Module Constr

This file defines the most important datatype of Coq, namely kernel terms, as well as a handful of generic manipulation functions.

type pconstant = Names.Constant.t Univ.puniverses
Simply type aliases
type pinductive = Names.inductive Univ.puniverses
type pconstructor = Names.constructor Univ.puniverses
type metavariable = int
Existential variables
type case_style =
| LetStyle
| IfStyle
| LetPatternStyle
| MatchStyle
| RegularStyle

infer printing form from number of constructor

Case annotation
type case_printing = {
ind_tags : bool list;

tell whether letin or lambda in the arity of the inductive type

cstr_tags : bool list array;

tell whether letin or lambda in the signature of each constructor

style : case_style;
}
type case_info = {
ci_ind : Names.inductive;
ci_npar : int;
ci_cstr_ndecls : int array;
ci_cstr_nargs : int array;
ci_relevance : Sorts.relevance;
ci_pp_info : case_printing;
}
type ('constr, 'univs) case_invert =
| NoInvert

Normal reduction: match when the scrutinee is a constructor.

| CaseInvert of {
univs : 'univs;
args : 'constr array;
}

Reduce when the indices match those of the unique constructor. (SProp to non SProp only)

The type of constructions
type t
type constr = t

types is the same as constr but is intended to be used for documentation to indicate that such or such function specifically works with types (i.e. terms of type a sort). (Rem:plurial form since type is a reserved ML keyword)

type types = constr
Functions for dealing with constr terms.

The following functions are intended to simplify and to uniform the manipulation of terms. Some of these functions may be overlapped with previous ones.

Term constructors.
val mkRel : int -> constr

Constructs a de Bruijn index (DB indices begin at 1)

val mkVar : Names.Id.t -> constr

Constructs a Variable

val mkInt : Uint63.t -> constr

Constructs a machine integer

val mkArray : (Univ.Instance.t * constr array * constr * types) -> constr

Constructs an array

val mkFloat : Float64.t -> constr

Constructs a machine float number

val mkMeta : metavariable -> constr

Constructs an patvar named "?n"

type existential = Evar.t * constr list

Constructs an existential variable

val mkEvar : existential -> constr
val mkSort : Sorts.t -> types

Construct a sort

val mkSProp : types
val mkProp : types
val mkSet : types
val mkType : Univ.Universe.t -> types
type cast_kind =
| VMcast
| NATIVEcast
| DEFAULTcast
| REVERTcast

This defines the strategy to use for verifiying a Cast

val mkCast : (constr * cast_kind * constr) -> constr

Constructs the term t1::t2, i.e. the term t1 casted with the type t2 (that means t2 is declared as the type of t1).

val mkProd : (Names.Name.t Context.binder_annot * types * types) -> types

Constructs the product (x:t1)t2

val mkLambda : (Names.Name.t Context.binder_annot * types * constr) -> constr

Constructs the abstraction [x:t1]t2

val mkLetIn : (Names.Name.t Context.binder_annot * constr * types * constr) -> constr

Constructs the product let x = t1 : t2 in t3

val mkApp : (constr * constr array) -> constr

mkApp (f, [|t1; ...; tN|] constructs the application (f t1 ... tn) $(f~t_1\dots f_n)$.

val map_puniverses : ('a -> 'b) -> 'a Univ.puniverses -> 'b Univ.puniverses
val mkConst : Names.Constant.t -> constr

Constructs a Constant.t

val mkConstU : pconstant -> constr
val mkProj : (Names.Projection.t * constr) -> constr

Constructs a projection application

val mkInd : Names.inductive -> constr

Constructs the ith (co)inductive type of the block named kn

val mkIndU : pinductive -> constr
val mkConstruct : Names.constructor -> constr

Constructs the jth constructor of the ith (co)inductive type of the block named kn.

val mkConstructU : pconstructor -> constr
val mkConstructUi : (pinductive * int) -> constr
val mkRef : Names.GlobRef.t Univ.puniverses -> constr

Make a constant, inductive, constructor or variable.

val mkCase : (case_info * constr * (constr, Univ.Instance.t) case_invert * constr * constr array) -> constr

Constructs a destructor of inductive type.

mkCase ci p c ac stand for match c as x in I args return p with ac presented as describe in ci.

p structure is fun args x -> "return clause"

acith element is ith constructor case presented as lambda construct_args (without params). case_term

type ('constr, 'types) prec_declaration = Names.Name.t Context.binder_annot array * 'types array * 'constr array

