TrapGameEA.lean

import Mathlib.Data.Fintype.Basic
import Mathlib.Probability.Kernel.Basic
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import LBTPreconditions

open MeasureTheory ProbabilityTheory
open scoped ENNReal
open Classical

/-!
# Trap Game Evolutionary Algorithm (EA) Base
This module formalizes the foundational state space for the trap game 
optimized by the CoEA. It defines the discrete measurable space of bit-strings,
the fitness function, the level-based partition sets, and the probability 
kernels for mutation and selection.

This module explicitly acts as the structural foundation necessary for verifying
the Level-Based Theorem (LBT) Preconditions (Task B6) and formulating the 
Stochastic Coupling argument (Task B4).
-/

namespace TrapGameEA

variable {n : ℕ} (hn : n > 0)

/-! ## State Space: Bit-Strings -/

/-- The search space X is the space of bit-strings of length n. -/
abbrev X (n : ℕ) := Fin n → Bool

-- Derive the necessary algebraic typeclasses for the search space to 
-- integrate seamlessly with the LBT generic framework.
instance : Fintype (X n) := inferInstance
instance : Nonempty (X n) := ⟨fun _ => false⟩
instance : MeasurableSpace (X n) := ⊤
instance : DiscreteMeasurableSpace (X n) := ⟨fun _ => trivial⟩

/-! ## Fitness Function and Levels -/

/-- The number of ones in a bit-string (its Hamming weight). -/
noncomputable def num_ones (x : X n) : ℕ :=
  (Finset.univ.filter (fun i => x i = true)).card

/-- 
The deterministic fitness function of the trap game.
The optimum is at `num_ones x = n`.
The trap deceptive gradient pulls toward `num_ones x = 0`.
-/
noncomputable def trap_fitness (x : X n) : ℝ :=
  if num_ones x = n then 
    (n : ℝ) + 1  -- Global optimum
  else 
    (n : ℝ) - (num_ones x : ℝ) -- Trap deceptive path

/--
The structural partition for the Level-Based Theorem.
Level `j` consists of all bit-strings with exactly `j` ones.
Because there are `n+1` possible weights (0 through n), we set `m = n + 1`.
-/
def A (j : Fin (n + 1)) : Set (X n) :=
  {x : X n | num_ones x = j.val}

/-! ## Transitions and Mutation Kernels (Scaffolding for B4/B6) -/

-- To be implemented: `μ_mut` (base mutation probability measure)
-- To be implemented: `f_ideal`, `f_CoEA` (deterministic transition functions)

end TrapGameEA