Algebraic universes everywhere #704
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Terminating but incorrect loop check Improved version, whose termination relies on the correctness of check_model Move back to template-coq folder Improve the algorithm after discussion with Marc Bezem Comment a bit Try enforcing new constraints Reorganize inner loop Cleaner inner_loop Finished inner loop Inner loop termination proven Before change of v_minus_w_bound Prove the well-foundedness of loop Inner loop termination proof finished Finished all proofs Finish proofs of auxilliary lemmas Finished all proofs with new invariants Comments and extraction setup Abstract LoopChecking on level / sets / maps implementation Move clauses.v to LoopChecking and TemplateLoopChecking Functorize the algorithm Simplified loops Cleaned up LoopChecking and TemplateLoopChecking Support correct quoting/unquoting of the universe graph/context. Also rename Constraint to LevelConstraint, preparing for a later move to general universe constraints Add a new (extracted) plugin to test the loop checking algorithm. Move back to subst_instance_cstr for level constraints Move live tests of loop_checking to the test-suite Revert changes to Universes.v Avoid repeateadly folding over clauses in inner loop Optimize a bit loop checking to avoid partitioning clause repeatedly Fix UnivConstraint -> LevelConstraint change Remove useless extraction file in template-coq MSetList is no longer needed
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…se checking inside Rocq.
…checker cannot run within Rocq yet
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May 10, 2022
This PR adds a new loop-checking algorithm to MetaCoq, which should allow to handle arbitrary universe constraints efficiently, e.g.
max (u + 1, v) = max (j, k+ 2). It is based on a constructive proof by Bezem et Coquand for the slightly more general problem of building a model inN^∞of a set of horn clauses of shapeu + k \/ ... \/ l + n -> l' + k'. This can be restricted to check if a loop follows from a set of constraints or if we have a model inN, which can in turn be seen as a valuation of the clauses making them all (non-vacuously) true.The algorithm is currently only shown to be terminating and to return a valid model when it finds one, but it might also answer
Loop, in which case we have not (yet) proven that there is indeed a loop.I added a new extracted plugin in the test-suite (based on plugin-demo), that provides a new command
MetaCoq Check Universesthat checks for a model using the new algorithm on all the constraints in the global environment at the point of the call. It currently takes < 0.5s to verify that there is a valuation with 4 distinct universes for all the universes and constraints in Coq's standard library.Nov 8, 2025
The PR is now complete with a correct and complete algorithm for MetaRocq handling all algebraic universes and an update of all the theory and quoting/denotation to support universe instances which actually contain arbitrary universes and not just levels.
It refines the algorithm of Bezem & Coquand to support a bottom universe (zero) and injective successor (so that i <= j <-> i + k <= j + k). The model is hence in Z to have a well behaved predecessor / substraction operation. We show that it is complete w.r.t. the same valuations used by MetaRocq earlier (with global universes
level s > 0, locallvar n >= 0and zero mapped to0%Z). Completeness is based on a presentation of semilattices as an equational theory and the construction of its initial model (syntax/term model). The files in LoopChecking have documented headers explaining the inner working of the algorithm.Re: changes to the API, for now I removed the consistent_extension_on support (checking we don't add global constraints), which needs redesign. The algorithm is written in highly dependent style to show termination, and does not compute inside Rocq itself. I think an extensionally equivalent fuel-based variant can be written easily for translations/examples that need computation inside Rocq.