(* Copyright (C) 2002, HELM Team. * * This file is part of HELM, an Hypertextual, Electronic * Library of Mathematics, developed at the Computer Science * Department, University of Bologna, Italy. * * HELM is free software; you can redistribute it and/or * modify it under the terms of the GNU General Public License * as published by the Free Software Foundation; either version 2 * of the License, or (at your option) any later version. * * HELM is distributed in the hope that it will be useful, * but WITHOUT ANY WARRANTY; without even the implied warranty of * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the * GNU General Public License for more details. * * You should have received a copy of the GNU General Public License * along with HELM; if not, write to the Free Software * Foundation, Inc., 59 Temple Place - Suite 330, Boston, * MA 02111-1307, USA. * * For details, see the HELM World-Wide-Web page, * http://cs.unibo.it/helm/. *) open ProofEngineHelpers open ProofEngineTypes exception NotAnInductiveTypeToEliminate exception NotTheRightEliminatorShape exception NoHypothesesFound exception WrongUriToVariable of string (* lambda_abstract newmeta ty *) (* returns a triple [bo],[context],[ty'] where *) (* [ty] = Pi/LetIn [context].[ty'] ([context] is a vector!) *) (* and [bo] = Lambda/LetIn [context].(Meta [newmeta]) *) (* So, lambda_abstract is the core of the implementation of *) (* the Intros tactic. *) let lambda_abstract metasenv context newmeta ty mk_fresh_name = let module C = Cic in let rec collect_context context = function C.Cast (te,_) -> collect_context context te | C.Prod (n,s,t) -> let n' = mk_fresh_name metasenv context n ~typ:s in let (context',ty,bo) = collect_context ((Some (n',(C.Decl s)))::context) t in (context',ty,C.Lambda(n',s,bo)) | C.LetIn (n,s,t) -> let (context',ty,bo) = collect_context ((Some (n,(C.Def (s,None))))::context) t in (context',ty,C.LetIn(n,s,bo)) | _ as t -> let irl = CicMkImplicit.identity_relocation_list_for_metavariable context in context, t, (C.Meta (newmeta,irl)) in collect_context context ty let eta_expand metasenv context t arg = let module T = CicTypeChecker in let module S = CicSubstitution in let module C = Cic in let rec aux n = function t' when t' = S.lift n arg -> C.Rel (1 + n) | C.Rel m -> if m <= n then C.Rel m else C.Rel (m+1) | C.Var (uri,exp_named_subst) -> let exp_named_subst' = aux_exp_named_subst n exp_named_subst in C.Var (uri,exp_named_subst') | C.Meta (i,l) -> let l' = List.map (function None -> None | Some t -> Some (aux n t)) l in C.Meta (i, l') | C.Sort _ | C.Implicit _ as t -> t | C.Cast (te,ty) -> C.Cast (aux n te, aux n ty) | C.Prod (nn,s,t) -> C.Prod (nn, aux n s, aux (n+1) t) | C.Lambda (nn,s,t) -> C.Lambda (nn, aux n s, aux (n+1) t) | C.LetIn (nn,s,t) -> C.LetIn (nn, aux n s, aux (n+1) t) | C.Appl l -> C.Appl (List.map (aux n) l) | C.Const (uri,exp_named_subst) -> let exp_named_subst' = aux_exp_named_subst n exp_named_subst in C.Const (uri,exp_named_subst') | C.MutInd (uri,i,exp_named_subst) -> let exp_named_subst' = aux_exp_named_subst n exp_named_subst in C.MutInd (uri,i,exp_named_subst') | C.MutConstruct (uri,i,j,exp_named_subst) -> let exp_named_subst' = aux_exp_named_subst n exp_named_subst in C.MutConstruct (uri,i,j,exp_named_subst') | C.MutCase (sp,i,outt,t,pl) -> C.MutCase (sp,i,aux n outt, aux n t, List.map (aux n) pl) | C.Fix (i,fl) -> let tylen = List.length fl in let substitutedfl = List.map (fun (name,i,ty,bo) -> (name, i, aux n ty, aux (n+tylen) bo)) fl in C.Fix (i, substitutedfl) | C.CoFix (i,fl) -> let tylen = List.length fl in let substitutedfl = List.map (fun (name,ty,bo) -> (name, aux n ty, aux (n+tylen) bo)) fl in C.CoFix (i, substitutedfl) and aux_exp_named_subst n = List.map (function uri,t -> uri,aux n t) in let argty,_ = T.type_of_aux' metasenv context arg CicUniv.empty_ugraph (* TASSI: FIXME *) in let fresh_name = FreshNamesGenerator.mk_fresh_name ~subst:[] metasenv context (Cic.Name "Heta") ~typ:argty in (C.Appl [C.Lambda (fresh_name,argty,aux 0 t) ; arg]) (*CSC: ma serve solamente la prima delle new_uninst e l'unione delle due!!! *) let classify_metas newmeta in_subst_domain subst_in metasenv = List.fold_right (fun (i,canonical_context,ty) (old_uninst,new_uninst) -> if in_subst_domain i then old_uninst,new_uninst else let ty' = subst_in canonical_context ty in let canonical_context' = List.fold_right (fun entry canonical_context' -> let entry' = match entry with Some (n,Cic.Decl s) -> Some (n,Cic.Decl (subst_in canonical_context' s)) | Some (n,Cic.Def (s,None)) -> Some (n,Cic.Def ((subst_in canonical_context' s),None)) | None -> None | Some (_,Cic.Def (_,Some _)) -> assert false in entry'::canonical_context' ) canonical_context [] in if i < newmeta then ((i,canonical_context',ty')::old_uninst),new_uninst else old_uninst,((i,canonical_context',ty')::new_uninst) ) metasenv ([],[]) (* Auxiliary function for apply: given a type (a backbone), it returns its *) (* head, a META environment in which there is new a META for each hypothesis,*) (* a list of arguments for the new applications and the indexes of the first *) (* and last new METAs introduced. The nth argument in the list of arguments *) (* is just the nth new META. *) let new_metasenv_for_apply newmeta proof context ty = let module C = Cic in let module S = CicSubstitution in let rec aux newmeta = function C.Cast (he,_) -> aux newmeta he (* CSC: patch to generate ?1 : ?2 : Type in place of ?1 : Type to simulate ?1 :< Type (* If the expected type is a Type, then also Set is OK ==> * we accept any term of type Type *) (*CSC: BUG HERE: in this way it is possible for the term of * type Type to be different from a Sort!!! *) | C.Prod (name,(C.Sort (C.Type _) as s),t) -> (* TASSI: ask CSC if BUG HERE refers to the C.Cast or C.Propd case *) let irl = CicMkImplicit.identity_relocation_list_for_metavariable context in let newargument = C.Meta (newmeta+1,irl) in let (res,newmetasenv,arguments,lastmeta) = aux (newmeta + 2) (S.subst newargument t) in res, (newmeta,[],s)::(newmeta+1,context,C.Meta (newmeta,[]))::newmetasenv, newargument::arguments,lastmeta *) | C.Prod (name,s,t) -> let irl = CicMkImplicit.identity_relocation_list_for_metavariable context in let newargument = C.Meta (newmeta,irl) in let (res,newmetasenv,arguments,lastmeta) = aux (newmeta + 1) (S.subst newargument t) in res,(newmeta,context,s)::newmetasenv,newargument::arguments,lastmeta | t -> t,[],[],newmeta in (* WARNING: here we are using the invariant that above the most *) (* recente new_meta() there are no used metas. *) let (res,newmetasenv,arguments,lastmeta) = aux newmeta ty in