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|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Machinery for typing rules %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
embed
{{ lem
let rec remove_one i l =
match l with
| [] -> []
| i2::l2 -> if i2 = i then l2 else i2::(remove_one i l2)
end
let rec remove_from l l2 =
match l2 with
| [] -> l
| i::l2' -> remove_from (remove_one i l) l2'
end
let disjoint s1 s2 = Set.null (s1 inter s2)
let rec disjoint_all sets =
match sets with
| [] -> true
| s1::[] -> true
| s1::s2::sets -> (disjoint s1 s2) && (disjoint_all (s2::sets))
end
}}
grammar
k :: 'Ki_' ::=
{{ com Internal kinds }}
| K_Typ :: :: typ
| K_Nat :: :: nat
| K_Ord :: :: ord
| K_Efct :: :: efct
| K_Lam ( k0 .. kn -> k' ) :: :: ctor
| K_infer :: :: infer {{ com Representing an unknown kind, inferred by context }}
t , u :: 'T_' ::=
{{ com Internal types }}
| x :: :: id
| ' x :: :: var
| t1 -> t2 effect :: :: fn
| ( t1 , .... , tn ) :: :: tup
| x < t_args > :: :: app
| t |-> t1 :: :: abbrev
| register < t_arg > :: S :: reg_app {{ ichlo T_app "register" [[t_arg]] }}
| range < ne ne' > :: S :: range_app {{ ichlo T_app "range" [[ [ ne ; ne' ] ]] }}
| atom < ne > :: S :: atom_app {{ ichlo T_app "atom" [ [[ne]] ] }}
| vector < ne ne' order t > :: S :: vector_app {{ ichlo T_app "vector" [[ [ ne; ne'; ord; t ] ]] }}
| list < t > :: S :: list_app {{ ichlo T_app "list" [[t]] }}
| reg < t > :: S :: box_app {{ ichlo T_app "reg" [[t]] }}
| implicit < ne > :: S :: implicit_app {{ ichlo T_app "implicit" [[ne]] }}
| bit :: S :: bit_typ {{ ichlo T_id "bit" }}
| string :: S :: string_typ {{ ichlo T_id "string" }}
| unit :: S :: unit_typ {{ ichlo T_id "unit" }}
| t [ t_arg1 / tid1 .. t_argn / tidn ] :: M :: subst {{ ichlo "todo" }}
optx :: '' ::= {{ phantom }} {{ lem maybe string }} {{ ocaml string option }}
| x :: :: optx_x
{{ lem (Just [[x]]) }} {{ ocaml (Some [[x]]) }}
| :: :: optx_none
{{ lem Nothing }} {{ ocaml None }}
tag :: 'Tag_' ::=
{{ com Data indicating where the identifier arises and thus information necessary in compilation }}
| None :: :: empty
| Global :: :: global {{ com Globally let-bound or enumeration based value/variable }}
| Ctor :: :: ctor {{ com Data constructor from a type union }}
| Extern optx :: :: extern {{ com External function, specied only with a val statement }}
| Default :: :: default {{ com Type has come from default declaration, identifier may not be bound locally }}
| Spec :: :: spec
| Enum :: :: enum
| Alias :: :: alias
ne :: 'Ne_' ::=
{{ com internal numeric expressions }}
| ' x :: :: var
| num :: :: const
| infinity :: :: inf
| ne1 * ne2 :: :: mult
| ne1 + ... + nen :: :: add
| 2 ** ne :: :: exp
| ( - ne ) :: :: unary
| zero :: S :: zero
{{ lem (Ne_const 0) }}
| one :: S :: one
{{ lem (Ne_const 1) }}
| bitlength ( bin ) :: M :: cbin
{{ ocaml (asssert false) }}
