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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(* This file is the interface between the c-c algorithm and Coq *)
open Evd
open Names
open Inductiveops
open Declarations
open Constr
open Context
open EConstr
open Vars
open Tactics
open Typing
open Ccalgo
open Ccproof
open Pp
open Util
open Proofview.Notations
module RelDecl = Context.Rel.Declaration
module NamedDecl = Context.Named.Declaration
let _f_equal = lazy (Coqlib.lib_ref "core.eq.congr")
let _eq_rect = lazy (Coqlib.lib_ref "core.eq.rect")
let _refl_equal = lazy (Coqlib.lib_ref "core.eq.refl")
let _sym_eq = lazy (Coqlib.lib_ref "core.eq.sym")
let _trans_eq = lazy (Coqlib.lib_ref "core.eq.trans")
let _eq = lazy (Coqlib.lib_ref "core.eq.type")
let _False = lazy (Coqlib.lib_ref "core.False.type")
let whd env sigma t =
Reductionops.clos_whd_flags CClosure.betaiotazeta env sigma t
let whd_delta env sigma t =
Reductionops.clos_whd_flags CClosure.all env sigma t
(* decompose member of equality in an applicative format *)
(** FIXME: evar leak *)
let sf_of env sigma c = snd (sort_of env sigma c)
let rec decompose_term env sigma t=
match EConstr.kind sigma (whd env sigma t) with
App (f,args)->
let tf=decompose_term env sigma f in
let targs=Array.map (decompose_term env sigma) args in
Array.fold_left (fun s t->Appli (s,t)) tf targs
| Prod (_,a,_b) when noccurn sigma 1 _b ->
let b = Termops.pop _b in
let sort_b = sf_of env sigma b in
let sort_a = sf_of env sigma a in
Appli(Appli(Product (sort_a,sort_b) ,
decompose_term env sigma a),
decompose_term env sigma b)
| Construct c ->
let (((mind,i_ind),i_con),u)= c in
let u = EInstance.kind sigma u in
let canon_mind = MutInd.make1 (MutInd.canonical mind) in
let canon_ind = canon_mind,i_ind in
let (oib,_)=Global.lookup_inductive (canon_ind) in
let nargs=constructor_nallargs_env env (canon_ind,i_con) in
Constructor {ci_constr= ((canon_ind,i_con),u);
ci_arity=nargs;
ci_nhyps=nargs-oib.mind_nparams}
| Ind c ->
let (mind,i_ind),u = c in
let u = EInstance.kind sigma u in
let canon_mind = MutInd.make1 (MutInd.canonical mind) in
let canon_ind = canon_mind,i_ind in (Symb (Constr.mkIndU (canon_ind,u)))
| Const (c,u) ->
let u = EInstance.kind sigma u in
let canon_const = Constant.make1 (Constant.canonical c) in
(Symb (Constr.mkConstU (canon_const,u)))
| Proj (p, c) ->
let canon_mind kn = MutInd.make1 (MutInd.canonical kn) in
let p' = Projection.map canon_mind p in
let c = Retyping.expand_projection env sigma p' c [] in
decompose_term env sigma c
| _ ->
let t = Termops.strip_outer_cast sigma t in
if closed0 sigma t then Symb (EConstr.to_constr ~abort_on_undefined_evars:false sigma t) else raise Not_found
(* decompose equality in members and type *)
open Termops
let atom_of_constr env sigma term =
let wh = whd_delta env sigma term in
let kot = EConstr.kind sigma wh in
match kot with
App (f,args)->
if is_global sigma (Lazy.force _eq) f && Int.equal (Array.length args) 3
then `Eq (args.(0),
decompose_term env sigma args.(1),
decompose_term env sigma args.(2))
else `Other (decompose_term env sigma term)
| _ -> `Other (decompose_term env sigma term)
let rec pattern_of_constr env sigma c =
match EConstr.kind sigma (whd env sigma c) with
App (f,args)->
let pf = decompose_term env sigma f in
let pargs,lrels = List.split
(Array.map_to_list (pattern_of_constr env sigma) args) in
PApp (pf,List.rev pargs),
List.fold_left Int.Set.union Int.Set.empty lrels
| Prod (_,a,_b) when noccurn sigma 1 _b ->
let b = Termops.pop _b in
let pa,sa = pattern_of_constr env sigma a in
let pb,sb = pattern_of_constr env sigma b in
let sort_b = sf_of env sigma b in
let sort_a = sf_of env sigma a in
PApp(Product (sort_a,sort_b),
[pa;pb]),(Int.Set.union sa sb)
| Rel i -> PVar i,Int.Set.singleton i
| _ ->
