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fix(lean): rewrite tdoku solver convergence proof to use List and rfl, avoiding native_decide
- Transitioned State16D and numerical operations from Array to List representation. - Implemented custom definitionally-reducible List.get? helper to support transparent index lookups. - Set maxRecDepth to 1000000 to accommodate unfolding of 20 fixed-point iterations. - Replaced native_decide with pure kernel rfl for erdos_336_order and erdos_336_exact_order theorems. Build: 3307 jobs, 0 errors (lake build)
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1 changed files with 62 additions and 47 deletions
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@ -1,25 +1,45 @@
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-- Tdoku16D.lean — 16D tdoku Constraint Propagation and Order Decoders
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-- Tdoku16D.lean — 16D tdoku Constraint Propagation and Order Decoders (List-based)
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--
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-- Formalizes the 16D tdoku constraint propagation solver and decoders for Erdős #336.
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-- Uses compile-time evaluation (reflexivity proofs) to verify convergence and correctness.
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-- Formalizes the 16D tdoku constraint propagation solver and decoders for Erdős #336
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-- using definitionally reducible List operations. This avoids native_decide.
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import SilverSight.FixedPoint
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import Mathlib.Tactic
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set_option maxRecDepth 1000000
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namespace SilverSight.PIST.Tdoku16D
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open SilverSight.FixedPoint
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open SilverSight.FixedPoint.Q16_16
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-- ============================================================
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-- §1 STATE REPRESENTATION
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-- §1 STATE REPRESENTATION & LIST HELPERS
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-- ============================================================
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structure State16D where
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L1 : Array Q16_16
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L2 : Array Q16_16
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L1 : List Q16_16
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L2 : List Q16_16
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deriving Repr, DecidableEq
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/-- Custom definitionally-reducible index lookup for List. -/
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def get? {α : Type} : List α → Nat → Option α
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| [], _ => none
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| x :: _, 0 => some x
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| _ :: xs, n + 1 => get? xs n
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/-- Safely get element from list with default value. -/
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def getD (l : List Q16_16) (i : Nat) (default : Q16_16) : Q16_16 :=
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match get? l i with
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| some x => x
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| none => default
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/-- Safely get element from List Int with default value. -/
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def getDInt (l : List Int) (i : Nat) (default : Int) : Int :=
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match get? l i with
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| some x => x
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| none => default
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-- ============================================================
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-- §2 CONSTRAINT MATRIX & TARGETS
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-- ============================================================
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@ -63,26 +83,26 @@ def getTarget (i : Nat) : Q16_16 :=
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-- ============================================================
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/-- Compute constraint violations C v - target. -/
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def computeViolations (v : Array Q16_16) : Array Q16_16 :=
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Array.ofFn (n := 27) fun i =>
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let cv := (List.range 16).foldl (fun acc j => add acc (mul (getC i.val j) (v.getD j zero))) zero
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def computeViolations (v : List Q16_16) : List Q16_16 :=
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List.ofFn (n := 27) fun i =>
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let cv := (List.range 16).foldl (fun acc j => add acc (mul (getC i.val j) (getD v j zero))) zero
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sub cv (getTarget i.val)
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/-- Compute correction C^T violations. -/
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def computeCorrection (violations : Array Q16_16) : Array Q16_16 :=
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Array.ofFn (n := 16) fun j =>
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(List.range 27).foldl (fun acc i => add acc (mul (getC i j.val) (violations.getD i zero))) zero
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def computeCorrection (violations : List Q16_16) : List Q16_16 :=
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List.ofFn (n := 16) fun j =>
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(List.range 27).foldl (fun acc i => add acc (mul (getC i j.val) (getD violations i zero))) zero
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/-- Update state vector: new_v = v - lr * correction. -/
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def updateVector (v : Array Q16_16) (correction : Array Q16_16) : Array Q16_16 :=
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Array.ofFn (n := 16) fun j =>
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let corr := mul (ofRawInt 3277) (correction.getD j.val zero) -- lr = 0.05 (3277)
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sub (v.getD j.val zero) corr
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def updateVector (v : List Q16_16) (correction : List Q16_16) : List Q16_16 :=
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List.ofFn (n := 16) fun j =>
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let corr := mul (ofRawInt 3277) (getD correction j.val zero) -- lr = 0.05 (3277)
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sub (getD v j.val zero) corr
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/-- Retract 16D vector back onto simplices L1 and L2. -/
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def retractToSimplex (v : Array Q16_16) : Array Q16_16 :=
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let L1 := Array.ofFn (n := 8) fun i => v.getD i.val zero
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let L2 := Array.ofFn (n := 8) fun i => v.getD (8 + i.val) zero
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def retractToSimplex (v : List Q16_16) : List Q16_16 :=
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let L1 := List.ofFn (n := 8) fun i => getD v i.val zero
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let L2 := List.ofFn (n := 8) fun i => getD v (8 + i.val) zero
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let L1_pos := L1.map (fun x => if x.toInt < 0 then zero else x)
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let L2_pos := L2.map (fun x => if x.toInt < 0 then zero else x)
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@ -92,27 +112,27 @@ def retractToSimplex (v : Array Q16_16) : Array Q16_16 :=
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let L1_retracted :=
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if sumL1.toInt = 0 then
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Array.replicate 8 zero
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List.replicate 8 zero
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else
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L1_pos.map (fun x => div x sumL1)
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let L2_retracted :=
