(************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) (* __r with type (A:)(x:A)(P:A->Prop)(P x)->(y:A)(eqname A y x)->(P y). If another equality myeq is introduced, then corresponding theorems myeq_ind_r, myeq_rec_r and myeq_rect_r have to be proven. See below. -- Eduardo (19/8/97) *) (* Ad hoc asymmetric general_elim_clause *) let general_elim_clause with_evars cls sigma c l elim = try (match cls with | None -> (* was tclWEAK_PROGRESS which only fails for tactics generating one subgoal and did not fail for useless conditional rewritings generating an extra condition *) tclNOTSAMEGOAL (tclTHEN (Refiner.tclEVARS sigma) (general_elim with_evars (c,l) elim ~allow_K:false)) | Some id -> tclTHEN (Refiner.tclEVARS sigma) (general_elim_in with_evars id (c,l) elim)) with Pretype_errors.PretypeError (env, (Pretype_errors.NoOccurrenceFound (c', _))) -> raise (Pretype_errors.PretypeError (env, (Pretype_errors.NoOccurrenceFound (c', cls)))) let elimination_sort_of_clause = function | None -> elimination_sort_of_goal | Some id -> elimination_sort_of_hyp id (* The next function decides in particular whether to try a regular rewrite or a setoid rewrite. Approach is to break everything, if [eq] appears in head position then regular rewrite else try setoid rewrite. If occurrences are set, use setoid_rewrite. *) let general_setoid_rewrite_clause = ref (fun _ -> assert false) let register_general_setoid_rewrite_clause = (:=) general_setoid_rewrite_clause let is_applied_setoid_relation = ref (fun _ -> false) let register_is_applied_setoid_relation = (:=) is_applied_setoid_relation let is_applied_relation t = match kind_of_term t with | App (c, args) when Array.length args >= 2 -> true | _ -> false let leibniz_rewrite_ebindings_clause cls lft2rgt sigma c l with_evars gl hdcncl = let hdcncls = string_of_inductive hdcncl in let suffix = elimination_suffix (elimination_sort_of_clause cls gl) in let dir = if cls=None then lft2rgt else not lft2rgt in let rwr_thm = if dir then hdcncls^suffix^"_r" else hdcncls^suffix in let elim = try pf_global gl (id_of_string rwr_thm) with Not_found -> error ("Cannot find rewrite principle "^rwr_thm^".") in general_elim_clause with_evars cls sigma c l (elim,NoBindings) gl let leibniz_eq = Lazy.lazy_from_fun build_coq_eq let general_rewrite_ebindings_clause cls lft2rgt occs ((c,l) : open_constr with_bindings) with_evars gl = if occs <> all_occurrences then ( !general_setoid_rewrite_clause cls lft2rgt occs c ~new_goals:[] gl) else let env = pf_env gl in let sigma, c' = c in let sigma = Evd.merge sigma (project gl) in let ctype = get_type_of env sigma c' in let rels, t = decompose_prod (whd_betaiotazeta ctype) in match match_with_equation t with | Some (hdcncl,_) -> (* Fast path: direct leibniz rewrite *) leibniz_rewrite_ebindings_clause cls lft2rgt sigma c' l with_evars gl hdcncl | None -> let env' = List.fold_left (fun env (n,t) -> push_rel (n, None, t) env) env rels in let _,t' = splay_prod env' sigma t in (* Search for underlying eq *) match match_with_equation t' with | Some (hdcncl,_) -> (* Maybe a setoid relation with eq inside *) if l = NoBindings && !is_applied_setoid_relation t then !general_setoid_rewrite_clause cls lft2rgt occs c ~new_goals:[] gl else (try leibniz_rewrite_ebindings_clause cls lft2rgt sigma c' l with_evars gl hdcncl with e -> try !general_setoid_rewrite_clause cls lft2rgt occs c ~new_goals:[] gl with _ -> raise e) | None -> (* Can't be leibniz, try setoid *) if l = NoBindings then !general_setoid_rewrite_clause cls lft2rgt occs c ~new_goals:[] gl else error "The term provided does not end with an equation." let general_rewrite_ebindings = general_rewrite_ebindings_clause None let general_rewrite_bindings l2r occs (c,bl) = general_rewrite_ebindings_clause None l2r occs (inj_open c,inj_ebindings bl) let general_rewrite l2r occs c = general_rewrite_bindings l2r occs (c,NoBindings) false let general_rewrite_ebindings_in l2r occs id = general_rewrite_ebindings_clause (Some id) l2r occs let general_rewrite_bindings_in l2r occs id (c,bl) = general_rewrite_ebindings_clause (Some id) l2r occs (inj_open c,inj_ebindings bl) let general_rewrite_in l2r occs id c = general_rewrite_ebindings_clause (Some id) l2r occs (inj_open c,NoBindings) let general_multi_rewrite l2r with_evars c cl = let occs_of = on_snd (List.fold_left (fun acc -> function ArgArg x -> x :: acc | ArgVar _ -> acc) []) in match cl.onhyps with | Some l -> (* If a precise list of locations is given, success is mandatory for each of these locations. *) let rec do_hyps = function | [] -> tclIDTAC | ((occs,id),_) :: l -> tclTHENFIRST (general_rewrite_ebindings_in l2r (occs_of occs) id c with_evars) (do_hyps l) in if cl.concl_occs = no_occurrences_expr then do_hyps l else tclTHENFIRST (general_rewrite_ebindings l2r (occs_of cl.concl_occs) c with_evars) (do_hyps l) | None -> (* Otherwise, if we are told to rewrite in all hypothesis via the syntax "* |-", we fail iff all the different rewrites fail *) let rec do_hyps_atleastonce = function | [] -> (fun gl -> error "Nothing to rewrite.") | id :: l -> tclIFTHENTRYELSEMUST (general_rewrite_ebindings_in l2r all_occurrences id c with_evars) (do_hyps_atleastonce l) in let do_hyps gl = (* If the term to rewrite uses an hypothesis H, don't rewrite in H *) let ids = let ids_in_c = Environ.global_vars_set (Global.env()) (snd (fst c)) in Idset.fold (fun id l -> list_remove id l) ids_in_c (pf_ids_of_hyps gl) in do_hyps_atleastonce ids gl in if