(** * Syntax of types *)

(** Compcert C types are similar to those of C.  They include numeric types,
  pointers, arrays, function types, and composite types (struct and
  union).  Numeric types (integers and floats) fully specify the
  bit size of the type.  An integer type is a pair of a signed/unsigned
  flag and a bit size: 8, 16, or 32 bits, or the special [IBool] size
  standing for the C99 [_Bool] type.  64-bit integers are treated separately. *)

Inductive signedness : Type :=
  | Signed: signedness
  | Unsigned: signedness.(** * Syntax of types *)

(** Compcert C types are similar to those of C.  They include numeric types,
  pointers, arrays, function types, and composite types (struct and
  union).  Numeric types (integers and floats) fully specify the
  bit size of the type.  An integer type is a pair of a signed/unsigned
  flag and a bit size: 8, 16, or 32 bits, or the special [IBool] size
  standing for the C99 [_Bool] type.  64-bit integers are treated separately. *)

Inductive signedness : Type :=
  | Signed: signedness
  | Unsigned: signedness.(** * Syntax of types *)

(** Compcert C types are similar to those of C.  They include numeric types,
  pointers, arrays, function types, and composite types (struct and
  union).  Numeric types (integers and floats) fully specify the
  bit size of the type.  An integer type is a pair of a signed/unsigned
  flag and a bit size: 8, 16, or 32 bits, or the special [IBool] size
  standing for the C99 [_Bool] type.  64-bit integers are treated separately. *)

Inductive signedness : Type :=
  | Signed: signedness
  | Unsigned: signedness.Inductive intsize : Type :=
  | I8: intsize
  | I16: intsize
  | I32: intsize
  | IBool: intsize.Inductive intsize : Type :=
  | I8: intsize
  | I16: intsize
  | I32: intsize
  | IBool: intsize.Inductive intsize : Type :=
  | I8: intsize
  | I16: intsize
  | I32: intsize
  | IBool: intsize.Inductive intsize : Type :=
  | I8: intsize
  | I16: intsize
  | I32: intsize
  | IBool: intsize.Inductive intsize : Type :=
  | I8: intsize
  | I16: intsize
  | I32: intsize
  | IBool: intsize.(** Float types come in two sizes: 32 bits (single precision)
  and 64-bit (double precision). *)

Inductive floatsize : Type :=
  | F32: floatsize
  | F64: floatsize.(** Float types come in two sizes: 32 bits (single precision)
  and 64-bit (double precision). *)

Inductive floatsize : Type :=
  | F32: floatsize
  | F64: floatsize.(** Float types come in two sizes: 32 bits (single precision)
  and 64-bit (double precision). *)

Inductive floatsize : Type :=
  | F32: floatsize
  | F64: floatsize.(** Every type carries a set of attributes.  Currently, only two
  attributes are modeled: [volatile] and [_Alignas(n)] (from ISO C 2011). *)

Record attr : Type := mk_attr {
  attr_volatile: bool;
  attr_alignas: option N         (**r log2 of required alignment *)
}.(** Every type carries a set of attributes.  Currently, only two
  attributes are modeled: [volatile] and [_Alignas(n)] (from ISO C 2011). *)

Record attr : Type := mk_attr {
  attr_volatile: bool;
  attr_alignas: option N         (**r log2 of required alignment *)
}.(** Every type carries a set of attributes.  Currently, only two
  attributes are modeled: [volatile] and [_Alignas(n)] (from ISO C 2011). *)

Record attr : Type := mk_attr {
  attr_volatile: bool;
  attr_alignas: option N         (**r log2 of required alignment *)
}.(** Every type carries a set of attributes.  Currently, only two
  attributes are modeled: [volatile] and [_Alignas(n)] (from ISO C 2011). *)

