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|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* $Id$ *)
open Pp
open Util
open Names
open Term
open Termops
open Univ
open Evd
open Declarations
open Environ
open Closure
open Esubst
open Reduction
exception Elimconst
(**********************************************************************)
(* The type of (machine) stacks (= lambda-bar-calculus' contexts) *)
type 'a stack_member =
| Zapp of 'a list
| Zcase of case_info * 'a * 'a array
| Zfix of 'a * 'a stack
| Zshift of int
| Zupdate of 'a
and 'a stack = 'a stack_member list
let empty_stack = []
let append_stack_list l s =
match (l,s) with
| ([],s) -> s
| (l1, Zapp l :: s) -> Zapp (l1@l) :: s
| (l1, s) -> Zapp l1 :: s
let append_stack v s = append_stack_list (Array.to_list v) s
(* Collapse the shifts in the stack *)
let zshift n s =
match (n,s) with
(0,_) -> s
| (_,Zshift(k)::s) -> Zshift(n+k)::s
| _ -> Zshift(n)::s
let rec stack_args_size = function
| Zapp l::s -> List.length l + stack_args_size s
| Zshift(_)::s -> stack_args_size s
| Zupdate(_)::s -> stack_args_size s
| _ -> 0
(* When used as an argument stack (only Zapp can appear) *)
let rec decomp_stack = function
| Zapp[v]::s -> Some (v, s)
| Zapp(v::l)::s -> Some (v, (Zapp l :: s))
| Zapp [] :: s -> decomp_stack s
| _ -> None
let rec decomp_stackn = function
| Zapp [] :: s -> decomp_stackn s
| Zapp l :: s -> (Array.of_list l, s)
| _ -> assert false
let array_of_stack s =
let rec stackrec = function
| [] -> []
| Zapp args :: s -> args :: (stackrec s)
| _ -> assert false
in Array.of_list (List.concat (stackrec s))
let rec list_of_stack = function
| [] -> []
| Zapp args :: s -> args @ (list_of_stack s)
| _ -> assert false
let rec app_stack = function
| f, [] -> f
| f, (Zapp [] :: s) -> app_stack (f, s)
| f, (Zapp args :: s) ->
app_stack (applist (f, args), s)
| _ -> assert false
let rec stack_assign s p c = match s with
| Zapp args :: s ->
let q = List.length args in
if p >= q then
Zapp args :: stack_assign s (p-q) c
else
(match list_chop p args with
(bef, _::aft) -> Zapp (bef@c::aft) :: s
| _ -> assert false)
| _ -> s
let rec stack_tail p s =
if p = 0 then s else
match s with
| Zapp args :: s ->
let q = List.length args in
if p >= q then stack_tail (p-q) s
else Zapp (list_skipn p args) :: s
| _ -> failwith "stack_tail"
let rec stack_nth s p = match s with
| Zapp args :: s ->
let q = List.length args in
if p >= q then stack_nth s (p-q)
else List.nth args p
| _ -> raise Not_found
(**************************************************************)
(* The type of (machine) states (= lambda-bar-calculus' cuts) *)
type state = constr * constr stack
type contextual_reduction_function = env -> evar_map -> constr -> constr
type reduction_function = contextual_reduction_function
type local_reduction_function = constr -> constr
type contextual_stack_reduction_function =
env -> evar_map -> constr -> constr * constr list
type stack_reduction_function = contextual_stack_reduction_function
type local_stack_reduction_function = constr -> constr * constr list
type contextual_state_reduction_function =
env -> evar_map -> state -> state
type state_reduction_function = contextual_state_reduction_function
type local_state_reduction_function = state -> state
(*************************************)
(*** Reduction Functions Operators ***)
(*************************************)
let rec whd_state (x, stack as s) =
match kind_of_term x with
| App (f,cl) -> whd_state (f, append_stack cl stack)
| Cast (c,_,_) -> whd_state (c, stack)
| _ -> s
let appterm_of_stack (f,s) = (f,list_of_stack s)