res,newmetasenv,arguments,lastmeta (* Useful only inside apply_tac *) let generalize_exp_named_subst_with_fresh_metas context newmeta uri exp_named_subst = let module C = Cic in let params = let o,_ = CicEnvironment.get_obj CicUniv.empty_ugraph uri in match o with C.Constant (_,_,_,params) | C.CurrentProof (_,_,_,_,params) | C.Variable (_,_,_,params) | C.InductiveDefinition (_,params,_) -> params in let exp_named_subst_diff,new_fresh_meta,newmetasenvfragment,exp_named_subst'= let next_fresh_meta = ref newmeta in let newmetasenvfragment = ref [] in let exp_named_subst_diff = ref [] in let rec aux = function [],[] -> [] | uri::tl,[] -> let ty = let o,_ = CicEnvironment.get_obj CicUniv.empty_ugraph uri in match o with C.Variable (_,_,ty,_) -> CicSubstitution.subst_vars !exp_named_subst_diff ty | _ -> raise (WrongUriToVariable (UriManager.string_of_uri uri)) in (* CSC: patch to generate ?1 : ?2 : Type in place of ?1 : Type to simulate ?1 :< Type (match ty with C.Sort (C.Type _) as s -> (* TASSI: ?? *) let fresh_meta = !next_fresh_meta in let fresh_meta' = fresh_meta + 1 in next_fresh_meta := !next_fresh_meta + 2 ; let subst_item = uri,C.Meta (fresh_meta',[]) in newmetasenvfragment := (fresh_meta,[],C.Sort (C.Type (CicUniv.fresh()))) :: (* TASSI: ?? *) (fresh_meta',[],C.Meta (fresh_meta,[])) :: !newmetasenvfragment ; exp_named_subst_diff := !exp_named_subst_diff @ [subst_item] ; subst_item::(aux (tl,[])) | _ -> *) let irl = CicMkImplicit.identity_relocation_list_for_metavariable context in let subst_item = uri,C.Meta (!next_fresh_meta,irl) in newmetasenvfragment := (!next_fresh_meta,context,ty)::!newmetasenvfragment ; exp_named_subst_diff := !exp_named_subst_diff @ [subst_item] ; incr next_fresh_meta ; subst_item::(aux (tl,[]))(*)*) | uri::tl1,((uri',_) as s)::tl2 -> assert (UriManager.eq uri uri') ; s::(aux (tl1,tl2)) | [],_ -> assert false in let exp_named_subst' = aux (params,exp_named_subst) in !exp_named_subst_diff,!next_fresh_meta, List.rev !newmetasenvfragment, exp_named_subst' in new_fresh_meta,newmetasenvfragment,exp_named_subst',exp_named_subst_diff ;; let apply_tac_verbose ~term (proof, goal) = (* Assumption: The term "term" must be closed in the current context *) let module T = CicTypeChecker in let module R = CicReduction in let module C = Cic in let (_,metasenv,_,_) = proof in let metano,context,ty = CicUtil.lookup_meta goal metasenv in let newmeta = new_meta_of_proof ~proof in let exp_named_subst_diff,newmeta',newmetasenvfragment,term' = match term with C.Var (uri,exp_named_subst) -> let newmeta',newmetasenvfragment,exp_named_subst',exp_named_subst_diff = generalize_exp_named_subst_with_fresh_metas context newmeta uri exp_named_subst in exp_named_subst_diff,newmeta',newmetasenvfragment, C.Var (uri,exp_named_subst') | C.Const (uri,exp_named_subst) -> let newmeta',newmetasenvfragment,exp_named_subst',exp_named_subst_diff = generalize_exp_named_subst_with_fresh_metas context newmeta uri exp_named_subst in exp_named_subst_diff,newmeta',newmetasenvfragment, C.Const (uri,exp_named_subst') | C.MutInd (uri,tyno,exp_named_subst) -> let newmeta',newmetasenvfragment,exp_named_subst',exp_named_subst_diff = generalize_exp_named_subst_with_fresh_metas context newmeta uri exp_named_subst in exp_named_subst_diff,newmeta',newmetasenvfragment, C.MutInd (uri,tyno,exp_named_subst') | C.MutConstruct (uri,tyno,consno,exp_named_subst) -> let newmeta',newmetasenvfragment,exp_named_subst',exp_named_subst_diff = generalize_exp_named_subst_with_fresh_metas context newmeta uri exp_named_subst in exp_named_subst_diff,newmeta',newmetasenvfragment, C.MutConstruct (uri,tyno,consno,exp_named_subst') | _ -> [],newmeta,[],term in let metasenv' = metasenv@newmetasenvfragment in let termty,_ = (* TASSI:FIXME *) CicTypeChecker.type_of_aux' metasenv' context term CicUniv.empty_ugraph in let termty = CicSubstitution.subst_vars exp_named_subst_diff termty in (* newmeta is the lowest index of the new metas introduced *) let (consthead,newmetas,arguments,_) = new_metasenv_for_apply newmeta' proof context termty in let newmetasenv = metasenv'@newmetas in let subst,newmetasenv',_ = (* TASSI:FIXME *) CicUnification.fo_unif newmetasenv context consthead ty CicUniv.empty_ugraph in let in_subst_domain i = List.exists (function (j,_) -> i=j) subst in let apply_subst = CicMetaSubst.apply_subst subst in let old_uninstantiatedmetas,new_uninstantiatedmetas = (* subst_in doesn't need the context. Hence the underscore. *) let subst_in _ = CicMetaSubst.apply_subst subst in classify_metas newmeta in_subst_domain subst_in newmetasenv' in let bo' = apply_subst (if List.length newmetas = 0 then term' else Cic.Appl (term'::arguments) ) in let newmetasenv'' = new_uninstantiatedmetas@old_uninstantiatedmetas in let subst_in = (* if we just apply the subtitution, the type is irrelevant: we may use Implicit, since it will be dropped *) CicMetaSubst.apply_subst ((metano,(context, bo', Cic.Implicit None))::subst) in let (newproof, newmetasenv''') = subst_meta_and_metasenv_in_proof proof metano subst_in newmetasenv'' in (subst_in,(newproof, List.map (function (i,_,_) -> i) new_uninstantiatedmetas)) let apply_tac ~term status = snd (apply_tac_verbose ~term status) let apply_tac_verbose ~term status = try apply_tac_verbose ~term status (* TODO cacciare anche altre eccezioni? *) with CicUnification.UnificationFailure _ as e -> raise (Fail (Printexc.to_string e)) (* TODO per implementare i tatticali e' necessario che tutte le tattiche sollevino _solamente_ Fail *) let apply_tac ~term = let apply_tac ~term status = try apply_tac ~term status (* TODO cacciare anche altre eccezioni? *) with CicUnification.UnificationFailure _ as e -> raise (Fail (Printexc.to_string e)) in mk_tactic (apply_tac ~term) let intros_tac ?(mk_fresh_name_callback = FreshNamesGenerator.mk_fresh_name ~subst:[]) ()= let intros_tac ?(mk_fresh_name_callback = (FreshNamesGenerator.mk_fresh_name ~subst:[])) () (proof, goal) = let module C = Cic in let module R = CicReduction in let (_,metasenv,_,_) = proof in let metano,context,ty = CicUtil.lookup_meta goal metasenv in let newmeta = new_meta_of_proof ~proof in let (context',ty',bo') = lambda_abstract metasenv context newmeta ty mk_fresh_name_callback in let (newproof, _) = subst_meta_in_proof proof metano bo' [newmeta,context',ty'] in (newproof, [newmeta]) in mk_tactic (intros_tac ~mk_fresh_name_callback ()) let cut_tac ?(mk_fresh_name_callback = FreshNamesGenerator.mk_fresh_name ~subst:[]) ~term= let cut_tac ?