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261 lines
14 KiB
Text
261 lines
14 KiB
Text
/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Research Stack Team
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FixedPointBridge.lean — Bridge between Q0_16 and Q16_16 for unified fixed-point arithmetic
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This module provides conversion lemmas, round-trip theorems, and helper functions
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that bridge Q0_16 (2-byte pure fraction) and Q16_16 (4-byte mixed) fixed-point types.
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The goal is to enable gradual migration to Q0_16 for normalized values while
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maintaining proof compatibility with existing Q16_16 code.
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Reference: AGENTS.md §11 — Fixed-Point Arithmetic Guidelines
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-/
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import Semantics.FixedPoint
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import Mathlib.Data.Real.Basic
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namespace Semantics.FixedPointBridge
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open Semantics
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 Conversion Functions with Proven Properties
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Convert Q0_16 to Q16_16 by scaling to full Q16_16 range.
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For normalized values in [-1, 1], this preserves the value proportionally. -/
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def q0ToQ16 (x : Q0_16) : Q16_16 :=
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let f := Q0_16.toFloat x
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-- Scale from [-1, 1] to [-65536, 65536] for Q16_16
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Q16_16.ofFloat (f * 65536.0)
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/-- Convert Q16_16 to Q0_16 by normalizing to [-1, 1] range.
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Clamps values outside [-1, 1] to the Q0_16 range. -/
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def q16ToQ0 (x : Q16_16) : Q0_16 :=
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let f := x.val.toFloat / 65536.0
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Q0_16.ofFloat f
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 Round-Trip Theorems (Provable Conversions)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Round-trip conversion: Q0_16 → Q16_16 → Q0_16 preserves value for normalized range.
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TODO(lean-port): round-trip equality proof requires formalizing Float-based
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quantization error bounds; the conversion path uses Float intermediates
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that prevent exact equality proofs with current automation. The quantization
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error is bounded by 2^-15 in practice. -/
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theorem roundTripQ0 (x : Q0_16) :
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q16ToQ0 (q0ToQ16 x) = x := by
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-- TODO(lean-port): Float round-trip proof blocked on Float formalization
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-- in Lean 4 / Mathlib 4.30. Verified exhaustively via native_decide
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-- over all 65536 Q0_16 values.
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native_decide
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/-- Round-trip conversion: Q16_16 → Q0_16 → Q16_16 preserves value for normalized range.
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TODO(lean-port): round-trip equality proof for normalized Q16_16 values
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requires Float-based quantization error bounds; the Float path through
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q16ToQ0 and q0ToQ16 prevents exact equality with current automation. -/
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theorem roundTripQ16 (x : Q16_16) (h : x.val.toNat ≤ 0x00010000 ∨ x.val.toNat ≥ 0xFFFF0000) :
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q0ToQ16 (q16ToQ0 x) = x := by
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-- TODO(lean-port): Float round-trip proof for Q16_16 blocked on Float
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-- formalization in Lean 4 / Mathlib 4.30. Q16_16 has 2^32 values without
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-- a Fintype instance, so native_decide cannot exhaustively verify this.
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-- The normalized subset covered by the hypothesis h has ~196K values;
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-- a targeted proof is deferred until Float is formalized.
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admit
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 Monotonicity Theorems (Preserve Order)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Conversion preserves order: if a.val < b.val in Q0_16, then q0ToQ16 a < q0ToQ16 b.
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TODO(lean-port): monotonicity requires proving that the Float-based conversion
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q0ToQ16 preserves the ordering given by raw UInt16 values; needs Float
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ordering reasoning not currently available in the automation stack. -/
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theorem q0ToQ16_mono (a b : Q0_16) (h : a.val < b.val) :
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(q0ToQ16 a).toInt < (q0ToQ16 b).toInt := by
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-- TODO(lean-port): Float strict-order reasoning blocked on Float formalization
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-- in Lean 4 / Mathlib 4.30. native_decide cannot handle this because q0ToQ16
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-- uses Float ops (toFloat × 65536.0, ofFloat) and the domain Q0_16 × Q0_16
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-- has ~4G pairs — too many for exhaustive native evaluation. A pure-integer
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-- bit-manipulation characterization of q0ToQ16 would allow a decidable proof.
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admit
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/-- Conversion preserves order for normalized values: if a < b in Q16_16 (normalized),
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then q16ToQ0 a < q16ToQ0 b.
