(* 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 _ | 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 in let fresh_name = FreshNamesGenerator.mk_fresh_name 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 | 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 = match CicEnvironment.get_obj uri 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 = match CicEnvironment.get_obj uri with C.Variable (_,_,ty,_) -> CicSubstitution.subst_vars !exp_named_subst_diff ty | _ -> raise (WrongUriToVariable (UriManager.string_of_uri uri)) in 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 prerr_endline ("@@@ " ^ CicPp.ppterm (Cic.Var (uri,exp_named_subst)) ^ " |--> " ^ CicPp.ppterm (Cic.Var (uri,exp_named_subst'))) ; new_fresh_meta,newmetasenvfragment,exp_named_subst',exp_named_subst_diff ;; let apply_tac ~term ~status:(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 prerr_endline ("^^^^^TERM': " ^ CicPp.ppterm term') ; let termty = CicSubstitution.subst_vars exp_named_subst_diff (CicTypeChecker.type_of_aux' metasenv' context term) in prerr_endline ("^^^^^TERMTY: " ^ CicPp.ppterm termty) ; (* 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' = CicUnification.fo_unif newmetasenv context consthead ty 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 prerr_endline ("XXXX " ^ CicPp.ppterm (if List.length newmetas = 0 then term' else Cic.Appl (term'::arguments)) ^ " |>>> " ^ CicPp.ppterm bo') ; let newmetasenv'' = new_uninstantiatedmetas@old_uninstantiatedmetas in let (newproof, newmetasenv''') = let subst_in = CicMetaSubst.apply_subst ((metano,bo')::subst) in subst_meta_and_metasenv_in_proof proof metano subst_in newmetasenv'' in (newproof, List.map (function (i,_,_) -> i) new_uninstantiatedmetas) (* TODO per implementare i tatticali e' necessario che tutte le tattiche sollevino _solamente_ Fail *) 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)) let intros_tac ?(mk_fresh_name_callback = FreshNamesGenerator.mk_fresh_name) () ~status:(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]) let cut_tac ?(mk_fresh_name_callback = FreshNamesGenerator.mk_fresh_name) term ~status:(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]) let letin_tac ?(mk_fresh_name_callback = FreshNamesGenerator.mk_fresh_name) term ~status:(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 _ = CicTypeChecker.type_of_aux' metasenv context term 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]) (** functional part of the "exact" tactic *) let exact_tac ~term ~status:(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 if R.are_convertible context (T.type_of_aux' metasenv context term) ty 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.") (* not really "primitive" tactics .... *) let elim_tac ~term ~status:(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,_,_) = proof in let metano,context,ty = CicUtil.lookup_meta goal metasenv in let termty = T.type_of_aux' metasenv context term in 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 = match CicEnvironment.get_obj uri with C.InductiveDefinition (tys,_,_) -> let (name,_,_,_) = List.nth tys typeno in name | _ -> assert false in let ext = match T.type_of_aux' metasenv context 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 in let newmeta = new_meta_of_proof ~proof in let (econclusion,newmetas,arguments,lastmeta) = new_metasenv_for_apply newmeta proof context ety in (* Here we assume that we have only one inductive hypothesis to *) (* eliminate and that it is the last hypothesis of the theorem. *) (* A better approach would be fingering the hypotheses in some *) (* way. *) let meta_of_corpse = let (_,canonical_context,_) = CicUtil.lookup_meta (lastmeta - 1) newmetas in let irl = CicMkImplicit.identity_relocation_list_for_metavariable canonical_context in Cic.Meta (lastmeta - 1, irl) in let newmetasenv = newmetas @ metasenv in let subst1,newmetasenv' = CicUnification.fo_unif newmetasenv context term meta_of_corpse in let ueconclusion = CicMetaSubst.apply_subst subst1 econclusion in (* The conclusion of our elimination principle is *) (* (?i farg1 ... fargn) *) (* The conclusion of our goal is ty. So, we can *) (* eta-expand ty w.r.t. farg1 .... fargn to get *) (* a new ty equal to (P farg1 ... fargn). Now *) (* ?i can be instantiated with P and we are ready *) (* to refine the term. *) let emeta, fargs = match ueconclusion with C.Appl ((C.Meta (emeta,_))::fargs) -> emeta,fargs | C.Meta (emeta,_) -> emeta,[] | _ -> raise NotTheRightEliminatorShape in let ty' = CicMetaSubst.apply_subst subst1 ty in let eta_expanded_ty = (*CSC: newmetasenv' era metasenv ??????????? *) List.fold_left (eta_expand newmetasenv' context) ty' fargs in let subst2,newmetasenv'' = (*CSC: passo newmetasenv', ma alcune variabili sono gia' state sostituite da subst1!!!! Dovrei rimuoverle o sono innocue?*) CicUnification.fo_unif newmetasenv' context ueconclusion eta_expanded_ty in let in_subst_domain i = let eq_to_i = function (j,_) -> i=j in List.exists eq_to_i subst1 || List.exists eq_to_i subst2 in (* When unwinding the META that corresponds to the elimination *) (* predicate (which is emeta), we must also perform one-step *) (* beta-reduction. apply_subst doesn't need the context. Hence *) (* the underscore. *) let apply_subst _ t = let t' = CicMetaSubst.apply_subst subst1 t in CicMetaSubst.apply_subst_reducing (Some (emeta,List.length fargs)) subst2 t' in let old_uninstantiatedmetas,new_uninstantiatedmetas = classify_metas newmeta in_subst_domain apply_subst newmetasenv'' in let arguments' = List.map (apply_subst context) arguments in let bo' = Cic.Appl (eliminator_ref::arguments') in let newmetasenv''' = new_uninstantiatedmetas@old_uninstantiatedmetas in let (newproof, newmetasenv'''') = (* When unwinding the META that corresponds to the *) (* elimination predicate (which is emeta), we must *) (* also perform one-step beta-reduction. *) (* The only difference w.r.t. apply_subst is that *) (* we also substitute metano with bo'. *) (*CSC: Nota: sostituire nuovamente subst1 e' superfluo, *) (*CSC: no? *) let apply_subst' t = let t' = CicMetaSubst.apply_subst subst1 t in CicMetaSubst.apply_subst_reducing (Some (emeta,List.length fargs)) ((metano,bo')::subst2) t' in subst_meta_and_metasenv_in_proof proof metano apply_subst' newmetasenv''' in (newproof, List.map (function (i,_,_) -> i) new_uninstantiatedmetas) ;; (* 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 ~status:(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 *) ignore (CicTypeChecker.type_of_aux' metasenv context with_what) ; if CicReduction.are_convertible context what with_what 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")