Module Constr

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

Simply type aliases

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 pcase_invert =
| NoInvert(*

Normal reduction: match when the scrutinee is a constructor.

*)
| CaseInvert of {
indices : '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 SList.t

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

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).

Constructs the product (x:t1)t2

Constructs the abstraction [x:t1]t2

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) .

val map_puniverses : ( 'a -> 'b ) -> 'a Univ.puniverses -> 'b Univ.puniverses
val mkConstU : pconstant -> constr

Constructs a Constant.t

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

Constructs a projection application

Inductive types

val mkIndU : pinductive -> constr

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

val mkConstructU : pconstructor -> constr

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

val mkConstructUi : (pinductive * int) -> constr

Make a constant, inductive, constructor or variable.

module UnsafeMonomorphic : sig ... end

Constructs a destructor of inductive type.

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

p structure is args x |- "return clause"

acith element is ith constructor case presented as construct_args |- case_term

type 'constr pcase_branch = Names.Name.t Context.binder_annot array * 'constr

Names of the indices + name of self

type 'types pcase_return = Names.Name.t Context.binder_annot array * 'types

Names of the branches

type ('constr, 'types, 'univs) pcase = case_info * 'univs * 'constr array * 'types pcase_return * 'constr pcase_invert * 'constr * 'constr pcase_branch array
type case_invert = constr pcase_invert
type case_return = types pcase_return
type case_branch = constr pcase_branch
type case = ( constr, types, Univ.Instance.t ) pcase
val mkCase : case -> constr
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

The array of int's tells for each component of the array of mutual fixpoints the number of lambdas to skip before finding the recursive argument (e.g., value is 2 in "fix f (x:A) (y:=t) (z:B) (v:=u) (w:I) struct w"), telling to skip x and z and that w is the recursive argument); The second component int tells which component of the block is returned

type ('constr, 'types) pcofixpoint = int * ( 'constr, 'types ) prec_declaration

The component int tells which component of the block of cofixpoint is returned

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 SList.t

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 * 'univs * 'constr array * 'types pcase_return * 'constr pcase_invert * 'constr * 'constr pcase_branch array(*

Case (ci,u,params,p,iv,c,brs) is a match c return p with brs expression. c lives in inductive ci.ci_ind at universe instance u and parameters params. If this match has case inversion (ie match on a 1 constructor SProp inductive with proof relevant return type) the indices are in iv.

The names in p are the names of the bound indices and inductive value (ie the in and as clauses).

The names in the brs are the names of the variables bound in the respective branch.

*)
| 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(*

Array (u,vals,def,t) is an array of vals in type t with default value def. u is a universe containing t.

*)

User view of constr. For App, it is ensured there is at least one argument and the function is not itself an applicative term

val kind_nocast_gen : ( 'v -> ( 'v, 'v, 'sort, 'univs ) kind_of_term ) -> 'v -> ( 'v, 'v, 'sort, 'univs ) 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

Destructor operations are partial functions and

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

Destructs the product $ (x:t_1)t_2 $

Destructs the abstraction $ x:t_1t_2 $

Destructs the let $ x:=b:t_1t_2 $

val destApp : constr -> constr * constr array

Destructs an application

val decompose_app_list : constr -> constr * constr list

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

val decompose_app : constr -> constr * constr array

Same as decompose_app, but returns an array.

Destructs a constant

val destEvar : constr -> existential

Destructs an existential variable

Destructs a (co)inductive type

Destructs a constructor

val destCase : constr -> case

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
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_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)

map_branches f br maps f on the immediate subterms of an array of "match" branches br in canonical eta-let-expanded form; it is not recursive and the order with which subterms are processed is not specified; it preserves sharing; the immediate subterms are the types and possibly terms occurring in the context of each branch as well as the body of each branch

val map_branches : ( constr -> constr ) -> case_branch array -> case_branch array

map_return_predicate f p maps f on the immediate subterms of a return predicate of a "match" in canonical eta-let-expanded form; it is not recursive and the order with which subterms are processed is not specified; it preserves sharing; the immediate subterms are the types and possibly terms occurring in the context of each branch as well as the body of the predicate

val map_return_predicate : ( constr -> constr ) -> case_return -> case_return

map_branches_with_binders f br maps f on the immediate subterms of an array of "match" branches br in canonical eta-let-expanded form; it carries an extra data n (typically a lift index) which is processed by g (which typically adds 1 to n) at each binder traversal; it is not recursive and the order with which subterms are processed is not specified; it preserves sharing; the immediate subterms are the types and possibly terms occurring in the context of the branch as well as the body of the branch

