(************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) (* assert false) let register_replace f = replace := f let general_rewrite = ref (fun _ _ -> assert false) let register_general_rewrite f = general_rewrite := f (* util function; it should be in util.mli *) let prlist_with_sepi sep elem = let rec aux n = function | [] -> mt () | [h] -> elem n h | h::t -> let e = elem n h and s = sep() and r = aux (n+1) t in e ++ s ++ r in aux 1 type relation = { rel_a: constr ; rel_aeq: constr; rel_refl: constr option; rel_sym: constr option; rel_trans : constr option; rel_quantifiers_no: int (* it helps unification *); rel_X_relation_class: constr; rel_Xreflexive_relation_class: constr } type 'a relation_class = Relation of 'a (* the rel_aeq of the relation or the relation *) | Leibniz of constr option (* the carrier (if eq is partially instantiated) *) type 'a morphism = { args : (bool option * 'a relation_class) list; output : 'a relation_class; lem : constr; morphism_theory : constr } type funct = { f_args : constr list; f_output : constr } type morphism_class = ACMorphism of relation morphism | ACFunction of funct let subst_mps_in_relation_class subst = function Relation t -> Relation (subst_mps subst t) | Leibniz t -> Leibniz (option_map (subst_mps subst) t) let subst_mps_in_argument_class subst (variance,rel) = variance, subst_mps_in_relation_class subst rel let constr_relation_class_of_relation_relation_class = function Relation relation -> Relation relation.rel_aeq | Leibniz t -> Leibniz t let constr_of c = Constrintern.interp_constr Evd.empty (Global.env()) c let constant dir s = Coqlib.gen_constant "Setoid_replace" ("Setoids"::dir) s let gen_constant dir s = Coqlib.gen_constant "Setoid_replace" dir s let reference dir s = Coqlib.gen_reference "Setoid_replace" ("Setoids"::dir) s let eval_reference dir s = EvalConstRef (destConst (constant dir s)) let eval_init_reference dir s = EvalConstRef (destConst (gen_constant ("Init"::dir) s)) let current_constant id = try global_reference id with Not_found -> anomaly ("Setoid: cannot find " ^ (string_of_id id)) (* From Setoid.v *) let coq_reflexive = lazy(gen_constant ["Relations"; "Relation_Definitions"] "reflexive") let coq_symmetric = lazy(gen_constant ["Relations"; "Relation_Definitions"] "symmetric") let coq_transitive = lazy(gen_constant ["Relations"; "Relation_Definitions"] "transitive") let coq_relation = lazy(gen_constant ["Relations"; "Relation_Definitions"] "relation") let coq_Relation_Class = lazy(constant ["Setoid"] "Relation_Class") let coq_Argument_Class = lazy(constant ["Setoid"] "Argument_Class") let coq_Setoid_Theory = lazy(constant ["Setoid"] "Setoid_Theory") let coq_Morphism_Theory = lazy(constant ["Setoid"] "Morphism_Theory") let coq_Build_Morphism_Theory= lazy(constant ["Setoid"] "Build_Morphism_Theory") let coq_Compat = lazy(constant ["Setoid"] "Compat") let coq_AsymmetricReflexive = lazy(constant ["Setoid"] "AsymmetricReflexive") let coq_SymmetricReflexive = lazy(constant ["Setoid"] "SymmetricReflexive") let coq_SymmetricAreflexive = lazy(constant ["Setoid"] "SymmetricAreflexive") let coq_AsymmetricAreflexive = lazy(constant ["Setoid"] "AsymmetricAreflexive") let coq_Leibniz = lazy(constant ["Setoid"] "Leibniz") let coq_RAsymmetric = lazy(constant ["Setoid"] "RAsymmetric") let coq_RSymmetric = lazy(constant ["Setoid"] "RSymmetric") let coq_RLeibniz = lazy(constant ["Setoid"] "RLeibniz") let coq_ASymmetric = lazy(constant ["Setoid"] "ASymmetric") let coq_AAsymmetric = lazy(constant ["Setoid"] "AAsymmetric") let coq_seq_refl = lazy(constant ["Setoid"] "Seq_refl") let coq_seq_sym = lazy(constant ["Setoid"] "Seq_sym") let coq_seq_trans = lazy(constant ["Setoid"] "Seq_trans") let coq_variance = lazy(constant ["Setoid"] "variance") let coq_Covariant = lazy(constant ["Setoid"] "Covariant") let coq_Contravariant = lazy(constant ["Setoid"] "Contravariant") let coq_Left2Right = lazy(constant ["Setoid"] "Left2Right") let coq_Right2Left = lazy(constant ["Setoid"] "Right2Left") let coq_MSNone = lazy(constant ["Setoid"] "MSNone") let coq_MSCovariant = lazy(constant ["Setoid"] "MSCovariant") let coq_MSContravariant = lazy(constant ["Setoid"] "MSContravariant") let coq_singl = lazy(constant ["Setoid"] "singl") let coq_cons = lazy(constant ["Setoid"] "necons") let coq_equality_morphism_of_asymmetric_areflexive_transitive_relation = lazy(constant ["Setoid"] "equality_morphism_of_asymmetric_areflexive_transitive_relation") let coq_equality_morphism_of_symmetric_areflexive_transitive_relation = lazy(constant ["Setoid"] "equality_morphism_of_symmetric_areflexive_transitive_relation") let coq_equality_morphism_of_asymmetric_reflexive_transitive_relation = lazy(constant ["Setoid"] "equality_morphism_of_asymmetric_reflexive_transitive_relation") let coq_equality_morphism_of_symmetric_reflexive_transitive_relation = lazy(constant ["Setoid"] "equality_morphism_of_symmetric_reflexive_transitive_relation") let coq_make_compatibility_goal = lazy(constant ["Setoid"] "make_compatibility_goal") let coq_make_compatibility_goal_eval_ref = lazy(eval_reference ["Setoid"] "make_compatibility_goal") let coq_make_compatibility_goal_aux_eval_ref = lazy(eval_reference ["Setoid"] "make_compatibility_goal_aux") let coq_App = lazy(constant ["Setoid"] "App") let coq_ToReplace = lazy(constant ["Setoid"] "ToReplace") let coq_ToKeep = lazy(constant ["Setoid"] "ToKeep") let coq_ProperElementToKeep = lazy(constant ["Setoid"] "ProperElementToKeep") let coq_fcl_singl = lazy(constant ["Setoid"] "fcl_singl") let coq_fcl_cons = lazy(constant ["Setoid"] "fcl_cons") let coq_setoid_rewrite = lazy(constant ["Setoid"] "setoid_rewrite") let coq_proj1 = lazy(gen_constant ["Init"; "Logic"] "proj1") let coq_proj2 = lazy(gen_constant ["Init"; "Logic"] "proj2") let coq_unit = lazy(gen_constant ["Init"; "Datatypes"] "unit") let coq_tt = lazy(gen_constant ["Init"; "Datatypes"] "tt") let coq_eq = lazy(gen_constant ["Init"; "Logic"] "eq") let coq_morphism_theory_of_function = lazy(constant ["Setoid"] "morphism_theory_of_function") let coq_morphism_theory_of_predicate = lazy(constant ["Setoid"] "morphism_theory_of_predicate") let coq_relation_of_relation_class = lazy(eval_reference ["Setoid"] "relation_of_relation_class") let coq_directed_relation_of_relation_class = lazy(eval_reference ["Setoid"] "directed_relation_of_relation_class") let coq_interp = lazy(eval_reference ["Setoid"] "interp") let coq_Morphism_Context_rect2 = lazy(eval_reference ["Setoid"] "Morphism_Context_rect2") let coq_iff = lazy(gen_constant ["Init";"Logic"] "iff") let coq_impl = lazy(constant ["Setoid"] "impl") (************************* Table of declared relations **********************) (* Relations are stored in a table which is synchronised with the Reset mechanism. *) let relation_table = ref Gmap.empty let relation_table_add (s,th) = relation_table := Gmap.add s th !relation_table let relation_table_find s = Gmap.find s !relation_table let relation_table_mem s = Gmap.mem s !relation_table let prrelation s = str "(" ++ pr_lconstr s.rel_a ++ str "," ++ pr_lconstr s.rel_aeq ++ str ")" let prrelation_class = function Relation eq -> (try prrelation (relation_table_find eq) with Not_found -> str "[[ Error: " ++ pr_lconstr eq ++ str " is not registered as a relation ]]") | Leibniz (Some ty) -> pr_lconstr ty | Leibniz None -> str "_" let prmorphism_argument_gen prrelation (variance,rel) = prrelation rel ++ match variance with None -> str " ==> " | Some true -> str " ++> " | Some false -> str " --> " let prargument_class = prmorphism_argument_gen prrelation_class let pr_morphism_signature (l,c) = prlist (prmorphism_argument_gen Ppconstr.pr_constr_expr) l ++ Ppconstr.pr_constr_expr c let prmorphism k m = pr_lconstr k ++ str ": " ++ prlist prargument_class m.args ++ prrelation_class m.output (* A function that gives back the only relation_class on a given carrier *) (*CSC: this implementation is really inefficient. I should define a new map to make it efficient. However, is this really worth of? *) let default_relation_for_carrier ?