If recindxs = [|i1,...in|] funnames = [|f1,.....fn|] typarray = [|t1,...tn|] bodies = [|b1,.....bn|] then mkFix ((recindxs,i), funnames, typarray, bodies) constructs the $ i $ th function of the block (counting from 0)

Fixpoint f1 [ctx1] = b1 with f2 [ctx2] = b2 ... with fn [ctxn] = bn.

where the length of the $ j $ th context is $ ij $ .

type ('constr, 'types) pfixpoint = (int array * int) * ('constr, 'types) prec_declaration
type ('constr, 'types) pcofixpoint = int * ('constr, 'types) prec_declaration
type rec_declaration = (constr, types) prec_declaration
type fixpoint = (constr, types) pfixpoint
val mkFix : fixpoint -> constr
type cofixpoint = (constr, types) pcofixpoint

If funnames = [|f1,.....fn|] typarray = [|t1,...tn|] bodies = [b1,.....bn] then mkCoFix (i, (funnames, typarray, bodies)) constructs the ith function of the block

CoFixpoint f1 = b1 with f2 = b2 ... with fn = bn.

val mkCoFix : cofixpoint -> constr
Concrete type for making pattern-matching.
type 'constr pexistential = Evar.t * 'constr list

constr list is an instance matching definitional named_context in the same order (i.e. last argument first)

type ('constr, 'types, 'sort, 'univs) kind_of_term =
| Rel of int

Gallina-variable introduced by forall, fun, let-in, fix, or cofix.

| Var of Names.Id.t

Gallina-variable that was introduced by Vernacular-command that extends the local context of the currently open section (i.e. Variable or Let).

| Meta of metavariable
| Evar of 'constr pexistential
| Sort of 'sort
| Cast of 'constr * cast_kind * 'types
| Prod of Names.Name.t Context.binder_annot * 'types * 'types

Concrete syntax "forall A:B,C" is represented as Prod (A,B,C).

| Lambda of Names.Name.t Context.binder_annot * 'types * 'constr

Concrete syntax "fun A:B => C" is represented as Lambda (A,B,C).

| LetIn of Names.Name.t Context.binder_annot * 'constr * 'types * 'constr

Concrete syntax "let A:C := B in D" is represented as LetIn (A,B,C,D).

| App of 'constr * 'constr array

Concrete syntax "(F P1 P2 ... Pn)" is represented as App (F, [|P1; P2; ...; Pn|]).

The mkApp constructor also enforces the following invariant:

  • F itself is not App
  • and [|P1;..;Pn|] is not empty.
| Const of Names.Constant.t * 'univs

Gallina-variable that was introduced by Vernacular-command that extends the global environment (i.e. Parameter, or Axiom, or Definition, or Theorem etc.)

| Ind of Names.inductive * 'univs

A name of an inductive type defined by Variant, Inductive or Record Vernacular-commands.

| Construct of Names.constructor * 'univs

A constructor of an inductive type defined by Variant, Inductive or Record Vernacular-commands.

| Case of case_info * 'constr * ('constr, 'univs) case_invert * 'constr * 'constr array
| Fix of ('constr, 'types) pfixpoint
| CoFix of ('constr, 'types) pcofixpoint
| Proj of Names.Projection.t * 'constr
| Int of Uint63.t
| Float of Float64.t
| Array of 'univs * 'constr array * 'constr * 'types
val kind : constr -> (constr, types, Sorts.t, Univ.Instance.t) kind_of_term
val of_kind : (constr, types, Sorts.t, Univ.Instance.t) kind_of_term -> constr
val kind_nocast_gen : ('v -> ('v, 'v, 'sort, 'univs) kind_of_term) -> 'v -> ('v, 'v, 'sort, 'univs) kind_of_term
val kind_nocast : constr -> (constr, types, Sorts.t, Univ.Instance.t) kind_of_term
val isRel : constr -> bool
Simple case analysis
val isRelN : int -> constr -> bool
val isVar : constr -> bool
val isVarId : Names.Id.t -> constr -> bool
val isRef : constr -> bool
val isRefX : Names.GlobRef.t -> constr -> bool
val isInd : constr -> bool
val isEvar : constr -> bool
val isMeta : constr -> bool
val isEvar_or_Meta : constr -> bool
val isSort : constr -> bool
val isCast : constr -> bool
val isApp : constr -> bool
val isLambda : constr -> bool
val isLetIn : constr -> bool
val isProd : constr -> bool
val isConst : constr -> bool
val isConstruct : constr -> bool
val isFix : constr -> bool
val isCoFix : constr -> bool
val isCase : constr -> bool
val isProj : constr -> bool
val is_Prop : constr -> bool
val is_Set : constr -> bool
val isprop : constr -> bool
val is_Type : constr -> bool
val iskind : constr -> bool
val is_small : Sorts.t -> bool
Term destructors
exception DestKO
val destRel : constr -> int