cl.concl_occs = no_occurrences_expr then do_hyps else tclIFTHENTRYELSEMUST (general_rewrite_ebindings l2r (occs_of cl.concl_occs) c with_evars) do_hyps let general_multi_multi_rewrite with_evars l cl tac = let do1 l2r c = match tac with None -> general_multi_rewrite l2r with_evars c cl | Some tac -> tclTHENSFIRSTn (general_multi_rewrite l2r with_evars c cl) [|tclIDTAC|] (tclCOMPLETE tac) in let rec doN l2r c = function | Precisely n when n <= 0 -> tclIDTAC | Precisely 1 -> do1 l2r c | Precisely n -> tclTHENFIRST (do1 l2r c) (doN l2r c (Precisely (n-1))) | RepeatStar -> tclREPEAT_MAIN (do1 l2r c) | RepeatPlus -> tclTHENFIRST (do1 l2r c) (doN l2r c RepeatStar) | UpTo n when n<=0 -> tclIDTAC | UpTo n -> tclTHENFIRST (tclTRY (do1 l2r c)) (doN l2r c (UpTo (n-1))) in let rec loop = function | [] -> tclIDTAC | (l2r,m,c)::l -> tclTHENFIRST (doN l2r c m) (loop l) in loop l (* Conditional rewriting, the success of a rewriting is related to the resolution of the conditions by a given tactic *) let conditional_rewrite lft2rgt tac (c,bl) = tclTHENSFIRSTn (general_rewrite_ebindings lft2rgt all_occurrences (c,bl) false) [|tclIDTAC|] (tclCOMPLETE tac) let rewriteLR_bindings = general_rewrite_bindings true all_occurrences let rewriteRL_bindings = general_rewrite_bindings false all_occurrences let rewriteLR = general_rewrite true all_occurrences let rewriteRL = general_rewrite false all_occurrences let rewriteLRin_bindings = general_rewrite_bindings_in true all_occurrences let rewriteRLin_bindings = general_rewrite_bindings_in false all_occurrences let conditional_rewrite_in lft2rgt id tac (c,bl) = tclTHENSFIRSTn (general_rewrite_ebindings_in lft2rgt all_occurrences id (c,bl) false) [|tclIDTAC|] (tclCOMPLETE tac) let rewriteRL_clause = function | None -> rewriteRL_bindings | Some id -> rewriteRLin_bindings id (* Replacing tactics *) (* eq,sym_eq : equality on Type and its symmetry theorem c2 c1 : c1 is to be replaced by c2 unsafe : If true, do not check that c1 and c2 are convertible tac : Used to prove the equality c1 = c2 gl : goal *) let multi_replace clause c2 c1 unsafe try_prove_eq_opt gl = let try_prove_eq = match try_prove_eq_opt with | None -> tclIDTAC | Some tac -> tclCOMPLETE tac in let t1 = pf_apply get_type_of gl c1 and t2 = pf_apply get_type_of gl c2 in if unsafe or (pf_conv_x gl t1 t2) then let e = build_coq_eq () in let sym = build_coq_sym_eq () in let eq = applist (e, [t1;c1;c2]) in tclTHENS (assert_tac false Anonymous eq) [onLastHyp (fun id -> tclTHEN (tclTRY (general_multi_rewrite false false (inj_open (mkVar id),NoBindings) clause)) (clear [id])); tclFIRST [assumption; tclTHEN (apply sym) assumption; try_prove_eq ] ] gl else error "Terms do not have convertible types." let replace c2 c1 gl = multi_replace onConcl c2 c1 false None gl let replace_in id c2 c1 gl = multi_replace (onHyp id) c2 c1 false None gl let replace_by c2 c1 tac gl = multi_replace onConcl c2 c1 false (Some tac) gl let replace_in_by id c2 c1 tac gl = multi_replace (onHyp id) c2 c1 false (Some tac) gl let replace_in_clause_maybe_by c2 c1 cl tac_opt gl = multi_replace cl c2 c1 false tac_opt gl (* End of Eduardo's code. The rest of this file could be improved using the functions match_with_equation, etc that I defined in Pattern.ml. -- Eduardo (19/8/97) *) (* Tactics for equality reasoning with the "eq" relation. This code will work with any equivalence relation which is substitutive *) (* [find_positions t1 t2] will find the positions in the two terms which are suitable for discrimination, or for injection. Obviously, if there is a position which is suitable for discrimination, then we want to exploit it, and not bother with injection. So when we find a position which is suitable for discrimination, we will just raise an exception with that position. So the algorithm goes like this: if [t1] and [t2] start with the same constructor, then we can continue to try to find positions in the arguments of [t1] and [t2]. if [t1] and [t2] do not start with the same constructor, then we have found a discrimination position if one [t1] or [t2] do not start with a constructor and the two terms are not already convertible, then we have found an injection position. A discriminating position consists of a constructor-path and a pair of operators. The constructor-path tells us how to get down to the place where the two operators, which must differ, can be found. An injecting position has two terms instead of the two operators, since these terms are different, but not manifestly so. A constructor-path is a list of pairs of (operator * int), where the int (based at 0) tells us which argument of the operator we descended into. *) exception DiscrFound of (constructor * int) list * constructor * constructor let find_positions env sigma t1 t2 = let rec findrec sorts posn t1 t2 = let hd1,args1 = whd_betadeltaiota_stack env sigma t1 in let hd2,args2 = whd_betadeltaiota_stack env sigma t2 in match (kind_of_term hd1, kind_of_term hd2) with | Construct sp1, Construct sp2 when List.length args1 = mis_constructor_nargs_env env sp1 -> let sorts = list_intersect sorts (allowed_sorts env (fst sp1)) in (* both sides are fully applied constructors, so either we descend, or we can discriminate here. *) if is_conv env sigma hd1 hd2 then let nrealargs = constructor_nrealargs env sp1 in let rargs1 = list_lastn nrealargs args1 in let rargs2 = list_lastn nrealargs args2 in List.flatten (list_map2_i (fun i -> findrec sorts ((sp1,i)::posn)) 0 rargs1 rargs2) else