{{ hol ARB }}
{{ lem (blength [[bin]]) }}
| bitlength ( hex ) :: M :: chex
{{ ocaml (assert false) }}
{{ hol ARB }}
{{ lem (hlength [[hex]]) }}
| count ( num0 ... numi ) :: M :: length {{ichlo "todo" }}
| length ( pat1 ... patn ) :: M :: cpat
{{ ocaml (assert false) }}
{{ hol ARB }}
{{ lem (Ne_const (List.length [[pat1...patn]])) }}
| length ( exp1 ... expn ) :: M :: cexp
{{ hol ARB }}
{{ ocaml (assert false) }}
{{ lem (Ne_const (List.length [[exp1...expn]])) }}
t_arg :: 't_arg_' ::=
{{ com Argument to type constructors }}
| t :: :: typ
| ne :: :: nexp
| effect :: :: effect
| order :: :: order
| fresh :: M :: freshvar {{ lem T_arg (T_var "fresh") }}
t_args :: '' ::= {{ lem list t_arg }}
{{ com Arguments to type constructors }}
| t_arg1 ... t_argn :: :: T_args
nec :: 'Nec_' ::=
{{ com Numeric expression constraints }}
| ne <= ne' :: :: lteq
| ne = ne' :: :: eq
| ne >= ne' :: :: gteq
| ' x 'IN' { num1 , ... , numn } :: :: in
| nec0 .. necn -> nec'0 ... nec'm :: :: cond
| nec0 ... necn :: :: branch
S_N {{ tex {\Sigma^{\textsc{N} } } }} :: '' ::= {{ phantom }}
{{ hol nec list }}
{{ lem list nec }}
{{ com nexp constraint lists }}
| { nec1 , .. , necn } :: :: Sn_concrete
{{ hol [[nec1 .. necn]] }}
{{ lem [[nec1 .. necn]] }}
| S_N1 u+ .. u+ S_Nn :: M :: SN_union
{{ hol (FOLDR FUNION FEMPTY [[S_N1..S_Nn]]) }}
{{ lem (List.foldr (++) [] [[S_N1..S_Nn]]) }}
{{ ocaml (assert false) }}
| consistent_increase ne1 ne'1 ... nen ne'n :: M :: SN_increasing
{{ com Generates constraints from pairs of constraints, where the first of each pair is always larger than the sum of the previous pair }}
{{ ocaml (assert false) }}
{{ ichl todo }}
| consistent_decrease ne1 ne'1 ... nen ne'n :: M :: SN_decreasing
{{ com Generates constraints from pairs of constraints, where the first of each pair is always smaller than the difference of the previous pair }}
{{ ocaml assert false }}
{{ ichl todo }}
| resolve ( S_N ) :: :: resolution
{{ lem [[S_N]] (* Write constraint solver *) }}
E_d {{ tex {\ottnt{E}^{\textsc{d} } } }} :: 'E_d_' ::= {{ phantom }}
{{ lem definition_env }}
{{ com Environments storing top level information, such as defined abbreviations, records, enumerations, and kinds }}
| < E_k , E_a , E_r , E_e > :: :: base
{{ hol arb }}
{{ lem (Denv [[E_k]] [[E_r]] [[E_e]]) }}
| empty :: :: empty
{{ hol arb }}
{{ lem DenvEmp }}
| E_d u+ E_d' :: :: union
{{ hol arb }}
{{ lem (denv_union [[E_d]] [[E_d']]) }}
kinf :: 'kinf_' ::=
{{ com Whether a kind is default or from a local binding }}
| k :: :: k
| k default :: :: def
tid :: 'tid_' ::=
{{ com A type identifier or type variable }}
| id :: :: id
| kid :: :: var
E_k {{ tex {\ottnt{E}^{\textsc{k} } } }} :: 'E_k_' ::= {{ phantom }}
{{ hol (tid-> kinf) }}
{{ lem (map tid kinf) }}
{{ com Kind environments }}
| { tid1 |-> kinf1 , .. , tidn |-> kinfn } :: :: concrete
