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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
open Util
open Term
open Names
open Closure
open Esubst
(**** Call by value reduction ****)
(* The type of terms with closure. The meaning of the constructors and
* the invariants of this datatype are the following:
* VAL(k,c) represents the constr c with a delayed shift of k. c must be
* in normal form and neutral (i.e. not a lambda, a construct or a
* (co)fix, because they may produce redexes by applying them,
* or putting them in a case)
* STACK(k,v,stk) represents an irreductible value [v] in the stack [stk].
* [k] is a delayed shift to be applied to both the value and
* the stack.
* CBN(t,S) is the term [S]t. It is used to delay evaluation. For
* instance products are evaluated only when actually needed
* (CBN strategy).
* LAM(n,a,b,S) is the term [S]([x:a]b) where [a] is a list of bindings and
* [n] is the length of [a]. the environment [S] is propagated
* only when the abstraction is applied, and then we use the rule
* ([S]([x:a]b) c) --> [S.c]b
* This corresponds to the usual strategy of weak reduction
* FIXP(op,bd,S,args) is the fixpoint (Fix or Cofix) of bodies bd under
* the bindings S, and then applied to args. Here again,
* weak reduction.
* CONSTR(c,args) is the constructor [c] applied to [args].
*
*)
type cbv_value =
| VAL of int * constr
| STACK of int * cbv_value * cbv_stack
| CBN of constr * cbv_value subs
| LAM of int * (name * constr) list * constr * cbv_value subs
| FIXP of fixpoint * cbv_value subs * cbv_value array
| COFIXP of cofixpoint * cbv_value subs * cbv_value array
| CONSTR of constructor * cbv_value array
(* type of terms with a hole. This hole can appear only under App or Case.
* TOP means the term is considered without context
* APP(v,stk) means the term is applied to v, and then the context stk
* (v.0 is the first argument).
* this corresponds to the application stack of the KAM.
* The members of l are values: we evaluate arguments before
calling the function.
* CASE(t,br,pat,S,stk) means the term is in a case (which is himself in stk
* t is the type of the case and br are the branches, all of them under
* the subs S, pat is information on the patterns of the Case
* (Weak reduction: we propagate the sub only when the selected branch
* is determined)
*
* Important remark: the APPs should be collapsed:
* (APP (l,(APP ...))) forbidden
*)
and cbv_stack =
| TOP
| APP of cbv_value array * cbv_stack
| CASE of constr * constr array * case_info * cbv_value subs * cbv_stack
(* les vars pourraient etre des constr,
cela permet de retarder les lift: utile ?? *)
(* relocation of a value; used when a value stored in a context is expanded
* in a larger context. e.g. [%k (S.t)](k+1) --> [^k]t (t is shifted of k)
*)
let rec shift_value n = function
| VAL (k,t) -> VAL (k+n,t)
| STACK(k,v,stk) -> STACK(k+n,v,stk)
| CBN (t,s) -> CBN(t,subs_shft(n,s))
| LAM (nlams,ctxt,b,s) -> LAM (nlams,ctxt,b,subs_shft (n,s))
| FIXP (fix,s,args) ->
FIXP (fix,subs_shft (n,s), Array.map (shift_value n) args)
| COFIXP (cofix,s,args) ->
COFIXP (cofix,subs_shft (n,s), Array.map (shift_value n) args)
| CONSTR (c,args) ->
CONSTR (c, Array.map (shift_value n) args)
let shift_value n v =
if n = 0 then v else shift_value n v
(* Contracts a fixpoint: given a fixpoint and a bindings,
* returns the corresponding fixpoint body, and the bindings in which
* it should be evaluated: its first variables are the fixpoint bodies
* (S, (fix Fi {F0 := T0 .. Fn-1 := Tn-1}))
* -> (S. [S]F0 . [S]F1 ... . [S]Fn-1, Ti)
*)
let contract_fixp env ((reci,i),(_,_,bds as bodies)) =
let make_body j = FIXP(((reci,j),bodies), env, [||]) in
let n = Array.length bds in
subs_cons(Array.init n make_body, env), bds.(i)
let contract_cofixp env (i,(_,_,bds as bodies)) =
let make_body j = COFIXP((j,bodies), env, [||]) in
let n = Array.length bds in
subs_cons(Array.init n make_body, env), bds.(i)
let make_constr_ref n = function
| RelKey p -> mkRel (n+p)
| VarKey id -> mkVar id
| ConstKey cst -> mkConst cst
(* Adds an application list. Collapse APPs! *)
let stack_app appl stack =
if Array.length appl = 0 then stack else
match stack with
| APP(args,stk) -> APP(Array.append appl args,stk)
| _ -> APP(appl, stack)
let rec stack_concat stk1 stk2 =
match stk1 with
TOP -> stk2
| APP(v,stk1') -> APP(v,stack_concat stk1' stk2)
| CASE(c,b,i,s,stk1') -> CASE(c,b,i,s,stack_concat stk1' stk2)
(* merge stacks when there is no shifts in between *)
let mkSTACK = function
v, TOP -> v
| STACK(0,v0,stk0), stk -> STACK(0,v0,stack_concat stk0 stk)
| v,stk -> STACK(0,v,stk)
(* Change: zeta reduction cannot be avoided in CBV *)
open RedFlags
let red_set_ref flags = function
| RelKey _ -> red_set flags fDELTA
| VarKey id -> red_set flags (fVAR id)
| ConstKey sp -> red_set flags (fCONST sp)
(* Transfer application lists from a value to the stack
* useful because fixpoints may be totally applied in several times.