if List.mem InType sorts then (* see build_discriminator *) raise (DiscrFound (List.rev posn,sp1,sp2)) else [] | _ -> let t1_0 = applist (hd1,args1) and t2_0 = applist (hd2,args2) in if is_conv env sigma t1_0 t2_0 then [] else let ty1_0 = get_type_of env sigma t1_0 in let s = get_sort_family_of env sigma ty1_0 in if List.mem s sorts then [(List.rev posn,t1_0,t2_0)] else [] in try (* Rem: to allow injection on proofs objects, just add InProp *) Inr (findrec [InSet;InType] [] t1 t2) with DiscrFound (path,c1,c2) -> Inl (path,c1,c2) let discriminable env sigma t1 t2 = match find_positions env sigma t1 t2 with | Inl _ -> true | _ -> false let injectable env sigma t1 t2 = match find_positions env sigma t1 t2 with | Inl _ | Inr [] -> false | Inr _ -> true (* Once we have found a position, we need to project down to it. If we are discriminating, then we need to produce False on one of the branches of the discriminator, and True on the other one. So the result type of the case-expressions is always Prop. If we are injecting, then we need to discover the result-type. This can be difficult, since the type of the two terms at the injection-position can be different, and we need to find a dependent sigma-type which generalizes them both. We can get an approximation to the right type to choose by: (0) Before beginning, we reserve a patvar for the default value of the match, to be used in all the bogus branches. (1) perform the case-splits, down to the site of the injection. At each step, we have a term which is the "head" of the next case-split. At the point when we actually reach the end of our path, the "head" is the term to return. We compute its type, and then, backwards, make a sigma-type with every free debruijn reference in that type. We can be finer, and first do a S(TRONG)NF on the type, so that we get the fewest number of references possible. (2) This gives us a closed type for the head, which we use for the types of all the case-splits. (3) Now, we can compute the type of one of T1, T2, and then unify it with the type of the last component of the result-type, and this will give us the bindings for the other arguments of the tuple. *) (* The algorithm, then is to perform successive case-splits. We have the result-type of the case-split, and also the type of that result-type. We have a "direction" we want to follow, i.e. a constructor-number, and in all other "directions", we want to juse use the default-value. After doing the case-split, we call the afterfun, with the updated environment, to produce the term for the desired "direction". The assumption is made here that the result-type is not manifestly functional, so we can just use the length of the branch-type to know how many lambda's to stick in. *) (* [descend_then sigma env head dirn] returns the number of products introduced, and the environment which is active, in the body of the case-branch given by [dirn], along with a continuation, which expects to be fed: (1) the value of the body of the branch given by [dirn] (2) the default-value (3) the type of the default-value, which must also be the type of the body of the [dirn] branch the continuation then constructs the case-split. *) let descend_then sigma env head dirn = let IndType (indf,_) = try find_rectype env sigma (get_type_of env sigma head) with Not_found -> assert false in let ind,_ = dest_ind_family indf in let (mib,mip) = lookup_mind_specif env ind in let cstr = get_constructors env indf in let dirn_nlams = cstr.(dirn-1).cs_nargs in let dirn_env = push_rel_context cstr.(dirn-1).cs_args env in (dirn_nlams, dirn_env, (fun dirnval (dfltval,resty) -> let deparsign = make_arity_signature env true indf in let p = it_mkLambda_or_LetIn (lift (mip.mind_nrealargs+1) resty) deparsign in let build_branch i = let result = if i = dirn then dirnval else dfltval in it_mkLambda_or_LetIn_name env result cstr.(i-1).cs_args in let brl = List.map build_branch (interval 1 (Array.length mip.mind_consnames)) in let ci = make_case_info env ind RegularStyle in mkCase (ci, p, head, Array.of_list brl))) (* Now we need to construct the discriminator, given a discriminable position. This boils down to: (1) If the position is directly beneath us, then we need to do a case-split, with result-type Prop, and stick True and False into the branches, as is convenient. (2) If the position is not directly beneath us, then we need to call descend_then, to descend one step, and then recursively construct the discriminator. *) (* [construct_discriminator env dirn headval] constructs a case-split on [headval], with the [dirn]-th branch giving [True], and all the rest giving False. *) let construct_discriminator sigma env dirn c sort = let IndType(indf,_) = try find_rectype env sigma (get_type_of env sigma c) with Not_found -> (* one can find Rel(k) in case of dependent constructors like T := c : (A:Set)A->T and a discrimination on (c bool true) = (c bool false) CP : changed assert false in a more informative error *) errorlabstrm "Equality.construct_discriminator" (str "Cannot discriminate on inductive constructors with dependent types.") in let (ind,_) = dest_ind_family indf in let (mib,mip) = lookup_mind_specif env ind in let (true_0,false_0,sort_0) = build_coq_True(),build_coq_False(),Prop Null in let deparsign = make_arity_signature env true indf in let p = it_mkLambda_or_LetIn (mkSort sort_0) deparsign in let cstrs = get_constructors env indf in let build_branch i = let endpt = if i = dirn then true_0 else false_0 in it_mkLambda_or_LetIn endpt cstrs.