{{ hol (FOLDR (\(k1,k2) E. E |+ (k1,k2)) FEMPTY [[tid1 kinf1 .. tidn kinfn]]) }}
{{ lem (List.foldr (fun (x,v) m -> Map.insert x v m) Map.empty [[tid1 kinf1 .. tidn kinfn]]) }}
| E_k1 u+ .. u+ E_kn :: M :: union
{{ com In a unioning kinf, {k default} u {k} results in {k} (i.e. the default is locally forgotten) }}
{{ hol (FOLDR FUNION FEMPTY [[E_k1..E_kn]]) }}
{{ lem (List.foldr (union) Map.empty [[E_k1..E_kn]]) }}
{{ ocaml (assert false) }}
| E_k u- E_k1 .. E_kn :: M :: multi_set_minus
{{ hol arb }}
{{ lem (Map.fromList (remove_from (Set_extra.toList (Map.toSet [[E_k]]))
(Set_extra.toList (Map.toSet (List.foldr (union) Map.empty [[E_k1..E_kn]]))))) }}
{{ ocaml assert false }}
tinf :: 'tinf_' ::=
{{ com Type variables, type, and constraints, bound to an identifier }}
| t :: :: typ
| E_k , S_N , tag , t :: :: quant_typ
tinflist :: 'tinfs_' ::=
{{ com In place so that a list of tinfs can be referred to without the dot form }}
| tinf1 ... tinfn :: :: ls
E_a {{ tex \ottnt{E}^{\textsc{a} } }} :: 'E_a_' ::= {{ phantom }}
{{ hol tid |-> tinf}}
{{ lem map tid tinf }}
| { tid1 |-> tinf1 , .. , tidn |-> tinfn } :: :: concrete
| E_a1 u+ .. u+ E_an :: :: union
field_typs :: 'FT_' ::= {{ phantom }}
{{ lem list (id * t) }}
{{ com Record fields }}
| id1 : t1 , .. , idn : tn :: :: fields
{{ lem [[id1 t1..idn tn]] }}
E_r {{ tex \ottnt{E}^{\textsc{r} } }} :: 'E_r_' ::= {{ phantom }}
{{ hol (id*t) |-> tinf) }}
{{ lem map (list (id*t)) tinf }}
{{ com Record environments }}
| { { field_typs1 } |-> tinf1 , .. , { field_typsn } |-> tinfn } :: :: concrete
{{ hol (FOLDR (\x E. E |+ x) FEMPTY) }}
{{ lem (List.foldr (fun (x,f) m -> Map.insert x f m) Map.empty [[field_typs1 tinf1..field_typsn tinfn]]) }}
| E_r1 u+ .. u+ E_rn :: M :: union
{{ hol (FOLDR FUNION FEMPTY [[E_r1..E_rn]]) }}
{{ lem (List.foldr (union) Map.empty [[E_r1..E_rn]]) }}
{{ ocaml (assert false) }}
enumerate_map :: '' ::= {{ phantom }}
{{ lem (list (nat*id)) }}
| { num1 |-> id1 ... numn |-> idn } :: :: enum_map
{{ lem [[num1 id1...numn idn]] }}
E_e {{ tex \ottnt{E}^{\textsc{e} } }} :: 'E_e_' ::= {{ phantom }}
{{ lem (map t (list (nat*id))) }}
{{ com Enumeration environments }}
| { t1 |-> enumerate_map1 , .. , tn |-> enumerate_mapn } :: :: base
{{ lem (List.foldr (fun (x,f) m -> Map.insert x f m) Map.empty [[t1 enumerate_map1..tn enumerate_mapn]]) }}
| E_e1 u+ .. u+ E_en :: :: union
{{ lem (List.foldr (union) Map.empty [[E_e1..E_en]]) }}
embed
{{ lem
type definition_env =
| DenvEmp
| Denv of (map tid kinf) * (map (list (id*t)) tinf) * (map t (list (nat*id)))
}}
grammar
E_t {{ tex {\ottnt{E}^{\textsc{t} } } }} :: 'E_t_' ::= {{ phantom }}
{{ hol (id |-> tinf) }}
{{ lem map id tinf }}
{{ com Type environments }}
| { id1 |-> tinf1 , .. , idn |-> tinfn } :: :: base
{{ hol (FOLDR (\x E. E |+ x) FEMPTY [[id1 tinf1 .. idn tinfn]]) }}
