composite_setoid_le_right
plain-language theorem explainer
Composite recognizer setoids refine the right component's setoid in the refinement preorder. Researchers formalizing the lattice of recognizers cite this lemma when constructing the meet operation ahead of Zorn applications. The term proof extracts the right conjunct from the composite setoid equivalence.
Claim. Let $r_1 : C → E_1$ and $r_2 : C → E_2$ be recognizers. The equivalence relation induced by the composite recognizer $r_1 ⊗ r_2$ refines the equivalence relation induced by $r_2$.
background
Recognition Geometry associates to each recognizer an equivalence relation on configuration space C via the RG3 indistinguishability axiom. The recognizerSetoid is this induced equivalence (synonym for indistinguishableSetoid), and composition ⊗ of recognizers corresponds to the meet in the refinement lattice of setoids. The module prepares these refinement properties for Zorn's lemma applications to maximal recognizers.
proof idea
One-line term proof that applies the composite_setoid_iff characterization and projects to the second component of the resulting conjunction.
why it matters
This lemma is a direct prerequisite for composite_setoid_is_inf, which establishes the infimum property, and for recognizer_meet_semilattice, which shows recognizer setoids form a meet-semilattice. It supplies the refinement facts needed to apply Zorn's lemma for existence of maximal recognizers, as referenced in the module documentation for Theorems 4.1 and 5.1.
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