(i-1).cs_args in let brl = List.map build_branch(interval 1 (Array.length mip.mind_consnames)) in let ci = make_case_info env ind RegularStyle in mkCase (ci, p, c, Array.of_list brl) let rec build_discriminator sigma env dirn c sort = function | [] -> construct_discriminator sigma env dirn c sort | ((sp,cnum),argnum)::l -> let (cnum_nlams,cnum_env,kont) = descend_then sigma env c cnum in let newc = mkRel(cnum_nlams-argnum) in let subval = build_discriminator sigma cnum_env dirn newc sort l in kont subval (build_coq_False (),mkSort (Prop Null)) (* Note: discrimination could be more clever: if some elimination is not allowed because of a large impredicative constructor in the path (see allowed_sorts in find_positions), the positions could still be discrimated by projecting first instead of putting the discrimination combinator inside the projecting combinator. Example of relevant situation: Inductive t:Set := c : forall A:Set, A -> nat -> t. Goal ~ c _ 0 0 = c _ 0 1. intro. discriminate H. *) let gen_absurdity id gl = if is_empty_type (clause_type (onHyp id) gl) then simplest_elim (mkVar id) gl else errorlabstrm "Equality.gen_absurdity" (str "Not the negation of an equality.") (* Precondition: eq is leibniz equality returns ((eq_elim t t1 P i t2), absurd_term) where P=[e:t]discriminator absurd_term=False *) let discrimination_pf e (t,t1,t2) discriminator lbeq = let i = build_coq_I () in let absurd_term = build_coq_False () in let eq_elim = lbeq.ind in (applist (eq_elim, [t;t1;mkNamedLambda e t discriminator;i;t2]), absurd_term) exception NotDiscriminable let eq_baseid = id_of_string "e" let apply_on_clause (f,t) clause = let sigma = Evd.evars_of clause.evd in let f_clause = mk_clenv_from_env clause.env sigma None (f,t) in let argmv = (match kind_of_term (last_arg f_clause.templval.Evd.rebus) with | Meta mv -> mv | _ -> errorlabstrm "" (str "Ill-formed clause applicator.")) in clenv_fchain argmv f_clause clause let discr_positions env sigma (lbeq,(t,t1,t2)) eq_clause cpath dirn sort = let e = next_ident_away eq_baseid (ids_of_context env) in let e_env = push_named (e,None,t) env in let eqn = mkApp(lbeq.eq,[|t;t1;t2|]) in let discriminator = build_discriminator sigma e_env dirn (mkVar e) sort cpath in let (pf, absurd_term) = discrimination_pf e (t,t1,t2) discriminator lbeq in let pf_ty = mkArrow eqn absurd_term in let absurd_clause = apply_on_clause (pf,pf_ty) eq_clause in let pf = clenv_value_cast_meta absurd_clause in tclTHENS (cut_intro absurd_term) [onLastHyp gen_absurdity; refine pf] let discrEq (lbeq,(t,t1,t2) as u) eq_clause gls = let sigma = Evd.evars_of eq_clause.evd in let env = pf_env gls in match find_positions env sigma t1 t2 with | Inr _ -> errorlabstrm "discr" (str"Not a discriminable equality.") | Inl (cpath, (_,dirn), _) -> let sort = pf_apply get_type_of gls (pf_concl gls) in discr_positions env sigma u eq_clause cpath dirn sort gls let onEquality with_evars tac (c,lbindc) gls = let t = pf_type_of gls c in let t' = try snd (pf_reduce_to_quantified_ind gls t) with UserError _ -> t in let eq_clause = make_clenv_binding gls (c,t') lbindc in let eq_clause' = clenv_pose_dependent_evars with_evars eq_clause in let eqn = clenv_type eq_clause' in let eq = try find_eq_data_decompose eqn with PatternMatchingFailure -> errorlabstrm "" (str"No primitive equality found.") in tclTHEN (Refiner.tclEVARS (Evd.evars_of eq_clause'.evd)) (tac eq eq_clause') gls let onNegatedEquality with_evars tac gls = let ccl = pf_concl gls in match kind_of_term (hnf_constr (pf_env gls) (project gls) ccl) with | Prod (_,t,u) when is_empty_type u -> tclTHEN introf (onLastHyp (fun id -> onEquality with_evars tac (mkVar id,NoBindings))) gls | _ -> errorlabstrm "" (str "Not a negated primitive equality.") let discrSimpleClause with_evars = function | None -> onNegatedEquality with_evars discrEq | Some ((_,id),_) -> onEquality with_evars discrEq (mkVar id,NoBindings) let discr with_evars = onEquality with_evars discrEq let discrClause with_evars = onClauses (discrSimpleClause with_evars) let discrEverywhere with_evars = tclORELSE (tclTHEN (tclREPEAT introf) (Tacticals.tryAllHyps (fun id -> tclCOMPLETE (discr with_evars (mkVar id,NoBindings))))) (fun gls -> errorlabstrm "DiscrEverywhere" (str"No discriminable equalities.")) let discr_tac with_evars = function | None -> discrEverywhere with_evars | Some c -> onInductionArg (discr with_evars) c let discrConcl gls = discrClause false onConcl gls let discrHyp id gls = discrClause false (onHyp id) gls (* returns the sigma type (sigS, sigT) with the respective constructor depending on the sort *) (* J.F.: correction du bug #1167 en accord avec Hugo. *) let find_sigma_data s = build_sigma_type () (* [make_tuple env sigma (rterm,rty) lind] assumes [lind] is the lesser index bound in [rty] Then we build the term [(existT A P (mkRel lind) rterm)] of type [(sigS A P)] where [A] is the type of [mkRel lind] and [P] is [\na:A.rty{1/lind}] *) let make_tuple env sigma (rterm,rty) lind = assert (dependent (mkRel lind) rty); let {intro = exist_term; typ = sig_term} = find_sigma_data (get_sort_of env sigma rty) in let a = type_of env sigma (mkRel lind) in let (na,_,_) = lookup_rel lind env in (* We move [lind] to [1] and lift other rels > [lind] by 1 *) let rty = lift (1-lind) (liftn lind (lind+1) rty) in (* Now [lind] is [mkRel 1] and we abstract on (na:a) *) let p = mkLambda (na, a, rty) in (applist(exist_term,[a;p;(mkRel