package lambdapi

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package lambdapi
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Evaluation and conversion.

Preliminary remarks. We define the head-structure of a term t as:

  • λx:_,h if t=λx:a,u and h is the head-structure of u
  • Π if t=Πx:a,u
  • h _ if t=uv and h is the head-structure of u
  • ? if t=?M.t1;...;tn (and ?M is not instantiated)
  • t itself otherwise (TYPE, KIND, x, f)

A term t is in head-normal form (hnf) if its head-structure is invariant by reduction.

A term t is in weak head-normal form (whnf) if it is an abstraction or if it is in hnf. In particular, a term in head-normal form is in weak head-normal form.

A term t is in strong normal form (snf) if it cannot be reduced further.

val eta_equality : bool Timed.ref

Flag indicating whether eta-reduction should be used or not.

type rw_tag = [
  1. | `NoBeta
    (*

    If true, no beta-reduction is performed.

    *)
  2. | `NoRw
    (*

    If true, no user-defined rewrite rule is used.

    *)
  3. | `NoExpand
    (*

    If true, definitions are not expanded.

    *)
]

Tags for rewriting configuration.

Functions that use the rewriting engine and accept an optional argument tags of type rw_tag list have the following behaviour.

  • If the argument is not given, then no tag is active and the rewrite engine is not constrained: it uses user defined reduction rules, it expands variable definitions (that are stored in the ctxt) and performs beta reductions.
  • Each tag if present disables some functionality of the rewrite engine. The descriptions of the functionalities are given in the documentation of rw_tag.

Reduction functions also accept an optional problem that is used to store metavariables that may be created while rewriting. Such metavariables may be created by particular rewrite rules (such as unification rules), but not by rules declared with rule t ↪ u;.

NOTE that all reduction functions, and eq_modulo, may reduce in-place some subterms of the reduced term.

val whnf : ?tags:rw_tag list -> Term.ctxt -> Term.term -> Term.term

whnf ?tags c t computes a whnf of the term t in context c.

val eq_modulo : ?tags:rw_tag list -> Term.ctxt -> Term.term -> Term.term -> bool

eq_modulo c a b tests the convertibility of a and b in context c.

val pure_eq_modulo : ?tags:rw_tag list -> Term.ctxt -> Term.term -> Term.term -> bool

pure_eq_modulo c a b tests the convertibility of a and b in context c with no side effects.

val snf : ?dtree:(Term.sym -> Term.dtree) -> ?tags:rw_tag list -> Term.ctxt -> Term.term -> Term.term

snf ~dtree c t computes a snf of t, unfolding the variables defined in the context c. The function dtree maps symbols to dtrees.

val hnf : ?tags:rw_tag list -> Term.ctxt -> Term.term -> Term.term

hnf ?tags c t computes a head-normal form of the term t in context c.

val simplify : Term.term -> Term.term

simplify t computes a beta whnf of t belonging to the set S such that:

  • terms of S are in beta whnf normal format
  • if t is a product, then both its domain and codomain are in S.
val unfold_sym : Term.sym -> Term.term -> Term.term

If s is a non-opaque symbol having a definition, unfold_sym s t replaces in t all the occurrences of s by its definition.

type strategy =
  1. | WHNF
    (*

    Reduce to weak head-normal form.

    *)
  2. | HNF
    (*

    Reduce to head-normal form.

    *)
  3. | SNF
    (*

    Reduce to strong normal form.

    *)
  4. | NONE
    (*

    Do nothing.

    *)

Dedukti evaluation strategies.

type strat = {
  1. strategy : strategy;
    (*

    Evaluation strategy.

    *)
  2. steps : int option;
    (*

    Max number of steps if given.

    *)
}
val eval : strat -> Term.ctxt -> Term.term -> Term.term

eval s c t evaluates the term t in the context c according to strategy s.