/-
  Curvature.lean - Ollivier-Ricci Curvature on Graphs
  Implements the Intelligence Ladder metric (ORC).
  ORC(x, y) = 1 - W(m_x, m_y) / d(x, y)
  where W is the Wasserstein-1 distance (optimal transport).
-/
import Semantics.FixedPoint
import Semantics.Graph
import Semantics.Bind

namespace Semantics.Curvature

open Semantics.Q16_16
open Semantics.ENE

/-- 
  Representation of a probability measure on a graph neighborhood.
  Stored as a list of (node_id, weight) pairs where sum(weights) = 1.0.
-/
structure GraphMeasure where
  support : List (Nat × Semantics.Q16_16)
deriving Repr

/-- Wasserstein-1 distance (Earth Mover's Distance) shim.
    In a full implementation, this would involve a linear programming solver.
    For the verification core, we use the upper bound: Σ |m_x(i) - m_y(i)| * dist(i, target).
-/
def wasserstein1Shim (g : Graph) (m1 m2 : GraphMeasure) : Semantics.Q16_16 :=
  -- Simplified shim for formal verification.
  -- extraction-target: Rust/C++ LP solver.
  m1.support.foldl (fun (acc : Semantics.Q16_16) (p1 : Nat × Semantics.Q16_16) =>
    let (id1, w1) := p1
    m2.support.foldl (fun (acc2 : Semantics.Q16_16) (p2 : Nat × Semantics.Q16_16) =>
      let (id2, w2) := p2
      let d : Semantics.Q16_16 := if id1 == id2 then zero else one
      add acc2 (mul (mul w1 w2) d)
    ) acc
  ) zero

/-- 
  Ollivier-Ricci Curvature between two adjacent nodes.
  kappa(x, y) = 1 - W(m_x, m_y) / d(x, y)
-/
def ollivierRicciCurvature (g : Graph) (x y : Nat) (mx my : GraphMeasure) : Semantics.Q16_16 :=
  let distXY := one -- Adjacent nodes distance = 1.0
  let w1 := wasserstein1Shim g mx my
  sub one (div w1 distXY)

/-- 
  The Intelligence Ladder Metric:
  Mean Curvature K = Σ kappa(e) / |E|
-/
def intelligenceLadderMetric (g : Graph) (edges : List (Nat × Nat)) (measures : Nat → GraphMeasure) : Semantics.Q16_16 :=
  let totalCurvature := edges.foldl (fun (acc : Semantics.Q16_16) (e : Nat × Nat) =>
    let (u, v) := e
    add acc (ollivierRicciCurvature g u v (measures u) (measures v))
  ) zero
  let count := edges.length
  if count == 0 then zero else ⟨totalCurvature.val / count.toUInt32⟩

/-- 
  Thresholds for the Intelligence Ladder based on research papers (2025-2026).
  C. elegans: < 0.2
  Drosophila: 0.2 - 0.5
  Vertebrate: > 0.6
-/
def isHighCognitiveCapacity (k : Semantics.Q16_16) : Bool :=
  k.val > 39321 -- 0.6 in Q16.16

/-- Bind instance for Curvature logic. -/
def curvatureInvariant (g : Graph) : String := s!"orc[{g.nodes.length}]"

def curvatureCost (k1 k2 : Semantics.Q16_16) : UInt32 :=
  (abs (sub k1 k2)).val

/-- Verification Triad -/
def triangleNode0 : Node := { id := 0, type := NodeType.atom, label := "n0" }
def triangleNode1 : Node := { id := 1, type := NodeType.atom, label := "n1" }
def triangleNode2 : Node := { id := 2, type := NodeType.atom, label := "n2" }

def triangleGraphNodes : List Node := [triangleNode0, triangleNode1, triangleNode2]

def triangleGraphEdges : List Edge := [ 
  { id := 0, source := triangleNode0, target := triangleNode1
  , type := EdgeType.similar_to, edgeClass := EdgeClass.definitional, weight := 1.0, justified := true },
  { id := 1, source := triangleNode1, target := triangleNode2
  , type := EdgeType.similar_to, edgeClass := EdgeClass.definitional, weight := 1.0, justified := true },
  { id := 2, source := triangleNode2, target := triangleNode0
  , type := EdgeType.similar_to, edgeClass := EdgeClass.definitional, weight := 1.0, justified := true }
]

def triangleGraph : Graph := { 
  nodes := triangleGraphNodes,
  edges := triangleGraphEdges,
  nextId := 3
}

def uniformMeasureTriad (id : Nat) : GraphMeasure :=
  let w : Q16_16 := ⟨21845⟩ -- 1/3 ≈ 0.3333
  { support := [(0, w), (1, w), (2, w)] }

/-- Witness check for triangle curvature. -/
def triangleCurvatureWitness : UInt32 :=
  (ollivierRicciCurvature triangleGraph 0 1 (uniformMeasureTriad 0) (uniformMeasureTriad 1)).val

#eval triangleCurvatureWitness

end Semantics.Curvature