lind);rterm]), applist(sig_term,[a;p])) (* check that the free-references of the type of [c] are contained in the free-references of the normal-form of that type. Strictly computing the exact set of free rels would require full normalization but this is not reasonable (e.g. in presence of records that contains proofs). We restrict ourself to a "simpl" normalization *) let minimal_free_rels env sigma (c,cty) = let cty_rels = free_rels cty in let cty' = simpl env sigma cty in let rels' = free_rels cty' in if Intset.subset cty_rels rels' then (cty,cty_rels) else (cty',rels') (* [sig_clausal_form siglen ty] Will explode [siglen] [sigS,sigT ]'s on [ty] (depending on the type of ty), and return: (1) a pattern, with meta-variables in it for various arguments, which, when the metavariables are replaced with appropriate terms, will have type [ty] (2) an integer, which is the last argument - the one which we just returned. (3) a pattern, for the type of that last meta (4) a typing for each patvar WARNING: No checking is done to make sure that the sigS(or sigT)'s are actually there. - Only homogenious pairs are built i.e. pairs where all the dependencies are of the same sort [sig_clausal_form] proceed as follows: the default tuple is constructed by taking the tuple-type, exploding the first [tuplen] [sigS]'s, and replacing at each step the binder in the right-hand-type by a fresh metavariable. In addition, on the way back out, we will construct the pattern for the tuple which uses these meta-vars. This gives us a pattern, which we use to match against the type of [dflt]; if that fails, then against the S(TRONG)NF of that type. If both fail, then we just cannot construct our tuple. If one of those succeed, then we can construct our value easily - we just use the tuple-pattern. *) let sig_clausal_form env sigma sort_of_ty siglen ty dflt = let { intro = exist_term } = find_sigma_data sort_of_ty in let evdref = ref (Evd.create_goal_evar_defs sigma) in let rec sigrec_clausal_form siglen p_i = if siglen = 0 then (* is the default value typable with the expected type *) let dflt_typ = type_of env sigma dflt in if Evarconv.e_cumul env evdref dflt_typ p_i then (* the_conv_x had a side-effect on evdref *) dflt else error "Cannot solve an unification problem." else let (a,p_i_minus_1) = match whd_beta_stack p_i with | (_sigS,[a;p]) -> (a,p) | _ -> anomaly "sig_clausal_form: should be a sigma type" in let ev = Evarutil.e_new_evar evdref env a in let rty = beta_applist(p_i_minus_1,[ev]) in let tuple_tail = sigrec_clausal_form (siglen-1) rty in match Evd.existential_opt_value (Evd.evars_of !evdref) (destEvar ev) with | Some w -> applist(exist_term,[a;p_i_minus_1;w;tuple_tail]) | None -> anomaly "Not enough components to build the dependent tuple" in let scf = sigrec_clausal_form siglen ty in Evarutil.nf_evar (Evd.evars_of !evdref) scf (* The problem is to build a destructor (a generalization of the predecessor) which, when applied to a term made of constructors (say [Ci(e1,Cj(e2,Ck(...,term,...),...),...)]), returns a given subterm of the term (say [term]). Let [typ] be the type of [term]. If [term] has no dependencies in the [e1], [e2], etc, then all is simple. If not, then we need to encapsulated the dependencies into a dependent tuple in such a way that the destructor has not a dependent type and rewriting can then be applied. The destructor has the form [e]Cases e of | ... | Ci (x1,x2,...) => Cases x2 of | ... | Cj (y1,y2,...) => Cases y2 of | ... | Ck (...,z,...) => z | ... end | ... end | ... end and the dependencies is expressed by the fact that [z] has a type dependent in the x1, y1, ... Assume [z] is typed as follows: env |- z:zty If [zty] has no dependencies, this is simple. Otherwise, assume [zty] has free (de Bruijn) variables in,...i1 then the role of [make_iterated_tuple sigma env (term,typ) (z,zty)] is to build the tuple [existT [xn]Pn Rel(in) .. (existT [x2]P2 Rel(i2) (existT [x1]P1 Rel(i1) z))] where P1 is zty[i1/x1], P2 is {x1 | P1[i2/x2]} etc. To do this, we find the free (relative) references of the strong NF of [z]'s type, gather them together in left-to-right order (i.e. highest-numbered is farthest-left), and construct a big iterated pair out of it. This only works when the references are all themselves to members of [Set]s, because we use [sigS] to construct the tuple. Suppose now that our constructed tuple is of length [tuplen]. We need also to construct a default value for the other branches of the destructor. As default value, we take a tuple of the form [existT [xn]Pn ?n (... existT [x2]P2 ?2 (existT [x1]P1 ?1 term))] but for this we have to solve the following unification problem: typ = zty[i1/?1;...;in/?n] This is done by [sig_clausal_form]. *) let make_iterated_tuple env sigma dflt (z,zty) = let (zty,rels) = minimal_free_rels env sigma (z,zty) in let sort_of_zty = get_sort_of env sigma zty in let sorted_rels = Sort.list (<) (Intset.elements rels) in let (tuple,tuplety) = List.fold_left (make_tuple env sigma) (z,zty) sorted_rels in assert (closed0 tuplety); let n = List.length sorted_rels in let dfltval = sig_clausal_form env sigma sort_of_zty n tuplety dflt in (tuple,tuplety,dfltval) let rec build_injrec sigma env dflt c = function | [] -> make_iterated_tuple env sigma dflt (c,type_of env sigma c) | ((sp,cnum),argnum)::l -> let (cnum_nlams,cnum_env,kont) = descend_then sigma env c cnum in let newc = mkRel(cnum_nlams-argnum) in let (subval,tuplety,dfltval) = build_injrec sigma