Finish the projection migration for reaching definitions by replacing the
accumulator-style runOfTrace*From definitions and their hand-rolled
re-association lemmas with a single analysis-agnostic projection:
Trace.steps / Traceₗ.steps, the chronological List of executed
(index, statement) pairs. Its four simp lemmas are one-line inductions,
with all re-association falling out of mathlib's List.append_assoc and
List.reverse_append.
Run is now an abbrev for List (State × BasicStmt) (latest-first, so
LastAssign keeps its first-match structure) and runOfTrace is just
steps.reverse.
Also hoist the generic reaches_final_post into Forward.lean, letting
analyze_correct' be stated directly about S.Post (prog.trace hrun).
Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
Introduce `tr₁ ++< he >++ tr₂` scoped notation for `Trace.concat`
(precedence 65, right-associative, mirroring `++`) and use it
throughout Properties.lean.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
This requires a few pieces:
* Make node tags use `Fin n` intead of natural numbers. This makes
it possible to build a finite lattice over AST nodes, and also
ensure automatic, total indexing from CFG nodes into the AST that
created them. For this, use the elaborator to derive the ordering
statements etc. where possible.
* Adjust the forward framework to enable proofs that don't just state
correctness on the environment, but also on an arbitrary additional
state accumulated from traversing the trace.
* State the reaching definition analysis's correctness in terms
of this new framework.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Convert every theorem to lemma (mathlib's default) except the headline results a
reader of each module seeks out: analyze_correct (Forward/Sign/Constant),
aFix_eq/aFix_le (Fixedpoint), trace (Language), and Stmt.cfg_sufficient
(Language/Properties). lemma and theorem are interchangeable keywords, so no
references change.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