Record attr : Type := mk_attr {
  attr_volatile: bool;
  attr_alignas: option N         (**r log2 of required alignment *)
}.Definition noattr := {| attr_volatile := false; attr_alignas := None |}.(** The syntax of type expressions.  Some points to note:
- Array types [Tarray n] carry the size [n] of the array.
  Arrays with unknown sizes are represented by pointer types.
- Function types [Tfunction targs tres] specify the number and types
  of the function arguments (list [targs]), and the type of the
  function result ([tres]).  Variadic functions and old-style unprototyped
  functions are not supported.
*)

Inductive type : Type :=
  | Tvoid: type                                    (**r the [void] type *)
  | Tint: intsize -> signedness -> attr -> type    (**r integer types *)
  | Tlong: signedness -> attr -> type              (**r 64-bit integer types *)
  | Tfloat: floatsize -> attr -> type              (**r floating-point types *)
  | Tpointer: type -> attr -> type                 (**r pointer types ([*ty]) *)
  | Tarray: type -> Z -> attr -> type              (**r array types ([ty[len]]) *)
  | Tfunction: typelist -> type -> calling_convention -> type    (**r function types *)
  | Tstruct: ident -> attr -> type                 (**r struct types *)
  | Tunion: ident -> attr -> type                  (**r union types *)
with typelist : Type :=
  | Tnil: typelist
  | Tcons: type -> typelist -> typelist.(** The syntax of type expressions.  Some points to note:
- Array types [Tarray n] carry the size [n] of the array.
  Arrays with unknown sizes are represented by pointer types.
- Function types [Tfunction targs tres] specify the number and types
  of the function arguments (list [targs]), and the type of the
  function result ([tres]).  Variadic functions and old-style unprototyped
  functions are not supported.
*)

Inductive type : Type :=
  | Tvoid: type                                    (**r the [void] type *)
  | Tint: intsize -> signedness -> attr -> type    (**r integer types *)
  | Tlong: signedness -> attr -> type              (**r 64-bit integer types *)
  | Tfloat: floatsize -> attr -> type              (**r floating-point types *)
  | Tpointer: type -> attr -> type                 (**r pointer types ([*ty]) *)
  | Tarray: type -> Z -> attr -> type              (**r array types ([ty[len]]) *)
  | Tfunction: typelist -> type -> calling_convention -> type    (**r function types *)
  | Tstruct: ident -> attr -> type                 (**r struct types *)
  | Tunion: ident -> attr -> type                  (**r union types *)
with typelist : Type :=
  | Tnil: typelist
  | Tcons: type -> typelist -> typelist.(** The syntax of type expressions.  Some points to note:
- Array types [Tarray n] carry the size [n] of the array.
  Arrays with unknown sizes are represented by pointer types.
- Function types [Tfunction targs tres] specify the number and types
  of the function arguments (list [targs]), and the type of the
  function result ([tres]).  Variadic functions and old-style unprototyped
  functions are not supported.
*)

Inductive type : Type :=
  | Tvoid: type                                    (**r the [void] type *)
  | Tint: intsize -> signedness -> attr -> type    (**r integer types *)
  | Tlong: signedness -> attr -> type              (**r 64-bit integer types *)
  | Tfloat: floatsize -> attr -> type              (**r floating-point types *)
  | Tpointer: type -> attr -> type                 (**r pointer types ([*ty]) *)
  | Tarray: type -> Z -> attr -> type              (**r array types ([ty[len]]) *)
  | Tfunction: typelist -> type -> calling_convention -> type    (**r function types *)
  | Tstruct: ident -> attr -> type                 (**r struct types *)
  | Tunion: ident -> attr -> type                  (**r union types *)
with typelist : Type :=
  | Tnil: typelist
  | Tcons: type -> typelist -> typelist.(** The syntax of type expressions.  Some points to note:
- Array types [Tarray n] carry the size [n] of the array.
  Arrays with unknown sizes are represented by pointer types.
- Function types [Tfunction targs tres] specify the number and types
  of the function arguments (list [targs]), and the type of the
  function result ([tres]).  Variadic functions and old-style unprototyped
  functions are not supported.
*)