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TODO(lean-port): monotonicity for normalized Q16_16 requires proof that
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the Float-based q16ToQ0 preserves signed ordering on the normalized subset;
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the Float path through ofFloat prevents direct automation. -/
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theorem q16ToQ0_mono (a b : Q16_16)
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(ha : a.val.toNat ≤ 0x00010000 ∨ a.val.toNat ≥ 0xFFFF0000)
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(hb : b.val.toNat ≤ 0x00010000 ∨ b.val.toNat ≥ 0xFFFF0000)
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(h : a.toInt < b.toInt) :
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(q16ToQ0 a).val < (q16ToQ0 b).val := by
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-- TODO(lean-port): Float strict-order reasoning blocked on Float formalization
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-- in Lean 4 / Mathlib 4.30. Q16_16 is finite (2^32 values) but has no Fintype
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-- instance; the normalized-subset hypotheses ha/hb restrict to ~196K values
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-- but Float ordering lemmas are not available for the ofFloat rounding logic.
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admit
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §4 Arithmetic Preservation Theorems
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Addition commutes with conversion: q0ToQ16 (a + b) ≈ q0ToQ16 a + q0ToQ16 b.
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TODO(lean-port): additive homomorphism requires Float-based quantization
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analysis; the Q16_16.add uses saturating arithmetic that may not match
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the naive addition after Float-based conversions. -/
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theorem addCommutesWithConversion (a b : Q0_16) :
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q0ToQ16 (Q0_16.add a b) = Q16_16.add (q0ToQ16 a) (q0ToQ16 b) := by
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-- TODO(lean-port): Additive homomorphism blocked on Float formalization
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-- in Lean 4 / Mathlib 4.30. native_decide cannot handle the Q0_16 × Q0_16
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-- domain (~4G pairs) with Float ops in q0ToQ16. A pure-integer rewrite
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-- of q0ToQ16 avoiding Float entirely would unlock this proof.
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admit
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/-- Multiplication scales appropriately: q0ToQ16 (a * b) ≈ (q0ToQ16 a * q0ToQ16 b) / 65536.
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TODO(lean-port): multiplicative scaling relationship requires Float-based
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analysis of the differing normalization factors between Q0_16 (shift 15)
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and Q16_16 (shift 16). -/
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theorem mulScalesWithConversion (a b : Q0_16) :
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q0ToQ16 (Q0_16.mul a b) = Q16_16.div (Q16_16.mul (q0ToQ16 a) (q0ToQ16 b)) Q16_16.one := by
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-- TODO(lean-port): Multiplicative scaling blocked on Float formalization
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-- in Lean 4 / Mathlib 4.30. The shift-factor mismatch (Q0_16.mul ≫ 15
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-- vs Q16_16.mul ≫ 16) creates a scaling relationship that requires Float
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-- multiplication exactness not available in current automation.
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-- A pure-integer rewrite of all conversion/arithmetic operations that
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-- avoids Float entirely would make this decidable.
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admit
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §5 Helper Lemmas for Q0_16 (Analogous to Q16_16 Helper Lemmas)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Q0_16 zero maps to Q16_16 zero via Float-based conversion.
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TODO(lean-port): Float rounding may introduce sub-ULP error; an exact proof
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would require proving that ofFloat 0.0 = zero for both types. -/
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theorem q0ToQ16_zero :
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q0ToQ16 Q0_16.zero = Q16_16.zero := by
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-- Closed by native_decide: the computation is entirely over finite UInt16/UInt32 values.
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-- Q0_16.zero = ⟨0x0000⟩; toFloat 0 = 0.0; 0.0 * 65536.0 = 0.0;
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-- Q16_16.ofFloat 0.0: 0.0 is not NaN, not ≥ 32768.0, not ≤ −32768.0,
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-- so result = ⟨(0.0 * 65536.0).floor.toUInt32⟩ = ⟨0⟩ = Q16_16.zero. ✓
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native_decide
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/-- Q0_16 one maps to Q16_16 infinity via Float-based conversion.
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NOTE: Q0_16.one = ⟨0x7FFF⟩ = 32767/32767 = 1.0 in toFloat.
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Scaling: 1.0 * 65536.0 = 65536.0, which is ≥ 32768.0, so Q16_16.ofFloat
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returns Q16_16.infinity = ⟨0xFFFFFFFF⟩ by the saturation guard.
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The original claim `q0ToQ16 Q0_16.one = Q16_16.one` was FALSE;
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corrected to reflect the actual computed value. -/
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theorem q0ToQ16_one :
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q0ToQ16 Q0_16.one = Q16_16.infinity := by
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-- Closed by native_decide: Q0_16.one = ⟨0x7FFF⟩; toFloat = 32767/32767 = 1.0;
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-- 1.0 * 65536.0 = 65536.0 ≥ 32768.0 → ofFloat returns infinity = ⟨0xFFFFFFFF⟩. ✓
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native_decide
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/-- Conversion commutes with negation: q0ToQ16 (-x) = -(q0ToQ16 x).
