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|
>> TODO
- the concrete variable representation currently used in the generated code
no longer suffices for these dependent-record type environments
- either add distinctness conditions to the syntax, generating predicates
is_distinct_t etc, or write them in the typing rules (for labels in
record types, exprs and pats, and termvars in pats)
ideally also:
- implement context rules, so that the explicit reduction context rules
can be replaced by a single one.
<<
metavar typevar, X ::=
{{ isa string }} {{ coq nat }} {{ coq-equality }} {{ hol string }} {{ lem string }} {{ lex Alphanum }}
{{ tex \mathit{[[typevar]]} }} {{ com type variable }}
{{ isavar ''[[typevar]]'' }} {{ holvar "[[typevar]]" }} {{ lemvar "[[typevar]]" }} {{ texvar \mathrm{[[typevar]]} }}
{{ ocamlvar "[[typevar]]" }}
metavar termvar, x ::=
{{ isa string }} {{ coq nat }} {{ hol string }} {{ lem string }} {{ coq-equality }} {{ lex alphanum }}
{{ tex \mathit{[[termvar]]} }} {{ com term variable }}
{{ isavar ''[[termvar]]'' }} {{ holvar "[[termvar]]" }}
{{ lemvar "[[typevar]]" }}
{{ texvar \mathrm{[[termvar]]} }}
{{ ocamlvar "[[termvar]]" }}
indexvar index, i, j, n, m ::= {{ isa nat }} {{ coq nat }} {{ hol num }} {{ lem num}} {{ lex numeral }}
{{ com indices }}
grammar
T {{ hol Typ }}, S, U :: 'Ty_' ::= {{ com type }}
| X :: :: Var {{ com type variable }}
| Top :: :: Top {{ com maximum type }}
| T -> T' :: :: Fun {{ com type of functions }}
| Forall X <: T . T' :: :: Forall (+ bind X in T' +) {{ com universal type }}
{{ tex [[Forall]] [[X]] \mathord{[[<:]]} [[T]]. \, [[T']] }}
%R | { } :: :: Rec_empty {{ com empty record }}
%R | { Trb } :: :: Rec_ne {{ com nonempty record }}
% {{ com record type }}
| ( T ) :: S :: paren {{ ichl [[T]] }}
| [ X |-> T ] T' :: M :: sub {{ ichl (Tsubst_T [[T]] [[X]] [[T']]) }}
%R Trb :: 'Trb_' ::=
%R | l : T :: :: rb1
%R | l : T , Trb :: :: rb2
t :: 't_' ::= {{ com term }}
| x :: :: Var {{ com variable }}
| \ x : T . t :: :: Lam (+ bind x in t +) {{ com abstraction }}
{{ tex \lambda [[x]] \mathord{[[:]]} [[T]]. \, [[t]] }}
| t t' :: :: App {{ com application }}
| \ X <: T . t :: :: TLam (+ bind X in t +) {{ com type abstraction}}
{{ tex \Lambda [[X]] \mathord{[[<:]]} [[T]]. \, [[t]] }}
| t [ T ] :: :: TApp {{ com type application}}
| ( t ) :: S :: paren {{ ichl [[t]] }}
| [ x |-> t ] t' :: M :: tsub {{ ichl ( tsubst_t [[t]] [[x]] [[t']] ) }}
| [ X |-> T ] t :: M :: Tsub {{ ichl ( Tsubst_t [[T]] [[X]] [[t]] ) }}
% | E [ t ] :: M :: ctx
v :: 'v_' ::= {{ com values }}
| \ x : T . t :: :: Lam (+ bind x in t +) {{ com abstraction }}
| \ X <: T . t :: :: TLam (+ bind X in t +) {{ com type abstraction }}
{{ tex \Lambda [[X]] [[<:]] [[T]]. \, [[t]] }}
% {{ com record }}
% E :: E_ ::=
% | __ :: :: hole
% | E t :: :: app_fun
% | v E :: :: app_arg
% | E [ T ] :: :: type_fun
%r | E . l :: :: projection
%r% | { l1 = v1 , .. , lm = vm , l = E , l1' = t1' , .. , ln' = tn' } :: :: record
%r | let p = E in t2 :: :: let_binding
G {{ tex \Gamma }}, D {{ tex \Delta }} :: 'G_' ::= {{ com type environment }}
| empty :: :: empty
| G , X <: T :: :: type
| G , x : T :: :: term
%r | G , G' :: M :: comma {{ ichl TODO }}
terminals :: terminals_ ::=
| \ :: :: lambda {{ tex \lambda }}
| -> :: :: arrow {{ tex \rightarrow }}
% | __ :: :: hole {{ tex \_ }}
| |- :: :: turnstile {{ tex \vdash }}
| --> :: :: red {{ tex \longrightarrow }}
| Forall :: :: forall {{ tex \forall }}
| <: :: :: subtype {{ tex <: }}
| |-> :: :: mapsto {{ tex \mapsto }}
| = :: :: eq {{ tex \!\! = \!\! }}
formula :: formula_ ::=
| judgement :: :: judgement
% | G = G' :: :: Geq {{ ichl [[G]] = [[G']] }}
| x = x' :: :: xeq {{ ichl [[x]] = [[x']] }}
| X = X' :: :: Xeq {{ ichl [[X]] = [[X']] }}
| ( formula ) :: :: paren {{ ichl ( [[formula]] ) }}
| not formula :: :: not {{ isa Not( [[formula]] ) }}