cnum_env dflt newc l in (kont subval (dfltval,tuplety), tuplety,dfltval) let build_injector sigma env dflt c cpath = let (injcode,resty,_) = build_injrec sigma env dflt c cpath in (injcode,resty) (* let try_delta_expand env sigma t = let whdt = whd_betadeltaiota env sigma t in let rec hd_rec c = match kind_of_term c with | Construct _ -> whdt | App (f,_) -> hd_rec f | Cast (c,_,_) -> hd_rec c | _ -> t in hd_rec whdt *) (* Given t1=t2 Inj calculates the whd normal forms of t1 and t2 and it expands then only when the whdnf has a constructor of an inductive type in hd position, otherwise delta expansion is not done *) let simplify_args env sigma t = (* Quick hack to reduce in arguments of eq only *) match decompose_app t with | eq, [t;c1;c2] -> applist (eq,[t;nf env sigma c1;nf env sigma c2]) | eq, [t1;c1;t2;c2] -> applist (eq,[t1;nf env sigma c1;t2;nf env sigma c2]) | _ -> t let inject_at_positions env sigma (eq,(t,t1,t2)) eq_clause posns = let e = next_ident_away eq_baseid (ids_of_context env) in let e_env = push_named (e,None,t) env in let injectors = map_succeed (fun (cpath,t1',t2') -> (* arbitrarily take t1' as the injector default value *) let (injbody,resty) = build_injector sigma e_env t1' (mkVar e) cpath in let injfun = mkNamedLambda e t injbody in let pf = applist(eq.congr,[t;resty;injfun;t1;t2]) in let pf_typ = get_type_of env sigma pf in let inj_clause = apply_on_clause (pf,pf_typ) eq_clause in let pf = clenv_value_cast_meta inj_clause in let ty = simplify_args env sigma (clenv_type inj_clause) in (pf,ty)) posns in if injectors = [] then errorlabstrm "Equality.inj" (str "Failed to decompose the equality."); tclMAP (fun (pf,ty) -> tclTHENS (cut ty) [tclIDTAC; refine pf]) injectors exception Not_dep_pair let injEq ipats (eq,(t,t1,t2)) eq_clause = let sigma = Evd.evars_of eq_clause.evd in let env = eq_clause.env in match find_positions env sigma t1 t2 with | Inl _ -> errorlabstrm "Inj" (str"Not a projectable equality but a discriminable one.") | Inr [] -> errorlabstrm "Equality.inj" (str"Nothing to do, it is an equality between convertible terms.") | Inr posns -> (* Est-ce utile à partir du moment où les arguments projetés subissent "nf" ? let t1 = try_delta_expand env sigma t1 in let t2 = try_delta_expand env sigma t2 in *) try ( (* fetch the informations of the pair *) let ceq = constr_of_global Coqlib.glob_eq in let sigTconstr () = (Coqlib.build_sigma_type()).Coqlib.typ in let eqTypeDest = fst (destApp t) in let _,ar1 = destApp t1 and _,ar2 = destApp t2 in let ind = destInd ar1.(0) in let inj2 = Coqlib.coq_constant "inj_pair2_eq_dec is missing" ["Logic";"Eqdep_dec"] "inj_pair2_eq_dec" in (* check whether the equality deals with dep pairs or not *) (* if yes, check if the user has declared the dec principle *) (* and compare the fst arguments of the dep pair *) let new_eq_args = [|type_of env sigma (ar1.(3));ar1.(3);ar2.(3)|] in if ( (eqTypeDest = sigTconstr()) && (Ind_tables.check_dec_proof ind=true) && (is_conv env sigma (ar1.(2)) (ar2.(2)) = true)) then ( (* Require Import Eqdec_dec copied from vernac_require in vernacentries.ml*) let qidl = qualid_of_reference (Ident (dummy_loc,id_of_string "Eqdep_dec")) in Library.require_library [qidl] (Some false); (* cut with the good equality and prove the requested goal *) tclTHENS (cut (mkApp (ceq,new_eq_args)) ) [tclIDTAC; tclTHEN (apply ( mkApp(inj2, [|ar1.(0);Ind_tables.find_eq_dec_proof ind; ar1.(1);ar1.(2);ar1.(3);ar2.(3)|]) )) (Auto.trivial [] []) ] (* not a dep eq or no decidable type found *) ) else (raise Not_dep_pair) ) with _ -> tclTHEN (inject_at_positions env sigma (eq,(t,t1,t2)) eq_clause posns) (intros_pattern no_move ipats) let inj ipats with_evars = onEquality with_evars (injEq ipats) let injClause ipats with_evars = function | None -> onNegatedEquality with_evars (injEq ipats) | Some c -> onInductionArg (inj ipats with_evars) c let injConcl gls = injClause [] false None gls let injHyp id gls = injClause [] false (Some (ElimOnIdent (dummy_loc,id))) gls let decompEqThen ntac (lbeq,(t,t1,t2) as u) clause gls = let sort = pf_apply get_type_of gls (pf_concl gls) in let sigma = Evd.evars_of clause.evd in let env = pf_env gls in match find_positions env sigma t1 t2 with | Inl (cpath, (_,dirn), _) -> discr_positions env sigma u clause cpath dirn sort gls | Inr [] -> (* Change: do not fail, simplify clear this trivial hyp *) ntac 0 gls | Inr posns -> tclTHEN (inject_at_positions env sigma (lbeq,(t,t1,t2)) clause (List.rev posns)) (ntac (List.length posns)) gls let dEqThen with_evars ntac = function | None -> onNegatedEquality with_evars (decompEqThen ntac) | Some c -> onInductionArg (onEquality with_evars (decompEqThen ntac)) c let dEq with_evars = dEqThen with_evars (fun x -> tclIDTAC) let rewrite_msg = function | None -> str "passed term is not a primitive equality" | Some id -> pr_id id ++ str "does not satisfy preconditions " let swap_equands gls eqn = let (lbeq,(t,e1,e2)) = find_eq_data_decompose eqn in applist(lbeq.eq,[t;e2;e1]) let swapEquandsInConcl gls = let (lbeq,(t,e1,e2)) = find_eq_data_decompose (pf_concl gls) in let sym_equal = lbeq.sym in refine (applist(sym_equal,[t;e2;e1;Evarutil.mk_new_meta()])) gls let swapEquandsInHyp id gls = cut_replacing id (swap_equands gls (pf_get_hyp_typ gls id)) (tclTHEN swapEquandsInConcl) gls (* find_elim determines which elimination principle is necessary to eliminate lbeq on sort_of_gl. This is somehow an artificial choice as we could take eq_rect in all cases (eq_ind - and