(filter=fun _ -> true) a = let rng = Gmap.rng !relation_table in match List.filter (fun ({rel_a=rel_a} as r) -> rel_a = a && filter r) rng with [] -> Leibniz (Some a) | relation::tl -> if tl <> [] then Options.if_warn msg_warning (str "There are several relations on the carrier \"" ++ pr_lconstr a ++ str "\". The relation " ++ prrelation relation ++ str " is chosen.") ; Relation relation let find_relation_class rel = try Relation (relation_table_find rel) with Not_found -> let rel = Reduction.whd_betadeltaiota (Global.env ()) rel in match kind_of_term rel with | App (eq,[|ty|]) when eq_constr eq (Lazy.force coq_eq) -> Leibniz (Some ty) | _ when eq_constr rel (Lazy.force coq_eq) -> Leibniz None | _ -> raise Not_found let coq_iff_relation = lazy (find_relation_class (Lazy.force coq_iff)) let coq_impl_relation = lazy (find_relation_class (Lazy.force coq_impl)) let relation_morphism_of_constr_morphism = let relation_relation_class_of_constr_relation_class = function Leibniz t -> Leibniz t | Relation aeq -> Relation (try relation_table_find aeq with Not_found -> assert false) in function mor -> let args' = List.map (fun (variance,rel) -> variance, relation_relation_class_of_constr_relation_class rel ) mor.args in let output' = relation_relation_class_of_constr_relation_class mor.output in {mor with args=args' ; output=output'} let subst_relation subst relation = let rel_a' = subst_mps subst relation.rel_a in let rel_aeq' = subst_mps subst relation.rel_aeq in let rel_refl' = option_map (subst_mps subst) relation.rel_refl in let rel_sym' = option_map (subst_mps subst) relation.rel_sym in let rel_trans' = option_map (subst_mps subst) relation.rel_trans in let rel_X_relation_class' = subst_mps subst relation.rel_X_relation_class in let rel_Xreflexive_relation_class' = subst_mps subst relation.rel_Xreflexive_relation_class in if rel_a' == relation.rel_a && rel_aeq' == relation.rel_aeq && rel_refl' == relation.rel_refl && rel_sym' == relation.rel_sym && rel_trans' == relation.rel_trans && rel_X_relation_class' == relation.rel_X_relation_class && rel_Xreflexive_relation_class'==relation.rel_Xreflexive_relation_class then relation else { rel_a = rel_a' ; rel_aeq = rel_aeq' ; rel_refl = rel_refl' ; rel_sym = rel_sym'; rel_trans = rel_trans'; rel_quantifiers_no = relation.rel_quantifiers_no; rel_X_relation_class = rel_X_relation_class'; rel_Xreflexive_relation_class = rel_Xreflexive_relation_class' } let equiv_list () = List.map (fun x -> x.rel_aeq) (Gmap.rng !relation_table) let _ = Summary.declare_summary "relation-table" { Summary.freeze_function = (fun () -> !relation_table); Summary.unfreeze_function = (fun t -> relation_table := t); Summary.init_function = (fun () -> relation_table := Gmap .empty); Summary.survive_module = false; Summary.survive_section = false } (* Declare a new type of object in the environment : "relation-theory". *) let (relation_to_obj, obj_to_relation)= let cache_set (_,(s, th)) = let th' = if relation_table_mem s then begin let old_relation = relation_table_find s in let th' = {th with rel_sym = match th.rel_sym with None -> old_relation.rel_sym | Some t -> Some t} in Options.if_warn msg_warning (strbrk "The relation " ++ prrelation th' ++ strbrk " is redeclared. The new declaration" ++ (match th'.rel_refl with None -> mt () | Some t -> strbrk " (reflexivity proved by " ++ pr_lconstr t) ++ (match th'.rel_sym with None -> mt () | Some t -> (if th'.rel_refl = None then strbrk " (" else strbrk " and ") ++ strbrk "symmetry proved by " ++ pr_lconstr t) ++ (if th'.rel_refl <> None && th'.rel_sym <> None then str ")" else str "") ++ strbrk " replaces the old declaration" ++ (match old_relation.rel_refl with None -> str "" | Some t -> strbrk " (reflexivity proved by " ++ pr_lconstr t) ++ (match old_relation.rel_sym with None -> str "" | Some t -> (if old_relation.rel_refl = None then strbrk " (" else strbrk " and ") ++ strbrk "symmetry proved by " ++ pr_lconstr t) ++ (if old_relation.rel_refl <> None && old_relation.rel_sym <> None then str ")" else str "") ++ str "."); th' end else th in relation_table_add (s,th') and subst_set (_,subst,(s,th as obj)) = let s' = subst_mps subst s in let th' = subst_relation subst th in if s' == s && th' == th then obj else (s',th') and export_set x = Some x in declare_object {(default_object "relation-theory") with cache_function = cache_set; load_function = (fun i o -> cache_set o); subst_function = subst_set; classify_function = (fun (_,x) -> Substitute x); export_function = export_set} (******************************* Table of declared morphisms ********************) (* Setoids are stored in a table which is synchronised with the Reset mechanism. *) let morphism_table = ref Gmap.empty let morphism_table_find m = Gmap.find m !morphism_table let morphism_table_add (m,c) = let old = try morphism_table_find m with Not_found -> [] in try let old_morph = List.find (function mor -> mor.args = c.args && mor.output = c.output) old in Options.if_warn msg_warning (strbrk "The morphism " ++ prmorphism m old_morph ++ strbrk " is redeclared. " ++ strbrk "The new declaration whose compatibility is proved by " ++ pr_lconstr c.lem ++ strbrk " replaces the old declaration whose" ++ strbrk " compatibility was proved by " ++ pr_lconstr old_morph.lem ++ str ".") with Not_found -> morphism_table := Gmap.add m (c::old) !morphism_table let default_morphism ?(filter=fun _ -> true) m = match List.filter filter (morphism_table_find m) with [] -> raise Not_found | m1::ml -> if ml <> [] then Options.if_warn msg_warning (strbrk "There are several morphisms associated to \"" ++ pr_lconstr m ++ strbrk "\". Morphism " ++ prmorphism m m1 ++ strbrk " is randomly chosen."); relation_morphism_of_constr_morphism m1 let subst_morph subst morph = let lem' = subst_mps subst morph.lem in let args' = list_smartmap (subst_mps_in_argument_class subst) morph.args in let output' = subst_mps_in_relation_class subst morph.output in let morphism_theory' = subst_mps subst morph.morphism_theory in if lem' == morph.lem && args' == morph.args && output' == morph.output && morphism_theory' == morph.morphism_theory then morph else { args = args' ; output = output' ; lem = lem' ; morphism_theory = morphism_theory' } let _ = Summary.declare_summary "morphism-table" { Summary.freeze_function = (fun () -> !morphism_table); Summary.unfreeze_function = (fun t -> morphism_table := t); Summary.init_function = (fun () -> morphism_table := Gmap .empty); Summary.survive_module = false; Summary.survive_section = false } (* Declare a new type of object in the environment : "morphism-definition". *) let (morphism_to_obj, obj_to_morphism)= let cache_set (_,(m, c)) = morphism_table_add (m, c) and subst_set (_,subst,(m,c as obj)) = let m' = subst_mps subst m in let c' = subst_morph subst c in if m' == m && c' == c then obj else (m',c') and export_set x = Some x in declare_object {(default_object "morphism-definition") with cache_function = cache_set; load_function = (fun i o -> cache_set o); subst_function = subst_set; classify_function = (fun (_,x) -> Substitute x); export_function = export_set} (************************** Printing relations and morphisms **********************) let print_setoids () = Gmap.iter (fun k relation -> assert (k=relation.rel_aeq) ; ppnl (str"Relation " ++ prrelation relation ++ str";" ++ (match relation.rel_refl with None -> str "" | Some t -> str" reflexivity proved by " ++ pr_lconstr t) ++ (match relation.rel_sym with None -> str "" | Some t -> str " symmetry proved by " ++ pr_lconstr t) ++ (match relation.rel_trans with None -> str "" | Some t -> str " transitivity proved by " ++ pr_lconstr t))) !relation_table ; Gmap.iter (fun k l -> List.iter (fun ({lem=lem} as mor) -> ppnl (str "Morphism " ++ prmorphism k mor ++ str ". Compatibility proved by " ++ pr_lconstr lem ++ str ".")) l) !morphism_table ;; (***************** Adding a morphism to the database ****************************) (* We maintain a table of the currently edited proofs of morphism lemma in order to add them in the morphism_table when the user does Save *) let edited = ref Gmap.empty let new_edited id m = edited := Gmap.add id m !edited let is_edited id = Gmap.mem id !edited let no_more_edited id = edited := Gmap.remove id !edited let what_edited id = Gmap.find id !edited (* also returns the triple (args_ty_quantifiers_rev,real_args_ty,real_output) where the args_ty and the output are delifted *) let check_is_dependent n args_ty output = let m = List.length args_ty - n in let args_ty_quantifiers, args_ty = Util.list_chop n args_ty in let rec aux m t = match kind_of_term t with Prod (n,s,t) when m > 0 -> if not (dependent (mkRel 1) t) then let args,out = aux (m - 1) (subst1 (mkRel 1) (* dummy *) t) in s::args,out else errorlabstrm "New Morphism" (str "The morphism is not a quantified non dependent product.") | _ -> [],t in let ty = compose_prod (List.rev args_ty) output in let args_ty, output = aux m ty in List.rev args_ty_quantifiers, args_ty, output let cic_relation_class_of_X_relation typ value = function {rel_a=rel_a; rel_aeq=rel_aeq; rel_refl=Some refl; rel_sym=None} -> mkApp ((Lazy.force coq_AsymmetricReflexive), [| typ ; value ; rel_a ; rel_aeq; refl |]) | {rel_a=rel_a; rel_aeq=rel_aeq; rel_refl=Some refl; rel_sym=Some sym} -> mkApp ((Lazy.force coq_SymmetricReflexive), [| typ ; rel_a ; rel_aeq; sym ; refl |]) | {rel_a=rel_a; rel_aeq=rel_aeq; rel_refl=None; rel_sym=None} -> mkApp ((Lazy.force coq_AsymmetricAreflexive), [| typ ; value ; rel_a ; rel_aeq |]) | {rel_a=rel_a; rel_aeq=rel_aeq; rel_refl=None; rel_sym=Some sym} -> mkApp ((Lazy.force coq_SymmetricAreflexive), [| typ ; rel_a ; rel_aeq; sym |]) let cic_relation_class_of_X_relation_class typ value = function Relation {rel_X_relation_class=x_relation_class} -> mkApp (x_relation_class, [| typ ; value |]) | Leibniz (Some t) -> mkApp ((Lazy.force coq_Leibniz), [| typ ; t |]) | Leibniz None -> assert false let cic_precise_relation_class_of_relation = function {rel_a=rel_a; rel_aeq=rel_aeq; rel_refl=Some refl; rel_sym=None} -> mkApp ((Lazy.force coq_RAsymmetric), [| rel_a ; rel_aeq; refl |]) | {rel_a=rel_a; rel_aeq=rel_aeq; rel_refl=Some refl; rel_sym=Some sym} -> mkApp ((Lazy.force coq_RSymmetric), [| rel_a ; rel_aeq; sym ; refl |]) | {rel_a=rel_a; rel_aeq=rel_aeq; rel_refl=None; rel_sym=None} -> mkApp ((Lazy.force coq_AAsymmetric), [| rel_a ; rel_aeq |]) | {rel_a=rel_a; rel_aeq=rel_aeq; rel_refl=None; rel_sym=Some sym} -> mkApp ((Lazy.force coq_ASymmetric), [| rel_a ; rel_aeq; sym |]) let cic_precise_relation_class_of_relation_class = function Relation {rel_aeq=rel_aeq; rel_Xreflexive_relation_class=lem; rel_refl=rel_refl } -> rel_aeq,lem,not(rel_refl=None) | Leibniz (Some t) -> mkApp ((Lazy.force coq_eq), [| t |]), mkApp ((Lazy.force coq_RLeibniz), [| t |]), true | Leibniz None -> assert false let cic_relation_class_of_relation_class rel = cic_relation_class_of_X_relation_class (Lazy.force coq_unit) (Lazy.force coq_tt) rel let cic_argument_class_of_argument_class (variance,arg) = let coq_variant_value = match variance with None -> (Lazy.force coq_Covariant) (* dummy value, it won't be used *) | Some true -> (Lazy.force coq_Covariant) | Some false -> (Lazy.force coq_Contravariant) in cic_relation_class_of_X_relation_class (Lazy.force coq_variance) coq_variant_value arg let cic_arguments_of_argument_class_list args = let rec aux = function [] -> assert false | [last] -> mkApp ((Lazy.force coq_singl), [| Lazy.force coq_Argument_Class; last |]) | he::tl -> mkApp ((Lazy.force coq_cons), [| Lazy.force coq_Argument_Class; he ; aux tl |]) in aux (List.map cic_argument_class_of_argument_class args) let gen_compat_lemma_statement quantifiers_rev output args m = let output = cic_relation_class_of_relation_class output in let args = cic_arguments_of_argument_class_list args in args, output, compose_prod quantifiers_rev (mkApp ((Lazy.force coq_make_compatibility_goal), [| args ; output ; m |])) let morphism_theory_id_of_morphism_proof_id id = id_of_string (string_of_id id ^ "_morphism_theory") (* apply_to_rels c [l1 ; ... ; ln] returns (c Rel1 ... reln) *) let apply_to_rels c l = if l = [] then c else let len = List.length l in applistc c (Util.list_map_i (fun i _ -> mkRel (len - i)) 0 l) let apply_to_relation subst rel = if Array.length subst = 0 then rel else let new_quantifiers_no = rel.rel_quantifiers_no - Array.length subst in assert (new_quantifiers_no >= 0) ; { rel_a = mkApp (rel.rel_a, subst) ; rel_aeq = mkApp (rel.rel_aeq, subst) ; rel_refl = option_map (fun c -> mkApp (c,subst)) rel.rel_refl ; rel_sym = option_map (fun c -> mkApp (c,subst)) rel.rel_sym; rel_trans = option_map (fun c -> mkApp (c,subst)) rel.rel_trans; rel_quantifiers_no = new_quantifiers_no; rel_X_relation_class = mkApp (rel.rel_X_relation_class, subst); rel_Xreflexive_relation_class = mkApp (rel.rel_Xreflexive_relation_class, subst) } let add_morphism lemma_infos mor_name (m,quantifiers_rev,args,output) = let lem = match lemma_infos with None -> (* the Morphism_Theory object has already been created *) let applied_args = let len = List.length quantifiers_rev in let subst = Array.of_list (Util.list_map_i (fun i _ -> mkRel (len - i)) 0 quantifiers_rev) in List.map (fun (v,rel) -> match rel with Leibniz (Some t) -> assert (subst=[||]); v, Leibniz (Some t) | Leibniz None -> assert (Array.length subst = 1); v, Leibniz (Some (subst.(0))) | Relation rel -> v, Relation (apply_to_relation subst rel)) args in compose_lam quantifiers_rev (mkApp (Lazy.force coq_Compat, [| cic_arguments_of_argument_class_list applied_args; cic_relation_class_of_relation_class output; apply_to_rels (current_constant mor_name) quantifiers_rev |])) | Some (lem_name,argsconstr,outputconstr) -> (* only the compatibility has been proved; we need to declare the Morphism_Theory object *) let mext = current_constant lem_name in ignore ( Declare.declare_internal_constant mor_name (DefinitionEntry {const_entry_body = compose_lam quantifiers_rev (mkApp ((Lazy.force coq_Build_Morphism_Theory), [| argsconstr; outputconstr; apply_to_rels m quantifiers_rev ; apply_to_rels mext quantifiers_rev |])); const_entry_type = None; const_entry_opaque = false; const_entry_boxed = Options.boxed_definitions()}, IsDefinition Definition)) ; mext in let mmor = current_constant mor_name in let args_constr = List.map (fun (variance,arg) -> variance, constr_relation_class_of_relation_relation_class arg) args in let output_constr = constr_relation_class_of_relation_relation_class output in Lib.add_anonymous_leaf (morphism_to_obj (m, { args = args_constr; output = output_constr; lem = lem; morphism_theory = mmor })); Options.if_verbose ppnl (pr_lconstr m ++ str " is registered as a morphism") (* first order matching with a bit of conversion *) let unify_relation_carrier_with_type env rel t = let raise_error quantifiers_no = errorlabstrm "New Morphism" (str "One morphism argument or its output has type " ++ pr_lconstr t ++ str " but the signature requires an argument of type \"" ++ pr_lconstr rel.rel_a ++ str " " ++ prvect_with_sep pr_spc (fun _ -> str "?") (Array.make quantifiers_no 0) ++ str "\"") in let args = match kind_of_term t with App (he',args') -> let argsno = Array.length args' - rel.rel_quantifiers_no in let args1 = Array.sub args' 0 argsno in let args2 = Array.sub args' argsno rel.rel_quantifiers_no in if is_conv env Evd.empty rel.rel_a (mkApp (he',args1)) then args2 else raise_error rel.rel_quantifiers_no | _ -> if rel.rel_quantifiers_no = 0 && is_conv env Evd.empty rel.rel_a t then [||] else let args = Clenv.clenv_conv_leq env Evd.empty t rel.rel_a rel.rel_quantifiers_no in Array.of_list args in apply_to_relation args rel let unify_relation_class_carrier_with_type env rel t = match rel with Leibniz (Some t') -> if is_conv env Evd.empty t t' then rel else errorlabstrm "New Morphism" (str "One morphism argument or its output has type " ++ pr_lconstr t ++ str " but the signature requires an argument of type " ++ pr_lconstr t') | Leibniz None -> Leibniz (Some t) | Relation rel -> Relation (unify_relation_carrier_with_type env rel t) exception Impossible (* first order matching with a bit of conversion *) (* Note: the type checking operations performed by the function could *) (* be done once and for all abstracting the morphism structure using *) (* the quantifiers. Would the new structure be more suited than the *) (* existent one for other tasks to? (e.g. pretty printing would expose *) (* much more information: is