C machine and focusing (internalized compositon)

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This is Literate Twelf
Code: here
Status: %% OK %%
Output: here.

C Machine and Focusing

Syntax

pos : type.  %name pos A+.
neg : type.  %name neg A-.

down : neg -> pos.
plus : pos -> pos -> pos.
times : pos -> pos -> pos.
zero  : pos.
one   : pos.

up   : pos -> neg.
arr  : pos -> neg -> neg.
with : neg -> neg -> neg.
top  : neg.

%abbrev conc = neg.

val+  : pos -> type.
cont- : neg -> conc -> type.
cont+ : pos -> conc -> type.
val-  : neg  -> type.
exp   : conc -> type.

%% positive values

mt+    : val+ one.
%% no rule for zero
pair+  : val+ A -> val+ B -> val+ (times A B).
inl    : val+ A -> val+ (plus A B).
inr    : val+ B -> val+ (plus A B).
delay- : exp A- -> val+ (down A-).

%% positive continuations (shallow matching)

%% positive conts
let1   : exp C -> cont+ one C.
%% no rule for zero
split  : (val+ A -> val+ B -> exp C) -> cont+ (times A B) C.
case   : (val+ A -> exp C) -> (val+ B -> exp C) -> cont+ (plus A B) C.
%% wait to peel off the shift until you're ready to focus.
force- : cont- A- C -> cont+ (down A-) C.
comp+- : cont+ A+ B- -> cont- B- C- -> cont+ A+ C-.

%% negative continuations

app    : val+ A+ -> cont- B C -> cont- (arr A+ B) C.
fst    : cont- A C -> cont- (with A B) C.
snd    : cont- B C -> cont- (with A B) C.
force+ : (val+ A+ -> exp C) -> cont- (up A+) C.
%% internalize identity so we don't need to eta-expand
id-    : cont- A A.
comp-- : cont- A- B- -> cont- B- C- -> cont- A- C-.

%% negative values (shallow)

lam    : (val+ A+ -> exp B-) -> val- (arr A+ B-).
pair-  : exp A- -> exp B- -> val- (with A- B-).
mt-    : val- top.
delay+ : val+ A+ -> val- (up A+).  %% wait to peel off shift until focus

%% neutral sequents

return : val- A -> exp A.

  %% canonical when the value is a variable;
  %% non-canonical otherwise
cut+   : val+ A+ -> cont+ A+ C -> exp C.
  %% non-canonical
cut-   : val- A- -> cont- A- C -> exp C.

compe- : exp A- -> cont- A- C- -> exp C-.

Stack Machine

%% the sum type that you'd define for progress is
%% (val C) + (exp C)
%% so let's just abuse exp C for this sum and write the
%% operational semantics as the progress proof.
%% the only downside is that you can't tell whether an expression
%% took a real step towards a value or already was a value.

step : exp C -> exp C -> type.
%mode step +S1 -S2.

step/return : step (return V) (return V).

step/let1   : step (cut+ mt+ (let1 E)) E.
step/split  : step (cut+ (pair+ V1 V2) (split E)) (E V1 V2).
step/case1  : step (cut+ (inl V1) (case E1 E2)) (E1 V1).
step/case2  : step (cut+ (inr V2) (case E1 E2)) (E2 V2).
step/force- : step (cut+ (delay- E) (force- K)) (compe- E K).
step/app    : step (cut- (lam E) (app V K)) (compe- (E V) K).
step/fst    : step (cut- (pair- E1 E2) (fst K)) (compe- E1 K).
step/snd    : step (cut- (pair- E1 E2) (snd K)) (compe- E2 K).
step/force+ : step (cut- (delay+ V) (force+ E)) (E V).
step/id     : step (cut- V id-) (return V).

%% find the redex
step/compe-/compe- : step (compe- (compe- E K1) K2) (compe- E (comp-- K1 K2)).
step/compe-/cut+   : step (compe- (cut+ V K1) K2) (cut+ V (comp+- K1 K2)).
step/compe-/cut-   : step (compe- (cut- V K1) K2) (cut- V (comp-- K1 K2)).
step/compe-/cut-   : step (compe- (return V) K2) (cut- V K2).

%% find the top of the stack
step/comp+-/let1   : step (cut+ V (comp+- (let1 E) K)) 
		          (cut+ V (let1 (compe- E K))).
step/comp+-/split  : step (cut+ V (comp+- (split E) K)) 
		          (cut+ V (split [x] [y] (compe- (E x y) K))).
step/comp+-/case   : step (cut+ V (comp+- (case E1 E2) K)) 
		          (cut+ V (case ([x] (compe- (E1 x) K)) ([x] (compe- (E2 x) K)))).
step/comp+-/force  : step (cut+ V (comp+- (force- K1) K2)) 
		          (cut+ V (force- (comp-- K1 K2))).
step/comp+-/force  : step (cut+ V (comp+- (comp+- K1 K2) K3)) 
		          (cut+ V (comp+- K1 (comp-- K2 K3))).
step/comp--/app    : step (cut- V1 (comp-- (app V2 K) K2))
		          (cut- V1 (app V2 (comp-- K K2))).
step/comp--/fst    : step (cut- V1 (comp-- (fst K) K2))
		          (cut- V1 (fst (comp-- K K2))).
step/comp--/snd    : step (cut- V1 (comp-- (snd K) K2))
		          (cut- V1 (snd (comp-- K K2))).
step/comp--/snd    : step (cut- V1 (comp-- (force+ E) K2))
		          (cut- V1 (force+ ([x] compe- (E x) K2))).
step/comp--/id     : step (cut- V1 (comp-- id- K))
		          (cut- V1 K).
step/comp--/id     : step (cut- V1 (comp-- (comp-- K1 K2) K))
		          (cut- V1 (comp-- K1 (comp-- K2 K))).

%worlds () (step _ _).
%total D (step D _).