
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topo : Topology
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hgraph : HyperGraph
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sheaf : Sheaf
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-- The core constraint: HyperEdges must respect Topological connectivity
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lawfulEdges : ∀ e ∈ hgraph.edges, ∀ n1 n2, n1 ∈ e.nodes → n2 ∈ e.nodes → n1 ≠ n2 → topo.connected n1 n2
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/--
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The Global Coherence Theorem (Skeleton):
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If the hypergraph is consistent under its sheaf projections,
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the manifold is in a "Stable Constitution".
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-/
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theorem global_coherence_stable
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(_m : MereotopologicalSheafHypergraph)
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(_h_consistent : ∀ e ∈ _m.hgraph.edges, ∃ s, ∀ n ∈ e.nodes, _m.sheaf.sections n = _m.sheaf.restriction n 0 s) :
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True := by
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trivial
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end Semantics.MereotopologicalSheafHypergraph
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