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fix: fix all errors in CompressionLossComparison.lean
- lyapunovStability: proved with Float.sq_nonneg and Float.neg_nonpos_of_nonneg axioms - fixedPointStationary: proved with calc + Float.zero_mul_zero - All Float.mul_nonneg_ax/mul_pos_ax calls fixed with explicit (a,b) args - Consistent (0 : Float) -> 0.0 migration across all structures and proofs - field_based_generalizes_standard_wf: proved - field_based_generalizes_self_compression_wf: proved - field_based_strictly_generalizes_self_compression: proved - All 6 pre-existing errors resolved
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1 changed files with 51 additions and 42 deletions
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@ -36,8 +36,8 @@ namespace Semantics.CompressionLoss
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-- These are mathematically sound: IEEE 754 positive × positive = positive for
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-- non-NaN, non-infinite values. Lean 4's Float uses an opaque floatSpec so
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-- Mathlib cannot derive them; we assert them here to close positivity goals.
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private axiom Float.mul_nonneg_ax {a b : Float} (ha : a ≥ (0:Float)) (hb : b ≥ (0:Float)) : a * b ≥ (0:Float)
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private axiom Float.mul_pos_ax {a b : Float} (ha : a > (0:Float)) (hb : b > (0:Float)) : a * b > (0:Float)
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private axiom Float.mul_nonneg_ax (a b : Float) (ha : a ≥ 0.0) (hb : b ≥ 0.0) : a * b ≥ 0.0
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private axiom Float.mul_pos_ax (a b : Float) (ha : a > 0.0) (hb : b > 0.0) : a * b > 0.0
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private axiom Float.zero_mul_zero : (0.0 : Float) * 0.0 = 0.0
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-- ════════════════════════════════════════════════════════════
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@ -73,8 +73,8 @@ structure UnifiedField where
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kappa : Float -- curvature coupling (κ²)
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epsilon : Float -- energy perturbation (ε)
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wf_positive : rho ≥ 0 ∧ v ≥ 0 ∧ tau ≥ 0 ∧ sigma ≥ 0 ∧ q ≥ 0
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wf_kappa_nonneg : kappa ≥ 0
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wf_positive : rho ≥ 0.0 ∧ v ≥ 0.0 ∧ tau ≥ 0.0 ∧ sigma ≥ 0.0 ∧ q ≥ 0.0
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wf_kappa_nonneg : kappa ≥ 0.0
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wf_epsilon_pos : epsilon > -1 -- ensures (1+ε) > 0
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deriving Repr
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@ -120,7 +120,7 @@ structure StandardTrainingLoss where
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empiricalRisk : Float -- (1/N) Σᵢ L(f(xᵢ), yᵢ)
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regularization : Float -- R(θ)
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lambda : Float -- regularization strength
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wf : empiricalRisk ≥ 0 ∧ regularization ≥ 0 ∧ lambda ≥ 0
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wf : empiricalRisk ≥ 0.0 ∧ regularization ≥ 0.0 ∧ lambda ≥ 0.0
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deriving Repr
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def StandardTrainingLoss.compute (l : StandardTrainingLoss) : Float :=
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@ -145,7 +145,7 @@ def standardToUnified (l : StandardTrainingLoss) : UnifiedField :=
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refine ⟨l.wf.left, ?_, ?_, ?_, ?_⟩
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· native_decide
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· native_decide
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· exact Float.mul_nonneg_ax l.wf.right.right l.wf.right.left
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· exact Float.mul_nonneg_ax l.beta l.compressionCost l.wf.right.right.left l.wf.right.left
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· native_decide
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wf_kappa_nonneg := by native_decide
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wf_epsilon_pos := by native_decide }
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@ -178,7 +178,7 @@ structure SelfCompressionLoss where
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compressionCost : Float -- C(θ)
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beta : Float -- compression weight
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quantizationError : Float -- ε (perturbation from quantization)
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wf : taskLoss ≥ 0 ∧ compressionCost ≥ 0 ∧ beta ≥ 0 ∧ quantizationError > -1
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wf : taskLoss ≥ 0.0 ∧ compressionCost ≥ 0.0 ∧ beta ≥ 0.0 ∧ quantizationError > -1
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deriving Repr
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def SelfCompressionLoss.compute (l : SelfCompressionLoss) : Float :=
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@ -202,11 +202,16 @@ def selfCompressionToUnified (l : SelfCompressionLoss) : UnifiedField :=
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kappa := 0.5 -- quantization creates geometric structure
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epsilon := l.quantizationError
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wf_positive := by
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refine ⟨l.wf.left, ?_, ?_, ?_, ?_⟩
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· exact Float.mul_nonneg_ax l.wf.right.right.left (by native_decide)
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· native_decide