val map_branches_with_binders : ( 'a -> 'a ) -> ( 'a -> constr -> constr ) -> 'a -> case_branch array -> case_branch array

map_return_predicate_with_binders f p maps f on the immediate subterms of a return predicate of a "match" in canonical eta-let-expanded form; it carries an extra data n (typically a lift index) which is processed by g (which typically adds 1 to n) at each binder traversal; it is not recursive and the order with which subterms are processed is not specified; it preserves sharing; the immediate subterms are the types and possibly terms occurring in the context of each branch as well as the body of the predicate

val map_return_predicate_with_binders : ( 'a -> 'a ) -> ( 'a -> constr -> constr ) -> 'a -> case_return -> case_return
Functionals working on the immediate subterm of a construction

fold f acc c folds f on the immediate subterms of c starting from acc and proceeding from left to right according to the usual representation of the constructions; it is not recursive

val fold : ( 'a -> constr -> 'a ) -> 'a -> constr -> 'a
val fold_invert : ( 'a -> 'b -> 'a ) -> 'a -> 'b pcase_invert -> 'a

map f c maps f on the immediate subterms of c; it is not recursive and the order with which subterms are processed is not specified

val map : ( constr -> constr ) -> constr -> constr
val map_invert : ( 'a -> 'a ) -> 'a pcase_invert -> 'a pcase_invert

Like map, but also has an additional accumulator.

val fold_map : ( 'a -> constr -> 'a * constr ) -> 'a -> constr -> 'a * constr
val fold_map_invert : ( 'a -> 'b -> 'a * 'b ) -> 'a -> 'b pcase_invert -> 'a * 'b pcase_invert

map_with_binders g f n c maps f n on the immediate subterms of c; it carries an extra data n (typically a lift index) which is processed by g (which typically add 1 to n) at each binder traversal; it is not recursive and the order with which subterms are processed is not specified

val map_with_binders : ( 'a -> 'a ) -> ( 'a -> constr -> constr ) -> 'a -> constr -> constr

iter f c iters f on the immediate subterms of c; it is not recursive and the order with which subterms are processed is not specified

val iter : ( constr -> unit ) -> constr -> unit
val iter_invert : ( 'a -> unit ) -> 'a pcase_invert -> unit

iter_with_binders g f n c iters f n on the immediate subterms of c; it carries an extra data n (typically a lift index) which is processed by g (which typically add 1 to n) at each binder traversal; it is not recursive and the order with which subterms are processed is not specified

val iter_with_binders : ( 'a -> 'a ) -> ( 'a -> constr -> unit ) -> 'a -> constr -> unit

iter_with_binders g f n c iters f n on the immediate subterms of c; it carries an extra data n (typically a lift index) which is processed by g (which typically add 1 to n) at each binder traversal; it is not recursive and the order with which subterms are processed is not specified

val fold_constr_with_binders : ( 'a -> 'a ) -> ( 'a -> 'b -> constr -> 'b ) -> 'a -> 'b -> constr -> 'b
type 'constr constr_compare_fn = int -> 'constr -> 'constr -> bool

compare_head f c1 c2 compare c1 and c2 using f to compare the immediate subterms of c1 of c2 if needed; Cast's, binders name and Cases annotations are not taken into account

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) .

compare_head_gen u s f c1 c2 compare c1 and c2 using f to compare the immediate subterms of c1 of c2 if needed, u to compare universe instances, s to compare sorts; Cast's, binders name and Cases annotations are not taken into account

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 pexistential -> 'v pexistential -> 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 pexistential -> 'v pexistential -> 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).

compare_head_gen_leq u s f fle c1 c2 compare c1 and c2 using f to compare the immediate subterms of c1 of c2 for conversion, fle for cumulativity, u to compare universe instances (the first boolean tells if they belong to a Constant.t), s to compare sorts for for subtyping; Cast's, binders name and Cases annotations are not taken into account

val eq_invert : ( 'a -> 'a -> bool ) -> 'a pcase_invert -> 'a pcase_invert -> bool
type 'a evar_expansion =
| EvarDefined of 'a
| EvarUndefined of Evar.t * 'a list
type 'constr evar_handler = {
evar_expand : 'constr pexistential -> 'constr evar_expansion;
evar_repack : (Evar.t * 'constr list) -> 'constr;
evar_relevant : 'constr pexistential -> bool;
qvar_relevant : Sorts.QVar.t -> bool;
}
val default_evar_handler : 'constr evar_handler
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
val mkConst : Names.Constant.t -> constr
val mkInd : Names.inductive -> constr
val mkConstruct : Names.constructor -> constr