🤖 AI Summary
This work presents the first complete formalization in Lean 4, based on mathlib, of Dana Scott’s 1972 theory of continuous lattices and its application to modeling the untyped λ-calculus. The project rigorously reproduces the 43 core results from the first four sections of Scott’s original paper, covering essential constructions such as T₀-space embeddings, the Scott topology, the way-below relation, function spaces, and inverse limits, while also incorporating Milner’s corrections to the original proofs. By introducing foundational infrastructure—including bases of Scott-open sets, step functions, towers of function spaces, and the i_∞/j_∞ mapping pair—the formalization establishes the self-embedding theorem D_∞ ≅ [D_∞ → D_∞] under classical logic (with the axiom of choice), propositional extensionality, and quotient type soundness assumptions. All results are verified machine-checked without any use of “sorry”.
📝 Abstract
We present a complete machine-checked formalization of Dana Scott's landmark 1972 paper \emph{Continuous Lattices} \textbf{[Sco72]}, carried out in Lean 4 against mathlib and including the March 1972 Milner correction in \textbf{[Sco72]} (pp.~135--136).
Scott's paper develops a model for \(λ\)-calculus from a topological starting point. He defines \emph{injective} \(T_0\)-spaces -- those with a strong extension property for continuous maps -- and shows that they are exactly the \emph{continuous lattices}: complete lattices whose Scott topology is determined by the order via the way-below relation (\(\ll\)). On this foundation he studies projections, retractions, products, function spaces, and inverse limits. The capstone (Theorem 4.4) constructs an inverse limit \(D_\infty\) of function-space approximants and proves \(D_\infty \cong [D_\infty \to D_\infty]\), yielding a purely mathematical model for Church's untyped \(λ\)-calculus.
Our development formalizes \textbf{43 numbered results} from Scott's Sections 1--4 (Propositions, Corollaries, Lemmas, and Theorems), each as a sorry-free Lean theorem, together with supporting infrastructure (step functions, the \(\Uparrow a\) basis of Scott opens, Milner's coarser-than-Scott hypothesis, the function-space tower, and the \(i_\infty\)/\(j_\infty\) pair). The formalization is \textbf{classical} (uses \texttt{Classical.choice} transitively) and follows Scott's proof dependency order. Where the Lean proof required choices not visible in the original -- or where dead ends were encountered -- we record detailed notes in Section 5. All proofs check with the standard footprint \(\texttt{[propext, Classical.choice, Quot.sound]}\).