package logtk

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package logtk
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Substitutions

Substitutions map (scoped) variables to terms/types.

They work on free variables (within a scope, so that the same variable can live within several scopes).

The concept of scope is to allow the same free variable to be used in several contexts without being renamed. A scope is kind of a namespace, where variables from distinct namespaces are always distinct.

type term = InnerTerm.t
type var = InnerTerm.t HVar.t

Renamings

A renaming is used to disambiguate variables that come from distinct scopes but have the same index. It is used to merge together several scopes, in a sound way, by ensuring variables from those scopes are mapped to distinct variables of the new scope. For instance, a given renaming applied to (X,0) and (X,1) will return two different variables, as if one of the X had been renamed prior to unification/binding.

module Renaming : sig ... end

Basics

type t

A substitution that binds term variables to other terms

type subst = t
val empty : t

The identity substitution

val is_empty : t -> bool

Is the substitution empty?

Operations on Substitutions

val find_exn : t -> var Scoped.t -> term Scoped.t

Lookup variable in substitution.

val find : t -> var Scoped.t -> term Scoped.t option
val deref : t -> term Scoped.t -> term Scoped.t

deref t s_t dereferences t as long as t is a variable bound in subst

val get_var : t -> var Scoped.t -> term Scoped.t option

Lookup recursively the var in the substitution, until it is not a variable anymore, or it is not bound.

  • returns

    None if the variable is not bound, Some (deref (t, sc_t)) if v is bound to t, sc_t

val mem : t -> var Scoped.t -> bool

Check whether the variable is bound by the substitution

exception InconsistentBinding of var Scoped.t * term Scoped.t * term Scoped.t
val bind : t -> var Scoped.t -> term Scoped.t -> t

Add v -> t to the substitution. Both terms have a context. It is important that the bound term is De-Bruijn-closed (assert).

val update : t -> var Scoped.t -> term Scoped.t -> t

Replace v -> ? by v -> t in the substitution. Both terms have a context. It is important that the bound term is De-Bruijn-closed (assert).

val merge : t -> t -> t

merge s1 s2 is the substitution that maps t to (s1 t) or to (s2 t).

val remove : t -> var Scoped.t -> t

Remove the given binding. No other variable should depend on it...

val filter_scope : t -> Scoped.scope -> t

Only keep bindings from this scope

Set operations

val domain : t -> var Scoped.t Iter.t

Domain of substitution

val codomain : t -> term Scoped.t Iter.t

Codomain (image terms) of substitution

val introduced : t -> var Scoped.t Iter.t

Variables introduced by the substitution (ie vars of codomain)

val normalize : t -> t

Normalize bindings that are in the same scope. E.g. x0 -> f(y0), y0 -> g(z0), z0->a becomes x0->f(g(a))0, y0->g(a)0, z0->g(z0)

val map : (term -> term) -> t -> t

Map on term

val filter : (var Scoped.t -> term Scoped.t -> bool) -> t -> t

Filter bindings

val is_renaming : t -> bool

Check whether the substitution is a variable renaming

val equal : t -> t -> bool
val compare : t -> t -> int
val hash : t -> int
include Interfaces.PRINT with type t := t
val to_string : t -> string
val pp_bindings : t CCFormat.printer

Only print the bindings, no box

val fold : ('a -> var Scoped.t -> term Scoped.t -> 'a) -> 'a -> t -> 'a
val iter : (var Scoped.t -> term Scoped.t -> unit) -> t -> unit
val to_iter : t -> (var Scoped.t * term Scoped.t) Iter.t
val to_list : t -> (var Scoped.t * term Scoped.t) list
val of_iter : ?init:t -> (var Scoped.t * term Scoped.t) Iter.t -> t
val of_list : ?init:t -> (var Scoped.t * term Scoped.t) list -> t

Applying a substitution

val apply : ?shift_vars:int -> Renaming.t -> t -> term Scoped.t -> term

Apply the substitution to the given term. This function assumes that all terms in the substitution are closed, and it will not perform De Bruijn indices shifting. For instance, applying {X -> f(db0)} (with db0 the De Bruijn index 0) to the term forall. p(X) will yield forall. p(f(db0)) (capturing) and not forall. p(f(db1)).

  • parameter renaming

    used to desambiguate free variables from distinct scopes

Specializations

module type SPECIALIZED = sig ... end
module Ty : SPECIALIZED with type term = Type.t
module FO : sig ... end

Projections for proofs

module Projection : sig ... end
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