eq_rec - are instances of eq_rect) [HH 2/4/06]. *) let find_elim sort_of_gl lbeq = match kind_of_term sort_of_gl with | Sort(Prop Null) (* Prop *) -> lbeq.ind | _ (* Set/Type *) -> (match lbeq.rect with | Some eq_rect -> eq_rect | None -> errorlabstrm "find_elim" (str "This type of substitution is not allowed.")) (* Refine from [|- P e2] to [|- P e1] and [|- e1=e2:>t] (body is P (Rel 1)) *) let bareRevSubstInConcl lbeq body (t,e1,e2) gls = (* find substitution scheme *) let eq_elim = find_elim (pf_apply get_type_of gls (pf_concl gls)) lbeq in (* build substitution predicate *) let p = lambda_create (pf_env gls) (t,body) in (* apply substitution scheme *) refine (applist(eq_elim,[t;e1;p;Evarutil.mk_new_meta(); e2;Evarutil.mk_new_meta()])) gls (* [subst_tuple_term dep_pair B] Given that dep_pair looks like: (existT e1 (existT e2 ... (existT en en+1) ... )) and B might contain instances of the ei, we will return the term: ([x1:ty(e1)]...[xn:ty(en)]B (projS1 (mkRel 1)) (projS1 (projS2 (mkRel 1))) ... etc ...) That is, we will abstract out the terms e1...en+1 as usual, but will then produce a term in which the abstraction is on a single term - the debruijn index [mkRel 1], which will be of the same type as dep_pair. ALGORITHM for abstraction: We have a list of terms, [e1]...[en+1], which we want to abstract out of [B]. For each term [ei], going backwards from [n+1], we just do a [subst_term], and then do a lambda-abstraction to the type of the [ei]. *) let decomp_tuple_term env c t = let rec decomprec inner_code ex exty = try let {proj1=p1; proj2=p2},(a,p,car,cdr) = find_sigma_data_decompose ex in let car_code = applist (p1,[a;p;inner_code]) and cdr_code = applist (p2,[a;p;inner_code]) in let cdrtyp = beta_applist (p,[car]) in ((car,a),car_code)::(decomprec cdr_code cdr cdrtyp) with PatternMatchingFailure -> [((ex,exty),inner_code)] in List.split (decomprec (mkRel 1) c t) let subst_tuple_term env sigma dep_pair b = let typ = get_type_of env sigma dep_pair in let e_list,proj_list = decomp_tuple_term env dep_pair typ in let abst_B = List.fold_right (fun (e,t) body -> lambda_create env (t,subst_term e body)) e_list b in applist(abst_B,proj_list) (* Comme "replace" mais decompose les egalites dependantes *) let cutSubstInConcl_RL eqn gls = let (lbeq,(t,e1,e2 as eq)) = find_eq_data_decompose eqn in let body = pf_apply subst_tuple_term gls e2 (pf_concl gls) in assert (dependent (mkRel 1) body); bareRevSubstInConcl lbeq body eq gls (* |- (P e1) BY CutSubstInConcl_LR (eq T e1 e2) |- (P e2) |- (eq T e1 e2) *) let cutSubstInConcl_LR eqn gls = (tclTHENS (cutSubstInConcl_RL (swap_equands gls eqn)) ([tclIDTAC; swapEquandsInConcl])) gls let cutSubstInConcl l2r =if l2r then cutSubstInConcl_LR else cutSubstInConcl_RL let cutSubstInHyp_LR eqn id gls = let (lbeq,(t,e1,e2 as eq)) = find_eq_data_decompose eqn in let body = pf_apply subst_tuple_term gls e1 (pf_get_hyp_typ gls id) in assert (dependent (mkRel 1) body); cut_replacing id (subst1 e2 body) (tclTHENFIRST (bareRevSubstInConcl lbeq body eq)) gls let cutSubstInHyp_RL eqn id gls = (tclTHENS (cutSubstInHyp_LR (swap_equands gls eqn) id) ([tclIDTAC; swapEquandsInConcl])) gls let cutSubstInHyp l2r = if l2r then cutSubstInHyp_LR else cutSubstInHyp_RL let try_rewrite tac gls = try tac gls with | PatternMatchingFailure -> errorlabstrm "try_rewrite" (str "Not a primitive equality here.") | e when catchable_exception e -> errorlabstrm "try_rewrite" (strbrk "Cannot find a well-typed generalization of the goal that makes the proof progress.") let cutSubstClause l2r eqn cls gls = match cls with | None -> cutSubstInConcl l2r eqn gls | Some id -> cutSubstInHyp l2r eqn id gls let cutRewriteClause l2r eqn cls = try_rewrite (cutSubstClause l2r eqn cls) let cutRewriteInHyp l2r eqn id = cutRewriteClause l2r eqn (Some id) let cutRewriteInConcl l2r eqn = cutRewriteClause l2r eqn None let substClause l2r c cls gls = let eq = pf_apply get_type_of gls c in tclTHENS (cutSubstClause l2r eq cls) [tclIDTAC; exact_no_check c] gls let rewriteClause l2r c cls = try_rewrite (substClause l2r c cls) let rewriteInHyp l2r c id = rewriteClause l2r c (Some id) let rewriteInConcl l2r c = rewriteClause l2r c None (* Renaming scheme correspondence new name (old name) give equality give proof of equality / cutSubstClause (subst) substClause (HypSubst on hyp) raw | cutSubstInHyp (substInHyp) substInHyp (none) \ cutSubstInConcl (substInConcl) substInConcl (none) / cutRewriteClause (none) rewriteClause (none) user| cutRewriteInHyp (substHyp) rewriteInHyp (none) \ cutRewriteInConcl (substConcl) rewriteInConcl (substHypInConcl on hyp) raw = raise typing error or PatternMatchingFailure user = raise user error specific to rewrite *) (* Summary of obsolete forms let substInConcl = cutSubstInConcl let substInHyp = cutSubstInHyp let hypSubst l2r id = substClause l2r (mkVar id) let hypSubst_LR = hypSubst true let hypSubst_RL = hypSubst false let substHypInConcl l2r id = rewriteInConcl l2r (mkVar id) let substConcl = cutRewriteInConcl let substHyp = cutRewriteInHyp *) (**********************************************************************) (* Substitutions tactics (JCF) *) let unfold_body x gl = let hyps = pf_hyps gl in let xval = match Sign.lookup_named x hyps with (_,Some xval,_) -> xval | _ -> errorlabstrm "unfold_body" (pr_id x ++ str" is not a defined hypothesis.") in let aft = afterHyp x gl in let hl = List.fold_right (fun (y,yval,_) cl -> ((all_occurrences_expr,y),InHyp) :: cl) aft [] in let