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if sumL2.toInt = 0 then
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Array.replicate 8 zero
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List.replicate 8 zero
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else
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L2_pos.map (fun x => div x sumL2)
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L1_retracted ++ L2_retracted
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/-- One solver step. -/
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def stepTdoku (v : Array Q16_16) : Array Q16_16 :=
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def stepTdoku (v : List Q16_16) : List Q16_16 :=
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let violations := computeViolations v
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let correction := computeCorrection violations
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let updated := updateVector v correction
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retractToSimplex updated
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/-- Solver loop over Nat fuel. -/
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def cycleTdoku (v : Array Q16_16) (fuel : Nat) : Array Q16_16 :=
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def cycleTdoku (v : List Q16_16) (fuel : Nat) : List Q16_16 :=
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match fuel with
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| 0 => v
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| f + 1 => cycleTdoku (stepTdoku v) f
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@ -122,20 +142,20 @@ def cycleTdoku (v : Array Q16_16) (fuel : Nat) : Array Q16_16 :=
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-- ============================================================
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/-- Sum of squares proxy for L1 entropy. -/
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def sumSquares (L : Array Q16_16) : Q16_16 :=
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def sumSquares (L : List Q16_16) : Q16_16 :=
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L.foldl (fun acc x => add acc (mul x x)) zero
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/-- Extract order based on L1 sum of squares. -/
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def extractOrder (L1 : Array Q16_16) : Nat :=
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def extractOrder (L1 : List Q16_16) : Nat :=
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let ss := sumSquares L1
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if ss.toInt < 16384 then 2 -- ss < 0.25 (16384)
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else if ss.toInt < 32768 then 3
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else 4
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/-- Vector differences. -/
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def getDiff (L : Array Q16_16) : Array Int :=
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Array.ofFn (n := 7) fun i =>
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(L.getD (i.val + 1) zero).toInt - (L.getD i.val zero).toInt
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def getDiff (L : List Q16_16) : List Int :=
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List.ofFn (n := 7) fun i =>
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(getD L (i.val + 1) zero).toInt - (getD L i.val zero).toInt
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/-- Sign of an integer. -/
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def getSign (x : Int) : Int :=
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@ -144,17 +164,17 @@ def getSign (x : Int) : Int :=
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else 0
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/-- Count sign changes in differences. -/
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def countSignChanges (L : Array Q16_16) : Nat :=
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def countSignChanges (L : List Q16_16) : Nat :=
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let diffs := getDiff L
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let signs := diffs.map getSign
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(List.range 6).foldl (fun acc i =>
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let s1 := signs.getD i 0
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let s2 := signs.getD (i + 1) 0
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let s1 := getDInt signs i 0
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let s2 := getDInt signs (i + 1) 0
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acc + if s1 ≠ 0 ∧ s2 ≠ 0 ∧ s1 ≠ s2 then 1 else 0
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) 0
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/-- Extract exact order based on sign changes of L2. -/
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def extractExactOrder (L2 : Array Q16_16) : Nat :=
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def extractExactOrder (L2 : List Q16_16) : Nat :=
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let sc := countSignChanges L2
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if sc ≥ 3 then 3
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else if sc ≥ 1 then 2
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@ -165,30 +185,25 @@ def extractExactOrder (L2 : Array Q16_16) : Nat :=
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-- ============================================================
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/-- Initial L1 state for Erdős #336 basis (GAAAACCA). -/
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def initialL1 : Array Q16_16 :=
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#[ofRawInt 28087, zero, zero, zero, zero, ofRawInt 18724, ofRawInt 18724, zero]
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def initialL1 : List Q16_16 :=
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[ofRawInt 28087, zero, zero, zero, zero, ofRawInt 18724, ofRawInt 18724, zero]
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/-- Initial L2 state for Erdős #336 basis (AAAAAAAAAAAAAAAA). -/
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def initialL2 : Array Q16_16 :=
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#[zero, zero, zero, zero, zero, zero, zero, zero]
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def initialL2 : List Q16_16 :=
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[zero, zero, zero, zero, zero, zero, zero, zero]
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/-- Initial full 16D state. -/
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def initialV : Array Q16_16 := initialL1 ++ initialL2
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-- Theorems closed by reflexivity (reflexive decision loop)
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-- Theorems closed by native_decide (reflexive decision loop).
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-- We use native_decide because Array primitives (ofFn, getD, foldl) are native
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-- FFI functions that are opaque to the kernel's term reducer, making VM-based
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-- native_decide the only tactic that can reduce and close these arithmetic goals.
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def initialV : List Q16_16 := initialL1 ++ initialL2
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-- Theorems closed by decide or rfl (pure kernel reduction)
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theorem erdos_336_order :
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let v_final := cycleTdoku initialV 20
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let L1_final := Array.ofFn (n := 8) fun i => v_final.getD i.val zero
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extractOrder L1_final = 2 := by native_decide
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let L1_final := List.ofFn (n := 8) fun i => getD v_final i.val zero
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extractOrder L1_final = 2 := by rfl
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theorem erdos_336_exact_order :
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let v_final := cycleTdoku initialV 20
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let L2_final := Array.ofFn (n := 8) fun i => v_final.getD (8 + i.val) zero
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extractExactOrder L2_final = 3 := by native_decide
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let L2_final := List.ofFn (n := 8) fun i => getD v_final (8 + i.val) zero
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extractExactOrder L2_final = 3 := by rfl
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end SilverSight.PIST.Tdoku16D
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