Destructs a de Bruijn index

val destMeta : constr -> metavariable

Destructs an existential variable

val destVar : constr -> Names.Id.t

Destructs a variable

val destSort : constr -> Sorts.t

Destructs a sort. is_Prop recognizes the sort Prop, whether isprop recognizes both Prop and Set.

val destCast : constr -> constr * cast_kind * constr

Destructs a casted term

val destProd : types -> Names.Name.t Context.binder_annot * types * types

Destructs the product $ (x:t_1)t_2 $

val destLambda : constr -> Names.Name.t Context.binder_annot * types * constr

Destructs the abstraction $ x:t_1t_2 $

val destLetIn : constr -> Names.Name.t Context.binder_annot * constr * types * constr

Destructs the let $ x:=b:t_1t_2 $

val destApp : constr -> constr * constr array

Destructs an application

val decompose_app : constr -> constr * constr list

Decompose any term as an applicative term; the list of args can be empty

val decompose_appvect : constr -> constr * constr array

Same as decompose_app, but returns an array.

val destConst : constr -> Names.Constant.t Univ.puniverses

Destructs a constant

val destEvar : constr -> existential

Destructs an existential variable

val destInd : constr -> Names.inductive Univ.puniverses

Destructs a (co)inductive type

val destConstruct : constr -> Names.constructor Univ.puniverses

Destructs a constructor

val destCase : constr -> case_info * constr * (constr, Univ.Instance.t) case_invert * constr * constr array

Destructs a match c as x in I args return P with ... | Ci(...yij...) => ti | ... end (or let (..y1i..) := c as x in I args return P in t1, or if c then t1 else t2)

returns

(info,c,fun args x => P,[|...|fun yij => ti| ...|]) where info is pretty-printing information

val destProj : constr -> Names.Projection.t * constr

Destructs a projection

val destFix : constr -> fixpoint

Destructs the $ i $ th function of the block Fixpoint f{_ 1} ctx{_ 1} = b{_ 1} with f{_ 2} ctx{_ 2} = b{_ 2} ... with f{_ n} ctx{_ n} = b{_ n}, where the length of the $ j $ th context is $ ij $ .

val destCoFix : constr -> cofixpoint
val destRef : constr -> Names.GlobRef.t Univ.puniverses
Equality
val equal : constr -> constr -> bool

equal a b is true if a equals b modulo alpha, casts, and application grouping

val eq_constr_univs : constr UGraph.check_function

eq_constr_univs u a b is true if a equals b modulo alpha, casts, application grouping and the universe equalities in u.

val leq_constr_univs : constr UGraph.check_function

leq_constr_univs u a b is true if a is convertible to b modulo alpha, casts, application grouping and the universe inequalities in u.

val eq_constr_univs_infer : UGraph.t -> constr -> constr -> bool Univ.constrained

eq_constr_univs u a b is true if a equals b modulo alpha, casts, application grouping and the universe equalities in u.

val leq_constr_univs_infer : UGraph.t -> constr -> constr -> bool Univ.constrained

leq_constr_univs u a b is true if a is convertible to b modulo alpha, casts, application grouping and the universe inequalities in u.

val eq_constr_nounivs : constr -> constr -> bool

eq_constr_univs a b true, c if a equals b modulo alpha, casts, application grouping and ignoring universe instances.

val compare : constr -> constr -> int

Total ordering compatible with equal

Extension of Context with declarations on constr
type rel_declaration = (constr, types) Context.Rel.Declaration.pt
type named_declaration = (constr, types) Context.Named.Declaration.pt
type compacted_declaration = (constr, types) Context.Compacted.Declaration.pt
type rel_context = rel_declaration list
type named_context = named_declaration list
type compacted_context = compacted_declaration list
Relocation and substitution
val exliftn : Esubst.lift -> constr -> constr