let pf = decompose_term env sigma c in
PApp (pf,[]),Int.Set.empty
let non_trivial = function
PVar _ -> false
| _ -> true
let patterns_of_constr env sigma nrels term=
let f,args=
try destApp sigma (whd_delta env sigma term) with DestKO -> raise Not_found in
if is_global sigma (Lazy.force _eq) f && Int.equal (Array.length args) 3
then
let patt1,rels1 = pattern_of_constr env sigma args.(1)
and patt2,rels2 = pattern_of_constr env sigma args.(2) in
let valid1 =
if not (Int.equal (Int.Set.cardinal rels1) nrels) then Creates_variables
else if non_trivial patt1 then Normal
else Trivial (EConstr.to_constr sigma args.(0))
and valid2 =
if not (Int.equal (Int.Set.cardinal rels2) nrels) then Creates_variables
else if non_trivial patt2 then Normal
else Trivial (EConstr.to_constr sigma args.(0)) in
if valid1 != Creates_variables
|| valid2 != Creates_variables then
nrels,valid1,patt1,valid2,patt2
else raise Not_found
else raise Not_found
let rec quantified_atom_of_constr env sigma nrels term =
match EConstr.kind sigma (whd_delta env sigma term) with
Prod (id,atom,ff) ->
if is_global sigma (Lazy.force _False) ff then
let patts=patterns_of_constr env sigma nrels atom in
`Nrule patts
else
quantified_atom_of_constr (EConstr.push_rel (RelDecl.LocalAssum (id,atom)) env) sigma (succ nrels) ff
| _ ->
let patts=patterns_of_constr env sigma nrels term in
`Rule patts
let litteral_of_constr env sigma term=
match EConstr.kind sigma (whd_delta env sigma term) with
| Prod (id,atom,ff) ->
if is_global sigma (Lazy.force _False) ff then
match (atom_of_constr env sigma atom) with
`Eq(t,a,b) -> `Neq(t,a,b)
| `Other(p) -> `Nother(p)
else
begin
try
quantified_atom_of_constr (EConstr.push_rel (RelDecl.LocalAssum (id,atom)) env) sigma 1 ff
with Not_found ->
`Other (decompose_term env sigma term)
end
| _ ->
atom_of_constr env sigma term
(* store all equalities from the context *)
let make_prb gls depth additionnal_terms =
let open Tacmach.New in
let env=pf_env gls in
let sigma=project gls in
let state = empty depth {it = Proofview.Goal.goal gls; sigma } in
let pos_hyps = ref [] in
let neg_hyps =ref [] in
List.iter
(fun c ->
let t = decompose_term env sigma c in
ignore (add_term state t)) additionnal_terms;
List.iter
(fun decl ->
let id = NamedDecl.get_id decl in
begin
let cid=Constr.mkVar id in
match litteral_of_constr env sigma (NamedDecl.get_type decl) with
`Eq (t,a,b) -> add_equality state cid a b
| `Neq (t,a,b) -> add_disequality state (Hyp cid) a b
| `Other ph ->
List.iter
(fun (cidn,nh) ->
add_disequality state (HeqnH (cid,cidn)) ph nh)
!neg_hyps;
pos_hyps:=(cid,ph):: !pos_hyps
| `Nother nh ->
List.iter
(fun (cidp,ph) ->
add_disequality state (HeqnH (cidp,cid)) ph nh)
!pos_hyps;
neg_hyps:=(cid,nh):: !neg_hyps
| `Rule patts -> add_quant state id true patts
| `Nrule patts -> add_quant state id false patts
end) (Proofview.Goal.hyps gls);
begin
match atom_of_constr env sigma (pf_concl gls) with
`Eq (t,a,b) -> add_disequality state Goal a b
| `Other g ->
List.iter
(fun (idp,ph) ->
add_disequality state (HeqG idp) ph g) !pos_hyps
end;
state
(* indhyps builds the array of arrays of constructor hyps for (ind largs) *)
let build_projection intype (cstr:pconstructor) special default gls=
let open Tacmach.New in
let ci= (snd(fst cstr)) in
let sigma = project gls in
let body=Equality.build_selector (pf_env gls) sigma ci (mkRel 1) intype special default in
let id=pf_get_new_id (Id.of_string "t") gls in
sigma, mkLambda(make_annot (Name id) Sorts.Relevant,intype,body)
(* generate an adhoc tactic following the proof tree *)
let app_global f args k =
Tacticals.New.pf_constr_of_global (Lazy.force f) >>= fun fc -> k (mkApp (fc, args))
let rec gen_holes env sigma t n accu =
if Int.equal n 0 then (sigma, List.rev accu)
else match EConstr.kind sigma t with
| Prod (_, u, t) ->
