DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Term denotation

Definition

For an $L$ -structure $M$ and assignment $s \in M^{ω}$ , define $⟦ v_{i} ⟧_{s} = s (i)$ , $⟦ c ⟧_{s} = c^{M}$ and

$⟦ f (t_{1}, \dots, t_{n}) ⟧_{s} = f^{M} (⟦ t_{1} ⟧_{s}, \dots, ⟦ t_{n} ⟧_{s}) .$

This is a simultaneous definition for all assignments: recurse on terms into the set $M^{(M^{ω})}$ of functions from assignments to $M$ . Variable leaves are projections, constant leaves are constant maps, and each function operation acts pointwise. Structural recursion gives a unique evaluation. The operations are total because the interpreted functions are total. Superscripts $M$ on brackets will distinguish structures when needed.

Conventions and prerequisites: Structures and variable assignments, Structural induction and recursion on syntax.

Depends on

Dependency tree · 2 levels

Sources

Moschovakis, Lecture Notes in Logic (2014) — 1C.7, p.11.