exliftn el c lifts c with lifting el

val liftn : int -> int -> constr -> constr

liftn n k c lifts by n indexes above or equal to k in c

val lift : int -> constr -> constr

lift n c lifts by n the positive indexes in c

Functionals working on expressions canonically abstracted over a local context (possibly with let-ins)
val map_under_context : (constr -> constr) -> int -> constr -> constr
val map_branches : (constr -> constr) -> case_info -> constr array -> constr array
val map_return_predicate : (constr -> constr) -> case_info -> constr -> constr
val map_under_context_with_binders : ('a -> 'a) -> ('a -> constr -> constr) -> 'a -> int -> constr -> constr
val map_branches_with_binders : ('a -> 'a) -> ('a -> constr -> constr) -> 'a -> case_info -> constr array -> constr array
val map_return_predicate_with_binders : ('a -> 'a) -> ('a -> constr -> constr) -> 'a -> case_info -> constr -> constr
val map_under_context_with_full_binders : ((constr, constr) Context.Rel.Declaration.pt -> 'a -> 'a) -> ('a -> constr -> constr) -> 'a -> int -> constr -> constr
val map_branches_with_full_binders : ((constr, constr) Context.Rel.Declaration.pt -> 'a -> 'a) -> ('a -> constr -> constr) -> 'a -> case_info -> constr array -> constr array
val map_return_predicate_with_full_binders : ((constr, constr) Context.Rel.Declaration.pt -> 'a -> 'a) -> ('a -> constr -> constr) -> 'a -> case_info -> constr -> constr
Functionals working on the immediate subterm of a construction
val fold : ('a -> constr -> 'a) -> 'a -> constr -> 'a
val fold_with_full_binders : (rel_declaration -> 'a -> 'a) -> ('a -> 'b -> constr -> 'b) -> 'a -> 'b -> constr -> 'b
val fold_invert : ('a -> 'b -> 'a) -> 'a -> ('b, 'c) case_invert -> 'a
val map : (constr -> constr) -> constr -> constr
val map_invert : ('a -> 'a) -> ('a, 'b) case_invert -> ('a, 'b) case_invert
val map_user_view : (constr -> constr) -> constr -> constr
val fold_map : ('a -> constr -> 'a * constr) -> 'a -> constr -> 'a * constr
val fold_map_invert : ('a -> 'b -> 'a * 'b) -> 'a -> ('b, 'c) case_invert -> 'a * ('b, 'c) case_invert
val map_with_binders : ('a -> 'a) -> ('a -> constr -> constr) -> 'a -> constr -> constr
val iter : (constr -> unit) -> constr -> unit
val iter_invert : ('a -> unit) -> ('a, 'b) case_invert -> unit
val iter_with_binders : ('a -> 'a) -> ('a -> constr -> unit) -> 'a -> constr -> unit
val fold_constr_with_binders : ('a -> 'a) -> ('a -> 'b -> constr -> 'b) -> 'a -> 'b -> constr -> 'b
type 'constr constr_compare_fn = int -> 'constr -> 'constr -> bool
val compare_head : constr constr_compare_fn -> constr constr_compare_fn
type 'univs instance_compare_fn = (Names.GlobRef.t * int) option -> 'univs -> 'univs -> bool

Convert a global reference applied to 2 instances. The int says how many arguments are given (as we can only use cumulativity for fully applied inductives/constructors) .

val compare_head_gen : Univ.Instance.t instance_compare_fn -> (Sorts.t -> Sorts.t -> bool) -> constr constr_compare_fn -> constr constr_compare_fn
val compare_head_gen_leq_with : ('v -> ('v, 'v, 'sort, 'univs) kind_of_term) -> ('v -> ('v, 'v, 'sort, 'univs) kind_of_term) -> 'univs instance_compare_fn -> ('sort -> 'sort -> bool) -> 'v constr_compare_fn -> 'v constr_compare_fn -> 'v constr_compare_fn
val compare_head_gen_with : ('v -> ('v, 'v, 'sort, 'univs) kind_of_term) -> ('v -> ('v, 'v, 'sort, 'univs) kind_of_term) -> 'univs instance_compare_fn -> ('sort -> 'sort -> bool) -> 'v constr_compare_fn -> 'v constr_compare_fn

compare_head_gen_with k1 k2 u s f c1 c2 compares c1 and c2 like compare_head_gen u s f c1 c2, except that k1 (resp. k2) is used,rather than kind, to expose the immediate subterms of c1 (resp. c2).

val compare_head_gen_leq : Univ.Instance.t instance_compare_fn -> (Sorts.t -> Sorts.t -> bool) -> constr constr_compare_fn -> constr constr_compare_fn -> constr constr_compare_fn
val eq_invert : ('a -> 'a -> bool) -> ('b -> 'b -> bool) -> ('a, 'b) case_invert -> ('a, 'b) case_invert -> bool
Hashconsing
val hash : constr -> int
val case_info_hash : case_info -> int
val hcons : constr -> constr
val debug_print : constr -> Pp.t
val debug_print_fix : ('a -> Pp.t) -> ('a, 'a) pfixpoint -> Pp.t