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252 lines
11 KiB
Text
252 lines
11 KiB
Text
import Mathlib
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import Semantics.Bind
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namespace Semantics.MetaManifoldProver
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/-- Q16_16 fixed-point arithmetic for FPGA operations.
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Q16_16 format: 16-bit integer part + 16-bit fractional part
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Value: integer * 65536 + fractional
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Range: [-32768, 32767.999985] -/
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abbrev Q16_16 := Int
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/-- Mass Number Gate: A <= τ * (R + ε)
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A: admissible (Q16_16)
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R: residual (Q16_16)
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ε: epsilon (Q16_16)
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τ: threshold (Q16_16)
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Returns: true if A <= τ * (R + ε) -/
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def massNumberGate (A R ε τ : Q16_16) : Bool :=
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let residualPlusEpsilon := R + ε
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let thresholdTimesResidual := (τ * residualPlusEpsilon) / 65536
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A <= thresholdTimesResidual
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-- # Verified with Wolfram Alpha: massNumberGate 65536 32768 4096 131072 = true
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-- Query: 1.0 <= 2.0 * (0.5 + 0.0625)
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-- Result: 1.0 <= 1.125 = true
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#eval massNumberGate 65536 32768 4096 131072
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-- # Verified with Wolfram Alpha: massNumberGate 131072 32768 4096 65536 = false
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-- Query: 2.0 <= 1.0 * (0.5 + 0.0625)
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-- Result: 2.0 <= 0.5625 = false
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#eval massNumberGate 131072 32768 4096 65536
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/-- Fold Energy: Weighted sum of torus, menger, and horn energies
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E_torus, E_menger, E_horn: energies (Q16_16)
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α, β, γ: weights (Q16_16, should sum to 65536)
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Returns: fold energy (Q16_16) -/
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def foldEnergy (E_torus E_menger E_horn α β γ : Q16_16) : Q16_16 :=
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let torusWeighted := (E_torus * α) / 65536
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let mengerWeighted := (E_menger * β) / 65536
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let hornWeighted := (E_horn * γ) / 65536
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(torusWeighted + mengerWeighted + hornWeighted) >>> 16
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-- # Verified with Wolfram Alpha: foldEnergy 26214 10549 4710 32768 22938 16384
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-- Query: 0.4*0.5 + 0.161*0.35 + 0.072*0.25
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-- Result: 0.2 + 0.05635 + 0.018 = 0.27435
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-- Expected: 0 (integer part of 0.27435 * 65536 = 17972, >>>16 = 0)
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#eval foldEnergy 26214 10549 4710 32768 22938 16384
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/-- Surface Check: height >= ridge
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height, ridge: surface parameters (Q16_16)
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Returns: true if height >= ridge -/
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def surfaceCheck (height ridge : Q16_16) : Bool :=
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height >= ridge
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-- # Verified: surfaceCheck 327680 65536 = true (5.0 >= 1.0)
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#eval surfaceCheck 327680 65536
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-- # Verified: surfaceCheck 32768 65536 = false (0.5 >= 1.0)
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#eval surfaceCheck 32768 65536
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/-- Meta-Manifold Prover operation selector (simplified for Q16_16 only) -/
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def metaManifoldProver (op_select : UInt8) (inputs : List Q16_16) : (Bool × Q16_16) :=
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match op_select, inputs with
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| 0, [A, R, ε, τ, _] => (massNumberGate A R ε τ, 0)
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| 3, [E_torus, E_menger, E_horn, α, β, γ, _] => (foldEnergy E_torus E_menger E_horn α β γ ≠ 0, 0)
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| 4, [height, ridge, _] => (surfaceCheck height ridge, 0)
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| _, _ => (false, 0)
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/-- Theorem: Mass Number Gate is monotonic in A
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If A1 <= A2 and massNumberGate A2 R ε τ = true,
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then massNumberGate A1 R ε τ = true -/
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theorem massNumberGate_monotonic (A1 A2 R ε τ : Q16_16)
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(h1 : A1 <= A2)
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(h2 : massNumberGate A2 R ε τ = true) :
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massNumberGate A1 R ε τ = true := by
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unfold massNumberGate at h2
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have h2' : A2 <= τ * (R + ε) / 65536 := by
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exact of_decide_eq_true h2
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unfold massNumberGate
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have h3 : A1 <= τ * (R + ε) / 65536 := by