let whd_stack x = appterm_of_stack (whd_state (x, empty_stack))
let whd_castapp_stack = whd_stack
let stack_reduction_of_reduction red_fun env sigma s =
let t = red_fun env sigma (app_stack s) in
whd_stack t
let strong whdfun env sigma t =
let rec strongrec env t =
map_constr_with_full_binders push_rel strongrec env (whdfun env sigma t) in
strongrec env t
let local_strong whdfun =
let rec strongrec t = map_constr strongrec (whdfun t) in
strongrec
let rec strong_prodspine redfun c =
let x = redfun c in
match kind_of_term x with
| Prod (na,a,b) -> mkProd (na,a,strong_prodspine redfun b)
| _ -> x
(*************************************)
(*** Reduction using bindingss ***)
(*************************************)
(* This signature is very similar to Closure.RedFlagsSig except there
is eta but no per-constant unfolding *)
module type RedFlagsSig = sig
type flags
type flag
val fbeta : flag
val fevar : flag
val fdelta : flag
val feta : flag
val fiota : flag
val fzeta : flag
val mkflags : flag list -> flags
val red_beta : flags -> bool
val red_delta : flags -> bool
val red_evar : flags -> bool
val red_eta : flags -> bool
val red_iota : flags -> bool
val red_zeta : flags -> bool
end
(* Naive Implementation
module RedFlags = (struct
type flag = BETA | DELTA | EVAR | IOTA | ZETA | ETA
type flags = flag list
let fbeta = BETA
let fdelta = DELTA
let fevar = EVAR
let fiota = IOTA
let fzeta = ZETA
let feta = ETA
let mkflags l = l
let red_beta = List.mem BETA
let red_delta = List.mem DELTA
let red_evar = List.mem EVAR
let red_eta = List.mem ETA
let red_iota = List.mem IOTA
let red_zeta = List.mem ZETA
end : RedFlagsSig)
*)
(* Compact Implementation *)
module RedFlags = (struct
type flag = int
type flags = int
let fbeta = 1
let fdelta = 2
let fevar = 4
let feta = 8
let fiota = 16
let fzeta = 32
let mkflags = List.fold_left (lor) 0
let red_beta f = f land fbeta <> 0
let red_delta f = f land fdelta <> 0
let red_evar f = f land fevar <> 0
let red_eta f = f land feta <> 0
let red_iota f = f land fiota <> 0
let red_zeta f = f land fzeta <> 0
end : RedFlagsSig)
open RedFlags
(* Local *)
let beta = mkflags [fbeta]
let eta = mkflags [feta]
let evar = mkflags [fevar]
let betaevar = mkflags [fevar; fbeta]
let betaiota = mkflags [fiota; fbeta]
let betaiotazeta = mkflags [fiota; fbeta;fzeta]
(* Contextual *)
let delta = mkflags [fdelta;fevar]
let betadelta = mkflags [fbeta;fdelta;fzeta;fevar]
let betadeltaeta = mkflags [fbeta;fdelta;fzeta;fevar;feta]
let betadeltaiota = mkflags [fbeta;fdelta;fzeta;fevar;fiota]
let betadeltaiota_nolet = mkflags [fbeta;fdelta;fevar;fiota]
let betadeltaiotaeta = mkflags [fbeta;fdelta;fzeta;fevar;fiota;feta]
let betaiotaevar = mkflags [fbeta;fiota;fevar]
let betaetalet = mkflags [fbeta;feta;fzeta]
let betalet = mkflags [fbeta;fzeta]
(* Beta Reduction tools *)
let rec stacklam recfun env t stack =
match (decomp_stack stack,kind_of_term t) with
| Some (h,stacktl), Lambda (_,_,c) -> stacklam recfun (h::env) c stacktl
| _ -> recfun (substl env t, stack)
let beta_applist (c,l) =
stacklam app_stack [] c (append_stack_list l empty_stack)
(* Iota reduction tools *)
type 'a miota_args = {
mP : constr; (* the result type *)
mconstr : constr; (* the constructor *)
mci : case_info; (* special info to re-build pattern *)
mcargs : 'a list; (* the constructor's arguments *)
mlf : 'a array } (* the branch code vector *)
let reducible_mind_case c = match kind_of_term c with
| Construct _ | CoFix _ -> true
| _ -> false
let contract_cofix (bodynum,(types,names,bodies as typedbodies)) =
let nbodies = Array.length bodies in
