drafting subtyping rules

README.md

@@ -10,7 +10,7 @@ s, t ::= s | t              (disjunction, a.k.a. union)
        | s => t             (implication)
        | s -> t             (arrow, a.k.a. function)
        | forall x . t       (forall, polymorphic)
-       | box t              (box, modular logic)
+       | box t              (box, c.f. modular logic)
        | a [t..]            (type operator application)
        | x                  (variable)
        | n                  (nominal)
@@ -27,33 +27,45 @@ a ∈ opname
 ## sappho subtyping rules
 
 ```
-s, t, γ ⊢ δ
+---- [identity]
+ρ; s, γ ⊢ s, δ
+
+---- [true-right]
+ρ; γ ⊢ true, δ
+
+---- [false-left]
+ρ; false, γ ⊢ δ
+
+ρ; s, t, γ ⊢ δ
 ---- [conj-left]
-s & t, γ ⊢ δ
+ρ; s & t, γ ⊢ δ
 
-γ ⊢ s, t, δ
+ρ; γ ⊢ s, t, δ
 ---- [disj-right]
-γ ⊢ s | t, δ
+ρ; γ ⊢ s | t, δ
 
-s, γ ⊢ δ
-t, γ ⊢ δ
+ρ; s, γ ⊢ δ
+ρ; t, γ ⊢ δ
 ---- [disj-left]
-s | t, γ ⊢ δ
+ρ; s | t, γ ⊢ δ
 
-γ ⊢ s, δ
-γ ⊢ t, δ
+ρ; γ ⊢ s, δ
+ρ; γ ⊢ t, δ
 ---- [conj-right]
-γ ⊢ s & t, δ
+ρ; γ ⊢ s & t, δ
 
 // XXX
-s => t, s, t, γ ⊢ δ
+// remove s => t? always make progress
+ρ; s => t, s, t, γ ⊢ δ
 ---- [impl-left]
-s => t, s, γ ⊢ δ
+ρ; s => t, s, γ ⊢ δ
 
 // discussion about implication below
-s, γ ⊢ t, δ
+// check how "normal sequent calculus handles this"
+ρ; s, γ ⊢ t, δ
 ---- [impl-right]
-γ ⊢ s => t, δ
+ρ; γ ⊢ s => t, δ
+
 
 
 // box works as a kind of "forall" for concrete types
@@ -62,22 +74,64 @@ box t, t, γ ⊢ δ
 box t, γ ⊢ δ
 
 // (.)-- filters the contexts.
-γ-- ⊢ t, δ--
+ρ; γ-- ⊢ t, δ--
 ---- [box-right]
-γ ⊢ box t, δ
+ρ; γ ⊢ box t, δ
+
+
+// recursion through operators are handled using the recursion context ρ
+// XXX compare with iso-recursive subtyping (double expansion)
+---- [op-discharge]
+a [t..], ρ; γ ⊢ a [t..], δ
+
+a [t..]*, ρ; γ ⊢ expand(a [t..]), δ
+---- [op-expand1]
+ρ; γ ⊢ a [t..], δ
+
+a [t..], ρ; γ ⊢ expand(a [t..]), δ
+---- [op-expand2]
+a [t..]*, ρ; γ ⊢ a [t..], δ
+
+
+
+// member types are filtered according to member name
+ρ; γ >> m ⊢ δ >> m
+---- [member]
+ρ; γ ⊢ δ
+
+
+// foralls are unwrapped and variable substituted with fresh name
+// i.e. we treat the bound variable nominally
+n fresh
+ρ; γ [n] ⊢ δ [n]
+---- [forall]
+ρ; γ ⊢ δ
+
 
 
 
 ```
 
-
 ### context operators
 
+* box filtering
 ```
-(t, γ)-- =def= t, γ--       if t == box s
+(t, γ)-- =def= t, γ--       if t == box s, for some s
                γ--          otherwise
 ```
 
+* member unwrapping
+```
+(t, γ) >> m =def= s, (γ >> m)       if t == { m : s }, for some s
+                  γ >> m            otherwise
+```
+
+* forall-unwrapping and substitution
+```
+(t, γ) [n] =def= s [n/x], (γ [n])   if t == forall x . s, for some x, s
+                 γ [n]              otherwise
+```
+
 What happens when box is buried under type operators, like box s & box t? It
 could be that the above definition filters out too much of the contexts.