(mk_fresh_name_callback = FreshNamesGenerator.mk_fresh_name ~subst:[]) term (proof, goal) = let module C = Cic in let curi,metasenv,pbo,pty = proof in let metano,context,ty = CicUtil.lookup_meta goal metasenv in let newmeta1 = new_meta_of_proof ~proof in let newmeta2 = newmeta1 + 1 in let fresh_name = mk_fresh_name_callback metasenv context (Cic.Name "Hcut") ~typ:term in let context_for_newmeta1 = (Some (fresh_name,C.Decl term))::context in let irl1 = CicMkImplicit.identity_relocation_list_for_metavariable context_for_newmeta1 in let irl2 = CicMkImplicit.identity_relocation_list_for_metavariable context in let newmeta1ty = CicSubstitution.lift 1 ty in let bo' = C.Appl [C.Lambda (fresh_name,term,C.Meta (newmeta1,irl1)) ; C.Meta (newmeta2,irl2)] in let (newproof, _) = subst_meta_in_proof proof metano bo' [newmeta2,context,term; newmeta1,context_for_newmeta1,newmeta1ty]; in (newproof, [newmeta1 ; newmeta2]) in mk_tactic (cut_tac ~mk_fresh_name_callback term) let letin_tac ?(mk_fresh_name_callback=FreshNamesGenerator.mk_fresh_name ~subst:[]) ~term= let letin_tac ?(mk_fresh_name_callback = FreshNamesGenerator.mk_fresh_name ~subst:[]) term (proof, goal) = let module C = Cic in let curi,metasenv,pbo,pty = proof in let metano,context,ty = CicUtil.lookup_meta goal metasenv in let _,_ = (* TASSI: FIXME *) CicTypeChecker.type_of_aux' metasenv context term CicUniv.empty_ugraph in let newmeta = new_meta_of_proof ~proof in let fresh_name = mk_fresh_name_callback metasenv context (Cic.Name "Hletin") ~typ:term in let context_for_newmeta = (Some (fresh_name,C.Def (term,None)))::context in let irl = CicMkImplicit.identity_relocation_list_for_metavariable context_for_newmeta in let newmetaty = CicSubstitution.lift 1 ty in let bo' = C.LetIn (fresh_name,term,C.Meta (newmeta,irl)) in let (newproof, _) = subst_meta_in_proof proof metano bo'[newmeta,context_for_newmeta,newmetaty] in (newproof, [newmeta]) in mk_tactic (letin_tac ~mk_fresh_name_callback term) (** functional part of the "exact" tactic *) let exact_tac ~term = let exact_tac ~term (proof, goal) = (* Assumption: the term bo must be closed in the current context *) let (_,metasenv,_,_) = proof in let metano,context,ty = CicUtil.lookup_meta goal metasenv in let module T = CicTypeChecker in let module R = CicReduction in let ty_term,u = T.type_of_aux' metasenv context term CicUniv.empty_ugraph in let b,_ = R.are_convertible context ty_term ty u in (* TASSI: FIXME *) if b then begin let (newproof, metasenv') = subst_meta_in_proof proof metano term [] in (newproof, []) end else raise (Fail "The type of the provided term is not the one expected.") in mk_tactic (exact_tac ~term) (* not really "primitive" tactics .... *) let elim_tac ~term = let elim_tac ~term (proof, goal) = let module T = CicTypeChecker in let module U = UriManager in let module R = CicReduction in let module C = Cic in let (curi,metasenv,proofbo,proofty) = proof in let metano,context,ty = CicUtil.lookup_meta goal metasenv in let termty,_ = T.type_of_aux' metasenv context term CicUniv.empty_ugraph in (* TASSI: FIXME *) let uri,exp_named_subst,typeno,args = match termty with C.MutInd (uri,typeno,exp_named_subst) -> (uri,exp_named_subst,typeno,[]) | C.Appl ((C.MutInd (uri,typeno,exp_named_subst))::args) -> (uri,exp_named_subst,typeno,args) | _ -> raise NotAnInductiveTypeToEliminate in let eliminator_uri = let buri = U.buri_of_uri uri in let name = let o,_ = CicEnvironment.get_obj CicUniv.empty_ugraph uri in match o with