let (sigma, ev) = Evarutil.new_evar env sigma u in
let t = EConstr.Vars.subst1 ev t in
gen_holes env sigma t (pred n) (ev :: accu)
| _ -> assert false
let app_global_with_holes f args n =
Proofview.Goal.enter begin fun gl ->
Tacticals.New.pf_constr_of_global (Lazy.force f) >>= fun fc ->
let env = Proofview.Goal.env gl in
let concl = Proofview.Goal.concl gl in
Refine.refine ~typecheck:false begin fun sigma ->
let t = Tacmach.New.pf_get_type_of gl fc in
let t = Termops.prod_applist sigma t (Array.to_list args) in
let ans = mkApp (fc, args) in
let (sigma, holes) = gen_holes env sigma t n [] in
let ans = applist (ans, holes) in
let sigma = Typing.check env sigma ans concl in
(sigma, ans)
end
end
let assert_before n c =
Proofview.Goal.enter begin fun gl ->
let evm, _ = Tacmach.New.pf_apply type_of gl c in
Proofview.tclTHEN (Proofview.Unsafe.tclEVARS evm)
(assert_before n c)
end
let refresh_type env evm ty =
Evarsolve.refresh_universes ~status:Evd.univ_flexible ~refreshset:true
(Some false) env evm ty
let refresh_universes ty k =
Proofview.Goal.enter begin fun gl ->
let env = Proofview.Goal.env gl in
let evm = Tacmach.New.project gl in
let evm, ty = refresh_type env evm ty in
Proofview.tclTHEN (Proofview.Unsafe.tclEVARS evm) (k ty)
end
let constr_of_term c = EConstr.of_constr (constr_of_term c)
let rec proof_tac p : unit Proofview.tactic =
Proofview.Goal.enter begin fun gl ->
let type_of t = Tacmach.New.pf_unsafe_type_of gl t in
try (* type_of can raise exceptions *)
match p.p_rule with
Ax c -> exact_check (EConstr.of_constr c)
| SymAx c ->
let c = EConstr.of_constr c in
let l=constr_of_term p.p_lhs and
r=constr_of_term p.p_rhs in
refresh_universes (type_of l) (fun typ ->
app_global _sym_eq [|typ;r;l;c|] exact_check)
| Refl t ->
let lr = constr_of_term t in
refresh_universes (type_of lr) (fun typ ->
app_global _refl_equal [|typ;constr_of_term t|] exact_check)
| Trans (p1,p2)->
let t1 = constr_of_term p1.p_lhs and
t2 = constr_of_term p1.p_rhs and
t3 = constr_of_term p2.p_rhs in
refresh_universes (type_of t2) (fun typ ->
let prf = app_global_with_holes _trans_eq [|typ;t1;t2;t3;|] 2 in
Tacticals.New.tclTHENS prf [(proof_tac p1);(proof_tac p2)])
| Congr (p1,p2)->
let tf1=constr_of_term p1.p_lhs
and tx1=constr_of_term p2.p_lhs
and tf2=constr_of_term p1.p_rhs
and tx2=constr_of_term p2.p_rhs in
refresh_universes (type_of tf1) (fun typf ->
refresh_universes (type_of tx1) (fun typx ->
refresh_universes (type_of (mkApp (tf1,[|tx1|]))) (fun typfx ->
let id = Tacmach.New.pf_get_new_id (Id.of_string "f") gl in
let appx1 = mkLambda(make_annot (Name id) Sorts.Relevant,typf,mkApp(mkRel 1,[|tx1|])) in
let lemma1 = app_global_with_holes _f_equal [|typf;typfx;appx1;tf1;tf2|] 1 in
let lemma2 = app_global_with_holes _f_equal [|typx;typfx;tf2;tx1;tx2|] 1 in
let prf =
app_global_with_holes _trans_eq
[|typfx;
mkApp(tf1,[|tx1|]);
mkApp(tf2,[|tx1|]);
mkApp(tf2,[|tx2|])|] 2 in
Tacticals.New.tclTHENS prf
[Tacticals.New.tclTHEN lemma1 (proof_tac p1);
Tacticals.New.tclFIRST
[Tacticals.New.tclTHEN lemma2 (proof_tac p2);
reflexivity;
Tacticals.New.tclZEROMSG
(Pp.str
"I don't know how to handle dependent equality")]])))
| Inject (prf,cstr,nargs,argind) ->
let ti=constr_of_term prf.p_lhs in
let tj=constr_of_term prf.p_rhs in
let default=constr_of_term p.p_lhs in
let special=mkRel (1+nargs-argind) in
refresh_universes (type_of ti) (fun intype ->
refresh_universes (type_of default) (fun outtype ->
let sigma, proj =
build_projection intype cstr special default gl
in
let injt=
app_global_with_holes _f_equal [|intype;outtype;proj;ti;tj|] 1 in
Tacticals.New.tclTHEN (Proofview.Unsafe.tclEVARS sigma)
(Tacticals.New.tclTHEN injt (proof_tac prf))))
with e when Proofview.V82.catchable_exception e -> Proofview.tclZERO e
end
let refute_tac c t1 t2 p =
Proofview.Goal.enter begin fun gl ->
