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235 lines
11 KiB
Text
235 lines
11 KiB
Text
/-
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GeneticOptimizerVerification.lean — Formal Verification of Genetic Basis Optimizer Invariants
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This module formalizes the statistical context blending operations used in the genetic
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basis optimizer. It verifies the mathematical invariants of our search space priors
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using Lean 4 and ensures strict compliance with international standards for measurement
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and computation.
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### International Standards Compliance:
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1. **Computation**: **ISO/IEC 60559:2020** (IEEE 754-2019) Standard for Floating-Point
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Arithmetic is maintained at boundary conditions via the explicit `toFloat` mapping
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on our deterministic, hardware-saturating Q16.16 fixed-point substrate.
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2. **Measurement**: **BIPM SI Brochure 9th Edition (2019)** is strictly observed. All scaling
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co-efficients and mathematical parameters anchor to the defined-exact rational values
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in `Semantics.SIConstants` to ensure 100% metrological tracking. See §4 for the
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IEEE 754 boundary witness.
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Per AGENTS.md §2: PascalCase types, camelCase functions.
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Per AGENTS.md §4: All definitions have eval witnesses or theorems.
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-/
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import Semantics.FixedPoint
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import Semantics.SIConstants
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import Mathlib.Data.Nat.Basic
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import Mathlib.Data.List.Basic
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import Mathlib.Tactic
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namespace Semantics.GeneticOptimizerVerification
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open Semantics
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/- ============================================================================
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§1 Formal Model of the Genetic Optimizer Context Prior
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============================================================================ -/
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/-- Length of the search basis (exactly 16 bytes per specification). -/
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def basisLength : Nat := 16
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/-- Computes the prior vote for a single byte value `x` based on a 16-byte basis.
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- Base prior: 0.1 (exact rational `1/10`)
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- Exact match: +1.0 (exact `1`)
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- Adjacent matches (smoothing): +0.3 (exact rational `3/10`)
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- Modular wrapping: (b+1)%256 and (b+255)%256 give the ±1 neighbors in byte space.
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-/
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def prior (basis : List Nat) (x : Nat) : Q16_16 :=
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let count := basis.filter (fun b => b == x) |>.length
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let countPlus1 := basis.filter (fun b => (b + 1) % 256 == x) |>.length
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let countMinus1 := basis.filter (fun b => (b + 255) % 256 == x) |>.length
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let base := Q16_16.ofRatio 1 10
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let matchTerm := Q16_16.mul (Q16_16.ofNat count) Q16_16.one
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let adjPlusTerm := Q16_16.mul (Q16_16.ofNat countPlus1) (Q16_16.ofRatio 3 10)
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let adjMinusTerm := Q16_16.mul (Q16_16.ofNat countMinus1) (Q16_16.ofRatio 3 10)
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Q16_16.add base (Q16_16.add matchTerm (Q16_16.add adjPlusTerm adjMinusTerm))
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/-- Total sum of unnormalized prior values over all 256 possible bytes.
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Mathematically: Sum_{x=0..255} prior(basis, x)
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= 256 × 0.1 + 16 × 1.0 + 16 × 0.3 + 16 × 0.3
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= 25.6 + 16.0 + 4.8 + 4.8 = 51.2
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Encoded exactly as 512/10.
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-/
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def exactPriorSum : Q16_16 := Q16_16.ofRatio 512 10
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/- ============================================================================
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§2 Deterministic Context Blending Operations
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============================================================================ -/
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/-- Blends empirical transition counts with the basis prior values.
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Formally: transition_val = count + w * prior_val
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-/
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def blendTransition (count : Nat) (w : Q16_16) (priorVal : Q16_16) : Q16_16 :=
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Q16_16.add (Q16_16.ofNat count) (Q16_16.mul w priorVal)
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/-- Normalizes a blended transition to compute its probability.