Inductive type : Type :=
  | Tvoid: type                                    (**r the [void] type *)
  | Tint: intsize -> signedness -> attr -> type    (**r integer types *)
  | Tlong: signedness -> attr -> type              (**r 64-bit integer types *)
  | Tfloat: floatsize -> attr -> type              (**r floating-point types *)
  | Tpointer: type -> attr -> type                 (**r pointer types ([*ty]) *)
  | Tarray: type -> Z -> attr -> type              (**r array types ([ty[len]]) *)
  | Tfunction: typelist -> type -> calling_convention -> type    (**r function types *)
  | Tstruct: ident -> attr -> type                 (**r struct types *)
  | Tunion: ident -> attr -> type                  (**r union types *)
with typelist : Type :=
  | Tnil: typelist
  | Tcons: type -> typelist -> typelist.(** The syntax of type expressions.  Some points to note:
- Array types [Tarray n] carry the size [n] of the array.
  Arrays with unknown sizes are represented by pointer types.
- Function types [Tfunction targs tres] specify the number and types
  of the function arguments (list [targs]), and the type of the
  function result ([tres]).  Variadic functions and old-style unprototyped
  functions are not supported.
*)

Inductive type : Type :=
  | Tvoid: type                                    (**r the [void] type *)
  | Tint: intsize -> signedness -> attr -> type    (**r integer types *)
  | Tlong: signedness -> attr -> type              (**r 64-bit integer types *)
  | Tfloat: floatsize -> attr -> type              (**r floating-point types *)
  | Tpointer: type -> attr -> type                 (**r pointer types ([*ty]) *)
  | Tarray: type -> Z -> attr -> type              (**r array types ([ty[len]]) *)
  | Tfunction: typelist -> type -> calling_convention -> type    (**r function types *)
  | Tstruct: ident -> attr -> type                 (**r struct types *)
  | Tunion: ident -> attr -> type                  (**r union types *)
with typelist : Type :=
  | Tnil: typelist
  | Tcons: type -> typelist -> typelist.(** The syntax of type expressions.  Some points to note:
- Array types [Tarray n] carry the size [n] of the array.
  Arrays with unknown sizes are represented by pointer types.
- Function types [Tfunction targs tres] specify the number and types
  of the function arguments (list [targs]), and the type of the
  function result ([tres]).  Variadic functions and old-style unprototyped
  functions are not supported.
*)

Inductive type : Type :=
  | Tvoid: type                                    (**r the [void] type *)
  | Tint: intsize -> signedness -> attr -> type    (**r integer types *)
  | Tlong: signedness -> attr -> type              (**r 64-bit integer types *)
  | Tfloat: floatsize -> attr -> type              (**r floating-point types *)
  | Tpointer: type -> attr -> type                 (**r pointer types ([*ty]) *)
  | Tarray: type -> Z -> attr -> type              (**r array types ([ty[len]]) *)
  | Tfunction: typelist -> type -> calling_convention -> type    (**r function types *)
  | Tstruct: ident -> attr -> type                 (**r struct types *)
  | Tunion: ident -> attr -> type                  (**r union types *)
with typelist : Type :=
  | Tnil: typelist
  | Tcons: type -> typelist -> typelist.(** The syntax of type expressions.  Some points to note:
- Array types [Tarray n] carry the size [n] of the array.
  Arrays with unknown sizes are represented by pointer types.
- Function types [Tfunction targs tres] specify the number and types
  of the function arguments (list [targs]), and the type of the
  function result ([tres]).  Variadic functions and old-style unprototyped
  functions are not supported.
*)

