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224e773 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 | import HyperKitty.QLG
import HyperKitty.QRA
/-! # Witness Evolution Formalization
This module formalizes the witness evolution system for QLG-certified tokens.
-/
-- Witness: vector of 3 glyphs
structure Witness : Type where
w : List Glyph
-- Canonical witness
abbrev canonicalWitness : Witness := ⟨[Glyph.Pi, Glyph.Gamma, Glyph.Delta]⟩
-- Witness evolution
abbrev evolveWitness (w : Witness) : Witness :=
let [a, b, c] := w.w else ⟨[]⟩;
⟨[a.next b, b.next c, c.next a]⟩
-- Exhaustion: evolve^2 reaches [Omega, Omega, Omega]
theorem canonical_exhaustion :
(evolveWitness (evolveWitness canonicalWitness)).w = [Glyph.Omega, Glyph.Omega, Glyph.Omega] := by
simp [evolveWitness, canonicalWitness, Glyph.next, Q]
decide
-- Invalid fixed point: [Lambda, Lambda, Lambda]
theorem invalid_fixed_point :
evolveWitness ⟨[Glyph.Lambda, Glyph.Lambda, Glyph.Lambda]⟩ = ⟨[Glyph.Lambda, Glyph.Lambda, Glyph.Lambda]⟩ := by
simp [evolveWitness, Glyph.next, Q]
decide
-- Witness is exhausted if all are Omega
def Witness.isExhausted (w : Witness) : Bool :=
w.w.all (· = Glyph.Omega)
-- Witness is at invalid fixed point
def Witness.isInvalidFixedPoint (w : Witness) : Bool :=
w.w.all (· = Glyph.Lambda)
-- Exhaustion property
theorem witness_exhaustion_in_2_steps :
∀ w : Witness, w = canonicalWitness →
(evolveWitness (evolveWitness w)).isExhausted := by
intro w hw
rw [hw]
simp [Witness.isExhausted, canonical_exhaustion]
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