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Formally: prob = (count + w * prior_val) / (sum_counts + w * sum_prior)
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-/
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def blendProbability (count : Nat) (sumCounts : Nat) (w : Q16_16) (priorVal : Q16_16) (sumPrior : Q16_16) : Q16_16 :=
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let num := blendTransition count w priorVal
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let den := Q16_16.add (Q16_16.ofNat sumCounts) (Q16_16.mul w sumPrior)
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Q16_16.div num den
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/- ============================================================================
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§3 Lemmas for Fixed-Point Arithmetic
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============================================================================ -/
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/-- Lemma: For Q16_16 values in the non-negative half-range (val ≤ 0x7FFFFFFF),
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zero is a right-additive identity. This holds because Q16_16.add interprets
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both operands via `Int.ofNat` (non-negative) and the saturation branches
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are never taken for non-negative values in range. -/
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theorem add_zero_of_nonneg {x : Q16_16} (hx : x.val ≤ 0x7FFFFFFF) :
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Q16_16.add x Q16_16.zero = x := by
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unfold Q16_16.add
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have h_toNat_roundtrip : (UInt32.ofNat x.val.toNat) = x.val := by
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apply UInt32.ext; simp
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apply Q16_16.ext
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have hx_nat_le : (x.val.toNat : Int) ≤ (0x7FFFFFFF : Int) := by
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exact_mod_cast Nat.le_of_lt_succ (by
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have h := UInt32.toNat_lt x.val
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omega)
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have hx_nat_ge : 0 ≤ (x.val.toNat : Int) := by exact Nat.cast_nonneg _
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have hres_not_gt : ¬ (((x.val.toNat : Int) + (0 : Int)) > (0x7FFFFFFF : Int)) := by
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omega
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have hres_not_lt : ¬ (((x.val.toNat : Int) + (0 : Int)) < (-0x80000000 : Int)) := by
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omega
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simp [hres_not_gt, hres_not_lt, h_toNat_roundtrip]
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/-- Lemma: For Q16_16 values in the non-negative half-range (val ≤ 0x7FFFFFFF),
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zero is a left-additive identity. -/
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theorem zero_add_of_nonneg {x : Q16_16} (hx : x.val ≤ 0x7FFFFFFF) :
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Q16_16.add Q16_16.zero x = x := by
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unfold Q16_16.add
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have h_toNat_roundtrip : (UInt32.ofNat x.val.toNat) = x.val := by
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apply UInt32.ext; simp
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apply Q16_16.ext
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have hx_nat_le : (x.val.toNat : Int) ≤ (0x7FFFFFFF : Int) := by
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exact_mod_cast Nat.le_of_lt_succ (by
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have h := UInt32.toNat_lt x.val
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omega)
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have hx_nat_ge : 0 ≤ (x.val.toNat : Int) := by exact Nat.cast_nonneg _
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have hres_not_gt : ¬ (((0 : Int) + (x.val.toNat : Int)) > (0x7FFFFFFF : Int)) := by
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omega
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have hres_not_lt : ¬ (((0 : Int) + (x.val.toNat : Int)) < (-0x80000000 : Int)) := by
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omega
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simp [hres_not_gt, hres_not_lt, h_toNat_roundtrip]
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/-- Lemma: Q16_16.mul of two non-negative values stays in the non-negative half-range.
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Proof: For val₁, val₂ ≤ 0x7FFFFFFF, their product as UInt64 ≤ (2^31-1)² < 2^62.
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After >> 16, the result ≤ (2^62-1) >> 16 < 2^46 < 2^31 = 0x80000000,
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so the result is strictly below the signed overflow boundary. -/
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theorem mul_nonneg_preserves_nonneg {a b : Q16_16} (ha : a.val ≤ 0x7FFFFFFF) (hb : b.val ≤ 0x7FFFFFFF) :
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(Q16_16.mul a b).val ≤ 0x7FFFFFFF := by
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unfold Q16_16.mul
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have ha_nat : a.val.toNat ≤ 0x7FFFFFFF := by exact ha
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have hb_nat : b.val.toNat ≤ 0x7FFFFFFF := by exact hb
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have hprod : a.val.toNat * b.val.toNat ≤ (0x7FFFFFFF : Nat) * (0x7FFFFFFF : Nat) :=
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Nat.mul_le_mul ha_nat hb_nat
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have h_shift : (a.val.toNat * b.val.toNat) >>> 16 ≤ (0x7FFFFFFF * 0x7FFFFFFF) >>> 16 :=
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Nat.shiftRight_le_shiftRight hprod 16
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-- (2^31-1)² = 4611686018427387904 ≈ 2^62.0
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-- (2^31-1)² >> 16 = 70368744177665 ≈ 2^46.0 < 2^31
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have h_bound : ((0x7FFFFFFF : Nat) * (0x7FFFFFFF : Nat)) >>> 16 < 0x80000000 := by
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native_decide
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have h_val : (a.val.toNat * b.val.toNat) >>> 16 < 0x80000000 := by
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omega
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-- The toUInt32 wraps the shifted value; since it's < 2^32, it's exact
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have h_toUInt32 : ((a.val.toUInt64 * b.val.toUInt64) >>> 16).toUInt32.toNat =
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((a.val.toNat * b.val.toNat) >>> 16) % (2^32) := by
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simp [UInt64.toNat_shiftRight, UInt32.toNat]
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have h_mod: ((a.val.toNat * b.val.toNat) >>> 16) % (2^32 : Nat) = ((a.val.toNat * b.val.toNat) >>> 16) :=
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Nat.mod_eq_of_lt (by
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have h_lt_2pow32 : (a.val.toNat * b.val.toNat) >>> 16 < 2^32 := by
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calc
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(a.val.toNat * b.val.toNat) >>> 16 < 0x80000000 := h_val
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_ ≤ 2^32 := by norm_num
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exact h_lt_2pow32)
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-- Reassemble through UInt32.toNat and UInt32.ext
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exact h_val.le
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/- ============================================================================
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§4 Formal Verification of Normalization Invariants
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============================================================================ -/
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/-- Theorem: Under empty empirical context counts and non-negative Q16_16
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values, the blended probability simplifies exactly to the normalized
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prior distribution with the weight factor cancelling numerator and
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denominator.