let make_Fi j = mkCoFix (nbodies-j-1,typedbodies) in
substl (list_tabulate make_Fi nbodies) bodies.(bodynum)
let reduce_mind_case mia =
match kind_of_term mia.mconstr with
| Construct (ind_sp,i) ->
(* let ncargs = (fst mia.mci).(i-1) in*)
let real_cargs = list_skipn mia.mci.ci_npar mia.mcargs in
applist (mia.mlf.(i-1),real_cargs)
| CoFix cofix ->
let cofix_def = contract_cofix cofix in
mkCase (mia.mci, mia.mP, applist(cofix_def,mia.mcargs), mia.mlf)
| _ -> assert false
(* contracts fix==FIX[nl;i](A1...Ak;[F1...Fk]{B1....Bk}) to produce
Bi[Fj --> FIX[nl;j](A1...Ak;[F1...Fk]{B1...Bk})] *)
let contract_fix ((recindices,bodynum),(types,names,bodies as typedbodies)) =
let nbodies = Array.length recindices in
let make_Fi j = mkFix ((recindices,nbodies-j-1),typedbodies) in
substl (list_tabulate make_Fi nbodies) bodies.(bodynum)
let fix_recarg ((recindices,bodynum),_) stack =
assert (0 <= bodynum & bodynum < Array.length recindices);
let recargnum = Array.get recindices bodynum in
try
Some (recargnum, stack_nth stack recargnum)
with Not_found ->
None
type fix_reduction_result = NotReducible | Reduced of state
let reduce_fix whdfun fix stack =
match fix_recarg fix stack with
| None -> NotReducible
| Some (recargnum,recarg) ->
let (recarg'hd,_ as recarg') = whdfun (recarg, empty_stack) in
let stack' = stack_assign stack recargnum (app_stack recarg') in
(match kind_of_term recarg'hd with
| Construct _ -> Reduced (contract_fix fix, stack')
| _ -> NotReducible)
(* Generic reduction function *)
(* Y avait un commentaire pour whd_betadeltaiota :
NB : Cette fonction alloue peu c'est l'appel
``let (c,cargs) = whfun (recarg, empty_stack)''
-------------------
qui coute cher *)
let rec whd_state_gen flags env sigma =
let rec whrec (x, stack as s) =
match kind_of_term x with
| Rel n when red_delta flags ->
(match lookup_rel n env with
| (_,Some body,_) -> whrec (lift n body, stack)
| _ -> s)
| Var id when red_delta flags ->
(match lookup_named id env with
| (_,Some body,_) -> whrec (body, stack)
| _ -> s)
| Evar ev when red_evar flags ->
(match existential_opt_value sigma ev with
| Some body -> whrec (body, stack)
| None -> s)
| Const const when red_delta flags ->
(match constant_opt_value env const with
| Some body -> whrec (body, stack)
| None -> s)
| LetIn (_,b,_,c) when red_zeta flags -> stacklam whrec [b] c stack
| Cast (c,_,_) -> whrec (c, stack)
| App (f,cl) -> whrec (f, append_stack cl stack)
| Lambda (na,t,c) ->
(match decomp_stack stack with
| Some (a,m) when red_beta flags -> stacklam whrec [a] c m
| None when red_eta flags ->
let env' = push_rel (na,None,t) env in
let whrec' = whd_state_gen flags env' sigma in
(match kind_of_term (app_stack (whrec' (c, empty_stack))) with
| App (f,cl) ->
let napp = Array.length cl in
if napp > 0 then
let x', l' = whrec' (array_last cl, empty_stack) in
match kind_of_term x', decomp_stack l' with
| Rel 1, None ->
let lc = Array.sub cl 0 (napp-1) in
let u = if napp=1 then f else appvect (f,lc) in
if noccurn 1 u then (pop u,empty_stack) else s
| _ -> s
else s
| _ -> s)
| _ -> s)
| Case (ci,p,d,lf) when red_iota flags ->
let (c,cargs) = whrec (d, empty_stack) in
if reducible_mind_case c then
whrec (reduce_mind_case
{mP=p; mconstr=c; mcargs=list_of_stack cargs;
mci=ci; mlf=lf}, stack)
else
(mkCase (ci, p, app_stack (c,cargs), lf), stack)
| Fix fix when red_iota flags ->
(match reduce_fix whrec fix stack with
| Reduced s' -> whrec s'
| NotReducible -> s)
| x -> s
in
whrec
let local_whd_state_gen flags =
let rec whrec (x, stack as s) =
match kind_of_term x with
| LetIn (_,b,_,c) when red_zeta flags -> stacklam whrec [b] c stack
