/- 
  MereotopologicalSheafHypergraph.lean
  
  Formalizes the intersection of Mereology (part-whole relations), 
  Topology (connectivity and closure), Sheaves (local-to-global consistency),
  and Hypergraphs (multi-node relations) for the informatic manifold.

  This module defines the "Constitutional Grammar" for node interactions
  in the Sovereign Informatic Manifold.
-/

import Semantics.FixedPoint

namespace Semantics.MereotopologicalSheafHypergraph

open Semantics.Q16_16

-- ============================================================
-- 1. MEREOTOPOLOGY (Part-Whole + Connectivity)
-- ============================================================

/-- Part-Of Relation Axioms -/
structure Mereology where
  partOf : Nat → Nat → Prop  -- Node A is part of Node B
  reflexive : ∀ a, partOf a a
  transitive : ∀ a b c, partOf a b → partOf b c → partOf a c
  antisymmetric : ∀ a b, partOf a b → partOf b a → a = b

/-- Connectivity on the Manifold -/
structure Topology where
  connected : Nat → Nat → Prop
  symmetric : ∀ a b, connected a b ↔ connected b a
  irreflexive : ∀ a, ¬ connected a a

-- ============================================================
-- 2. HYPERGRAPH REWRITING
-- ============================================================

/-- A HyperEdge connects a set of nodes -/
structure HyperEdge where
  nodes : List Nat
  weight : Q16_16
  label : String

/-- A HyperGraph is a set of nodes and hyperedges -/
structure HyperGraph where
  nodes : List Nat
  edges : List HyperEdge

/-- HyperGraph Rewriting Production -/
structure RewriteRule where
  lhs : HyperGraph
  rhs : HyperGraph
  canApply : HyperGraph → Prop

-- ============================================================
-- 3. SHEAF CONSISTENCY (Local-to-Global)
-- ============================================================

/-- A Section represents the data (Value) at a specific Node or Region -/
structure Section where
  data : Array Q16_16
  entropy : Q16_16

/-- 
  Consistency check between two sections.
  Used to ensure the "Gluing" axiom holds across node boundaries.
-/
def isConsistent (s1 s2 : Section) (overlap : Q16_16) : Prop :=
  -- Simplification: L1 distance of data is bounded by overlap threshold
  True -- Placeholder for formal bounded distance proof

/-- 
  The Sheaf condition: local sections can be uniquely glued 
  if they are consistent on their overlaps.
-/
structure Sheaf where
  sections : Nat → Section
  restriction : Nat → Nat → Section → Section  -- Maps section at node B to section at part A
  consistency : ∀ a b, isConsistent (sections a) (sections b) (Q16_16.ofFloat 0.1)

-- ============================================================
-- 4. UNIFIED STRUCTURE
-- ============================================================

structure MereotopologicalSheafHypergraph where
  mereo : Mereology
  topo : Topology
  hgraph : HyperGraph
  sheaf : Sheaf
  
  -- The core constraint: HyperEdges must respect Topological connectivity
  lawfulEdges : ∀ e ∈ hgraph.edges, ∀ n1 n2, n1 ∈ e.nodes → n2 ∈ e.nodes → n1 ≠ n2 → topo.connected n1 n2

/-- 
  The Global Coherence Theorem (Skeleton):
  If the hypergraph is consistent under its sheaf projections, 
  the manifold is in a "Stable Constitution".
-/
theorem global_coherence_stable
  (_m : MereotopologicalSheafHypergraph)
  (_h_consistent : ∀ e ∈ _m.hgraph.edges, ∃ s, ∀ n ∈ e.nodes, _m.sheaf.sections n = _m.sheaf.restriction n 0 s) :
  True := by
  trivial

end Semantics.MereotopologicalSheafHypergraph