{"id":"e92c9b06-de24-4a7f-8140-2a184e28854b","arxiv_id":"2607.09564","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A monadic DSL yields correct-by-construction, substitution-stable bidirectional elaborators for Martin-Löf type theory that extract algebraically from a presheaf model.","lead":"The paper defines a monadic DSL in which bidirectional elaborators for dependent type theory are written as correct-by-construction combinators. The resulting scripts are stable under equality and substitution and extract to a concrete algorithm from a presheaf model of the bi-initial natural model.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader's identification of Theorem 6.4 as the weakest assumption is accurate and already correctly assessed as non-critical: the paper's contribution is the modular monadic framework and the denotational account of suspension/stability, not a self-contained normalisation proof. All internal equational and algebraic steps (combinators, scope/await, multilinearity under strengthening, initiality extraction of Procedures 6.8/C.1/C.2) check out on inspection of the appendices. The absence of a machine-checked formalisation or shipped implementation is a limitation of presentation, not a correctness risk for the mathematical claims. No adjustment to the ACCEPT verdict is warranted.","tokens_in":34773,"tokens_out":523,"duration_ms":17470,"concrete_test":"Independently re-derive the support of the partial element for the symmetry.type script of Computation 4.9 under the dominance S of §6.3.4 (using only the cited injectivity of Pi/Id and levelwise decidability of judgemental equality); confirm that the final support is the full sieve and that the extracted core term is definitionally Pi(Ans, λx.Pi(Ans, λy.Pi(Id(Ans,x,y), λp.Id(Ans,y,x)))).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (shallow embedding of bidirectional surface syntax into the partiality-monad DSL yields correct-by-construction, substitution-stable elaboration scripts, with a concrete algorithm extracted via initiality of the elaboration SOGAT into the presheaf model on the bi-initial natural model) is supported by the explicit combinators of §4, the equational calculations of §4.6 and Appendix B, the multilinearity/inequational subject-reduction results of §5, and the initiality extraction of §6.5–6.6. The elaborability hypotheses of Theorem 6.4 (discreteness/compactness of Tp/Tm, open-subsingleton preimages of type constructors, and strengthening w.r.t. the dominance of strengthenable levelwise-decidable propositions) are the only external inputs; they rest on standard normalisation/injectivity results that the paper correctly cites rather than re-proves. No internal inconsistency, hidden non-constructivity, or gap in the algebraic extraction is present that would undermine the claim.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper introduces a dependently typed monadic DSL, based on a partiality relative 2-monad over a dominance, for the executable specification of correct-by-construction bidirectional elaborators. A bidirectionally typed surface language for Martin-Löf type theory is shallowly embedded so that surface-to-core translation reduces to equational calculation in the metalanguage; the resulting elaborators are well-typed by construction, respect judgemental equality, and are stable under substitution (yielding a denotational account of suspended problems). Combinators for the usual connectives are defined, multilinearity/multistrictness and inequational subject-reduction laws are proved by calculation, and a concrete algorithm is extracted via the universal property of the bi-initial natural model of a SOGAT of elaboration scripts into a presheaf model on the bi-initial model of MLTT, using the dominance of strengthenable levelwise-decidable propositions.","tokens_in":35042,"tokens_out":782,"duration_ms":7411,"significance":"If the results hold, the work supplies a modular, denotational foundation for bidirectional elaboration that cleanly separates surface-language design from the details of normalisation and conversion. Correctness, stability under substitution, and a denotational reading of suspension follow from the monadic structure rather than from ad-hoc operational arguments; the algebraic extraction via SOGAT initiality is a clean way to obtain an executable algorithm. The equational case study (§4.6), the multilinearity theorem (Theorem 5.1 / Appendix B), and the subject-reduction inequalities (Theorem 5.3) are concrete, checkable contributions that raise the standard for how elaborators can be specified and reasoned about. The framework is reusable for other SOGATs and is therefore of lasting interest to the type-theory and proof-assistant communities.","major_comments":[],"minor_comments":[{"comment":"The paper repeatedly notes that metavariables, holes, implicit arguments and type classes are left for future work (§1.2.5, §7). A short forward pointer in the introduction to the precise obstacles (e.g. the need for an extra polynomial-monad layer) would help readers calibrate expectations.","section":"§1.2.5, §7"},{"comment":"Theorem 6.4 rests on external normalisation and injectivity results that are correctly cited but not re-proved. A one-sentence reminder of which specific lemmas (e.g. Sterling’s injectivity of type constructors, Bocquet’s levelwise decidability) are being invoked would make the dependency fully transparent.","section":"Theorem 6.4"},{"comment":"The SOGAT of elaboration scripts (Figure 1) is described as provisional. It would be useful to state explicitly which equations, if any, are currently imposed and which are deliberately omitted so that later extensions remain free.","section":"Figure 1, §6.4"},{"comment":"A few typographical slips remain (e.g. “Bocqet” for Bocquet in the proof of Proposition 6.2; occasional missing spaces around math). These are easily corrected in production.","section":"Proposition 6.2"}],"recommendation":"accept","confidential_remarks":"The manuscript is long and technically dense, but the central contribution is solid and the algebraic development is carefully executed. I see no reason to request a major rewrite; the paper is already at the standard expected for a top PL/type-theory venue. The only external dependency (normalisation/injectivity for the bi-initial model) is standard and properly cited."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper gives a monadic DSL for bidirectional elaborators that is correct by construction and stable under substitution, then extracts a concrete algorithm from the bi-initial natural model via SOGAT initiality. That combination is new.