finiteLatticeEnumerationCert_holds
plain-language theorem explainer
The theorem constructs a FiniteLatticeEnumerationCert for any admissible-state predicate on a carrier and any generator action that preserves admissibility. Researchers advancing the DREAM completeness program in Recognition Science ethics would cite it to obtain constructive reachability witnesses on finite lattices. The proof is a direct term construction that populates the certificate's three fields from identity and singleton reachability witnesses plus trivial search closure.
Claim. Let $adm : α → Prop$ be an admissible-state predicate on a type $α$, let $act : G → α → α$ be a generator action on a type $G$, and let $h_{pres}$ be a proof that $act$ preserves admissibility. Then the structure FiniteLatticeEnumerationCert$(α, G, adm, act, h_{pres})$ holds, asserting that the identity map is a reachable transition, every singleton generator action is a reachable transition, and the search is closed for the empty generator list.
background
The module supplies constructive search infrastructure for the SigmaPreservingIsReachable residual hypothesis inside the DREAM completeness program. It works on abstract carriers $α$ and generator sets $G$ to avoid prior bit-rot in golden-ratio references. An admissible predicate selects valid states; a generator action maps a generator and state to a new state; preservation requires that admissible states remain admissible after any generator step.
proof idea
The term proof builds the FiniteLatticeEnumerationCert structure by direct field assignment. The identity_reachable field is supplied by the identity_witness_reachable lemma applied to adm and act. The singleton_reachable field is obtained by lambda-abstraction over generators followed by singleton_witness_reachable using the preservation hypothesis. The trivial_closed field is filled by the trivial_search_closed witness.
why it matters
This result packages the base certificate that lets the finite-lattice enumeration argument discharge the SigmaPreservingIsReachable hypothesis for the DREAM completeness program. It advances the self-contained stub layer by treating the 14 or 15 DREAM virtues as parameters of an abstract framework rather than a hard-coded list. The construction applies uniformly to any finite carrier with a finite generator set.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.