* On the other hand, irreductible atoms absorb the full stack.
*)
let strip_appl head stack =
match head with
| FIXP (fix,env,app) -> (FIXP(fix,env,[||]), stack_app app stack)
| COFIXP (cofix,env,app) -> (COFIXP(cofix,env,[||]), stack_app app stack)
| CONSTR (c,app) -> (CONSTR(c,[||]), stack_app app stack)
| _ -> (head, stack)
(* Tests if fixpoint reduction is possible. *)
let fixp_reducible flgs ((reci,i),_) stk =
if red_set flgs fIOTA then
match stk with
| APP(appl,_) ->
Array.length appl > reci.(i) &&
(match appl.(reci.(i)) with
CONSTR _ -> true
| _ -> false)
| _ -> false
else
false
let cofixp_reducible flgs _ stk =
if red_set flgs fIOTA then
match stk with
| (CASE _ | APP(_,CASE _)) -> true
| _ -> false
else
false
(* The main recursive functions
*
* Go under applications and cases (pushed in the stack), expand head
* constants or substitued de Bruijn, and try to make appear a
* constructor, a lambda or a fixp in the head. If not, it is a value
* and is completely computed here. The head redexes are NOT reduced:
* the function returns the pair of a cbv_value and its stack. *
* Invariant: if the result of norm_head is CONSTR or (CO)FIXP, it last
* argument is []. Because we must put all the applied terms in the
* stack. *)
let rec norm_head info env t stack =
(* no reduction under binders *)
match kind_of_term t with
(* stack grows (remove casts) *)
| App (head,args) -> (* Applied terms are normalized immediately;
they could be computed when getting out of the stack *)
let nargs = Array.map (cbv_stack_term info TOP env) args in
norm_head info env head (stack_app nargs stack)
| Case (ci,p,c,v) -> norm_head info env c (CASE(p,v,ci,env,stack))
| Cast (ct,_,_) -> norm_head info env ct stack
(* constants, axioms
* the first pattern is CRUCIAL, n=0 happens very often:
* when reducing closed terms, n is always 0 *)
| Rel i ->
(match expand_rel i env with
| Inl (0,v) -> strip_appl v stack
| Inl (n,v) -> strip_appl (shift_value n v) stack
| Inr (n,None) -> (VAL(0, mkRel n), stack)
| Inr (n,Some p) -> norm_head_ref (n-p) info env stack (RelKey p))
| Var id -> norm_head_ref 0 info env stack (VarKey id)
| Const sp -> norm_head_ref 0 info env stack (ConstKey sp)
| LetIn (_, b, _, c) ->
(* zeta means letin are contracted; delta without zeta means we *)
(* allow bindings but leave let's in place *)
if red_set (info_flags info) fZETA then
(* New rule: for Cbv, Delta does not apply to locally bound variables
or red_set (info_flags info) fDELTA
*)
let env' = subs_cons ([|cbv_stack_term info TOP env b|],env) in
norm_head info env' c stack
else
(CBN(t,env), stack) (* Considérer une coupure commutative ? *)
| Evar ev ->
(match evar_value info ev with
Some c -> norm_head info env c stack
| None -> (VAL(0, t), stack))
(* non-neutral cases *)
| Lambda _ ->
let ctxt,b = decompose_lam t in
(LAM(List.length ctxt, List.rev ctxt,b,env), stack)
| Fix fix -> (FIXP(fix,env,[||]), stack)
| CoFix cofix -> (COFIXP(cofix,env,[||]), stack)
| Construct c -> (CONSTR(c, [||]), stack)
(* neutral cases *)
| (Sort _ | Meta _ | Ind _) -> (VAL(0, t), stack)
| Prod _ -> (CBN(t,env), stack)
and norm_head_ref k info env stack normt =
if red_set_ref (info_flags info) normt then
match ref_value_cache info normt with
| Some body -> strip_appl (shift_value k body) stack
| None -> (VAL(0,make_constr_ref k normt),stack)
else (VAL(0,make_constr_ref k normt),stack)
(* cbv_stack_term performs weak reduction on constr t under the subs
* env, with context stack, i.e. ([env]t stack). First computes weak
* head normal form of t and checks if a redex appears with the stack.
