%%% Mapping evaluations to CLS computations.
%%% This expresses the completeness proof for the CLS machine.
%%% Author: Frank Pfenning, based on [Hannan & Pfenning 92]

%{
Lemma: For every
  C :: st Ks P S =>* st Ks' P' S'
and
  C' :: st Ks P S =>* st Ks' P' S'
there exists a computation
  C'' :: st Ks P S =>* st Ks'' P'' S''.

Proof: By induction on the structure of C.
}%

append : st Ks P S =>* st Ks' P' S'
       -> st Ks' P' S' =>* st Ks'' P'' S''
       -> st Ks P S =>* st Ks'' P'' S''
       -> type.

apd_id   : append (id) C' C'.
apd_step : append (R ~ C) C' (R ~ C'')
	    <- append C C' C''.

%{
Subcomputation Lemma:  For every
  D :: feval K F W
environment stack Ks, program P, and stack S there exists a computation
  C :: st (Ks ;; K) (ev F & P) S =>* st Ks P (S ; W)

Proof: By induction on the structure of D
}%

subcomp : feval K F W
	   -> st (Ks ;; K) (ev F & P) S =>* st Ks P (S ; W)
	   -> type.

% Variables

sc_1  : subcomp (fev_1) (c_1 ~ id).
sc_^  : subcomp (fev_^ D1) (c_^ ~ C1)
	 <- subcomp D1 C1.
sc_1+ : subcomp (fev_1+ D1) (c_1+ ~ C1)
	 <- subcomp D1 C1.
sc_^+ : subcomp (fev_^+ D1) (c_^+ ~ C1)
	 <- subcomp D1 C1.

% Natural Numbers

sc_z     : subcomp (fev_z) (c_z ~ id).
sc_s     : subcomp (fev_s D1) C
	    <- subcomp D1 C1
	    <- append (c_s ~ C1) (c_add1 ~ id) C.
sc_branch_z : subcomp (fev_case_z D2 D1) C
	       <- subcomp D1 C1
	       <- subcomp D2 C2
	       <- append (c_case ~ C1) (c_branch_z ~ C2) C.
sc_branch_s : subcomp (fev_case_s D3 D1) C
	       <- subcomp D1 C1
	       <- subcomp D3 C3
	       <- append (c_case ~ C1) (c_branch_s ~ C3) C.

% Pairs

sc_pair : subcomp (fev_pair D2 D1) C
	   <- subcomp D1 C1
	   <- subcomp D2 C2
	   <- append (c_pair ~ C1) C2 C'
	   <- append C' (c_mkpair ~ id) C.

sc_fst : subcomp (fev_fst D1) C
	  <- subcomp D1 C1
	  <- append (c_fst ~ C1) (c_getfst ~ id) C.

sc_snd : subcomp (fev_snd D1) C
	  <- subcomp D1 C1
	  <- append (c_snd ~ C1) (c_getsnd ~ id) C.

% Functions

sc_lam : subcomp (fev_lam) (c_lam ~ id).

sc_app : subcomp (fev_app D3 D2 D1) C
	   <- subcomp D1 C1
	   <- subcomp D2 C2
	   <- subcomp D3 C3
	   <- append (c_app ~ C1) C2 C'
	   <- append C' (c_apply ~ C3) C.

% Definitions

sc_letv : subcomp (fev_letv D2 D1) C
	   <- subcomp D1 C1
	   <- subcomp D2 C2
	   <- append (c_letv ~ C1) (c_bind ~ C2) C.

sc_letn : subcomp (fev_letn D2) (c_letn ~ C2)
	   <- subcomp D2 C2.

% Recursion

sc_fix : subcomp (fev_fix D1) (c_fix ~ C1)
	  <- subcomp D1 C1.


%{
Completeness Theorem: For every evaluation 
 D :: feval K F W
there exists a complete computation
 E :: ceval K F W

Proof: Immediately from the subcomputation lemma.
}%

cev_complete : feval K F W -> ceval K F W -> type.

cevc : cev_complete D (run C) <- subcomp D C.