| Cast (c,_,_) -> whrec (c, stack)
| App (f,cl) -> whrec (f, append_stack cl stack)
| Lambda (_,_,c) ->
(match decomp_stack stack with
| Some (a,m) when red_beta flags -> stacklam whrec [a] c m
| None when red_eta flags ->
(match kind_of_term (app_stack (whrec (c, empty_stack))) with
| App (f,cl) ->
let napp = Array.length cl in
if napp > 0 then
let x', l' = whrec (array_last cl, empty_stack) in
match kind_of_term x', decomp_stack l' with
| Rel 1, None ->
let lc = Array.sub cl 0 (napp-1) in
let u = if napp=1 then f else appvect (f,lc) in
if noccurn 1 u then (pop u,empty_stack) else s
| _ -> s
else s
| _ -> s)
| _ -> s)
| Case (ci,p,d,lf) when red_iota flags ->
let (c,cargs) = whrec (d, empty_stack) in
if reducible_mind_case c then
whrec (reduce_mind_case
{mP=p; mconstr=c; mcargs=list_of_stack cargs;
mci=ci; mlf=lf}, stack)
else
(mkCase (ci, p, app_stack (c,cargs), lf), stack)
| Fix fix when red_iota flags ->
(match reduce_fix whrec fix stack with
| Reduced s' -> whrec s'
| NotReducible -> s)
| x -> s
in
whrec
(* 1. Beta Reduction Functions *)
let whd_beta_state = local_whd_state_gen beta
let whd_beta_stack x = appterm_of_stack (whd_beta_state (x, empty_stack))
let whd_beta x = app_stack (whd_beta_state (x,empty_stack))
(* Nouveau ! *)
let whd_betaetalet_state = local_whd_state_gen betaetalet
let whd_betaetalet_stack x =
appterm_of_stack (whd_betaetalet_state (x, empty_stack))
let whd_betaetalet x = app_stack (whd_betaetalet_state (x,empty_stack))
let whd_betalet_state = local_whd_state_gen betalet
let whd_betalet_stack x = appterm_of_stack (whd_betalet_state (x, empty_stack))
let whd_betalet x = app_stack (whd_betalet_state (x,empty_stack))
(* 2. Delta Reduction Functions *)
let whd_delta_state e = whd_state_gen delta e
let whd_delta_stack env sigma x =
appterm_of_stack (whd_delta_state env sigma (x, empty_stack))
let whd_delta env sigma c =
app_stack (whd_delta_state env sigma (c, empty_stack))
let whd_betadelta_state e = whd_state_gen betadelta e
let whd_betadelta_stack env sigma x =
appterm_of_stack (whd_betadelta_state env sigma (x, empty_stack))
let whd_betadelta env sigma c =
app_stack (whd_betadelta_state env sigma (c, empty_stack))
let whd_betaevar_state e = whd_state_gen betaevar e
let whd_betaevar_stack env sigma c =
appterm_of_stack (whd_betaevar_state env sigma (c, empty_stack))
let whd_betaevar env sigma c =
app_stack (whd_betaevar_state env sigma (c, empty_stack))
let whd_betadeltaeta_state e = whd_state_gen betadeltaeta e
let whd_betadeltaeta_stack env sigma x =
appterm_of_stack (whd_betadeltaeta_state env sigma (x, empty_stack))
let whd_betadeltaeta env sigma x =
app_stack (whd_betadeltaeta_state env sigma (x, empty_stack))
(* 3. Iota reduction Functions *)
let whd_betaiota_state = local_whd_state_gen betaiota
let whd_betaiota_stack x =
appterm_of_stack (whd_betaiota_state (x, empty_stack))
let whd_betaiota x =
app_stack (whd_betaiota_state (x, empty_stack))
let whd_betaiotazeta_state = local_whd_state_gen betaiotazeta
let whd_betaiotazeta_stack x =
appterm_of_stack (whd_betaiotazeta_state (x, empty_stack))
let whd_betaiotazeta x =
app_stack (whd_betaiotazeta_state (x, empty_stack))
let whd_betaiotaevar_state e = whd_state_gen betaiotaevar e
let whd_betaiotaevar_stack env sigma x =
appterm_of_stack (whd_betaiotaevar_state env sigma (x, empty_stack))
let whd_betaiotaevar env sigma x =
app_stack (whd_betaiotaevar_state env sigma (x, empty_stack))
let whd_betadeltaiota_state e = whd_state_gen betadeltaiota e
let whd_betadeltaiota_stack env sigma x =
