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Primitive objects

Primitive Integers

The language of terms features 63-bit machine integers as values. The type of such a value is axiomatized; it is declared through the following sentence (excerpt from the Int63 module):

Primitive int := #int63_type.

This type is equipped with a few operators, that must be similarly declared. For instance, equality of two primitive integers can be decided using the Int63.eqb function, declared and specified as follows:

Primitive eqb := #int63_eq. Notation "m '==' n" := (eqb m n) (at level 70, no associativity) : int63_scope. Axiom eqb_correct : forall i j, (i == j)%int63 = true -> i = j.

The complete set of such operators can be obtained looking at the Int63 module.

These primitive declarations are regular axioms. As such, they must be trusted and are listed by the Print Assumptions command, as in the following example.

From Coq Require Import Int63.
[Loading ML file newring_plugin.cmxs ... done] [Loading ML file zify_plugin.cmxs ... done] [Loading ML file micromega_plugin.cmxs ... done] [Loading ML file omega_plugin.cmxs ... done] [Loading ML file int63_syntax_plugin.cmxs ... done]
Lemma one_minus_one_is_zero : (1 - 1 = 0)%int63.
1 subgoal ============================ (1 - 1)%int63 = 0%int63
Proof. apply eqb_correct; vm_compute; reflexivity. Qed.
No more subgoals.
Print Assumptions one_minus_one_is_zero.
Axioms: sub : int -> int -> int eqb_correct : forall i j : int, (i == j)%int63 = true -> i = j eqb : int -> int -> bool

The reduction machines implement dedicated, efficient rules to reduce the applications of these primitive operations.

The extraction of these primitives can be customized similarly to the extraction of regular axioms (see Program extraction). Nonetheless, the ExtrOCamlInt63 module can be used when extracting to OCaml: it maps the Coq primitives to types and functions of a Uint63 module. Said OCaml module is not produced by extraction. Instead, it has to be provided by the user (if they want to compile or execute the extracted code). For instance, an implementation of this module can be taken from the kernel of Coq.

Literal values (at type Int63.int) are extracted to literal OCaml values wrapped into the Uint63.of_int (resp. Uint63.of_int64) constructor on 64-bit (resp. 32-bit) platforms. Currently, this cannot be customized (see the function Uint63.compile from the kernel).

Primitive Floats

The language of terms features Binary64 floating-point numbers as values. The type of such a value is axiomatized; it is declared through the following sentence (excerpt from the PrimFloat module):

Primitive float := #float64_type.

This type is equipped with a few operators, that must be similarly declared. For instance, the product of two primitive floats can be computed using the PrimFloat.mul function, declared and specified as follows:

Primitive mul := #float64_mul. Notation "x * y" := (mul x y) : float_scope. Axiom mul_spec : forall x y, Prim2SF (x * y)%float = SF64mul (Prim2SF x) (Prim2SF y).

where Prim2SF is defined in the FloatOps module.

The set of such operators is described in section Floats library.

These primitive declarations are regular axioms. As such, they must be trusted, and are listed by the Print Assumptions command.

The reduction machines (vm_compute, native_compute) implement dedicated, efficient rules to reduce the applications of these primitive operations, using the floating-point processor operators that are assumed to comply with the IEEE 754 standard for floating-point arithmetic.

The extraction of these primitives can be customized similarly to the extraction of regular axioms (see Program extraction). Nonetheless, the ExtrOCamlFloats module can be used when extracting to OCaml: it maps the Coq primitives to types and functions of a Float64 module. Said OCaml module is not produced by extraction. Instead, it has to be provided by the user (if they want to compile or execute the extracted code). For instance, an implementation of this module can be taken from the kernel of Coq.

Literal values (of type Float64.t) are extracted to literal OCaml values (of type float) written in hexadecimal notation and wrapped into the Float64.of_float constructor, e.g.: Float64.of_float (0x1p+0).

Primitive Arrays

The language of terms features persistent arrays as values. The type of such a value is axiomatized; it is declared through the following sentence (excerpt from the PArray module):

Primitive array := #array_type.

This type is equipped with a few operators, that must be similarly declared. For instance, elements in an array can be accessed and updated using the PArray.get and PArray.set functions, declared and specified as follows:

Primitive get := #array_get. Primitive set := #array_set. Notation "t .[ i ]" := (get t i). Notation "t .[ i <- a ]" := (set t i a). Axiom get_set_same : forall A t i (a:A), (i < length t) = true -> t.[i<-a].[i] = a. Axiom get_set_other : forall A t i j (a:A), i <> j -> t.[i<-a].[j] = t.[j].

The complete set of such operators can be obtained looking at the PArray module.

These primitive declarations are regular axioms. As such, they must be trusted and are listed by the Print Assumptions command.

The reduction machines (vm_compute, native_compute) implement dedicated, efficient rules to reduce the applications of these primitive operations.

The extraction of these primitives can be customized similarly to the extraction of regular axioms (see Program extraction). Nonetheless, the ExtrOCamlPArray module can be used when extracting to OCaml: it maps the Coq primitives to types and functions of a Parray module. Said OCaml module is not produced by extraction. Instead, it has to be provided by the user (if they want to compile or execute the extracted code). For instance, an implementation of this module can be taken from the kernel of Coq (see kernel/parray.ml).

Primitive arrays expose a functional interface, but they are internally implemented using a persistent data structure [CF07]. Update and access to an element in the most recent copy of an array are constant time operations.