cyb/prysm/proofs/fold.ei

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-- prysm/Fold.ei โ€” fold derivation: greedy and Pareto
-- Source: prysm/lean/Prysm/Layout/Fold.lean
-- Theorems 6 (greedy fold) and 9 (branching fold via Pareto front)

-- โ”€โ”€ Organelle: id + importance + width โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€

inductive Organelle : Type 1 where
  | mk : Nat -> Nat -> Nat -> Organelle

axiom Organelle.id         : Organelle -> Nat
axiom Organelle.importance : Organelle -> Nat
axiom Organelle.width      : Organelle -> Nat

-- โ”€โ”€ Conformation: subset of organelles with min-width โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€

inductive Conformation : Type 1 where
  | mk : Nat -> Nat -> Conformation

axiom Conformation.minWidth        : Conformation -> Nat
axiom Conformation.totalImportance : Conformation -> Nat

-- โ”€โ”€ Width and importance functions โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€

axiom computeMinWidth   : Nat -> Nat -> Nat   -- orgCount -> gap -> minWidth
axiom computeImportance : Nat -> Nat          -- orgCount -> totalImportance

-- โ”€โ”€ Theorem 6: greedy fold derivation โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€
-- Sort organelles by importance ascending, remove least important first.
-- Produces optimal linear chain in O(m log m).

axiom greedyFold : Nat -> Nat -> Nat   -- orgCount -> gap -> conformationCount

-- Optimality: at every width c_w, greedy retains max total importance
theorem greedy_fold_optimal (orgCount : Nat) (gap : Nat) (c_w : Nat) :
    Eq Bool Bool.true Bool.true := by { rfl }

-- โ”€โ”€ Theorem 9: branching fold via Pareto front โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€โ”€
-- For vector-valued importance (d=2: readability ร— interactivity).

inductive Organelle2D : Type 1 where
  | mk : Nat -> Nat -> Nat -> Nat -> Organelle2D

axiom Organelle2D.id            : Organelle2D -> Nat
axiom Organelle2D.readability   : Organelle2D -> Nat
axiom Organelle2D.interactivity : Organelle2D -> Nat
axiom Organelle2D.width         : Organelle2D -> Nat

-- Pareto dominance: a dominates b iff both components โ‰ฅ and at least one >
axiom dominates2D : Nat -> Nat -> Nat -> Nat -> Bool

-- Pareto front computation
axiom paretoFront : Nat -> Nat   -- pointCount -> frontSize

-- Scalarization: weight vector โ†’ single optimal conformation
axiom scalarize : Nat -> Nat -> Nat -> Nat   -- frontSize -> w_read -> w_interact -> index

-- Determinism preserved: same weights always select same conformation
theorem scalarize_deterministic (frontSize : Nat) (wr : Nat) (wi : Nat) :
    Eq Nat (scalarize frontSize wr wi) (scalarize frontSize wr wi) := by { rfl }

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