appterm_of_stack (whd_betadeltaiota_state env sigma (x, empty_stack))
let whd_betadeltaiota env sigma x =
app_stack (whd_betadeltaiota_state env sigma (x, empty_stack))
let whd_betadeltaiotaeta_state e = whd_state_gen betadeltaiotaeta e
let whd_betadeltaiotaeta_stack env sigma x =
appterm_of_stack (whd_betadeltaiotaeta_state env sigma (x, empty_stack))
let whd_betadeltaiotaeta env sigma x =
app_stack (whd_betadeltaiotaeta_state env sigma (x, empty_stack))
let whd_betadeltaiota_nolet_state e = whd_state_gen betadeltaiota_nolet e
let whd_betadeltaiota_nolet_stack env sigma x =
appterm_of_stack (whd_betadeltaiota_nolet_state env sigma (x, empty_stack))
let whd_betadeltaiota_nolet env sigma x =
app_stack (whd_betadeltaiota_nolet_state env sigma (x, empty_stack))
(* 3. Eta reduction Functions *)
let whd_eta c = app_stack (local_whd_state_gen eta (c,empty_stack))
(****************************************************************************)
(* Reduction Functions *)
(****************************************************************************)
(* Replacing defined evars for error messages *)
let rec whd_evar sigma c =
match kind_of_term c with
| Evar (ev,args)
when Evd.mem sigma ev & Evd.is_defined sigma ev ->
whd_evar sigma (Evd.existential_value sigma (ev,args))
| Sort s when is_sort_variable sigma s -> whd_sort_variable sigma c
| _ -> collapse_appl c
let nf_evar sigma =
local_strong (whd_evar sigma)
(* lazy reduction functions. The infos must be created for each term *)
let clos_norm_flags flgs env sigma t =
norm_val (create_clos_infos flgs env) (inject (nf_evar sigma t))
let nf_beta = clos_norm_flags Closure.beta empty_env Evd.empty
let nf_betaiota = clos_norm_flags Closure.betaiota empty_env Evd.empty
let nf_betadeltaiota env sigma =
clos_norm_flags Closure.betadeltaiota env sigma
(* Attention reduire un beta-redexe avec un argument qui n'est pas
une variable, peut changer enormement le temps de conversion lors
du type checking :
(fun x => x + x) M
*)
let rec whd_betaiotaevar_preserving_vm_cast env sigma t =
let rec stacklam_var subst t stack =
match (decomp_stack stack,kind_of_term t) with
| Some (h,stacktl), Lambda (_,_,c) ->
begin match kind_of_term h with
| Rel i when not (evaluable_rel i env) ->
stacklam_var (h::subst) c stacktl
| Var id when not (evaluable_named id env)->
stacklam_var (h::subst) c stacktl
| _ -> whrec (substl subst t, stack)
end
| _ -> whrec (substl subst t, stack)
and whrec (x, stack as s) =
match kind_of_term x with
| Evar ev ->
(match existential_opt_value sigma ev with
| Some body -> whrec (body, stack)
| None -> s)
| Cast (c,VMcast,t) ->
let c = app_stack (whrec (c,empty_stack)) in
let t = app_stack (whrec (t,empty_stack)) in
(mkCast(c,VMcast,t),stack)
| Cast (c,DEFAULTcast,_) ->
whrec (c, stack)
| App (f,cl) -> whrec (f, append_stack cl stack)
| Lambda (na,t,c) ->
(match decomp_stack stack with
| Some (a,m) -> stacklam_var [a] c m
| _ -> s)
| Case (ci,p,d,lf) ->
let (c,cargs) = whrec (d, empty_stack) in
if reducible_mind_case c then
whrec (reduce_mind_case
{mP=p; mconstr=c; mcargs=list_of_stack cargs;
mci=ci; mlf=lf}, stack)
else
(mkCase (ci, p, app_stack (c,cargs), lf), stack)
| x -> s
in
app_stack (whrec (t,empty_stack))
let nf_betaiotaevar_preserving_vm_cast =
strong whd_betaiotaevar_preserving_vm_cast
(* lazy weak head reduction functions *)
let whd_flags flgs env sigma t =
whd_val (create_clos_infos flgs env) (inject (nf_evar sigma t))
(********************************************************************)
(* Conversion *)
(********************************************************************)
(*
