import Mathlib.Data.List.Basic
import Mathlib.Logic.Equiv.Basic
import Semantics.BraidedField

/-!
# Braided Field Sets & Virtual Information Paths

This module formalizes the topology of virtual information paths
created by braiding polaron-polariton quasiparticles (anyons).
Instead of tracking particle locations in Cartesian space, information
is stored in the topological equivalence class of the braid sequence.

This is a finite braid-word scaffold. It does not prove physical topological
protection, material feasibility, or local-noise immunity.
-/

namespace Semantics.BraidedField

/--
A single generator `σ_i` representing the virtual operation of
swapping adjacent quasiparticles `i` and `i+1` in a 2D manifold.
-/
structure Swap (n : Nat) where
  i : Nat
  valid : i + 1 < n
  deriving Repr, DecidableEq

/--
A Virtual Information Path is a temporal sequence of swaps.
In formal topology, this is known as a braid word.
This acts as an encoding system where data is written into spacetime history.
-/
abbrev VirtualPath (n : Nat) := List (Swap n)

/--
Topological equivalence of virtual paths.

Two paths are treated as the same candidate path if they can be deformed into
one another via the Artin braid relations. This is a structural equivalence
relation for the virtual-path model, not a physical immunity claim.
-/
inductive TopologicallyEquivalent {n : Nat} : VirtualPath n → VirtualPath n → Prop where
  | refl (p : VirtualPath n) : TopologicallyEquivalent p p

  | symm {p q : VirtualPath n} :
      TopologicallyEquivalent p q → TopologicallyEquivalent q p

  | trans {p q r : VirtualPath n} :
      TopologicallyEquivalent p q → TopologicallyEquivalent q r → TopologicallyEquivalent p r

  /--
  Far-commutation: Swapping particles on opposite sides of the field
  are independent operations. (σ_i σ_j = σ_j σ_i if |i - j| ≥ 2)
  -/
  | commute (i j : Swap n) (left right : VirtualPath n)
      (h_dist : i.i + 1 < j.i ∨ j.i + 1 < i.i) :
      TopologicallyEquivalent (left ++ [i, j] ++ right) (left ++ [j, i] ++ right)

  /--
  The Yang-Baxter Equation / braid relation.
  The core braid-word equivalence of the virtual-path encoding system.
  (σ_i σ_{i+1} σ_i = σ_{i+1} σ_i σ_{i+1})
  -/
  | braid_rel (i j : Swap n) (left right : VirtualPath n)
      (h_adj : j.i = i.i + 1) :
      TopologicallyEquivalent (left ++ [i, j, i] ++ right) (left ++ [j, i, j] ++ right)

/--
A basic theorem demonstrating that the relation is explicitly symmetric.
-/
theorem path_integrity_preserved {n : Nat} (p1 p2 : VirtualPath n)
  (h : TopologicallyEquivalent p1 p2) :
  TopologicallyEquivalent p2 p1 := by
  exact TopologicallyEquivalent.symm h

namespace Examples

def s0 : Swap 4 := ⟨0, by decide⟩
def s1 : Swap 4 := ⟨1, by decide⟩
def s2 : Swap 4 := ⟨2, by decide⟩

def farLeft : VirtualPath 4 := [s0, s2]
def farRight : VirtualPath 4 := [s2, s0]

def yangBaxterLeft : VirtualPath 4 := [s0, s1, s0]
def yangBaxterRight : VirtualPath 4 := [s1, s0, s1]

def pathLength {n : Nat} (p : VirtualPath n) : Nat :=
  p.length

def generatorIndexSum {n : Nat} (p : VirtualPath n) : Nat :=
  p.foldl (fun acc swap => acc + swap.i) 0

theorem far_commute_example : TopologicallyEquivalent farLeft farRight := by
  exact TopologicallyEquivalent.commute s0 s2 [] [] (by decide)

theorem yang_baxter_example : TopologicallyEquivalent yangBaxterLeft yangBaxterRight := by
  exact TopologicallyEquivalent.braid_rel s0 s1 [] [] (by decide)

theorem far_commute_symmetric : TopologicallyEquivalent farRight farLeft := by
  exact path_integrity_preserved farLeft farRight far_commute_example

#eval pathLength farLeft
#eval generatorIndexSum yangBaxterLeft
#eval generatorIndexSum yangBaxterRight

end Examples

end Semantics.BraidedField