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Add the canonical map `PFunctor.W.toM` and identify `P.W` with the well-founded trees of `P.M`, together with Lambek's lemma `M.destEquiv` and an induction principle for `PFunctor.W` through `W.mk`. Identify `P.FreeM α` with the W-type of `C α + P`. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Add `PFunctor.Resumption P α`, the M-type of `C α + P`: possibly non-terminating programs that return an `α` or perform an operation of `P`. It has a corecursor, a bisimulation principle and a lawful monad structure, and `FreeM.toResumption` is an injective monad morphism whose image is exactly the well-founded resumptions. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Use `liftBind` as the implementation-detail constructor with simp-normal form `(lift a).bind k`, as for `PFunctor.FreeM`, and name the cases and lemmas accordingly (`lift_bind`, `liftBind_bind`, `dest_lift_bind`, `map_bind`, `bind_pure_comp`). Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Add `M.toResumption`, `Resumption.toMOfIsEmpty` and `equivMOfIsEmpty`, mirroring the W-type embedding of free programs, and show that embedding W-trees commutes with these maps. Describe the four tree types and the maps between them in the module documentation. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
dtumad
requested review from
SamuelSchlesinger,
arademaker,
chenson2018,
crei,
fmontesi and
sorrachai
as code owners
October 5, 2026 23:56
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Completes the polynomial tree types with coinductive resumptions and relates the four representations.
PFunctor.WandPFunctor.Mare the initial algebra and final coalgebra ofP;PFunctor.FreeM P αand the newPFunctor.Resumption P αare those ofX ↦ α ⊕ P X, the W- and M-types ofC α + P(FreeM.equivW). The canonical mapsW.toMandFreeM.toResumptionare injective with image the well-founded trees, the latter a monad morphism, and they commute with the embeddings of trees that never return (W.toFreeM,M.toResumption), which are equivalences when nothing can be returned.Resumptions have a corecursor, a bisimulation principle and a lawful monad structure whose API mirrors
PFunctor.FreeM. Overythey are Capretta's delay monad; they give semantics to loops whose termination is not structural, such as rejection sampling or machine execution. Adapted from PolyFun.AI agents were used to adapt VCVio definitions/proofs to Cslib definitions and conventions.