C.InductiveDefinition (tys,_,_) -> let (name,_,_,_) = List.nth tys typeno in name | _ -> assert false in let ty_ty,_ = T.type_of_aux' metasenv context ty CicUniv.empty_ugraph in (* TASSI: FIXME *) let ext = match ty_ty with C.Sort C.Prop -> "_ind" | C.Sort C.Set -> "_rec" | C.Sort C.CProp -> "_rec" | C.Sort (C.Type _)-> "_rect" | _ -> assert false in U.uri_of_string (buri ^ "/" ^ name ^ ext ^ ".con") in let eliminator_ref = C.Const (eliminator_uri,exp_named_subst) in let ety,_ = T.type_of_aux' metasenv context eliminator_ref CicUniv.empty_ugraph in let rec find_args_no = function C.Prod (_,_,t) -> 1 + find_args_no t | C.Cast (s,_) -> find_args_no s | C.LetIn (_,_,t) -> 0 + find_args_no t | _ -> 0 in let args_no = find_args_no ety in let term_to_refine = let rec make_tl base_case = function 0 -> [base_case] | n -> (C.Implicit None)::(make_tl base_case (n - 1)) in C.Appl (eliminator_ref :: make_tl term (args_no - 1)) in let metasenv', term_to_refine' = CicMkImplicit.expand_implicits metasenv [] context term_to_refine in let refined_term,_,metasenv'',_ = (* TASSI: FIXME *) CicRefine.type_of_aux' metasenv' context term_to_refine' CicUniv.empty_ugraph in let new_goals = ProofEngineHelpers.compare_metasenvs ~oldmetasenv:metasenv ~newmetasenv:metasenv'' in let proof' = curi,metasenv'',proofbo,proofty in let proof'', new_goals' = apply_tactic (apply_tac ~term:refined_term) (proof',goal) in (* The apply_tactic can have closed some of the new_goals *) let patched_new_goals = let (_,metasenv''',_,_) = proof'' in List.filter (function i -> List.exists (function (j,_,_) -> j=i) metasenv''' ) new_goals @ new_goals' in proof'', patched_new_goals in mk_tactic (elim_tac ~term) ;; (* The simplification is performed only on the conclusion *) let elim_intros_simpl_tac ~term = Tacticals.then_ ~start:(elim_tac ~term) ~continuation: (Tacticals.thens ~start:(intros_tac ()) ~continuations: [ReductionTactics.simpl_tac ~also_in_hypotheses:false ~terms:None]) ;; exception NotConvertible (*CSC: Bug (or feature?). [with_what] is parsed in the context of the goal, *) (*CSC: while [what] can have a richer context (because of binders) *) (*CSC: So it is _NOT_ possible to use those binders in the [with_what] term. *) (*CSC: Is that evident? Is that right? Or should it be changed? *) let change_tac ~what ~with_what = let change_tac ~what ~with_what (proof, goal) = let curi,metasenv,pbo,pty = proof in let metano,context,ty = CicUtil.lookup_meta goal metasenv in (* are_convertible works only on well-typed terms *) let _,u = CicTypeChecker.type_of_aux' metasenv context with_what CicUniv.empty_ugraph in (* TASSI: FIXME *) let b,_ = CicReduction.are_convertible context what with_what u in if b then begin let replace = ProofEngineReduction.replace ~equality:(==) ~what:[what] ~with_what:[with_what] in let ty' = replace ty in let context' = List.map (function Some (name,Cic.Def (t,None))-> Some (name,Cic.Def ((replace t),None)) | Some (name,Cic.Decl t) -> Some (name,Cic.Decl (replace t)) | None -> None | Some (_,Cic.Def (_,Some _)) -> assert false ) context in let metasenv' = List.map (function (n,_,_) when n = metano -> (metano,context',ty') | _ as t -> t ) metasenv in (curi,metasenv',pbo,pty), [metano] end else raise (ProofEngineTypes.Fail "Not convertible") in mk_tactic (change_tac ~what ~with_what)