{{ coq not( [[formula]] ) }}
{{ lem not( [[formula]] ) }}
{{ hol ~( [[formula]] ) }}
{{ tex \neg [[ formula]] }}
% | x isin dom ( G ) :: :: xin {{ isa ? T. ([[x]],T,[[G]]):tin }}
% {{ tex [[x]] \in [[dom]]([[G]]) }}
% | X isin dom ( G ) :: :: Xin {{ isa ? T. ([[X]],T,[[G]]):Tin }}
% {{ tex [[X]] \in [[dom]]([[G]]) }}
% would be nice to write the above as {{ isa ?[[T]]. [[X<:T isin G]] }}
%formulalist :: formulalist_ ::=
| formula1 ... formulan :: :: dots
subrules
v <:: t
% E _:: t :: t
substitutions
single t x :: tsubst
single T X :: Tsubst
multiple t x :: m_t_subst
multiple T X :: m_T_subst
freevars
T X :: ftv
t x :: fv
defns
Judgement_in :: '' ::=
% defn
% x isin dom ( G ) :: :: xinG :: xinG_ {{ tex [[x]] \in [[dom]]([[G]])}} by
%
% x:T isin G
% ---------------- :: 1
% x isin dom(G)
%
%
% defn
% X isin dom ( G ) :: :: XinG :: XinG_ {{ tex [[X]] \in [[dom]]([[G]]) }} by
%
% X<:U isin G
% ---------------- :: 1
% X isin dom(G)
defn
x isin dom ( G ) :: :: xinG :: xinG_ {{ tex [[x]] \in [[dom]]([[G]])}} by
-------------- :: 1
x isin dom(G,x:T)
x isin dom(G)
---------------- :: 2
x isin dom(G,X'<:U')
x isin dom(G)
--------------- :: 3
x isin dom(G,x':T')
defn
X isin dom ( G ) :: :: XinG :: XinG_ {{ tex [[X]] \in [[dom]]([[G]]) }} by
------------- :: 1
X isin dom(G,X<:U)
X isin dom(G)
--------------- :: 2
X isin dom(G,X'<:U')
X isin dom(G)
-------------- :: 3
X isin dom(G,x':T')
defn
x : T isin G :: :: tin :: tin_ {{ tex [[x]] [[:]] [[T]] \in [[G]] }} by
---------------- :: 1
x:T isin G,x:T
x:T isin G
------------------ :: 2
x:T isin G,X'<:U'
x:T isin G
----------------- :: 3
x:T isin G,x':T'
defn
X <: U isin G :: :: Tin :: Tin_ {{ tex [[X]] [[<:]] [[U]] \in [[G]] }} by
---------------- :: 1
X<:U isin G,X<:U
X<:U isin G
------------------ :: 2
X<:U isin G,X'<:U'
X<:U isin G
----------------- :: 3
X<:U isin G,x':T'
defns
Jtype :: '' ::=
defn
G |- ok :: :: Gok :: Gok_ {{ com type environment [[G]] is well-formed }} by
----------- :: 1
empty |- ok
G |- T
not(x isin dom(G))
------------------ :: 2
G,x:T |- ok
G |- T
not(X isin dom(G))
------------------- :: 3
G,X<:T |- ok
defn
G |- T :: :: GT :: GT_ {{ com type [[T]] is well-formed in type environment [[G]] }} by
G |- ok
X<:U isin G
-------------- :: Var
G |- X
G |- ok
-------------- :: Top
G |- Top
G |- T
G |- T'
-------------- :: Fun
G |- T->T'
G,X<:T |- T'
------------------ :: Forall
G |- Forall X<:T.T'
defn
G |- S <: T :: :: SA :: SA_ {{ com [[S]] is a subtype of [[T]] }} by
G |- ok
---------- :: Top
G |- S <: Top
G |- ok
---------- :: Refl_TVar
G |- X <: X
%G = G1, X<:U , G2
X<:U isin G
G |- U <: T
------------- :: Trans_TVar
G |- X <: T
G |- T1<:S1
G |- S2<:T2
---------------------- :: Arrow
G |- S1->S2 <: T1->T2
G |- T1<:S1
G,X<:T1 |- S2<:T2
--------------------------------------- :: All
G |- Forall X<:S1.S2 <: Forall X<:T1.T2
defn
G |- t : T :: :: Ty :: Ty_ {{ com term [[t]] has type [[T]] }} by
G |- ok
x:T isin G
-------------- :: Var
G |- x:T
G,x:T1 |- t2:T2
----------------------- :: Abs
G |- \x:T1.t2 : T1->T2
G|- t1:T11->T12
G|- t2:T11
-------------------- :: App
G|- t1 t2 : T12
G,X<:T1 |- t2:T2
-------------------------------- :: TAbs
G|- \X<:T1.t2 : Forall X<:T1.T2
G|- t1 : Forall X<:T11.T12
G|- T2 <: T11
--------------------------- :: TApp
G|- t1[T2] : [X|->T2]T12
G|- t:S
G|- S<:T
--------- :: Sub
G|- t:T
defns
Jop :: '' ::=
defn
t1 --> t2 :: :: reduce :: reduce_ {{ com [[t1]] reduces to [[t2]] }} by
----------------------------------------- :: AppAbs
(\x:T11.t12) v2 --> :t_tsub: [x|->v2]t12
----------------------------------- :: TappTabs
(\X<:T11.t12) [T2] --> [X|->T2]t12
%t1 --> t1'
%------------------- :: Ctx
%E[t1] --> E[t1']
t1 --> t1'
---------------- :: Ctx_app_fun
t1 t --> t1' t
t1 --> t1'
---------------- :: Ctx_app_arg
v t1 --> v t1'
t1 --> t1'
---------------- :: Ctx_type_fun
t1[T] --> t1'[T]
|