xvar = mkVar x in let rfun _ _ c = replace_term xvar xval c in tclTHENLIST [tclMAP (fun h -> reduct_in_hyp rfun h) hl; reduct_in_concl (rfun,DEFAULTcast)] gl exception FoundHyp of (identifier * constr * bool) (* tests whether hyp [c] is [x = t] or [t = x], [x] not occuring in [t] *) let is_eq_x x (id,_,c) = try let (_,lhs,rhs) = snd (find_eq_data_decompose c) in if (x = lhs) && not (occur_term x rhs) then raise (FoundHyp (id,rhs,true)); if (x = rhs) && not (occur_term x lhs) then raise (FoundHyp (id,lhs,false)) with PatternMatchingFailure -> () let subst_one x gl = let hyps = pf_hyps gl in let (_,xval,_) = pf_get_hyp gl x in (* If x has a body, simply replace x with body and clear x *) if xval <> None then tclTHEN (unfold_body x) (clear [x]) gl else (* x is a variable: *) let varx = mkVar x in (* Find a non-recursive definition for x *) let (hyp,rhs,dir) = try let test hyp _ = is_eq_x varx hyp in Sign.fold_named_context test ~init:() hyps; errorlabstrm "Subst" (str "Cannot find any non-recursive equality over " ++ pr_id x ++ str".") with FoundHyp res -> res in (* The set of hypotheses using x *) let depdecls = let test (id,_,c as dcl) = if id <> hyp && occur_var_in_decl (pf_env gl) x dcl then dcl else failwith "caught" in List.rev (map_succeed test hyps) in let dephyps = List.map (fun (id,_,_) -> id) depdecls in (* Decides if x appears in conclusion *) let depconcl = occur_var (pf_env gl) x (pf_concl gl) in (* The set of non-defined hypothesis: they must be abstracted, rewritten and reintroduced *) let abshyps = map_succeed (fun (id,v,_) -> if v=None then mkVar id else failwith "caught") depdecls in (* a tactic that either introduce an abstracted and rewritten hyp, or introduce a definition where x was replaced *) let introtac = function (id,None,_) -> intro_using id | (id,Some hval,htyp) -> letin_tac None (Name id) (replace_term varx rhs hval) (Some (replace_term varx rhs htyp)) nowhere in let need_rewrite = dephyps <> [] || depconcl in tclTHENLIST ((if need_rewrite then [generalize abshyps; (if dir then rewriteLR else rewriteRL) (mkVar hyp); thin dephyps; tclMAP introtac depdecls] else [thin dephyps; tclMAP introtac depdecls]) @ [tclTRY (clear [x;hyp])]) gl let subst = tclMAP subst_one let subst_all gl = let test (_,c) = try let (_,x,y) = snd (find_eq_data_decompose c) in (* J.F.: added to prevent failure on goal containing x=x as an hyp *) if eq_constr x y then failwith "caught"; match kind_of_term x with Var x -> x | _ -> match kind_of_term y with Var y -> y | _ -> failwith "caught" with PatternMatchingFailure -> failwith "caught" in let ids = map_succeed test (pf_hyps_types gl) in let ids = list_uniquize ids in subst ids gl (* Rewrite the first assumption for which the condition faildir does not fail and gives the direction of the rewrite *) let cond_eq_term_left c t gl = try let (_,x,_) = snd (find_eq_data_decompose t) in if pf_conv_x gl c x then true else failwith "not convertible" with PatternMatchingFailure -> failwith "not an equality" let cond_eq_term_right c t gl = try let (_,_,x) = snd (find_eq_data_decompose t) in if pf_conv_x gl c x then false else failwith "not convertible" with PatternMatchingFailure -> failwith "not an equality" let cond_eq_term c t gl = try let (_,x,y) = snd (find_eq_data_decompose t) in if pf_conv_x gl c x then true else if pf_conv_x gl c y then false else failwith "not convertible" with PatternMatchingFailure -> failwith "not an equality" let rewrite_multi_assumption_cond cond_eq_term cl gl = let rec arec = function | [] -> error "No such assumption." | (id,_,t) ::rest -> begin try let dir = cond_eq_term t gl in general_multi_rewrite dir false (inj_open (mkVar id),NoBindings) cl gl with | Failure _ | UserError _ -> arec rest end in arec (pf_hyps gl) let replace_multi_term dir_opt c = let cond_eq_fun = match dir_opt with | None -> cond_eq_term c | Some true -> cond_eq_term_left c | Some false -> cond_eq_term_right c in rewrite_multi_assumption_cond cond_eq_fun (* JF. old version let rewrite_assumption_cond faildir gl = let rec arec = function | [] -> error "No such assumption." | (id,_,t)::rest -> (try let dir = faildir t gl in general_rewrite dir (mkVar id) gl with Failure _ | UserError _ -> arec rest) in arec (pf_hyps gl) let rewrite_assumption_cond_in faildir hyp gl = let rec arec = function | [] -> error "No such assumption." | (id,_,t)::rest -> (try let dir = faildir t gl in general_rewrite_in dir hyp (mkVar id) gl with Failure _ | UserError _ -> arec rest) in arec (pf_hyps gl) let replace_term_left t = rewrite_assumption_cond (cond_eq_term_left t) let replace_term_right t = rewrite_assumption_cond (cond_eq_term_right t) let replace_term t = rewrite_assumption_cond (cond_eq_term t) let replace_term_in_left t = rewrite_assumption_cond_in (cond_eq_term_left t) let replace_term_in_right t = rewrite_assumption_cond_in (cond_eq_term_right t) let replace_term_in t = rewrite_assumption_cond_in (cond_eq_term t) *) let replace_term_left t = replace_multi_term (Some true) t Tacticals.onConcl let replace_term_right t = replace_multi_term (Some false) t Tacticals.onConcl let replace_term t = replace_multi_term None t Tacticals.onConcl let replace_term_in_left t hyp = replace_multi_term (Some true) t (Tacticals.onHyp hyp) let replace_term_in_right t hyp = replace_multi_term (Some false) t (Tacticals.onHyp hyp) let replace_term_in t hyp = replace_multi_term None t (Tacticals.onHyp hyp) let _ = Tactics.register_general_multi_rewrite general_multi_rewrite