it ok or is it too much information?) *) let unify_morphism_with_arguments gl (c,av) {args=args; output=output; lem=lem; morphism_theory=morphism_theory} t = let avlen = Array.length av in let argsno = List.length args in if avlen < argsno then raise Impossible; (* partial application *) let al = Array.to_list av in let quantifiers,al' = Util.list_chop (avlen - argsno) al in let quantifiersv = Array.of_list quantifiers in let c' = mkApp (c,quantifiersv) in if dependent t c' then raise Impossible; (* these are pf_type_of we could avoid *) let al'_type = List.map (Tacmach.pf_type_of gl) al' in let args' = List.map2 (fun (var,rel) ty -> var,unify_relation_class_carrier_with_type (pf_env gl) rel ty) args al'_type in (* this is another pf_type_of we could avoid *) let ty = Tacmach.pf_type_of gl (mkApp (c,av)) in let output' = unify_relation_class_carrier_with_type (pf_env gl) output ty in let lem' = mkApp (lem,quantifiersv) in let morphism_theory' = mkApp (morphism_theory,quantifiersv) in ({args=args'; output=output'; lem=lem'; morphism_theory=morphism_theory'}, c',Array.of_list al') let new_morphism m signature id hook = if Nametab.exists_cci (Lib.make_path id) or is_section_variable id then errorlabstrm "New Morphism" (pr_id id ++ str " already exists") else let env = Global.env() in let typeofm = Typing.type_of env Evd.empty m in let typ = clos_norm_flags Closure.betaiotazeta empty_env Evd.empty typeofm in let argsrev, output = match signature with None -> decompose_prod typ | Some (_,output') -> (* the carrier of the relation output' can be a Prod ==> we must uncurry on the fly output. E.g: A -> B -> C vs A -> (B -> C) args output args output *) let rel = try find_relation_class output' with Not_found -> errorlabstrm "Add Morphism" (str "Not a valid signature: " ++ pr_lconstr output' ++ str " is neither a registered relation nor the Leibniz " ++ str " equality.") in let rel_a,rel_quantifiers_no = match rel with Relation rel -> rel.rel_a, rel.rel_quantifiers_no | Leibniz (Some t) -> t, 0 | Leibniz None -> assert false in let rel_a_n = clos_norm_flags Closure.betaiotazeta empty_env Evd.empty rel_a in try let _,output_rel_a_n = decompose_lam_n rel_quantifiers_no rel_a_n in let argsrev,_ = decompose_prod output_rel_a_n in let n = List.length argsrev in let argsrev',_ = decompose_prod typ in let m = List.length argsrev' in decompose_prod_n (m - n) typ with UserError(_,_) -> (* decompose_lam_n failed. This may happen when rel_a is an axiom, a constructor, an inductive type, etc. *) decompose_prod typ in let args_ty = List.rev argsrev in let args_ty_len = List.length (args_ty) in let args_ty_quantifiers_rev,args,args_instance,output,output_instance = match signature with None -> if args_ty = [] then errorlabstrm "New Morphism" (str "The term " ++ pr_lconstr m ++ str " has type " ++ pr_lconstr typeofm ++ str " that is not a product.") ; ignore (check_is_dependent 0 args_ty output) ; let args = List.map (fun (_,ty) -> None,default_relation_for_carrier ty) args_ty in let output = default_relation_for_carrier output in [],args,args,output,output | Some (args,output') -> assert (args <> []); let number_of_arguments = List.length args in let number_of_quantifiers = args_ty_len - number_of_arguments in if number_of_quantifiers < 0 then errorlabstrm "New Morphism" (str "The morphism " ++ pr_lconstr m ++ str " has type " ++ pr_lconstr typeofm ++ str " that attends at most " ++ int args_ty_len ++ str " arguments. The signature that you specified requires " ++ int number_of_arguments ++ str " arguments.") else begin (* the real_args_ty returned are already delifted *) let args_ty_quantifiers_rev, real_args_ty, real_output = check_is_dependent number_of_quantifiers args_ty output in let quantifiers_rel_context = List.map (fun (n,t) -> n,None,t) args_ty_quantifiers_rev in let env = push_rel_context quantifiers_rel_context env in let find_relation_class t real_t = try let rel = find_relation_class t in rel, unify_relation_class_carrier_with_type env rel real_t with Not_found -> errorlabstrm "Add Morphism" (str "Not a valid signature: " ++ pr_lconstr t ++ str " is neither a registered relation nor the Leibniz " ++ str " equality.") in let find_relation_class_v (variance,t) real_t = let relation,relation_instance = find_relation_class t real_t in match relation, variance with Leibniz _, None | Relation {rel_sym = Some _}, None | Relation {rel_sym = None}, Some _ -> (variance, relation), (variance, relation_instance) | Relation {rel_sym = None},None -> errorlabstrm "Add Morphism" (str "You must specify the variance in each argument " ++ str "whose relation is asymmetric.") | Leibniz _, Some _ | Relation {rel_sym = Some _}, Some _ -> errorlabstrm "Add Morphism" (str "You cannot specify the variance of an argument " ++ str "whose relation is symmetric.") in let args, args_instance = List.split (List.map2 find_relation_class_v args real_args_ty) in let output,output_instance= find_relation_class output' real_output in args_ty_quantifiers_rev, args, args_instance, output, output_instance end in let argsconstr,outputconstr,lem = gen_compat_lemma_statement args_ty_quantifiers_rev output_instance args_instance (apply_to_rels m args_ty_quantifiers_rev) in (* "unfold make_compatibility_goal" *) let lem = Reductionops.clos_norm_flags (Closure.unfold_red (Lazy.force coq_make_compatibility_goal_eval_ref)) env Evd.empty lem in (* "unfold make_compatibility_goal_aux" *) let lem = Reductionops.clos_norm_flags (Closure.unfold_red(Lazy.force coq_make_compatibility_goal_aux_eval_ref)) env Evd.empty lem in (* "simpl" *) let lem = Tacred.nf env Evd.empty lem in if Lib.is_modtype () then begin ignore (Declare.declare_internal_constant id (ParameterEntry lem, IsAssumption Logical)) ; let mor_name = morphism_theory_id_of_morphism_proof_id id in let lemma_infos = Some (id,argsconstr,outputconstr) in add_morphism lemma_infos mor_name (m,args_ty_quantifiers_rev,args,output) end else begin new_edited id (m,args_ty_quantifiers_rev,args,argsconstr,output,outputconstr); Pfedit.start_proof id (Global, Proof Lemma) (Declare.clear_proofs (Global.named_context ())) lem hook; Options.if_verbose msg (Printer.pr_open_subgoals ()); end let morphism_hook _ ref = let pf_id = id_of_global ref in let mor_id = morphism_theory_id_of_morphism_proof_id pf_id in let (m,quantifiers_rev,args,argsconstr,output,outputconstr) = what_edited pf_id in if (is_edited pf_id) then begin add_morphism (Some (pf_id,argsconstr,outputconstr)) mor_id (m,quantifiers_rev,args,output) ; no_more_edited pf_id end type morphism_signature = (bool option * Topconstr.constr_expr) list * Topconstr.constr_expr let new_named_morphism id m sign = let sign = match sign with None -> None | Some (args,out) -> Some (List.map (fun (variance,ty) -> variance, constr_of ty) args, constr_of out) in new_morphism (constr_of m) sign id morphism_hook (************************** Adding a relation to the database *********************) let check_a env a = let typ = Typing.type_of env Evd.empty a in let a_quantifiers_rev,_ = Reduction.dest_arity env typ in a_quantifiers_rev let check_eq env a_quantifiers_rev a aeq = let typ = Sign.it_mkProd_or_LetIn (mkApp ((Lazy.force coq_relation),[| apply_to_rels a a_quantifiers_rev |])) a_quantifiers_rev in if not (is_conv env Evd.empty (Typing.type_of env Evd.empty aeq) typ) then errorlabstrm "Add Relation Class" (pr_lconstr aeq ++ str " should have type (" ++ pr_lconstr typ ++ str ")") let check_property env a_quantifiers_rev a aeq strprop coq_prop t = if not (is_conv env Evd.empty (Typing.type_of env Evd.empty t) (Sign.it_mkProd_or_LetIn (mkApp ((Lazy.force coq_prop), [| apply_to_rels a a_quantifiers_rev ; apply_to_rels aeq a_quantifiers_rev |])) a_quantifiers_rev)) then errorlabstrm "Add Relation Class" (str "Not a valid proof of " ++ str strprop ++ str ".") let check_refl env a_quantifiers_rev a aeq refl = check_property env a_quantifiers_rev a aeq "reflexivity" coq_reflexive refl let check_sym env a_quantifiers_rev a aeq sym = check_property env a_quantifiers_rev a aeq "symmetry" coq_symmetric sym let check_trans env a_quantifiers_rev a aeq trans = check_property env a_quantifiers_rev a aeq "transitivity" coq_transitive trans let check_setoid_theory env a_quantifiers_rev a aeq th = if not (is_conv env Evd.empty (Typing.type_of env Evd.empty th) (Sign.it_mkProd_or_LetIn (mkApp ((Lazy.force coq_Setoid_Theory), [| apply_to_rels a a_quantifiers_rev ; apply_to_rels aeq a_quantifiers_rev |])) a_quantifiers_rev)) then errorlabstrm "Add Relation Class" (str "Not a valid proof of symmetry") let int_add_relation id a aeq refl sym trans = let env = Global.env () in let a_quantifiers_rev = check_a env a in check_eq env a_quantifiers_rev a aeq ; option_iter (check_refl env a_quantifiers_rev a aeq) refl ; option_iter (check_sym env a_quantifiers_rev a aeq) sym ; option_iter (check_trans env a_quantifiers_rev a aeq) trans ; let quantifiers_no = List.length a_quantifiers_rev in let aeq_rel = { rel_a = a; rel_aeq = aeq; rel_refl = refl; rel_sym = sym; rel_trans = trans; rel_quantifiers_no = quantifiers_no; rel_X_relation_class = mkProp; (* dummy value, overwritten below *) rel_Xreflexive_relation_class = mkProp (* dummy value, overwritten below *) } in let x_relation_class = let subst = let len = List.length a_quantifiers_rev in Array.of_list (Util.list_map_i (fun i _ -> mkRel (len - i + 2)) 0 a_quantifiers_rev) in cic_relation_class_of_X_relation (mkRel 2) (mkRel 1) (apply_to_relation subst aeq_rel) in let _ = Declare.declare_internal_constant id (DefinitionEntry {const_entry_body = Sign.it_mkLambda_or_LetIn x_relation_class ([ Name (id_of_string "v"),None,mkRel 1; Name (id_of_string "X"),None,mkType (Termops.new_univ ())] @ a_quantifiers_rev); const_entry_type = None; const_entry_opaque = false; const_entry_boxed = Options.boxed_definitions()}, IsDefinition Definition) in let id_precise = id_of_string (string_of_id id ^ "_precise_relation_class") in let xreflexive_relation_class = let subst = let len = List.length a_quantifiers_rev in Array.of_list (Util.list_map_i (fun i _ -> mkRel (len - i)) 0 a_quantifiers_rev) in cic_precise_relation_class_of_relation (apply_to_relation subst aeq_rel) in let _ = Declare.declare_internal_constant id_precise (DefinitionEntry {const_entry_body = Sign.it_mkLambda_or_LetIn xreflexive_relation_class a_quantifiers_rev; const_entry_type = None; const_entry_opaque = false; const_entry_boxed = Options.boxed_definitions() }, IsDefinition Definition) in let aeq_rel = { aeq_rel with rel_X_relation_class = current_constant id; rel_Xreflexive_relation_class = current_constant id_precise } in Lib.add_anonymous_leaf (relation_to_obj (aeq, aeq_rel)) ; Options.if_verbose ppnl (pr_lconstr aeq ++ str " is registered as a relation"); match trans with None -> () | Some trans -> let mor_name = id_of_string (string_of_id id ^ "_morphism") in let a_instance = apply_to_rels a a_quantifiers_rev in let aeq_instance = apply_to_rels aeq a_quantifiers_rev in let sym_instance = option_map (fun x -> apply_to_rels x a_quantifiers_rev) sym in let refl_instance = option_map (fun x -> apply_to_rels x a_quantifiers_rev) refl in let trans_instance = apply_to_rels trans a_quantifiers_rev in let aeq_rel_class_and_var1, aeq_rel_class_and_var2, lemma, output = match sym_instance, refl_instance with None, None -> (Some false, Relation aeq_rel), (Some true, Relation aeq_rel), mkApp ((Lazy.force coq_equality_morphism_of_asymmetric_areflexive_transitive_relation), [| a_instance ; aeq_instance ; trans_instance |]), Lazy.force coq_impl_relation | None, Some refl_instance -> (Some false, Relation aeq_rel), (Some true, Relation aeq_rel), mkApp ((Lazy.force coq_equality_morphism_of_asymmetric_reflexive_transitive_relation), [| a_instance ; aeq_instance ; refl_instance ; trans_instance |]), Lazy.force coq_impl_relation | Some sym_instance, None -> (None, Relation aeq_rel), (None, Relation aeq_rel), mkApp ((Lazy.force coq_equality_morphism_of_symmetric_areflexive_transitive_relation), [| a_instance ; aeq_instance ; sym_instance ; trans_instance |]), Lazy.force coq_iff_relation | Some sym_instance, Some refl_instance -> (None, Relation aeq_rel), (None, Relation aeq_rel), mkApp ((Lazy.force coq_equality_morphism_of_symmetric_reflexive_transitive_relation), [| a_instance ; aeq_instance ; refl_instance ; sym_instance ; trans_instance |]), Lazy.force coq_iff_relation in let _ = Declare.declare_internal_constant mor_name (DefinitionEntry {const_entry_body = Sign.it_mkLambda_or_LetIn lemma a_quantifiers_rev; const_entry_type = None; const_entry_opaque = false; const_entry_boxed = Options.boxed_definitions()}, IsDefinition Definition) in let a_quantifiers_rev = List.map (fun (n,b,t) -> assert (b = None); n,t) a_quantifiers_rev in add_morphism None mor_name (aeq,a_quantifiers_rev,[aeq_rel_class_and_var1; aeq_rel_class_and_var2], output) (* The vernac command "Add Relation ..." *) let add_relation id a aeq refl sym trans = int_add_relation id (constr_of a) (constr_of aeq) (option_map constr_of refl) (option_map constr_of sym) (option_map constr_of trans) (************************ Add Setoid ******************************************) (* The vernac command "Add Setoid" *) let add_setoid id a aeq th = let a = constr_of a in let aeq = constr_of aeq in let th = constr_of th in let env = Global.env () in let a_quantifiers_rev = check_a env a in check_eq env a_quantifiers_rev a aeq ; check_setoid_theory env a_quantifiers_rev a aeq th ; let a_instance = apply_to_rels a a_quantifiers_rev in let aeq_instance = apply_to_rels aeq a_quantifiers_rev in let th_instance = apply_to_rels th a_quantifiers_rev in let refl = Sign.it_mkLambda_or_LetIn (mkApp ((Lazy.force coq_seq_refl), [| a_instance; aeq_instance; th_instance |])) a_quantifiers_rev in let sym = Sign.it_mkLambda_or_LetIn (mkApp ((Lazy.force coq_seq_sym), [| a_instance; aeq_instance; th_instance |])) a_quantifiers_rev in let trans = Sign.it_mkLambda_or_LetIn (mkApp ((Lazy.force coq_seq_trans), [| a_instance; aeq_instance; th_instance |])) a_quantifiers_rev in int_add_relation id a aeq (Some refl) (Some sym) (Some trans) (****************************** The tactic itself *******************************) type direction = Left2Right | Right2Left let prdirection = function Left2Right -> str "->" | Right2Left -> str "<-" type constr_with_marks = | MApp of constr * morphism_class * constr_with_marks array * direction | ToReplace | ToKeep of constr * relation relation_class * direction let is_to_replace = function | ToKeep _ -> false | ToReplace -> true | MApp _ -> true let get_mark a = Array.fold_left (||) false (Array.map is_to_replace a) let cic_direction_of_direction = function Left2Right -> Lazy.force coq_Left2Right | Right2Left -> Lazy.force coq_Right2Left let opposite_direction = function Left2Right -> Right2Left | Right2Left -> Left2Right let direction_of_constr_with_marks hole_direction = function MApp (_,_,_,dir) -> cic_direction_of_direction dir | ToReplace -> hole_direction | ToKeep (_,_,dir) -> cic_direction_of_direction dir type argument = Toapply of constr (* apply the function to the argument *) | Toexpand of name * types (* beta-expand the function w.r.t. an argument of this type *) let beta_expand c args_rev = let rec to_expand = function [] -> [] | (Toapply _)::tl -> to_expand tl | (Toexpand (name,s))::tl -> (name,s)::(to_expand tl) in let rec aux n = function [] -> [] | (Toapply arg)::tl -> arg::(aux n tl) | (Toexpand _)::tl -> (mkRel n)::(aux (n + 1) tl) in compose_lam (to_expand args_rev) (mkApp (c, Array.of_list (List.rev (aux 1 args_rev)))) exception Optimize (* used to fall-back on the tactic for Leibniz equality *) let relation_class_that_matches_a_constr caller_name new_goals hypt = let (heq, hargs) = decompose_app hypt in let rec get_all_but_last_two = function [] | [_] -> errorlabstrm caller_name (pr_lconstr hypt ++ str " is not a registered relation.") | [_;_] -> [] | he::tl -> he::(get_all_but_last_two tl) in let all_aeq_args = get_all_but_last_two hargs in let rec find_relation l subst = let aeq = mkApp (heq,(Array.of_list l)) in try let rel = find_relation_class aeq in match rel,new_goals with Leibniz _,[] -> assert (subst = []); raise Optimize (* let's optimize the proof term size *) | Leibniz (Some _), _ -> assert (subst = []); rel | Leibniz None, _ -> (* for well-typedness