(* Title: ZF/ind_syntax.ML ID: $Id: ind_syntax.ML,v 1.42 2007/10/07 19:19:33 wenzelm Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1993 University of Cambridge Abstract Syntax functions for Inductive Definitions. *) (*The structure protects these items from redeclaration (somewhat!). The datatype definitions in theory files refer to these items by name! *) structure Ind_Syntax = struct (*Print tracing messages during processing of "inductive" theory sections*) val trace = ref false; fun traceIt msg thy t = if !trace then (tracing (msg ^ Sign.string_of_term thy t); t) else t; (** Abstract syntax definitions for ZF **) val iT = Type("i",[]); val mem_const = @{term mem}; (*Creates All(%v.v:A --> P(v)) rather than Ball(A,P) *) fun mk_all_imp (A,P) = FOLogic.all_const iT $ Abs("v", iT, FOLogic.imp $ (mem_const $ Bound 0 $ A) $ Term.betapply(P, Bound 0)); val Part_const = Const("Part", [iT,iT-->iT]--->iT); val apply_const = @{term apply}; val Vrecursor_const = Const("Univ.Vrecursor", [[iT,iT]--->iT, iT]--->iT); val Collect_const = Const("Collect", [iT, iT-->FOLogic.oT] ---> iT); fun mk_Collect (a,D,t) = Collect_const $ D $ absfree(a, iT, t); (*simple error-checking in the premises of an inductive definition*) fun chk_prem rec_hd (Const("op &",_) $ _ $ _) = error"Premises may not be conjuctive" | chk_prem rec_hd (Const(@{const_name mem},_) $ t $ X) = (Logic.occs(rec_hd,t) andalso error "Recursion term on left of member symbol"; ()) | chk_prem rec_hd t = (Logic.occs(rec_hd,t) andalso error "Recursion term in side formula"; ()); (*Return the conclusion of a rule, of the form t:X*) fun rule_concl rl = let val Const("Trueprop",_) $ (Const(@{const_name mem},_) $ t $ X) = Logic.strip_imp_concl rl in (t,X) end; (*As above, but return error message if bad*) fun rule_concl_msg sign rl = rule_concl rl handle Bind => error ("Ill-formed conclusion of introduction rule: " ^ Sign.string_of_term sign rl); (*For deriving cases rules. CollectD2 discards the domain, which is redundant; read_instantiate replaces a propositional variable by a formula variable*) val equals_CollectD = read_instantiate [("W","?Q")] (make_elim (@{thm equalityD1} RS @{thm subsetD} RS @{thm CollectD2})); (** For datatype definitions **) (*Constructor name, type, mixfix info; internal name from mixfix, datatype sets, full premises*) type constructor_spec = ((string * typ * mixfix) * string * term list * term list); fun dest_mem (Const(@{const_name mem},_) $ x $ A) = (x,A) | dest_mem _ = error "Constructor specifications must have the form x:A"; (*read a constructor specification*) fun read_construct sign (id, sprems, syn) = let val prems = map (Sign.simple_read_term sign FOLogic.oT) sprems val args = map (#1 o dest_mem) prems val T = (map (#2 o dest_Free) args) ---> iT handle TERM _ => error "Bad variable in constructor specification" val name = Syntax.const_name id syn (*handle infix constructors*) in ((id,T,syn), name, args, prems) end; val read_constructs = map o map o read_construct; (*convert constructor specifications into introduction rules*) fun mk_intr_tms sg (rec_tm, constructs) = let fun mk_intr ((id,T,syn), name, args, prems) = Logic.list_implies (map FOLogic.mk_Trueprop prems, FOLogic.mk_Trueprop (mem_const $ list_comb (Const (Sign.full_name sg name, T), args) $ rec_tm)) in map mk_intr constructs end; fun mk_all_intr_tms sg arg = List.concat (ListPair.map (mk_intr_tms sg) arg); fun mk_Un (t1, t2) = Const(@{const_name Un}, [iT,iT]--->iT) $ t1 $ t2; val empty = Const("0", iT) and univ = Const("Univ.univ", iT-->iT) and quniv = Const("QUniv.quniv", iT-->iT); (*Make a datatype's domain: form the union of its set parameters*) fun union_params (rec_tm, cs) = let val (_,args) = strip_comb rec_tm fun is_ind arg = (type_of arg = iT) in case List.filter is_ind (args @ cs) of [] => empty | u_args => BalancedTree.make mk_Un u_args end; (*univ or quniv constitutes the sum domain for mutual recursion; it is applied to the datatype parameters and to Consts occurring in the definition other than Nat.nat and the datatype sets themselves. FIXME: could insert all constant set expressions, e.g. nat->nat.*) fun data_domain co (rec_tms, con_ty_lists) = let val rec_hds = map head_of rec_tms val dummy = assert_all is_Const rec_hds (fn t => "Datatype set not previously declared as constant: " ^ Sign.string_of_term @{theory IFOL} t); val rec_names = (*nat doesn't have to be added*) "Nat.nat" :: map (#1 o dest_Const) rec_hds val u = if co then quniv else univ val cs = (fold o fold) (fn (_, _, _, prems) => prems |> (fold o fold_aterms) (fn t as Const (a, _) => if a mem_string rec_names then I else insert (op =) t | _ => I)) con_ty_lists []; in u $ union_params (hd rec_tms, cs) end; (*Could go to FOL, but it's hardly general*) val def_swap_iff = prove_goal (the_context ()) "a==b ==> a=c <-> c=b" (fn [def] => [(rewtac def), (rtac iffI 1), (REPEAT (etac sym 1))]); val def_trans = prove_goal (the_context ()) "[| f==g; g(a)=b |] ==> f(a)=b" (fn [rew,prem] => [ rewtac rew, rtac prem 1 ]); (*Delete needless equality assumptions*) val refl_thin = prove_goal (the_context ()) "!!P. [| a=a; P |] ==> P" (fn _ => [assume_tac 1]); (*Includes rules for succ and Pair since they are common constructions*) val elim_rls = [asm_rl, FalseE, thm "succ_neq_0", sym RS thm "succ_neq_0", thm "Pair_neq_0", sym RS thm "Pair_neq_0", thm "Pair_inject", make_elim (thm "succ_inject"), refl_thin, conjE, exE, disjE]; (*From HOL/ex/meson.ML: raises exception if no rules apply -- unlike RL*) fun tryres (th, rl::rls) = (th RS rl handle THM _ => tryres(th,rls)) | tryres (th, []) = raise THM("tryres", 0, [th]); fun gen_make_elim elim_rls rl = standard (tryres (rl, elim_rls @ [revcut_rl])); (*Turns iff rules into safe elimination rules*) fun mk_free_SEs iffs = map (gen_make_elim [conjE,FalseE]) (iffs RL [iffD1]); end; (*For convenient access by the user*) val trace_induct = Ind_Syntax.trace;