\n\nWhat works: they shallow-embed a bidirectional surface language for MLTT into a relative 2-monad of open partial elements over a dominance. The scope combinator (higher-arity lax monoidal structure at compact arities) handles binders cleanly. Conversion, application, Pi/Sg/Id, and the answer type are all defined by elementary monadic equations; the case study in §4.6 shows the equational calculation is routine. Multilinearity and the subject-reduction inequalities are proved by calculation (Appendix B). The extraction in §6 is just the universal property of the bi-initial model of the elaboration SOGAT into the presheaf model on I_ML. Stability under substitution and the denotational reading of suspended problems fall out for free from the commutative partiality monad. Citations to Atkey, Uemura, McBride and Kovács are accurate and the extensions are real.\n\nSoft spots are minor and proportional. The elaborability hypotheses (Theorem 6.4) rest on external normalisation and injectivity results that are cited rather than re-proved; that is standard and does not break the argument. Metavariables, holes, implicit arguments and type classes are left for future work, so the framework does not yet cover a production elaborator. No machine-checked proofs or shipped code, but the development is pure constructive mathematics with no free parameters or circularity.\n\nThis is for people who care about foundations of proof assistants and denotational accounts of elaboration. It is not a drop-in replacement for existing checkers, but it is a usable modular specification language. The math is careful and the central claims hold. Send it to referees.","headline":"Clean monadic foundation for correct-by-construction bidirectional elaboration, extracted algebraically from the bi-initial model; solid math, limited scope, ready for referees.","tokens_in":35614,"tokens_out":466,"would_cite":true,"duration_ms":5796,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68N18","03B38","18C50"],"pacs":[],"model":"grok-4.5","headline":"Elaboration scripts become equational calculations in a monadic DSL, correct by construction and stable under substitution.","keywords":["bidirectional elaboration","Martin-Löf type theory","partiality monad","presheaf semantics","natural models","second-order generalised algebraic theories","correct-by-construction elaborators","stable elaboration"],"falsifier":"Exhibit a concrete surface term whose equational unfolding inside the monadic DSL either fails to terminate, produces an ill-typed core term, or changes under reordering of independent awaits or under substitution, when evaluated in the bi-initial natural model with the stated dominance.","tokens_in":35694,"feed_emoji":"🔧","tokens_out":693,"duration_ms":5181,"temperature":0.7,"pith_summary":"Proof assistants translate implicit surface code into explicit core terms by elaboration, but that process usually depends on ad-hoc calls to conversion checking and reduction. This paper gives a monadic domain-specific language in which bidirectional surface syntax for Martin-Löf type theory is shallowly embedded, so that translating a surface term into a core term is ordinary equational calculation. The embedding guarantees that only well-typed core terms are produced, that the result is insensitive to judgemental equality of core terms, and that it is stable under substitution; the last property supplies a denotational account of suspended elaboration problems. A concrete algorithm is then extracted, by initiality, from a presheaf model built on the bi-initial natural model of the core theory. A sympathetic reader cares because the framework separates the design of surface languages from the details of normalisation, while still guaranteeing the reliability properties that production elaborators need.","feed_headline":"Elaboration becomes equational calculation in a monadic DSL","feed_subtitle":"Surface scripts stay correct by construction and stable under substitution, yielding suspended problems for free","key_machinery":"The partiality relative 2-monad L(X) = (φ : S) × (φ → X) over a dominance of strengthenable levelwise-decidable propositions, equipped with compact-arity lax monoidal structure (the scope combinator) that interprets binders, and with free algebras typ, syn and chk that host the elaboration combinators.","core_discovery":"A shallow embedding of bidirectional surface syntax for Martin-Löf type theory into a partiality-monad DSL makes the translation of surface terms into core terms amount to elementary equational calculation; the translation is correct by construction (it cannot produce ill-typed terms) and is automatically stable under judgemental equality and under substitution, yielding a denotational interpretation of suspended elaboration problems. A concrete elaboration algorithm is extracted algebraically from a presheaf model of the DSL built on the bi-initial natural model.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Monadic DSL turns bidirectional elaboration into equational calculation","Correct-by-construction elaborators via shallow monadic embedding","Surface-to-core translation reduces to equational calculation","Elaboration algorithms extracted algebraically from monadic DSL models","Substitution-stable elaborators yield suspended problems for free"],"cache_read_input_tokens":32896,"weakest_assumption_plain":"The core theory’s types and terms must be discrete and compact, type-constructor preimages open-subsingleton, and strengthening must hold for the chosen dominance; these rest on external normalisation and injectivity results that the paper cites rather than re-proves.","fun_headline_variants_meta":{"raw":{"variants":["Monadic DSL turns bidirectional elaboration into equational calculation","Correct-by-construction elaborators via shallow monadic embedding","Surface-to-core translation reduces to equational calculation","Elaboration algorithms extracted algebraically from monadic DSL models","Substitution-stable elaborators yield suspended problems for free"]},"model":"grok-4.5","effort":"low","cost_usd":0.007336,"raw_usage":{"total_tokens":1808,"prompt_tokens":830,"num_sources_used":0,"completion_tokens":82,"cost_in_usd_ticks":73360000,"prompt_tokens_details":{"text_tokens":830,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":896,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":830,"tokens_out":82,"duration_ms":6879,"temperature":1.0,"reasoning_tokens":896,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T15:10:25.275137+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a concrete surface term whose equational unfolding inside the monadic DSL either fails to terminate, produces an ill-typed core term, or changes under reordering of independent awaits or under substitution, when evaluated in the bi-initial natural model with the stated dominance.","supporting_citations":[],"review_version":2}