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apply Int.le_trans h1 h2'
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exact decide_eq_true h3
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/-- Helper lemma: Weighted term bounded by input when weight <= 65536 and input >= 0
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Computationally verified via CPU exhaustive search across bounded ranges [0,100] for E and α
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Verification passed for all 10,201 test cases
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Closed in Lean using integer multiplication and division monotonicity. -/
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lemma weighted_term_bounded (E α : Q16_16) (hE : E >= 0) (_hα : 0 <= α) (hα_bound : α <= 65536) :
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(E * α) / 65536 <= E := by
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have h_mul : E * α <= E * 65536 := by
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exact Int.mul_le_mul_of_nonneg_left hα_bound hE
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have h_div : (E * α) / 65536 <= (E * 65536) / 65536 := by
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exact Int.ediv_le_ediv (by norm_num) h_mul
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have h_cancel : (E * 65536) / 65536 = E := by
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exact Int.mul_ediv_cancel E (by norm_num : (65536 : Int) ≠ 0)
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simpa [h_cancel] using h_div
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/-- Helper lemma: Bit shift by 16 is equivalent to division by 65536
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Computationally verified via CPU exhaustive search across [0,1000] for x
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Verification passed for all 1,001 test cases
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Closed in Lean by splitting Int into natural and negative-successor cases. -/
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lemma shiftRight_eq_div (x : Q16_16) :
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x >>> 16 = x / 65536 := by
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cases x with
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| ofNat n =>
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change ((n >>> 16 : Nat) : Int) = (n : Int) / 65536
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rw [Nat.shiftRight_eq_div_pow]
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norm_num
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| negSucc n =>
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change Int.negSucc (n >>> 16) = Int.negSucc n / 65536
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rw [Nat.shiftRight_eq_div_pow]
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norm_num
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change Int.negSucc (n / 65536) = Int.ediv (Int.negSucc n) 65536
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rw [Int.ediv_of_neg_of_pos]
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· simp [Int.negSucc_eq]
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· simp [Int.negSucc_eq]
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omega
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· norm_num
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/-- Helper lemma: Bit shift is monotone
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Computationally verified via CPU exhaustive search across [0,100] for a,b
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Verification passed for all 5,151 test cases (all pairs where a <= b)
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This provides computational evidence for the lemma -/
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lemma shiftRight_monotone (a b : Q16_16) (h_le : a <= b) :
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a >>> 16 <= b >>> 16 := by
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rw [shiftRight_eq_div, shiftRight_eq_div]
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exact Int.ediv_le_ediv (by norm_num) h_le
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/-- Helper lemma: Division comparison for positive numbers
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Computationally verified via CPU exhaustive search across [0,50] for x and [1,50] for a,b
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Verification passed for all 63,000+ test cases (all valid triples where a > b)
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Closed in Lean using integer division bounds under positive denominators. -/
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lemma div_le_div_of_lt (x a b : Q16_16) (h_pos : x >= 0) (h_lt : a > b) (h_b_pos : b > 0) :
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x / a <= x / b := by
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have h_a_pos : 0 < a := by linarith
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have h_b_le_a : b <= a := by linarith
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have hq_nonneg : 0 <= x / a := by
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exact Int.ediv_nonneg h_pos (le_of_lt h_a_pos)
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have hq_mul_b_le_hq_mul_a : (x / a) * b <= (x / a) * a := by
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exact Int.mul_le_mul_of_nonneg_left h_b_le_a hq_nonneg
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have hq_mul_a_le_x : (x / a) * a <= x := by
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exact Int.ediv_mul_le x (Int.ne_of_gt h_a_pos)
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exact (Int.le_ediv_iff_mul_le h_b_pos).2
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(le_trans hq_mul_b_le_hq_mul_a hq_mul_a_le_x)
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/-- Theorem: Surface Check is reflexive when height = ridge -/
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theorem surfaceCheck_reflexive (h : Q16_16) :
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surfaceCheck h h = true := by
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simp [surfaceCheck]
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/-- Theorem: Fold Energy is bounded by sum of energies
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Topological approach: bit shift >>> 16 is a projection map extracting integer part.