{{ lem (List.foldr (fun (x,f) m -> Map.insert x f m) Map.empty [[id1 tinf1 .. idn tinfn]]) }}
| { id |-> overload tinf : tinf1 , ... , tinfn } :: :: overload
| ( E_t1 u+ .... u+ E_tn ) :: M :: union
{{ hol (FOLDR FUNION FEMPTY [[E_t1....E_tn]]) }}
{{ lem (List.foldr (union) Map.empty [[E_t1....E_tn]]) }}
{{ ocaml (assert false) }}
| u+ E_t1 .. E_tn :: M :: multi_union
{{ hol arb }}
{{ lem (List.foldr (union) Map.empty [[E_t1..E_tn]]) }}
{{ ocaml assert false }}
| E_t u- E_t1 .. E_tn :: M :: multi_set_minus
{{ hol arb }}
{{ lem (Map.fromList (remove_from (Set_extra.toList (Map.toSet [[E_t]]))
(Set_extra.toList (Map.toSet (List.foldr (union) Map.empty [[E_t1..E_tn]]))))) }}
{{ ocaml assert false }}
| ( E_t1 inter .... inter E_tn ) :: M :: intersect
{{ hol arb }}
{{ lem (List.foldr (fun a b -> (Map.fromList (Set_extra.toList ((Map.toSet a) inter (Map.toSet b))))) Map.empty [[E_t1....E_tn]]) }}
{{ ocaml (assert false) }}
| inter E_t1 .. E_tn :: M :: multi_inter
{{ hol arb }}
{{ lem (List.foldr (fun a b -> (Map.fromList (Set_extra.toList ((Map.toSet a) inter (Map.toSet b))))) Map.empty [[E_t1..E_tn]]) }}
{{ ocaml assert false }}
ts :: ts_ ::= {{ phantom }}
{{ lem list t }}
| t1 , .. , tn :: :: lst
embed
{{ lem
let blength (bit) = Ne_const 8
let hlength (bit) = Ne_const 8
type env =
| EnvEmp
| Env of (map id tinf) * definition_env
type inf =
| Iemp
| Inf of (list nec) * effect
val denv_union : definition_env -> definition_env -> definition_env
let denv_union de1 de2 =
match (de1,de2) with
| (DenvEmp,de2) -> de2
| (de1,DenvEmp) -> de1
| ((Denv ke1 re1 ee1),(Denv ke2 re2 ee2)) ->
Denv (ke1 union ke2) (re1 union re2) (ee1 union ee2)
end
val env_union : env -> env -> env
let env_union e1 e2 =
match (e1,e2) with
| (EnvEmp,e2) -> e2
| (e1,EnvEmp) -> e1
| ((Env te1 de1),(Env te2 de2)) ->
Env (te1 union te2) (denv_union de1 de2)
end
let inf_union i1 i2 =
match (i1,i2) with
| (Iemp,i2) -> i2
| (i1,Iemp) -> i1
| (Inf n1 e1,Inf n2 e2) -> (Inf (n1++n2) (effect_union e1 e2))
end
let fresh_kid denv = Var "x" (*TODO When strings can be manipulated, this should actually build a fresh string*)
}}
grammar
E :: '' ::=
{{ hol ((string,env_body) fmaptree) }}
{{ lem env }}
{{ com Definition environment and lexical environment }}
| < E_t , E_d > :: :: E
{{ hol arb }}
{{ lem (Env [[E_t]] [[E_d]]) }}
| empty :: M :: E_empty
{{ hol arb }}
{{ lem EnvEmp }}
{{ ocaml assert false }}
| E u+ E' :: :: E_union
{{ lem (env_union [[E]] [[E']]) }}
I :: '' ::= {{ lem inf }}
{{ com Information given by type checking an expression }}
| < S_N , effect > :: :: I
{{ lem (Inf [[S_N]] [[effect]]) }}
| Ie :: :: Iempty {{ com Empty constraints, effect }} {{ tex {\ottnt{I}_{\epsilon} } }}
{{ lem Iemp }}
| ( I1 u+ I2 ) :: :: singleunion {{ tex [[I1]] [[u+]] [[I2]] }}
| I1 u+ .. u+ In :: :: Iunion {{ com Unions the constraints and effect }}
{{ lem (List.foldr inf_union Iemp [[I1..In]]) }}
|