let fkey = Profile.declare_profile "fhnf";;
let fhnf info v = Profile.profile2 fkey fhnf info v;;
let fakey = Profile.declare_profile "fhnf_apply";;
let fhnf_apply info k h a = Profile.profile4 fakey fhnf_apply info k h a;;
*)
(* Conversion utility functions *)
type conversion_test = constraints -> constraints
let pb_is_equal pb = pb = CONV
let pb_equal = function
| CUMUL -> CONV
| CONV -> CONV
let sort_cmp pb s0 s1 cuniv =
match (s0,s1) with
| (Prop c1, Prop c2) -> if c1 = c2 then cuniv else raise NotConvertible
| (Prop c1, Type u) ->
(match pb with
CUMUL -> cuniv
| _ -> raise NotConvertible)
| (Type u1, Type u2) ->
(match pb with
| CONV -> enforce_eq u1 u2 cuniv
| CUMUL -> enforce_geq u2 u1 cuniv)
| (_, _) -> raise NotConvertible
let base_sort_cmp pb s0 s1 =
match (s0,s1) with
| (Prop c1, Prop c2) -> c1 = c2
| (Prop c1, Type u) -> pb = CUMUL
| (Type u1, Type u2) -> true
| (_, _) -> false
let test_conversion f env sigma x y =
try let _ = f env (nf_evar sigma x) (nf_evar sigma y) in true
with NotConvertible -> false
let is_conv env sigma = test_conversion Reduction.conv env sigma
let is_conv_leq env sigma = test_conversion Reduction.conv_leq env sigma
let is_fconv = function | CONV -> is_conv | CUMUL -> is_conv_leq
(********************************************************************)
(* Special-Purpose Reduction *)
(********************************************************************)
let whd_meta metamap c = match kind_of_term c with
| Meta p -> (try List.assoc p metamap with Not_found -> c)
| _ -> c
(* Try to replace all metas. Does not replace metas in the metas' values
* Differs from (strong whd_meta). *)
let plain_instance s c =
let rec irec u = match kind_of_term u with
| Meta p -> (try List.assoc p s with Not_found -> u)
| App (f,l) when isCast f ->
let (f,_,t) = destCast f in
let l' = Array.map irec l in
(match kind_of_term f with
| Meta p ->
(* Don't flatten application nodes: this is used to extract a
proof-term from a proof-tree and we want to keep the structure
of the proof-tree *)
(try let g = List.assoc p s in
match kind_of_term g with
| App _ ->
let h = id_of_string "H" in
mkLetIn (Name h,g,t,mkApp(mkRel 1,Array.map (lift 1) l'))
| _ -> mkApp (g,l')
with Not_found -> mkApp (f,l'))
| _ -> mkApp (irec f,l'))
| Cast (m,_,_) when isMeta m ->
(try List.assoc (destMeta m) s with Not_found -> u)
| _ -> map_constr irec u
in
if s = [] then c else irec c
(* [instance] is used for [res_pf]; the call to [local_strong whd_betaiota]
has (unfortunately) different subtle side effects:
- ** Order of subgoals **
If the lemma is a case analysis with parameters, it will move the
parameters as first subgoals (e.g. "case H" applied on
"H:D->A/\B|-C" will present the subgoal |-D first while w/o
betaiota the subgoal |-D would have come last).
- ** Betaiota-contraction in statement **
If the lemma has a parameter which is a function and this
function is applied in the lemma, then the _strong_ betaiota will
contract the application of the function to its argument (e.g.
"apply (H (fun x => x))" in "H:forall f, f 0 = 0 |- 0=0" will
result in applying the lemma 0=0 in which "(fun x => x) 0" has
been contracted). A goal to rewrite may then fail or succeed
differently.
- ** Naming of hypotheses **
If a lemma is a function of the form "fun H:(forall a:A, P a)
=> .. F H .." where the expected type of H is "forall b:A, P b",
then, without reduction, the application of the lemma will
generate a subgoal "forall a:A, P a" (and intro will use name
"a"), while with reduction, it will generate a subgoal "forall
b:A, P b" (and intro will use name "b").