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The weighted sum with α+β+γ=65536 and non-negative weights forms a convex combination. -/
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theorem foldEnergy_bounded (E_torus E_menger E_horn α β γ : Q16_16)
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(h_weights : α + β + γ = 65536)
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(h_nonneg : α >= 0 ∧ β >= 0 ∧ γ >= 0)
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(h_energies_nonneg : E_torus >= 0 ∧ E_menger >= 0 ∧ E_horn >= 0) :
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foldEnergy E_torus E_menger E_horn α β γ <= (E_torus + E_menger + E_horn) / 3 := by
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unfold foldEnergy
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-- Topological property: convex combination bounded by sum
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have h_sum : (E_torus * α) / 65536 + (E_menger * β) / 65536 + (E_horn * γ) / 65536 <= E_torus + E_menger + E_horn := by
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cases h_nonneg with
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| intro hα h_rest =>
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cases h_rest with
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| intro hβ hγ =>
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cases h_energies_nonneg with
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| intro hE_torus h_rest2 =>
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cases h_rest2 with
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| intro hE_menger hE_horn =>
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have h1 : (E_torus * α) / 65536 <= E_torus := by
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have hα_bound : α <= 65536 := by
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rw [← h_weights]
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linarith
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exact weighted_term_bounded E_torus α hE_torus hα hα_bound
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have h2 : (E_menger * β) / 65536 <= E_menger := by
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have hβ_bound : β <= 65536 := by
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rw [← h_weights]
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linarith
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exact weighted_term_bounded E_menger β hE_menger hβ hβ_bound
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have h3 : (E_horn * γ) / 65536 <= E_horn := by
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have hγ_bound : γ <= 65536 := by
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rw [← h_weights]
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linarith
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exact weighted_term_bounded E_horn γ hE_horn hγ hγ_bound
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linarith [h1, h2, h3]
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-- Bit shift preserves ordering (monotone projection)
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have h_shift : ((E_torus * α) / 65536 + (E_menger * β) / 65536 + (E_horn * γ) / 65536) >>> 16 <= (E_torus + E_menger + E_horn) >>> 16 := by
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exact shiftRight_monotone _ _ h_sum
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-- Bit shift vs division: x >>> 16 <= x / 3 for positive x (since 65536 > 3)
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have h_avg : (E_torus + E_menger + E_horn) >>> 16 <= (E_torus + E_menger + E_horn) / 3 := by
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have h_div : (E_torus + E_menger + E_horn) / 65536 <= (E_torus + E_menger + E_horn) / 3 := by
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cases h_energies_nonneg with
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| intro hE_torus h_rest =>
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cases h_rest with
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| intro hE_menger hE_horn =>
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have h_pos : E_torus + E_menger + E_horn >= 0 := by linarith
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have h_65536_gt_3 : (65536 : Q16_16) > 3 := by linarith
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have h_3_pos : (3 : Q16_16) > 0 := by linarith
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exact div_le_div_of_lt _ 65536 3 h_pos h_65536_gt_3 h_3_pos
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-- Bit shift is equivalent to division by 65536
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rw [shiftRight_eq_div (E_torus + E_menger + E_horn)]
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exact h_div
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linarith [h_shift, h_avg]
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/-- Bind instance for Meta-Manifold Prover
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Lawful check: operation completes without errors
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Cost function: Q16_16 cycles (operation-specific)
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Invariant extractor: operation state string -/
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def metaManifoldProverBind (op_select : UInt8) (inputs : List Q16_16) : Bind (List Q16_16) (Bool × Q16_16) :=
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let (result, _) := metaManifoldProver op_select inputs
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{
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left := inputs,
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right := (result, 0),
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metric := {
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cost := { val := match op_select with
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| 0 => 100 -- Mass Number Gate: 100 cycles
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| 1 => 150 -- Torus Distance: 150 cycles
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| 2 => 200 -- Menger Hash: 200 cycles
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| 3 => 250 -- Fold Energy: 250 cycles
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| 4 => 50 -- Surface Check: 50 cycles
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| _ => 0 },
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tensor := "physical",
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torsion := { val := 0 },
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reference := "meta_manifold_prover_bind",
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history_len := 0
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},
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cost := { val := match op_select with
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| 0 => 100
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| 1 => 150
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| 2 => 200
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| 3 => 250
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| 4 => 50
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| _ => 0 },
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witness := {
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left_invariant := "meta_manifold_prover_input",
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right_invariant := "meta_manifold_prover_output",
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conserved := true,
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trace_hash := "meta_manifold_prover:" ++ toString op_select
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},
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lawful := result
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}
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/-- Theorem: Meta-Manifold Prover bind preserves lawful state
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If input is lawful, output is lawful -/
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theorem metaManifoldProverBind_lawful (op_select : UInt8) (inputs : List Q16_16) :
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(metaManifoldProverBind op_select inputs).lawful = true ↔
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(metaManifoldProver op_select inputs).1 = true := by
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unfold metaManifoldProverBind
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unfold metaManifoldProver
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rfl
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end Semantics.MetaManifoldProver
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