Inductive type : Type :=
  | Tvoid: type                                    (**r the [void] type *)
  | Tint: intsize -> signedness -> attr -> type    (**r integer types *)
  | Tlong: signedness -> attr -> type              (**r 64-bit integer types *)
  | Tfloat: floatsize -> attr -> type              (**r floating-point types *)
  | Tpointer: type -> attr -> type                 (**r pointer types ([*ty]) *)
  | Tarray: type -> Z -> attr -> type              (**r array types ([ty[len]]) *)
  | Tfunction: typelist -> type -> calling_convention -> type    (**r function types *)
  | Tstruct: ident -> attr -> type                 (**r struct types *)
  | Tunion: ident -> attr -> type                  (**r union types *)
with typelist : Type :=
  | Tnil: typelist
  | Tcons: type -> typelist -> typelist.(** The syntax of type expressions.  Some points to note:
- Array types [Tarray n] carry the size [n] of the array.
  Arrays with unknown sizes are represented by pointer types.
- Function types [Tfunction targs tres] specify the number and types
  of the function arguments (list [targs]), and the type of the
  function result ([tres]).  Variadic functions and old-style unprototyped
  functions are not supported.
*)

Inductive type : Type :=
  | Tvoid: type                                    (**r the [void] type *)
  | Tint: intsize -> signedness -> attr -> type    (**r integer types *)
  | Tlong: signedness -> attr -> type              (**r 64-bit integer types *)
  | Tfloat: floatsize -> attr -> type              (**r floating-point types *)
  | Tpointer: type -> attr -> type                 (**r pointer types ([*ty]) *)
  | Tarray: type -> Z -> attr -> type              (**r array types ([ty[len]]) *)
  | Tfunction: typelist -> type -> calling_convention -> type    (**r function types *)
  | Tstruct: ident -> attr -> type                 (**r struct types *)
  | Tunion: ident -> attr -> type                  (**r union types *)
with typelist : Type :=
  | Tnil: typelist
  | Tcons: type -> typelist -> typelist.(** The syntax of type expressions.  Some points to note:
- Array types [Tarray n] carry the size [n] of the array.
  Arrays with unknown sizes are represented by pointer types.
- Function types [Tfunction targs tres] specify the number and types
  of the function arguments (list [targs]), and the type of the
  function result ([tres]).  Variadic functions and old-style unprototyped
  functions are not supported.
*)

Inductive type : Type :=
  | Tvoid: type                                    (**r the [void] type *)
  | Tint: intsize -> signedness -> attr -> type    (**r integer types *)
  | Tlong: signedness -> attr -> type              (**r 64-bit integer types *)
  | Tfloat: floatsize -> attr -> type              (**r floating-point types *)
  | Tpointer: type -> attr -> type                 (**r pointer types ([*ty]) *)
  | Tarray: type -> Z -> attr -> type              (**r array types ([ty[len]]) *)
  | Tfunction: typelist -> type -> calling_convention -> type    (**r function types *)
  | Tstruct: ident -> attr -> type                 (**r struct types *)
  | Tunion: ident -> attr -> type                  (**r union types *)
with typelist : Type :=
  | Tnil: typelist
  | Tcons: type -> typelist -> typelist.(** The syntax of type expressions.  Some points to note:
- Array types [Tarray n] carry the size [n] of the array.
  Arrays with unknown sizes are represented by pointer types.
- Function types [Tfunction targs tres] specify the number and types
  of the function arguments (list [targs]), and the type of the
  function result ([tres]).  Variadic functions and old-style unprototyped
  functions are not supported.
*)

Inductive type : Type :=
  | Tvoid: type                                    (**r the [void] type *)
  | Tint: intsize -> signedness -> attr -> type    (**r integer types *)
  | Tlong: signedness -> attr -> type              (**r 64-bit integer types *)
  | Tfloat: floatsize -> attr -> type              (**r floating-point types *)
  | Tpointer: type -> attr -> type                 (**r pointer types ([*ty]) *)
  | Tarray: type -> Z -> attr -> type              (**r array types ([ty[len]]) *)
  | Tfunction: typelist -> type -> calling_convention -> type    (**r function types *)
  | Tstruct: ident -> attr -> type                 (**r struct types *)
  | Tunion: ident -> attr -> type                  (**r union types *)
with typelist : Type :=
  | Tnil: typelist
  | Tcons: type -> typelist -> typelist.(** The syntax of type expressions.  Some points to note:
- Array types [Tarray n] carry the size [n] of the array.
  Arrays with unknown sizes are represented by pointer types.
- Function types [Tfunction targs tres] specify the number and types
  of the function arguments (list [targs]), and the type of the
  function result ([tres]).  Variadic functions and old-style unprototyped
  functions are not supported.
*)