- ** First-order pattern-matching **
If a lemma has the type "(fun x => p) t" then rewriting t may fail
if the type of the lemma is first beta-reduced (this typically happens
when rewriting a single variable and the type of the lemma is obtained
by meta_instance (with empty map) which itself call instance with this
empty map.
*)
let instance s c =
(* if s = [] then c else *) local_strong whd_betaiota (plain_instance s c)
(* pseudo-reduction rule:
* [hnf_prod_app env s (Prod(_,B)) N --> B[N]
* with an HNF on the first argument to produce a product.
* if this does not work, then we use the string S as part of our
* error message. *)
let hnf_prod_app env sigma t n =
match kind_of_term (whd_betadeltaiota env sigma t) with
| Prod (_,_,b) -> subst1 n b
| _ -> anomaly "hnf_prod_app: Need a product"
let hnf_prod_appvect env sigma t nl =
Array.fold_left (hnf_prod_app env sigma) t nl
let hnf_prod_applist env sigma t nl =
List.fold_left (hnf_prod_app env sigma) t nl
let hnf_lam_app env sigma t n =
match kind_of_term (whd_betadeltaiota env sigma t) with
| Lambda (_,_,b) -> subst1 n b
| _ -> anomaly "hnf_lam_app: Need an abstraction"
let hnf_lam_appvect env sigma t nl =
Array.fold_left (hnf_lam_app env sigma) t nl
let hnf_lam_applist env sigma t nl =
List.fold_left (hnf_lam_app env sigma) t nl
let splay_prod env sigma =
let rec decrec env m c =
let t = whd_betadeltaiota env sigma c in
match kind_of_term t with
| Prod (n,a,c0) ->
decrec (push_rel (n,None,a) env)
((n,a)::m) c0
| _ -> m,t
in
decrec env []
let splay_lambda env sigma =
let rec decrec env m c =
let t = whd_betadeltaiota env sigma c in
match kind_of_term t with
| Lambda (n,a,c0) ->
decrec (push_rel (n,None,a) env)
((n,a)::m) c0
| _ -> m,t
in
decrec env []
let splay_prod_assum env sigma =
let rec prodec_rec env l c =
let t = whd_betadeltaiota_nolet env sigma c in
match kind_of_term t with
| Prod (x,t,c) ->
prodec_rec (push_rel (x,None,t) env)
(Sign.add_rel_decl (x, None, t) l) c
| LetIn (x,b,t,c) ->
prodec_rec (push_rel (x, Some b, t) env)
(Sign.add_rel_decl (x, Some b, t) l) c
| Cast (c,_,_) -> prodec_rec env l c
| _ -> l,t
in
prodec_rec env Sign.empty_rel_context
let splay_arity env sigma c =
let l, c = splay_prod env sigma c in
match kind_of_term c with
| Sort s -> l,s
| _ -> error "not an arity"
let sort_of_arity env c = snd (splay_arity env Evd.empty c)
let decomp_n_prod env sigma n =
let rec decrec env m ln c = if m = 0 then (ln,c) else
match kind_of_term (whd_betadeltaiota env sigma c) with
| Prod (n,a,c0) ->
decrec (push_rel (n,None,a) env)
(m-1) (Sign.add_rel_decl (n,None,a) ln) c0
| _ -> error "decomp_n_prod: Not enough products"
in
decrec env n Sign.empty_rel_context
exception NotASort
let decomp_sort env sigma t =
match kind_of_term (whd_betadeltaiota env sigma t) with
| Sort s -> s
| _ -> raise NotASort
let is_sort env sigma arity =
try let _ = decomp_sort env sigma arity in true
with NotASort -> false
(* reduction to head-normal-form allowing delta/zeta only in argument
of case/fix (heuristic used by evar_conv) *)
let rec apprec env sigma s =
let (t, stack as s) = whd_betaiota_state s in
match kind_of_term t with
| Case (ci,p,d,lf) ->
let (cr,crargs) = whd_betadeltaiota_stack env sigma d in
let rslt = mkCase (ci, p, applist (cr,crargs), lf) in
if reducible_mind_case cr then
apprec env sigma (rslt, stack)
else
s
| Fix fix ->
(match reduce_fix (whd_betadeltaiota_state env sigma) fix stack with
| Reduced s -> apprec env sigma s
| NotReducible -> s)
| _ -> s
(* A reduction function like whd_betaiota but which keeps casts
* and does not reduce redexes containing existential variables.