reasons it should have been catched by the previous guard in the previous iteration. *) assert false | Relation rel,_ -> Relation (apply_to_relation (Array.of_list subst) rel) with Not_found -> if l = [] then errorlabstrm caller_name (pr_lconstr (mkApp (aeq, Array.of_list all_aeq_args)) ++ str " is not a registered relation.") else let last,others = Util.list_sep_last l in find_relation others (last::subst) in find_relation all_aeq_args [] (* rel1 is a subrelation of rel2 whenever forall x1 x2, rel1 x1 x2 -> rel2 x1 x2 The Coq part of the tactic, however, needs rel1 == rel2. Hence the third case commented out. Note: accepting user-defined subrelations seems to be the last useful generalization that does not go against the original spirit of the tactic. *) let subrelation gl rel1 rel2 = match rel1,rel2 with Relation {rel_aeq=rel_aeq1}, Relation {rel_aeq=rel_aeq2} -> Tacmach.pf_conv_x gl rel_aeq1 rel_aeq2 | Leibniz (Some t1), Leibniz (Some t2) -> Tacmach.pf_conv_x gl t1 t2 | Leibniz None, _ | _, Leibniz None -> assert false (* This is the commented out case (see comment above) | Leibniz (Some t1), Relation {rel_a=t2; rel_refl = Some _} -> Tacmach.pf_conv_x gl t1 t2 *) | _,_ -> false (* this function returns the list of new goals opened by a constr_with_marks *) let rec collect_new_goals = function MApp (_,_,a,_) -> List.concat (List.map collect_new_goals (Array.to_list a)) | ToReplace | ToKeep (_,Leibniz _,_) | ToKeep (_,Relation {rel_refl=Some _},_) -> [] | ToKeep (c,Relation {rel_aeq=aeq; rel_refl=None},_) -> [mkApp(aeq,[|c ; c|])] (* two marked_constr are equivalent if they produce the same set of new goals *) let marked_constr_equiv_or_more_complex to_marked_constr gl c1 c2 = let glc1 = collect_new_goals (to_marked_constr c1) in let glc2 = collect_new_goals (to_marked_constr c2) in List.for_all (fun c -> List.exists (fun c' -> pf_conv_x gl c c') glc1) glc2 let pr_new_goals i c = let glc = collect_new_goals c in str " " ++ int i ++ str ") side conditions:" ++ (if glc = [] then str " no side conditions" else (pr_fnl () ++ str " " ++ prlist_with_sep (fun () -> str "\n ") (fun c -> str " ... |- " ++ pr_lconstr c) glc)) (* given a list of constr_with_marks, it returns the list where constr_with_marks than open more goals than simpler ones in the list are got rid of *) let elim_duplicates gl to_marked_constr = let rec aux = function [] -> [] | he:: tl -> if List.exists (marked_constr_equiv_or_more_complex to_marked_constr gl he) tl then aux tl else he::aux tl in aux let filter_superset_of_new_goals gl new_goals l = List.filter (fun (_,_,c) -> List.for_all (fun g -> List.exists (pf_conv_x gl g) (collect_new_goals c)) new_goals) l (* given the array of lists [| l1 ; ... ; ln |] it returns the list of arrays [ c1 ; ... ; cn ] that is the cartesian product of the sets l1, ..., ln *) let cartesian_product gl a = let rec aux = function [] -> assert false | [he] -> List.map (fun e -> [e]) he | he::tl -> let tl' = aux tl in List.flatten (List.map (function e -> List.map (function l -> e :: l) tl') he) in List.map Array.of_list (aux (List.map (elim_duplicates gl identity) (Array.to_list a))) let mark_occur gl ~new_goals t in_c input_relation input_direction = let rec aux output_relation output_directions in_c = if eq_constr t in_c then if List.mem input_direction output_directions && subrelation gl input_relation output_relation then [ToReplace] else [] else match kind_of_term in_c with | App (c,al) -> let mors_and_cs_and_als = let mors_and_cs_and_als = let morphism_table_find c = try morphism_table_find c with Not_found -> [] in let rec aux acc = function [] -> let c' = mkApp (c, Array.of_list acc) in let al' = [||] in List.map (fun m -> m,c',al') (morphism_table_find c') | (he::tl) as l -> let c' = mkApp (c, Array.of_list acc) in let al' = Array.of_list l in let acc' = acc @ [he] in (List.map (fun m -> m,c',al') (morphism_table_find c')) @ (aux acc' tl) in aux [] (Array.to_list al) in let mors_and_cs_and_als = List.map (function (m,c,al) -> relation_morphism_of_constr_morphism m, c, al) mors_and_cs_and_als in let mors_and_cs_and_als = List.fold_left (fun l (m,c,al) -> try (unify_morphism_with_arguments gl (c,al) m t) :: l with Impossible -> l ) [] mors_and_cs_and_als in List.filter (fun (mor,_,_) -> subrelation gl mor.output output_relation) mors_and_cs_and_als in (* First we look for well typed morphisms *) let res_mors = List.fold_left (fun res (mor,c,al) -> let a = let arguments = Array.of_list mor.args in let apply_variance_to_direction = function None -> [Left2Right;Right2Left] | Some true -> output_directions | Some false -> List.map opposite_direction output_directions in Util.array_map2 (fun a (variance,relation) -> (aux relation (apply_variance_to_direction variance) a) ) al arguments in let a' = cartesian_product gl a in List.flatten (List.map (fun output_direction -> (List.map (function a -> if not (get_mark a) then ToKeep (in_c,output_relation,output_direction) else MApp (c,ACMorphism mor,a,output_direction)) a')) output_directions) @ res ) [] mors_and_cs_and_als in (* Then we look for well typed functions *) let res_functions = (* the tactic works only if the function type is made of non-dependent products only. However, here we can cheat a bit by partially instantiating c to match the requirement when the arguments to be replaced are bound by non-dependent products only. *) let typeofc = Tacmach.pf_type_of gl c in let typ = nf_betaiota typeofc in let rec find_non_dependent_function env c c_args_rev typ f_args_rev a_rev = function [] -> if a_rev = [] then List.map (fun output_direction -> ToKeep (in_c,output_relation,output_direction)) output_directions else let a' = cartesian_product gl (Array.of_list (List.rev a_rev)) in List.fold_left (fun res a -> if not (get_mark a) then List.map (fun output_direction -> (ToKeep (in_c,output_relation,output_direction))) output_directions @ res else let err = match output_relation with Leibniz (Some typ') when pf_conv_x gl typ typ' -> false | Leibniz None -> assert false | _ when output_relation = Lazy.force coq_iff_relation -> false | _ -> true in if err then res else let mor = ACFunction{f_args=List.rev f_args_rev;f_output=typ} in let func = beta_expand c c_args_rev in List.map (fun output_direction -> (MApp (func,mor,a,output_direction))) output_directions @ res ) [] a' | (he::tl) as a-> let typnf = Reduction.whd_betadeltaiota env typ in match kind_of_term typnf with Cast (typ,_,_) -> find_non_dependent_function env c c_args_rev typ f_args_rev a_rev a | Prod (name,s,t) -> let env' = push_rel (name,None,s) env in let he = (aux (Leibniz (Some s)) [Left2Right;Right2Left] he) in if he = [] then [] else let he0 = List.hd he in begin match noccurn 1 t, he0 with _, ToKeep (arg,_,_) -> (* invariant: if he0 = ToKeep (t,_,_) then every element in he is = ToKeep (t,_,_) *) assert (List.for_all (function ToKeep(arg',_,_) when pf_conv_x gl arg arg' -> true | _ -> false) he) ; (* generic product, to keep *) find_non_dependent_function env' c ((Toapply arg)::c_args_rev) (subst1 arg t) f_args_rev a_rev tl | true, _ -> (* non-dependent product, to replace *) find_non_dependent_function env' c ((Toexpand (name,s))::c_args_rev) (lift 1 t) (s::f_args_rev) (he::a_rev) tl | false, _ -> (* dependent product, to replace *) (* This limitation is due to the reflexive implementation and it is hard to lift *) errorlabstrm "Setoid_replace" (str "Cannot rewrite in the argument of a " ++ str "dependent product. If you need this " ++ str "feature, please report to the author.") end | _ -> assert false in find_non_dependent_function (Tacmach.pf_env gl) c [] typ [] [] (Array.to_list al) in elim_duplicates gl identity (res_functions @ res_mors) | Prod (_, c1, c2) -> if (dependent (mkRel 1) c2) then if (occur_term t c2) then errorlabstrm "Setoid_replace" (str "Cannot rewrite in the type of a variable bound " ++ str "in a dependent product.") else List.map (fun output_direction -> ToKeep (in_c,output_relation,output_direction)) output_directions else let typeofc1 = Tacmach.pf_type_of gl c1 in if not (Tacmach.pf_conv_x gl typeofc1 mkProp) then (* to avoid this error we should introduce an impl relation whose first