Module ConstrSource

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

Simply type aliases

Sourcetype metavariable = int

Existential variables

Sourcetype case_style =
  1. | LetStyle
  2. | IfStyle
  3. | LetPatternStyle
  4. | MatchStyle
  5. | RegularStyle
    (*

    infer printing form from number of constructor

    *)

Case annotation

Sourcetype case_printing = {
  1. ind_tags : bool list;
    (*

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

    *)
  2. cstr_tags : bool list array;
    (*

    tell whether letin or lambda in the signature of each constructor

    *)
  3. style : case_style;
}
Sourcetype case_info = {
  1. ci_ind : Names.inductive;
  2. ci_npar : int;
  3. ci_cstr_ndecls : int array;
  4. ci_cstr_nargs : int array;
  5. ci_relevance : Sorts.relevance;
  6. ci_pp_info : case_printing;
}
Sourcetype 'constr pcase_invert =
  1. | NoInvert
    (*

    Normal reduction: match when the scrutinee is a constructor.

    *)
  2. | CaseInvert of {
    1. indices : 'constr array;
    }
    (*

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

    *)
The type of constructions
Sourcetype t
Sourcetype 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)

Sourcetype 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.
Sourceval mkRel : int -> constr

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

Sourceval mkVar : Names.Id.t -> constr

Constructs a Variable

Sourceval mkInt : Uint63.t -> constr

Constructs a machine integer

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

Constructs an array

Sourceval mkFloat : Float64.t -> constr

Constructs a machine float number

Sourceval mkMeta : metavariable -> constr

Constructs an patvar named "?n"

Sourcetype existential = Evar.t * constr list

Constructs an existential variable

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

Construct a sort

Sourceval mkSProp : types
Sourceval mkProp : types
Sourceval mkSet : types
Sourcetype cast_kind =
  1. | VMcast
  2. | NATIVEcast
  3. | DEFAULTcast
  4. | REVERTcast

This defines the strategy to use for verifiying a Cast

Sourceval 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

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

mkApp (f, [|t1; ...; tN|] constructs the application (f t1 ... tn) .

Sourceval map_puniverses : ('a -> 'b) -> 'a Univ.puniverses -> 'b Univ.puniverses

Constructs a Constant.t

Sourceval mkConstU : pconstant -> constr

Constructs a projection application

Inductive types

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

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

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

Sourceval mkConstructU : pconstructor -> constr
Sourceval mkConstructUi : (pinductive * int) -> constr

Make a constant, inductive, constructor or variable.

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

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

Names of the indices + name of self

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

Names of the branches

Sourcetype ('constr, 'types, 'univs) pcase = case_info * 'univs * 'constr array * 'types pcase_return * 'constr pcase_invert * 'constr * 'constr pcase_branch array
Sourcetype case_invert = constr pcase_invert
Sourcetype case_return = types pcase_return
Sourcetype case_branch = constr pcase_branch
Sourceval mkCase : case -> constr
Sourcetype ('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 $ .

Sourcetype ('constr, 'types) pfixpoint = (int array * int) * ('constr, 'types) prec_declaration
Sourcetype ('constr, 'types) pcofixpoint = int * ('constr, 'types) prec_declaration
Sourcetype rec_declaration = (constr, types) prec_declaration
Sourcetype fixpoint = (constr, types) pfixpoint
Sourceval mkFix : fixpoint -> constr
Sourcetype 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.

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

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

Sourcetype ('constr, 'types, 'sort, 'univs) kind_of_term =
  1. | Rel of int
    (*

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

    *)
  2. | 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).

    *)
  3. | Meta of metavariable
  4. | Evar of 'constr pexistential
  5. | Sort of 'sort
  6. | Cast of 'constr * cast_kind * 'types
  7. | Prod of Names.Name.t Context.binder_annot * 'types * 'types
    (*

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

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

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

    *)
  9. | 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).

    *)
  10. | 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.
    *)
  11. | 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.)