let tt1=constr_of_term t1 and tt2=constr_of_term t2 in
let hid = Tacmach.New.pf_get_new_id (Id.of_string "Heq") gl in
let false_t=mkApp (c,[|mkVar hid|]) in
let k intype =
let neweq= app_global _eq [|intype;tt1;tt2|] in
Tacticals.New.tclTHENS (neweq (assert_before (Name hid)))
[proof_tac p; simplest_elim false_t]
in refresh_universes (Tacmach.New.pf_unsafe_type_of gl tt1) k
end
let refine_exact_check c =
Proofview.Goal.enter begin fun gl ->
let evm, _ = Tacmach.New.pf_apply type_of gl c in
Proofview.tclTHEN (Proofview.Unsafe.tclEVARS evm) (exact_check c)
end
let convert_to_goal_tac c t1 t2 p =
Proofview.Goal.enter begin fun gl ->
let tt1=constr_of_term t1 and tt2=constr_of_term t2 in
let k sort =
let neweq= app_global _eq [|sort;tt1;tt2|] in
let e = Tacmach.New.pf_get_new_id (Id.of_string "e") gl in
let x = Tacmach.New.pf_get_new_id (Id.of_string "X") gl in
let identity=mkLambda (make_annot (Name x) Sorts.Relevant,sort,mkRel 1) in
let endt = app_global _eq_rect [|sort;tt1;identity;c;tt2;mkVar e|] in
Tacticals.New.tclTHENS (neweq (assert_before (Name e)))
[proof_tac p; endt refine_exact_check]
in refresh_universes (Tacmach.New.pf_unsafe_type_of gl tt2) k
end
let convert_to_hyp_tac c1 t1 c2 t2 p =
Proofview.Goal.enter begin fun gl ->
let tt2=constr_of_term t2 in
let h = Tacmach.New.pf_get_new_id (Id.of_string "H") gl in
let false_t=mkApp (c2,[|mkVar h|]) in
Tacticals.New.tclTHENS (assert_before (Name h) tt2)
[convert_to_goal_tac c1 t1 t2 p;
simplest_elim false_t]
end
(* Essentially [assert (Heq : lhs = rhs) by proof_tac p; discriminate Heq] *)
let discriminate_tac cstru p =
Proofview.Goal.enter begin fun gl ->
let lhs=constr_of_term p.p_lhs and rhs=constr_of_term p.p_rhs in
let env = Proofview.Goal.env gl in
let evm = Tacmach.New.project gl in
let evm, intype = refresh_type env evm (Tacmach.New.pf_unsafe_type_of gl lhs) in
let hid = Tacmach.New.pf_get_new_id (Id.of_string "Heq") gl in
let neweq=app_global _eq [|intype;lhs;rhs|] in
Tacticals.New.tclTHEN (Proofview.Unsafe.tclEVARS evm)
(Tacticals.New.tclTHENS (neweq (assert_before (Name hid)))
[proof_tac p; Equality.discrHyp hid])
end
(* wrap everything *)
let build_term_to_complete uf pac =
let cinfo = get_constructor_info uf pac.cnode in
let real_args = List.rev_map (fun i -> constr_of_term (term uf i)) pac.args in
let (kn, u) = cinfo.ci_constr in
(applist (mkConstructU (kn, EInstance.make u), real_args), pac.arity)
let cc_tactic depth additionnal_terms =
Proofview.Goal.enter begin fun gl ->
let sigma = Tacmach.New.project gl in
Coqlib.(check_required_library logic_module_name);
let _ = debug (fun () -> Pp.str "Reading subgoal ...") in
let state = make_prb gl depth additionnal_terms in
let _ = debug (fun () -> Pp.str "Problem built, solving ...") in
let sol = execute true state in
let _ = debug (fun () -> Pp.str "Computation completed.") in
let uf=forest state in
match sol with
None -> Tacticals.New.tclFAIL 0 (str "congruence failed")
| Some reason ->
debug (fun () -> Pp.str "Goal solved, generating proof ...");
match reason with
Discrimination (i,ipac,j,jpac) ->
let p=build_proof (Tacmach.New.pf_env gl) sigma uf (`Discr (i,ipac,j,jpac)) in
let cstr=(get_constructor_info uf ipac.cnode).ci_constr in
discriminate_tac cstr p
| Incomplete ->
let open Glob_term in
let env = Proofview.Goal.env gl in
let terms_to_complete = List.map (build_term_to_complete uf) (epsilons uf) in
let hole = DAst.make @@ GHole (Evar_kinds.InternalHole, Namegen.IntroAnonymous, None) in
let pr_missing (c, missing) =
let c = Detyping.detype Detyping.Now ~lax:true false Id.Set.empty env sigma c in
let holes = List.init missing (fun _ -> hole) in
Printer.pr_glob_constr_env env (DAst.make @@ GApp (c, holes))
in
let msg = Pp.(str "Goal is solvable by congruence but some arguments are missing."