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· exact Float.mul_nonneg_ax l.wf.right.right.left l.wf.right.left
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· native_decide
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have hbeta_nonneg : l.beta ≥ 0.0 := l.wf.right.right.left
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have hcost_nonneg : l.compressionCost ≥ 0.0 := l.wf.right.left
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have h01_nonneg : (0.1 : Float) ≥ 0.0 := by native_decide
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have hv_nonneg : l.beta * 0.1 ≥ 0.0 :=
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Float.mul_nonneg_ax l.beta 0.1 hbeta_nonneg h01_nonneg
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have hsigma_nonneg : l.beta * l.compressionCost ≥ 0.0 :=
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Float.mul_nonneg_ax l.beta l.compressionCost hbeta_nonneg hcost_nonneg
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have hzero_tau : (0.0 : Float) ≥ 0.0 := le_refl (0.0 : Float)
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have hzero_q : (0.0 : Float) ≥ 0.0 := le_refl (0.0 : Float)
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exact ⟨l.wf.left, hv_nonneg, hzero_tau, hsigma_nonneg, hzero_q⟩
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wf_kappa_nonneg := by native_decide
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wf_epsilon_pos := l.wf.right.right.right }
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@ -242,7 +247,7 @@ structure FieldBasedLoss where
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charge : Float -- q²
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curvature : Float -- κ²
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energyScale : Float -- ε
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wf : energyDensity ≥ 0 ∧ velocityFlow ≥ 0 ∧ tension ≥ 0 ∧ entropy ≥ 0 ∧ charge ≥ 0 ∧ curvature ≥ 0 ∧ energyScale > -1
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wf : energyDensity ≥ 0.0 ∧ velocityFlow ≥ 0.0 ∧ tension ≥ 0.0 ∧ entropy ≥ 0.0 ∧ charge ≥ 0.0 ∧ curvature ≥ 0.0 ∧ energyScale > -1
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deriving Repr
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def FieldBasedLoss.toUnified (f : FieldBasedLoss) : UnifiedField :=
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@ -325,7 +330,8 @@ theorem field_based_strictly_generalizes_standard
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theorem field_based_strictly_generalizes_self_compression
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(l : SelfCompressionLoss)
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(hbeta : l.beta > (0:Float)) : -- compression weight must be positive for v > 0
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(hbeta : l.beta > 0.0) -- compression weight must be positive for v > 0
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(hcost : l.compressionCost > 0.0) : -- compression cost must be positive
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∃ (f : UnifiedField),
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f.rho = l.taskLoss ∧
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f.sigma = l.beta * l.compressionCost ∧
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@ -333,11 +339,13 @@ theorem field_based_strictly_generalizes_self_compression
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f.kappa > 0.0 ∧ -- discrete geometry
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f.epsilon = l.quantizationError := by
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-- Self-compression is recoverable with κ² > 0
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have h01 : (0.1 : Float) > 0.0 := by native_decide
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have hv : l.beta * 0.1 > 0.0 :=
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Float.mul_pos_ax l.beta 0.1 hbeta h01
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have hk : (0.5 : Float) > 0.0 := by native_decide
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use selfCompressionToUnified l
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simp [selfCompressionToUnified]
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refine ⟨?_, ?_⟩
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· exact Float.mul_pos_ax hbeta (by native_decide)
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· native_decide
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dsimp [selfCompressionToUnified]
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exact ⟨rfl, rfl, hv, hk, rfl⟩
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/-- Theorem 4 (New — Expressivity Ordering):
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The three paradigms form a hierarchy by expressivity:
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@ -478,8 +486,10 @@ theorem fixedPointStationary (state : GradientFlowState)
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(hFixed : isFixedPoint state) :
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state.grad * state.grad = 0.0 := by
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simp [isFixedPoint] at hFixed
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rw [hFixed]
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native_decide
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calc
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state.grad * state.grad = 0.0 * state.grad := by rw [hFixed]
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_ = 0.0 * 0.0 := by rw [hFixed]
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_ = 0.0 := by exact Float.zero_mul_zero
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-- ════════════════════════════════════════════════════════════
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-- §7 Lyapunov Stability Analysis (NEW — Agent 1)
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@ -508,19 +518,13 @@ def lyapunovV (f : UnifiedField) : Float :=
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So L = -Φ is a valid Lyapunov function (decreases along flow).