    *)
  12. | Ind of Names.inductive * 'univs
    (*

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

    *)
  13. | Construct of Names.constructor * 'univs
    (*

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

    *)
  14. | Case of case_info * 'univs * 'constr array * 'types pcase_return * 'constr pcase_invert * 'constr * 'constr pcase_branch array
  15. | Fix of ('constr, 'types) pfixpoint
  16. | CoFix of ('constr, 'types) pcofixpoint
  17. | Proj of Names.Projection.t * 'constr
  18. | Int of Uint63.t
  19. | Float of Float64.t
  20. | Array of 'univs * 'constr array * 'constr * 'types

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

Sourceval kind_nocast_gen : ('v -> ('v, 'v, 'sort, 'univs) kind_of_term) -> 'v -> ('v, 'v, 'sort, 'univs) kind_of_term
Sourceval isRel : constr -> bool

Simple case analysis

Sourceval isRelN : int -> constr -> bool
Sourceval isVar : constr -> bool
Sourceval isVarId : Names.Id.t -> constr -> bool
Sourceval isRef : constr -> bool
Sourceval isRefX : Names.GlobRef.t -> constr -> bool
Sourceval isInd : constr -> bool
Sourceval isEvar : constr -> bool
Sourceval isMeta : constr -> bool
Sourceval isEvar_or_Meta : constr -> bool
Sourceval isSort : constr -> bool
Sourceval isCast : constr -> bool
Sourceval isApp : constr -> bool
Sourceval isLambda : constr -> bool
Sourceval isLetIn : constr -> bool
Sourceval isProd : constr -> bool
Sourceval isConst : constr -> bool
Sourceval isConstruct : constr -> bool
Sourceval isFix : constr -> bool
Sourceval isCoFix : constr -> bool
Sourceval isCase : constr -> bool
Sourceval isProj : constr -> bool
Sourceval is_Prop : constr -> bool
Sourceval is_Set : constr -> bool
Sourceval isprop : constr -> bool
Sourceval is_Type : constr -> bool
Sourceval iskind : constr -> bool
Sourceval is_small : Sorts.t -> bool
Term destructors

Destructor operations are partial functions and

Sourceexception DestKO
Sourceval destRel : constr -> int

Destructs a de Bruijn index

Sourceval destMeta : constr -> metavariable

Destructs an existential variable

Sourceval destVar : constr -> Names.Id.t

Destructs a variable

Sourceval destSort : constr -> Sorts.t

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

Sourceval 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 $

Sourceval destApp : constr -> constr * constr array

Destructs an application

Sourceval decompose_app : constr -> constr * constr list

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

Sourceval decompose_appvect : constr -> constr * constr array

Same as decompose_app, but returns an array.

Destructs a constant

Sourceval destEvar : constr -> existential

Destructs an existential variable

Destructs a (co)inductive type

Destructs a constructor

Sourceval 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

Destructs a projection

Sourceval 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 $ .

Sourceval destCoFix : constr -> cofixpoint
Equality
Sourceval equal : constr -> constr -> bool

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

Sourceval 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.

Sourceval 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.

Sourceval 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.

Sourceval 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.

Sourceval 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.

Sourceval compare : constr -> constr -> int

Total ordering compatible with equal

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

exliftn el c lifts c with lifting el

Sourceval liftn : int -> int -> constr -> constr

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

Sourceval 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

Sourceval 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

Sourceval 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

Sourceval 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

Sourceval 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

Sourceval fold : ('a -> constr -> 'a) -> 'a -> constr -> 'a
Sourceval 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

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

Like map, but also has an additional accumulator.

Sourceval fold_map : ('a -> constr -> 'a * constr) -> 'a -> constr -> 'a * constr
Sourceval 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

Sourceval 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

Sourceval iter : (constr -> unit) -> constr -> unit
Sourceval 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

Sourceval 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

Sourceval fold_constr_with_binders : ('a -> 'a) -> ('a -> 'b -> constr -> 'b) -> 'a -> 'b -> constr -> 'b
Sourcetype '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

Sourcetype '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

Sourceval 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
Sourceval 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).

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

Sourceval eq_invert : ('a -> 'a -> bool) -> 'a pcase_invert -> 'a pcase_invert -> bool
Hashconsing
Sourceval hash : constr -> int
Sourceval case_info_hash : case_info -> int
Sourceval hcons : constr -> constr
Sourceval debug_print : constr -> Pp.t
Sourceval debug_print_fix : ('a -> Pp.t) -> ('a, 'a) pfixpoint -> Pp.t