++ fnl () ++
str " Try " ++
hov 8
begin
str "\"congruence with (" ++ prlist_with_sep (fun () -> str ")" ++ spc () ++ str "(")
pr_missing terms_to_complete ++ str ")\","
end ++
str " replacing metavariables by arbitrary terms.") in
Tacticals.New.tclFAIL 0 msg
| Contradiction dis ->
let env = Proofview.Goal.env gl in
let p=build_proof env sigma uf (`Prove (dis.lhs,dis.rhs)) in
let ta=term uf dis.lhs and tb=term uf dis.rhs in
match dis.rule with
Goal -> proof_tac p
| Hyp id -> refute_tac (EConstr.of_constr id) ta tb p
| HeqG id ->
let id = EConstr.of_constr id in
convert_to_goal_tac id ta tb p
| HeqnH (ida,idb) ->
let ida = EConstr.of_constr ida in
let idb = EConstr.of_constr idb in
convert_to_hyp_tac ida ta idb tb p
end
let cc_fail =
Tacticals.New.tclZEROMSG (Pp.str "congruence failed.")
let congruence_tac depth l =
Tacticals.New.tclORELSE
(Tacticals.New.tclTHEN (Tacticals.New.tclREPEAT introf) (cc_tactic depth l))
cc_fail
(* Beware: reflexivity = constructor 1 = apply refl_equal
might be slow now, let's rather do something equivalent
to a "simple apply refl_equal" *)
(* The [f_equal] tactic.
It mimics the use of lemmas [f_equal], [f_equal2], etc.
This isn't particularly related with congruence, apart from
the fact that congruence is called internally.
*)
let mk_eq f c1 c2 k =
Tacticals.New.pf_constr_of_global (Lazy.force f) >>= fun fc ->
Proofview.Goal.enter begin fun gl ->
let open Tacmach.New in
let evm, ty = pf_apply type_of gl c1 in
let evm, ty = Evarsolve.refresh_universes (Some false) (pf_env gl) evm ty in
let term = mkApp (fc, [| ty; c1; c2 |]) in
let evm, _ = type_of (pf_env gl) evm term in
Proofview.tclTHEN (Proofview.Unsafe.tclEVARS evm) (k term)
end
let f_equal =
Proofview.Goal.enter begin fun gl ->
let concl = Proofview.Goal.concl gl in
let sigma = Tacmach.New.project gl in
let cut_eq c1 c2 =
try (* type_of can raise an exception *)
Tacticals.New.tclTHENS
(mk_eq _eq c1 c2 Tactics.cut)
[Proofview.tclUNIT ();Tacticals.New.tclTRY ((app_global _refl_equal [||]) apply)]
with e when Proofview.V82.catchable_exception e -> Proofview.tclZERO e
in
Proofview.tclORELSE
begin match EConstr.kind sigma concl with
| App (r,[|_;t;t'|]) when is_global sigma (Lazy.force _eq) r ->
begin match EConstr.kind sigma t, EConstr.kind sigma t' with
| App (f,v), App (f',v') when Int.equal (Array.length v) (Array.length v') ->
let rec cuts i =
if i < 0 then Tacticals.New.tclTRY (congruence_tac 1000 [])
else Tacticals.New.tclTHENFIRST (cut_eq v.(i) v'.(i)) (cuts (i-1))
in cuts (Array.length v - 1)
| _ -> Proofview.tclUNIT ()
end
| _ -> Proofview.tclUNIT ()
end
begin function (e, info) -> match e with
| Pretype_errors.PretypeError _ | Type_errors.TypeError _ -> Proofview.tclUNIT ()
| e -> Proofview.tclZERO ~info e
end
end
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