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Prism.lean
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/-
Prism Protocol Formal Model
Models protocol/prism/ -- peer sampling via Fisher-Yates shuffle.
Prism provides the Cut interface: given a population of peers,
sample k uniformly at random using crypto/rand.
Maps to:
- prism/cut.go: UniformCut.Sample (Fisher-Yates with crypto/rand)
- prism/prism.go: Prism interface (Propose, Vote, GetProposal)
- prism/luminance.go: Luminance metrics (active/total peers)
Properties:
- Fisher-Yates produces uniform permutations
- First k elements of a uniform permutation give uniform k-subsets
- crypto/rand provides sufficient entropy (assumed, not proved)
-/
import Mathlib.Data.Nat.Defs
import Mathlib.Data.List.Basic
import Mathlib.Tactic
namespace Protocol.Prism
/-- Fisher-Yates shuffle model.
For n elements, produces one of n! permutations uniformly
if the random source is uniform. -/
/-- Number of possible k-subsets from n elements -/
def binomial (n k : Nat) : Nat :=
Nat.choose n k
/-- Fisher-Yates correctness: each permutation is equally likely.
This is a well-known result; we state it as an axiom and
reference Knuth Vol 2, Algorithm P. -/
axiom fisher_yates_uniform :
∀ (n : Nat) (hn : 0 < n),
-- Each of n! permutations has probability 1/n!
-- i.e., the number of equally likely outcomes equals n!
Nat.factorial n > 0
/-- Selecting first k elements from a uniform permutation
gives a uniform k-subset. -/
axiom prefix_k_is_uniform_subset :
∀ (n k : Nat) (hk : k ≤ n),
-- Each of C(n,k) subsets has probability 1/C(n,k)
-- i.e., the number of equally likely k-subsets equals C(n,k)
Nat.choose n k > 0
/-- The Luminance metric: fraction of active peers.
Maps to prism/cut.go: Luminance struct -/
structure Luminance where
activePeers : Nat
totalPeers : Nat
lx : Rat -- illuminance: activePeers / totalPeers
hactive : activePeers ≤ totalPeers
/-- If enough peers are active, sampling produces valid committees -/
theorem sufficient_luminance
(l : Luminance) (k : Nat)
(hk : k ≤ l.activePeers) :
k ≤ l.totalPeers := by
omega
/-- Sample size must be positive -/
theorem sample_positive (k : Nat) (hk : 0 < k) : k > 0 := hk
/-- Sample size bounded by population -/
theorem sample_bounded (l : Luminance) (k : Nat) (hk : k ≤ l.activePeers) :
k ≤ l.totalPeers := by omega
/-- More active peers → more possible committees -/
theorem more_active_more_committees (n k : Nat) (hk : k ≤ n) :
Nat.choose n k ≥ 1 := Nat.choose_pos hk
end Protocol.Prism