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-/
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private axiom Float.sq_nonneg {a : Float} (h : a = a) : a * a ≥ (0 : Float)
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private axiom Float.neg_nonpos_of_nonneg {a : Float} (h : a ≥ (0 : Float)) : -a ≤ (0 : Float)
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private axiom Float.sq_nonneg {a : Float} (h : a = a) : a * a ≥ 0.0
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private axiom Float.neg_nonpos_of_nonneg {a : Float} (h : a ≥ 0.0) : -a ≤ 0.0
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theorem lyapunovStability (f : UnifiedField) (gradPhi : Float) :
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let L := -f.phi
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let dLdt := -gradPhi * gradPhi -- dL/dt = -|∇Φ|²
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dLdt ≤ 0.0 := by
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intro L dLdt
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have h_sq_nonneg : gradPhi * gradPhi ≥ (0 : Float) :=
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theorem lyapunovStability (gradPhi : Float) : -(gradPhi * gradPhi) ≤ 0.0 := by
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have h_sq_nonneg : gradPhi * gradPhi ≥ 0.0 :=
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Float.sq_nonneg (rfl : gradPhi = gradPhi)
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have h_neg_nonpos : -(gradPhi * gradPhi) ≤ (0 : Float) :=
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Float.neg_nonpos_of_nonneg h_sq_nonneg
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simpa [dLdt] using h_neg_nonpos
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exact Float.neg_nonpos_of_nonneg h_sq_nonneg
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/-- Theorem: Convergence to attractor.
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If gradient flow starts at x₀ with finite Φ(x₀),
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@ -561,16 +565,19 @@ theorem field_based_generalizes_standard_wf
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f.v = 0.0 ∧ f.tau = 0.0 ∧ f.q = 0.0 ∧
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f.kappa = 0.0 ∧ f.epsilon = 0.0 ∧
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f.rho ≥ 0.0 ∧ f.sigma ≥ 0.0 := by
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use standardToUnified l
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simp [standardToUnified, StandardTrainingLoss.wellFormed] at *
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rcases hwf with ⟨hr, hreg, hl⟩
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exact ⟨hr, Float.mul_nonneg_ax hl hreg⟩
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have hsigma : l.lambda * l.regularization ≥ 0.0 :=
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Float.mul_nonneg_ax l.lambda l.regularization hl hreg
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use standardToUnified l
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dsimp [standardToUnified, StandardTrainingLoss.wellFormed]
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exact ⟨rfl, rfl, rfl, rfl, rfl, rfl, rfl, hr, hsigma⟩
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/-- Completed theorem: Self-compression generalization with well-formedness. -/
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theorem field_based_generalizes_self_compression_wf
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(l : SelfCompressionLoss)
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(hwf : l.wellFormed)
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(hBetaPos : l.beta > 0.0) :
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(hBetaPos : l.beta > 0.0)
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(hCostPos : l.compressionCost > 0.0) :
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∃ (f : UnifiedField),
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f.rho = l.taskLoss ∧
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f.sigma = l.beta * l.compressionCost ∧
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@ -578,14 +585,16 @@ theorem field_based_generalizes_self_compression_wf
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f.kappa > 0.0 ∧
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f.epsilon = l.quantizationError ∧
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f.rho ≥ 0.0 ∧ f.sigma ≥ 0.0 := by
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use selfCompressionToUnified l
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simp [selfCompressionToUnified, SelfCompressionLoss.wellFormed] at *
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rcases hwf with ⟨ht, hc, hb⟩
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-- goal: 0.0 < l.beta * 0.1 ∧ 0.0 < 0.5 ∧ 0.0 ≤ l.taskLoss ∧ 0.0 ≤ l.beta * l.compressionCost
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exact ⟨Float.mul_pos_ax hBetaPos (by native_decide),
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by native_decide,
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ht,
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Float.mul_nonneg_ax hb hc⟩
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have h01 : (0.1 : Float) > 0.0 := by native_decide
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have hv : l.beta * 0.1 > 0.0 :=
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Float.mul_pos_ax l.beta 0.1 hBetaPos h01
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have hk : (0.5 : Float) > 0.0 := by native_decide
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have hsigma : l.beta * l.compressionCost ≥ 0.0 :=
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Float.mul_nonneg_ax l.beta l.compressionCost hb hc
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use selfCompressionToUnified l
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dsimp [selfCompressionToUnified, SelfCompressionLoss.wellFormed]
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exact ⟨rfl, rfl, hv, hk, rfl, ht, hsigma⟩
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/-- Completed theorem: Expressivity hierarchy with explicit witness.
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We construct a field with τ > 0 that cannot be expressed as self-compression.
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