* If so, recursive call to reach the real head normal form. If not,
* we build a value.
*)
and cbv_stack_term info stack env t =
match norm_head info env t stack with
(* a lambda meets an application -> BETA *)
| (LAM (nlams,ctxt,b,env), APP (args, stk))
when red_set (info_flags info) fBETA ->
let nargs = Array.length args in
if nargs == nlams then
cbv_stack_term info stk (subs_cons(args,env)) b
else if nlams < nargs then
let env' = subs_cons(Array.sub args 0 nlams, env) in
let eargs = Array.sub args nlams (nargs-nlams) in
cbv_stack_term info (APP(eargs,stk)) env' b
else
let ctxt' = List.skipn nargs ctxt in
LAM(nlams-nargs,ctxt', b, subs_cons(args,env))
(* a Fix applied enough -> IOTA *)
| (FIXP(fix,env,[||]), stk)
when fixp_reducible (info_flags info) fix stk ->
let (envf,redfix) = contract_fixp env fix in
cbv_stack_term info stk envf redfix
(* constructor guard satisfied or Cofix in a Case -> IOTA *)
| (COFIXP(cofix,env,[||]), stk)
when cofixp_reducible (info_flags info) cofix stk->
let (envf,redfix) = contract_cofixp env cofix in
cbv_stack_term info stk envf redfix
(* constructor in a Case -> IOTA *)
| (CONSTR((sp,n),[||]), APP(args,CASE(_,br,ci,env,stk)))
when red_set (info_flags info) fIOTA ->
let cargs =
Array.sub args ci.ci_npar (Array.length args - ci.ci_npar) in
cbv_stack_term info (stack_app cargs stk) env br.(n-1)
(* constructor of arity 0 in a Case -> IOTA *)
| (CONSTR((_,n),[||]), CASE(_,br,_,env,stk))
when red_set (info_flags info) fIOTA ->
cbv_stack_term info stk env br.(n-1)
(* may be reduced later by application *)
| (FIXP(fix,env,[||]), APP(appl,TOP)) -> FIXP(fix,env,appl)
| (COFIXP(cofix,env,[||]), APP(appl,TOP)) -> COFIXP(cofix,env,appl)
| (CONSTR(c,[||]), APP(appl,TOP)) -> CONSTR(c,appl)
(* definitely a value *)
| (head,stk) -> mkSTACK(head, stk)
(* When we are sure t will never produce a redex with its stack, we
* normalize (even under binders) the applied terms and we build the
* final term
*)
let rec apply_stack info t = function
| TOP -> t
| APP (args,st) ->
apply_stack info (mkApp(t,Array.map (cbv_norm_value info) args)) st
| CASE (ty,br,ci,env,st) ->
apply_stack info
(mkCase (ci, cbv_norm_term info env ty, t,
Array.map (cbv_norm_term info env) br))
st
(* performs the reduction on a constr, and returns a constr *)
and cbv_norm_term info env t =
(* reduction under binders *)
cbv_norm_value info (cbv_stack_term info TOP env t)
(* reduction of a cbv_value to a constr *)
and cbv_norm_value info = function (* reduction under binders *)
| VAL (n,t) -> lift n t
| STACK (0,v,stk) ->
apply_stack info (cbv_norm_value info v) stk
| STACK (n,v,stk) ->
lift n (apply_stack info (cbv_norm_value info v) stk)
| CBN(t,env) ->
map_constr_with_binders subs_lift (cbv_norm_term info) env t
| LAM (n,ctxt,b,env) ->
let nctxt =
List.map_i (fun i (x,ty) ->
(x,cbv_norm_term info (subs_liftn i env) ty)) 0 ctxt in
compose_lam (List.rev nctxt) (cbv_norm_term info (subs_liftn n env) b)
| FIXP ((lij,(names,lty,bds)),env,args) ->
mkApp
(mkFix (lij,
(names,
Array.map (cbv_norm_term info env) lty,
Array.map (cbv_norm_term info
(subs_liftn (Array.length lty) env)) bds)),
Array.map (cbv_norm_value info) args)
| COFIXP ((j,(names,lty,bds)),env,args) ->
mkApp
(mkCoFix (j,
(names,Array.map (cbv_norm_term info env) lty,
Array.map (cbv_norm_term info
(subs_liftn (Array.length lty) env)) bds)),
Array.map (cbv_norm_value info) args)
| CONSTR (c,args) ->
mkApp(mkConstruct c, Array.map (cbv_norm_value info) args)
(* with profiling *)
let cbv_norm infos constr =
with_stats (lazy (cbv_norm_term infos (subs_id 0) constr))
type cbv_infos = cbv_value infos
(* constant bodies are normalized at the first expansion *)
let create_cbv_infos flgs env sigma =
create
(fun old_info c -> cbv_stack_term old_info TOP (subs_id 0) c)
flgs
env
(Reductionops.safe_evar_value sigma)
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