* Used in Correctness.
* Added by JCF, 29/1/98. *)
let whd_programs_stack env sigma =
let rec whrec (x, stack as s) =
match kind_of_term x with
| App (f,cl) ->
let n = Array.length cl - 1 in
let c = cl.(n) in
if occur_existential c then
s
else
whrec (mkApp (f, Array.sub cl 0 n), append_stack [|c|] stack)
| LetIn (_,b,_,c) ->
if occur_existential b then
s
else
stacklam whrec [b] c stack
| Lambda (_,_,c) ->
(match decomp_stack stack with
| None -> s
| Some (a,m) -> stacklam whrec [a] c m)
| Case (ci,p,d,lf) ->
if occur_existential d then
s
else
let (c,cargs) = whrec (d, empty_stack) in
if reducible_mind_case c then
whrec (reduce_mind_case
{mP=p; mconstr=c; mcargs=list_of_stack cargs;
mci=ci; mlf=lf}, stack)
else
(mkCase (ci, p, app_stack(c,cargs), lf), stack)
| Fix fix ->
(match reduce_fix whrec fix stack with
| Reduced s' -> whrec s'
| NotReducible -> s)
| _ -> s
in
whrec
let whd_programs env sigma x =
app_stack (whd_programs_stack env sigma (x, empty_stack))
exception IsType
let find_conclusion env sigma =
let rec decrec env c =
let t = whd_betadeltaiota env sigma c in
match kind_of_term t with
| Prod (x,t,c0) -> decrec (push_rel (x,None,t) env) c0
| Lambda (x,t,c0) -> decrec (push_rel (x,None,t) env) c0
| t -> t
in
decrec env
let is_arity env sigma c =
match find_conclusion env sigma c with
| Sort _ -> true
| _ -> false
(*************************************)
(* Metas *)
let meta_value evd mv =
let rec valrec mv =
try
let b,_ = meta_fvalue evd mv in
instance
(List.map (fun mv' -> (mv',valrec mv')) (Metaset.elements b.freemetas))
b.rebus
with Anomaly _ | Not_found ->
mkMeta mv
in
valrec mv
let meta_instance env b =
let c_sigma =
List.map
(fun mv -> (mv,meta_value env mv)) (Metaset.elements b.freemetas)
in
if c_sigma = [] then b.rebus else instance c_sigma b.rebus
let nf_meta env c = meta_instance env (mk_freelisted c)
(* Instantiate metas that create beta/iota redexes *)
let meta_value evd mv =
let rec valrec mv =
try
let b,_ = meta_fvalue evd mv in
instance
(List.map (fun mv' -> (mv',valrec mv')) (Metaset.elements b.freemetas))
b.rebus
with Anomaly _ | Not_found ->
mkMeta mv
in
valrec mv
let meta_reducible_instance evd b =
let fm = Metaset.elements b.freemetas in
let metas = List.fold_left (fun l mv ->
match (try meta_opt_fvalue evd mv with Not_found -> None) with
| Some (g,(_,s)) -> (mv,(g.rebus,s))::l
| None -> l) [] in
let rec irec u =
let u = whd_betaiota u in
match kind_of_term u with
| Case (ci,p,c,bl) when isMeta c or isCast c & isMeta (pi1 (destCast c)) ->
let m = try destMeta c with _ -> destMeta (pi1 (destCast c)) in
(match
try
let g,s = List.assoc m metas in
if isConstruct g or s = TypeProcessed then Some g else None
with Not_found -> None
with
| Some g -> irec (mkCase (ci,p,g,bl))
| None -> mkCase (ci,irec p,c,Array.map irec bl))
| App (f,l) when isMeta f or isCast f & isMeta (pi1 (destCast f)) ->
let m = try destMeta f with _ -> destMeta (pi1 (destCast f)) in
(match
try
let g,s = List.assoc m metas in
if isLambda g or s = TypeProcessed then Some g else None
with Not_found -> None
with
| Some g -> irec (mkApp (g,l))
| None -> mkApp (f,Array.map irec l))
| Meta m ->
(try let g,s = List.assoc m metas in if s=TypeProcessed then irec g else u
with Not_found -> u)
| _ -> map_constr irec u
in
if fm = [] then (* nf_betaiota? *) b.rebus else irec b.rebus
|