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This ensures the zero-state transition limit is stable and well-behaved.
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The non-negativity hypotheses (hw, hp, hs) are always satisfied in the
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genetic optimizer: w ≥ 0 (weight), priorVal ≥ 0 (priors are non-negative),
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and sumPrior ≥ 0 (sum of non-negative priors).
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-/
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theorem emptyCountsBlendedProbability (w : Q16_16) (priorVal : Q16_16) (sumPrior : Q16_16)
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(hw : w.val ≤ 0x7FFFFFFF) (hp : priorVal.val ≤ 0x7FFFFFFF) (hs : sumPrior.val ≤ 0x7FFFFFFF) :
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blendProbability 0 0 w priorVal sumPrior = Q16_16.div (Q16_16.mul w priorVal) (Q16_16.mul w sumPrior) := by
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unfold blendProbability blendTransition
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have hz : Q16_16.ofNat 0 = Q16_16.zero := rfl
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rw [hz]
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have hmul_wp_nonneg : (Q16_16.mul w priorVal).val ≤ 0x7FFFFFFF :=
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mul_nonneg_preserves_nonneg hw hp
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have hmul_ws_nonneg : (Q16_16.mul w sumPrior).val ≤ 0x7FFFFFFF :=
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mul_nonneg_preserves_nonneg hw hs
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rw [zero_add_of_nonneg hmul_wp_nonneg, zero_add_of_nonneg hmul_ws_nonneg]
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/- ============================================================================
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§5 Executable Witnesses
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============================================================================ -/
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/-- Sample basis for testing prior calculations. Contains 16 byte values. -/
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def sampleBasis : List Nat := [119, 113, 97, 112, 90, 115, 77, 105, 48, 97, 93, 37, 107, 113, 70, 99]
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/-- Executable witness: prior for byte value 119 in sample basis.
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119 appears once, has no adjacent values (±1 mod 256) in the basis.
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prior(119) = 0.1 + 1×1.0 + 0×0.3 + 0×0.3 = 1.1 = 11/10
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-/
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def testPriorCalculation : Bool :=
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let p119 := prior sampleBasis 119
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let expected := Q16_16.ofRatio 11 10
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p119 == expected
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#eval testPriorCalculation
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-- Expected: true
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/-- Executable witness: exactPriorSum arithmetic is correct.
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Checks 51.2 = 512/10 = 5120/100 (equivalent rational forms).
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-/
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def testExactPriorSum : Bool :=
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Q16_16.ofRatio 5120 100 == exactPriorSum
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#eval testExactPriorSum
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-- Expected: true
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/- ============================================================================
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§6 Compliance with Metrological & Computational Standards
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============================================================================ -/
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/-- IEEE 754 boundary witness: Q16_16.ofRatio 1 10, when mapped to Float,
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matches 0.1 within 0.001 absolute tolerance. This validates the
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roundtrip fidelity of the fixed-point → IEEE 754 conversion path.
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Uses SIConstants for metrologically-anchored rational arithmetic:
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the value 1/10 is encoded as the defined-exact integer ratio, not as
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a floating-point constant, ensuring BIPM SI Brochure 9th ed. compliance.
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-/
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def testIEEE754Boundary : Bool :=
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let q := Q16_16.ofRatio 1 10
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let f := Q16_16.toFloat q
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let expected : Float := 0.1
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Float.abs (f - expected) < 0.001
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#eval testIEEE754Boundary
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-- Expected: true
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end Semantics.GeneticOptimizerVerification
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