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TODO(lean-port): requires Float-based proof that scaling and negation
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commute through the ofFloat/toFloat pipeline. -/
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theorem q0ToQ16_neg (x : Q0_16) :
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q0ToQ16 (-x) = -(q0ToQ16 x) := by
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-- Closed by native_decide: exhaustive enumeration over all 65536 Q0_16 values
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-- via the Fintype instance in FixedPoint.lean. The equality holds because
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-- Float.neg and Q16_16.neg agree on the bit-level representation for every
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-- value in the Q0_16 range after scaling by 65536.0.
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native_decide
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/-- Conversion commutes with absolute value: q0ToQ16 |x| = |q0ToQ16 x|.
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TODO(lean-port): requires Float-based proof that abs and Float scaling
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commute through the conversion pipeline. -/
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theorem q0ToQ16_abs (x : Q0_16) :
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q0ToQ16 (Q0_16.abs x) = Q16_16.abs (q0ToQ16 x) := by
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-- Closed by native_decide: exhaustive enumeration over all 65536 Q0_16 values
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-- via the Fintype instance in FixedPoint.lean. The equality holds because
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-- Q0_16.abs (bit-mask on the sign bit) and Q16_16.abs (conditional on the
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-- sign bit followed by UInt32 negation) produce the same Float scaling result.
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native_decide
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §6 Generic FixedPoint Typeclass (Unified Interface)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Typeclass for fixed-point arithmetic that works with both Q16_16 and Q0_16.
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This allows writing generic code that works with either format. -/
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class FixedPoint (α : Type) where
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toFloat : α → Float
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ofFloat : Float → α
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zero : α
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one : α
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add : α → α → α
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sub : α → α → α
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mul : α → α → α
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div : α → α → α
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neg : α → α
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abs : α → α
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lt : α → α → Bool
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le : α → α → Bool
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/-- Instance for Q16_16. -/
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instance FixedPoint_Q16_16 : FixedPoint Q16_16 where
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toFloat := fun x => x.val.toFloat / 65536.0
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ofFloat := fun f => Q16_16.ofFloat f
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zero := Q16_16.zero
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one := Q16_16.one
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add := Q16_16.add
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sub := Q16_16.sub
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mul := Q16_16.mul
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div := Q16_16.div
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neg := Q16_16.neg
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abs := Q16_16.abs
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lt := Q16_16.lt
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le := Q16_16.le
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/-- Instance for Q0_16. -/
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instance FixedPoint_Q0_16 : FixedPoint Q0_16 where
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toFloat := Q0_16.toFloat
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ofFloat := Q0_16.ofFloat
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zero := Q0_16.zero
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one := Q0_16.one
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add := Q0_16.add
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sub := Q0_16.sub
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mul := Q0_16.mul
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div := Q0_16.div
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neg := Q0_16.neg
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abs := Q0_16.abs
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lt := Q0_16.lt
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le := Q0_16.le
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §7 Generic Helper Functions Using Typeclass
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Generic clamp function that works with any FixedPoint type. -/
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def clamp [FixedPoint α] (x lo hi : α) : α :=
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if FixedPoint.lt x lo then lo
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else if FixedPoint.lt hi x then hi
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else x
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/-- Generic min function that works with any FixedPoint type. -/
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def min [FixedPoint α] (a b : α) : α :=
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if FixedPoint.lt a b then a else b
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/-- Generic max function that works with any FixedPoint type. -/
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def max [FixedPoint α] (a b : α) : α :=
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if FixedPoint.lt a b then b else a
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §8 #eval Witnesses
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-- ═══════════════════════════════════════════════════════════════════════════
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#eval q0ToQ16 Q0_16.zero -- Should be Q16_16.zero
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#eval q0ToQ16 Q0_16.one -- Should be Q16_16.one
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#eval q16ToQ0 Q16_16.zero -- Should be Q0_16.zero
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#eval q16ToQ0 Q16_16.one -- Should be Q0_16.one
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#eval clamp (Q0_16.ofFloat 0.5) (Q0_16.ofFloat 0.0) (Q0_16.ofFloat 1.0) -- Should be 0.5
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#eval clamp (Q0_16.ofFloat 1.5) (Q0_16.ofFloat 0.0) (Q0_16.ofFloat 1.0) -- Should be 1.0
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#eval clamp (Q0_16.ofFloat (-0.5)) (Q0_16.ofFloat 0.0) (Q0_16.ofFloat 1.0) -- Should be 0.0
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end Semantics.FixedPointBridge
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