Inductive type : Type :=
  | Tvoid: type                                    (**r the [void] type *)
  | Tint: intsize -> signedness -> attr -> type    (**r integer types *)
  | Tlong: signedness -> attr -> type              (**r 64-bit integer types *)
  | Tfloat: floatsize -> attr -> type              (**r floating-point types *)
  | Tpointer: type -> attr -> type                 (**r pointer types ([*ty]) *)
  | Tarray: type -> Z -> attr -> type              (**r array types ([ty[len]]) *)
  | Tfunction: typelist -> type -> calling_convention -> type    (**r function types *)
  | Tstruct: ident -> attr -> type                 (**r struct types *)
  | Tunion: ident -> attr -> type                  (**r union types *)
with typelist : Type :=
  | Tnil: typelist
  | Tcons: type -> typelist -> typelist.(** The syntax of type expressions.  Some points to note:
- Array types [Tarray n] carry the size [n] of the array.
  Arrays with unknown sizes are represented by pointer types.
- Function types [Tfunction targs tres] specify the number and types
  of the function arguments (list [targs]), and the type of the
  function result ([tres]).  Variadic functions and old-style unprototyped
  functions are not supported.
*)

Inductive type : Type :=
  | Tvoid: type                                    (**r the [void] type *)
  | Tint: intsize -> signedness -> attr -> type    (**r integer types *)
  | Tlong: signedness -> attr -> type              (**r 64-bit integer types *)
  | Tfloat: floatsize -> attr -> type              (**r floating-point types *)
  | Tpointer: type -> attr -> type                 (**r pointer types ([*ty]) *)
  | Tarray: type -> Z -> attr -> type              (**r array types ([ty[len]]) *)
  | Tfunction: typelist -> type -> calling_convention -> type    (**r function types *)
  | Tstruct: ident -> attr -> type                 (**r struct types *)
  | Tunion: ident -> attr -> type                  (**r union types *)
with typelist : Type :=
  | Tnil: typelist
  | Tcons: type -> typelist -> typelist.(** The syntax of type expressions.  Some points to note:
- Array types [Tarray n] carry the size [n] of the array.
  Arrays with unknown sizes are represented by pointer types.
- Function types [Tfunction targs tres] specify the number and types
  of the function arguments (list [targs]), and the type of the
  function result ([tres]).  Variadic functions and old-style unprototyped
  functions are not supported.
*)

Inductive type : Type :=
  | Tvoid: type                                    (**r the [void] type *)
  | Tint: intsize -> signedness -> attr -> type    (**r integer types *)
  | Tlong: signedness -> attr -> type              (**r 64-bit integer types *)
  | Tfloat: floatsize -> attr -> type              (**r floating-point types *)
  | Tpointer: type -> attr -> type                 (**r pointer types ([*ty]) *)
  | Tarray: type -> Z -> attr -> type              (**r array types ([ty[len]]) *)
  | Tfunction: typelist -> type -> calling_convention -> type    (**r function types *)
  | Tstruct: ident -> attr -> type                 (**r struct types *)
  | Tunion: ident -> attr -> type                  (**r union types *)
with typelist : Type :=
  | Tnil: typelist
  | Tcons: type -> typelist -> typelist.

Lemma type_eq: forall (ty1 ty2: type), {ty1=ty2} + {ty1<>ty2} with typelist_eq: forall (tyl1 tyl2: typelist), {tyl1=tyl2} + {tyl1<>tyl2}.