argument is Type instead of Prop. However, the type of the new impl would be Type -> Prop -> Prop that is no longer a Relation_Definitions.relation. Thus the Coq part of the tactic should be heavily modified. *) errorlabstrm "Setoid_replace" (str "Rewriting in a product A -> B is possible only when A " ++ str "is a proposition (i.e. A is of type Prop). The type " ++ pr_lconstr c1 ++ str " has type " ++ pr_lconstr typeofc1 ++ str " that is not convertible to Prop.") else aux output_relation output_directions (mkApp ((Lazy.force coq_impl), [| c1 ; subst1 (mkRel 1 (*dummy*)) c2 |])) | _ -> if occur_term t in_c then errorlabstrm "Setoid_replace" (str "Trying to replace " ++ pr_lconstr t ++ str " in " ++ pr_lconstr in_c ++ str " that is not an applicative context.") else List.map (fun output_direction -> ToKeep (in_c,output_relation,output_direction)) output_directions in let aux2 output_relation output_direction = List.map (fun res -> output_relation,output_direction,res) (aux output_relation [output_direction] in_c) in let res = (aux2 (Lazy.force coq_iff_relation) Right2Left) @ (* [Left2Right] is the case of a prop of signature A ++> iff or B --> iff *) (aux2 (Lazy.force coq_iff_relation) Left2Right) @ (aux2 (Lazy.force coq_impl_relation) Right2Left) in let res = elim_duplicates gl (function (_,_,t) -> t) res in let res' = filter_superset_of_new_goals gl new_goals res in match res' with [] when res = [] -> errorlabstrm "Setoid_rewrite" (str "Either the term " ++ pr_lconstr t ++ str " that must be " ++ str "rewritten occurs in a covariant position or the goal is not " ++ str "made of morphism applications only. You can replace only " ++ str "occurrences that are in a contravariant position and such that " ++ str "the context obtained by abstracting them is made of morphism " ++ str "applications only.") | [] -> errorlabstrm "Setoid_rewrite" (str "No generated set of side conditions is a superset of those " ++ str "requested by the user. The generated sets of side conditions " ++ str "are: " ++ pr_fnl () ++ prlist_with_sepi pr_fnl (fun i (_,_,mc) -> pr_new_goals i mc) res) | [he] -> he | he::_ -> Options.if_warn msg_warning (strbrk "The application of the tactic is subject to one of " ++ strbrk "the following set of side conditions that the user needs " ++ strbrk "to prove:" ++ pr_fnl () ++ prlist_with_sepi pr_fnl (fun i (_,_,mc) -> pr_new_goals i mc) res' ++ pr_fnl () ++ strbrk "The first set is randomly chosen. Use the syntax " ++ strbrk "\"setoid_rewrite ... generate side conditions ...\" to choose " ++ strbrk "a different set.") ; he let cic_morphism_context_list_of_list hole_relation hole_direction out_direction = let check = function (None,dir,dir') -> mkApp ((Lazy.force coq_MSNone), [| dir ; dir' |]) | (Some true,dir,dir') -> assert (dir = dir'); mkApp ((Lazy.force coq_MSCovariant), [| dir |]) | (Some false,dir,dir') -> assert (dir <> dir'); mkApp ((Lazy.force coq_MSContravariant), [| dir |]) in let rec aux = function [] -> assert false | [(variance,out),(value,direction)] -> mkApp ((Lazy.force coq_singl), [| Lazy.force coq_Argument_Class ; out |]), mkApp ((Lazy.force coq_fcl_singl), [| hole_relation; hole_direction ; out ; direction ; out_direction ; check (variance,direction,out_direction) ; value |]) | ((variance,out),(value,direction))::tl -> let outtl, valuetl = aux tl in mkApp ((Lazy.force coq_cons), [| Lazy.force coq_Argument_Class ; out ; outtl |]), mkApp ((Lazy.force coq_fcl_cons), [| hole_relation ; hole_direction ; out ; outtl ; direction ; out_direction ; check (variance,direction,out_direction) ; value ; valuetl |]) in aux let rec cic_type_nelist_of_list = function [] -> assert false | [value] -> mkApp ((Lazy.force coq_singl), [| mkType (Termops.new_univ ()) ; value |]) | value::tl -> mkApp ((Lazy.force coq_cons), [| mkType (Termops.new_univ ()); value; cic_type_nelist_of_list tl |]) let syntactic_but_representation_of_marked_but hole_relation hole_direction = let rec aux out (rel_out,precise_out,is_reflexive) = function MApp (f, m, args, direction) -> let direction = cic_direction_of_direction direction in let morphism_theory, relations = match m with ACMorphism { args = args ; morphism_theory = morphism_theory } -> morphism_theory,args | ACFunction { f_args = f_args ; f_output = f_output } -> let mt = if eq_constr out (cic_relation_class_of_relation_class (Lazy.force coq_iff_relation)) then mkApp ((Lazy.force coq_morphism_theory_of_predicate), [| cic_type_nelist_of_list f_args; f|]) else mkApp ((Lazy.force coq_morphism_theory_of_function), [| cic_type_nelist_of_list f_args; f_output; f|]) in mt,List.map (fun x -> None,Leibniz (Some x)) f_args in let cic_relations = List.map (fun (variance,r) -> variance, r, cic_relation_class_of_relation_class r, cic_precise_relation_class_of_relation_class r ) relations in let cic_args_relations,argst = cic_morphism_context_list_of_list hole_relation hole_direction direction (List.map2 (fun (variance,trel,t,precise_t) v -> (variance,cic_argument_class_of_argument_class (variance,trel)), (aux t precise_t v, direction_of_constr_with_marks hole_direction v) ) cic_relations (Array.to_list args)) in mkApp ((Lazy.force coq_App), [|hole_relation ; hole_direction ; cic_args_relations ; out ; direction ; morphism_theory ; argst|]) | ToReplace -> mkApp ((Lazy.force coq_ToReplace), [| hole_relation ; hole_direction |]) | ToKeep (c,_,direction) -> let direction = cic_direction_of_direction direction in if is_reflexive then mkApp ((Lazy.force coq_ToKeep), [| hole_relation ; hole_direction ; precise_out ; direction ; c |]) else let c_is_proper = let typ = mkApp (rel_out, [| c ; c |]) in mkCast (Evarutil.mk_new_meta (),DEFAULTcast, typ) in mkApp ((Lazy.force coq_ProperElementToKeep), [| hole_relation ; hole_direction; precise_out ; direction; c ; c_is_proper |]) in aux let apply_coq_setoid_rewrite hole_relation prop_relation c1 c2 (direction,h) prop_direction m = let hole_relation = cic_relation_class_of_relation_class hole_relation in let hyp,hole_direction = h,cic_direction_of_direction direction in let cic_prop_relation = cic_relation_class_of_relation_class prop_relation in let precise_prop_relation = cic_precise_relation_class_of_relation_class prop_relation in mkApp ((Lazy.force coq_setoid_rewrite), [| hole_relation ; hole_direction ; cic_prop_relation ; prop_direction ; c1 ; c2 ; syntactic_but_representation_of_marked_but hole_relation hole_direction cic_prop_relation precise_prop_relation m ; hyp |]) let check_evar_map_of_evars_defs evd = let metas = Evd.meta_list evd in let check_freemetas_is_empty rebus = Evd.Metaset.iter (fun m -> if Evd.meta_defined evd m then () else raise (Logic.RefinerError (Logic.OccurMetaGoal rebus))) in List.iter (fun (_,binding) -> match binding with Evd.Cltyp (_,{Evd.rebus=rebus; Evd.freemetas=freemetas}) -> check_freemetas_is_empty rebus freemetas | Evd.Clval (_,({Evd.rebus=rebus1; Evd.freemetas=freemetas1},_), {Evd.rebus=rebus2; Evd.freemetas=freemetas2}) -> check_freemetas_is_empty rebus1 freemetas1 ; check_freemetas_is_empty rebus2 freemetas2 ) metas (* For a correct meta-aware "rewrite in", we split unification apart from the actual rewriting (Pierre L, 05/04/06) *) (* [unification_rewrite] searchs a match for [c1] in [but] and then returns the modified objects (in particular [c1] and [c2]) *) let unification_rewrite c1 c2 cl but gl = let (env',c1) = try (* ~mod_delta:false to allow to mark occurences that must not be rewritten simply by replacing them with let-defined definitions in the context *) w_unify_to_subterm ~mod_delta:false (pf_env gl) (c1,but) cl.env with Pretype_errors.PretypeError _ -> (* ~mod_delta:true to make Ring work (since it really exploits conversion) *) w_unify_to_subterm ~mod_delta:true (pf_env gl) (c1,but) cl.env in let cl' = {cl with env = env' } in let c2 = Clenv.clenv_nf_meta cl' c2 in check_evar_map_of_evars_defs env' ; env',Clenv.clenv_value cl', c1, c2 (* no unification is performed in this function. [sigma] is the substitution obtained from an earlier unification. *) let relation_rewrite_no_unif c1 c2 hyp ~new_goals sigma gl = let but = pf_concl gl in try let input_relation = relation_class_that_matches_a_constr "Setoid_rewrite" new_goals (Typing.mtype_of (pf_env gl) sigma (snd hyp)) in let output_relation,output_direction,marked_but = mark_occur gl ~new_goals c1 but input_relation (fst hyp) in let cic_output_direction = cic_direction_of_direction output_direction in let if_output_relation_is_iff gl = let th = apply_coq_setoid_rewrite input_relation output_relation c1 c2 hyp cic_output_direction marked_but in let new_but = Termops.replace_term c1 c2 but in let hyp1,hyp2,proj = match output_direction with Right2Left -> new_but, but, Lazy.force coq_proj1 | Left2Right -> but, new_but, Lazy.force coq_proj2 in let impl1 = mkProd (Anonymous, hyp2, lift 1 hyp1) in let impl2 = mkProd (Anonymous, hyp1, lift 1 hyp2) in let th' = mkApp (proj, [|impl2; impl1; th|]) in Tactics.refine (mkApp (th',[|mkCast (Evarutil.mk_new_meta(), DEFAULTcast, new_but)|])) gl in let if_output_relation_is_if gl = let th = apply_coq_setoid_rewrite input_relation output_relation c1 c2 hyp cic_output_direction marked_but in let new_but = Termops.replace_term c1 c2 but in Tactics.refine (mkApp (th, [|mkCast (Evarutil.mk_new_meta(), DEFAULTcast, new_but)|])) gl in if output_relation = (Lazy.force coq_iff_relation) then if_output_relation_is_iff gl else if_output_relation_is_if gl with Optimize -> !general_rewrite (fst hyp = Left2Right) (snd hyp) gl let relation_rewrite c1 c2 (input_direction,cl) ~new_goals gl = let (sigma,cl,c1,c2) = unification_rewrite c1 c2 cl (pf_concl gl) gl in relation_rewrite_no_unif c1 c2 (input_direction,cl) ~new_goals sigma gl let analyse_hypothesis gl c = let ctype = pf_type_of gl c in let eqclause = Clenv.make_clenv_binding gl (c,ctype) Rawterm.NoBindings in let (equiv, args) = decompose_app (Clenv.clenv_type eqclause) in let rec split_last_two = function | [c1;c2] -> [],(c1, c2) | x::y::z -> let l,res = split_last_two (y::z) in x::l, res | _ -> error "The term provided is not an equivalence" in let others,(c1,c2) = split_last_two args in eqclause,mkApp (equiv, Array.of_list others),c1,c2 let general_s_rewrite lft2rgt c ~new_goals gl = let eqclause,_,c1,c2 = analyse_hypothesis gl c in if lft2rgt then relation_rewrite c1 c2 (Left2Right,eqclause) ~new_goals gl else relation_rewrite c2 c1 (Right2Left,eqclause) ~new_goals gl let relation_rewrite_in id c1 c2 (direction,eqclause) ~new_goals gl = let hyp = pf_type_of gl (mkVar id) in (* first, we find a match for c1 in the hyp *) let (sigma,cl,c1,c2) = unification_rewrite c1 c2 eqclause hyp gl in (* since we will actually rewrite in the opposite direction, we also need to replace every occurrence of c2 (resp. c1) in hyp with something that is convertible but not syntactically equal. To this aim we introduce a let-in and then we will use the intro tactic to get rid of it. Quite tricky to do properly since c1 can occur in c2 or vice-versa ! *) let mangled_new_hyp = let hyp = lift 2 hyp in (* first, we backup every occurences of c1 in newly allocated (Rel 1) *) let hyp = Termops.replace_term (lift 2 c1) (mkRel 1) hyp in (* then, we factorize every occurences of c2 into (Rel 2) *) let hyp = Termops.replace_term (lift 2 c2) (mkRel 2) hyp in (* Now we substitute (Rel 1) (i.e. c1) for c2 *) let hyp = subst1 (lift 1 c2) hyp in (* Since subst1 has killed Rel 1 and decreased the other Rels, Rel 1 is now coding for c2, we can build the let-in factorizing c2 *) mkLetIn (Anonymous,c2,pf_type_of gl c2,hyp) in let new_hyp = Termops.replace_term c1 c2 hyp in let oppdir = opposite_direction direction in cut_replacing id new_hyp (tclTHENLAST (tclTHEN (change_in_concl None mangled_new_hyp) (tclTHEN intro (relation_rewrite_no_unif c2 c1 (oppdir,cl) ~new_goals sigma)))) gl let general_s_rewrite_in id lft2rgt c ~new_goals gl = let eqclause,_,c1,c2 = analyse_hypothesis gl c in if lft2rgt then relation_rewrite_in id c1 c2 (Left2Right,eqclause) ~new_goals gl else relation_rewrite_in id c2 c1 (Right2Left,eqclause) ~new_goals gl (* [general_setoid_replace rewrite_tac try_prove_eq_tac_opt relation c1 c2 ~new_goals ] common part of [setoid_replace] and [setoid_replace_in] (distinction is done using rewrite_tac). Algorith sketch: 1- find the (setoid) relation [rel] between [c1] and [c2] using [relation] 2- assert [H:rel c2 c1] 3- replace [c1] with [c2] using [rewrite_tac] (should be [general_s_rewrite] if we want to replace in the goal, and [general_s_rewrite_in id] if we want to replace in the hypothesis [id]). Possibly generate new_goals if asked (cf general_s_rewrite) 4- if [try_prove_eq_tac_opt] is [Some tac] try to complete [rel c2 c1] using tac and do nothing if [try_prove_eq_tac_opt] is [None] *) let general_setoid_replace rewrite_tac try_prove_eq_tac_opt relation c1 c2 ~new_goals gl = let try_prove_eq_tac = match try_prove_eq_tac_opt with | None -> Tacticals.tclIDTAC | Some tac -> Tacticals.tclTRY (Tacticals.tclCOMPLETE tac ) in try let relation = match relation with Some rel -> (try match find_relation_class rel with Relation sa -> sa | Leibniz _ -> raise Optimize with Not_found -> errorlabstrm "Setoid_rewrite" (pr_lconstr rel ++ str " is not a registered relation.")) | None -> match default_relation_for_carrier (pf_type_of gl c1) with Relation sa -> sa | Leibniz _ -> raise Optimize in let eq_left_to_right = mkApp (relation.rel_aeq, [| c1 ; c2 |]) in let eq_right_to_left = mkApp (relation.rel_aeq, [| c2 ; c1 |]) in let replace dir eq = tclTHENS (assert_tac false Anonymous eq) [onLastHyp (fun id -> tclTHEN (rewrite_tac dir (mkVar id) ~new_goals) (clear [id])); try_prove_eq_tac] in tclORELSE (replace true eq_left_to_right) (replace false eq_right_to_left) gl with Optimize -> (* (!replace tac_opt c1 c2) gl *) let eq = mkApp (Lazy.force coq_eq, [| pf_type_of gl c1;c2 ; c1 |]) in tclTHENS (assert_tac false Anonymous eq) [onLastHyp (fun id -> tclTHEN (rewrite_tac false (mkVar id) ~new_goals) (clear [id])); try_prove_eq_tac] gl let setoid_replace = general_setoid_replace general_s_rewrite let setoid_replace_in tac_opt id relation c1 c2 ~new_goals gl = general_setoid_replace (general_s_rewrite_in id) tac_opt relation c1 c2 ~new_goals gl (* [setoid_]{reflexivity,symmetry,transitivity} tactics *) let setoid_reflexivity gl = try let relation_class = relation_class_that_matches_a_constr "Setoid_reflexivity" [] (pf_concl gl) in match relation_class with Leibniz _ -> assert false (* since [] is empty *) | Relation rel -> match rel.rel_refl with None -> errorlabstrm "Setoid_reflexivity" (str "The relation " ++ prrelation rel ++ str " is not reflexive.") | Some refl -> apply refl gl with Optimize -> reflexivity gl let setoid_symmetry gl = try let relation_class = relation_class_that_matches_a_constr "Setoid_symmetry" [] (pf_concl gl) in match relation_class with Leibniz _ -> assert false (* since [] is empty *) | Relation rel -> match rel.rel_sym with None -> errorlabstrm "Setoid_symmetry" (str "The relation " ++ prrelation rel ++ str " is not symmetric.") | Some sym -> apply sym gl with Optimize -> symmetry gl let setoid_symmetry_in id gl = let new_hyp = let _,he,c1,c2 = analyse_hypothesis gl (mkVar id) in mkApp (he, [| c2 ; c1 |]) in cut_replacing id new_hyp (tclTHEN setoid_symmetry) gl let setoid_transitivity c gl = try let relation_class = relation_class_that_matches_a_constr "Setoid_transitivity" [] (pf_concl gl) in match relation_class with Leibniz _ -> assert false (* since [] is empty *) | Relation rel -> let ctyp = pf_type_of gl c in let rel' = unify_relation_carrier_with_type (pf_env gl) rel ctyp in match rel'.rel_trans with None -> errorlabstrm "Setoid_transitivity" (str "The relation " ++ prrelation rel ++ str " is not transitive.") | Some trans -> let transty = nf_betaiota (pf_type_of gl trans) in let argsrev, _ = Reductionops.decomp_n_prod (pf_env gl) Evd.empty 2 transty in let binder = match List.rev argsrev with _::(Name n2,None,_)::_ -> Rawterm.NamedHyp n2 | _ -> assert false in apply_with_bindings (trans, Rawterm.ExplicitBindings [ dummy_loc, binder, c ]) gl with Optimize -> transitivity c gl ;; Tactics.register_setoid_reflexivity setoid_reflexivity;; Tactics.register_setoid_symmetry setoid_symmetry;; Tactics.register_setoid_symmetry_in setoid_symmetry_in;; Tactics.register_setoid_transitivity setoid_transitivity;;