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Quellcode-Bibliothek© Kompilation durch diese Firma[Weder Korrektheit noch Funktionsfähigkeit der Software werden zugesichert.]Datei: inductive_set.ML Sprache: SMLOriginal von: Isabelle© |
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(* Title: HOL/Tools/inductive_set.ML
Author: Stefan Berghofer, TU Muenchen Wrapper for defining inductive sets using package for inductive predicates, including infrastructure for converting between predicates and sets. *) signature INDUCTIVE_SET = sig val to_set_att: thm list -> attribute val to_pred_att: thm list -> attribute val to_pred : thm list -> Context.generic -> thm -> thm val pred_set_conv_att: attribute val add_inductive: Inductive.flags -> ((binding * typ) * mixfix) list -> (string * typ) list -> (Attrib.binding * term) list -> thm list -> local_theory -> Inductive.result * local_theory val add_inductive_cmd: bool -> bool -> (binding * string option * mixfix) list -> (binding * string option * mixfix) list -> Specification.multi_specs_cmd -> (Facts.ref * Token.src list) list -> local_theory -> Inductive.result * local_theory val mono_add: attribute val mono_del: attribute end; structure Inductive_Set: INDUCTIVE_SET = struct (******************************************************************************* (* simplifies (%x y. (x, y) : S & P x y) to (%x y. (x, y) : S Int {(x, y). P x y}) *) (* and (%x y. (x, y) : S | P x y) to (%x y. (x, y) : S Un {(x, y). P x y}) *) (* used for converting "strong" (co)induction rules *) (******************************************************************************* val anyt = Free ("t", TFree ("'t", [])); fun strong_ind_simproc tab = Simplifier.make_simproc \<^context> "strong_ind" {lhss = [\<^term>\<open>x::'a::{}\ proc = fn _ => fn ctxt => fn ct => let fun close p t f = let val vs = Term.add_vars t [] in Thm.instantiate' [] (rev (map (SOME o Thm.cterm_of ctxt o Var) vs)) (p (fold (Logic.all o Var) vs t) f) end; fun mkop \<^const_name>\<open>HOL.conj\<close> T x = SOME (Const (\<^const_name>\<open>Lattices.inf\<close>, T --> T --> T), x) | mkop \<^const_name>\<open>HOL.disj\<close> T x = SOME (Const (\<^const_name>\<open>Lattices.sup\<close>, T --> T --> T), x) | mkop _ _ _ = NONE; fun mk_collect p T t = let val U = HOLogic.dest_setT T in HOLogic.Collect_const U $ HOLogic.mk_ptupleabs (HOLogic.flat_tuple_paths p) U HOLogic.boolT t end; fun decomp (Const (s, _) $ ((m as Const (\<^const_name>\<open>Set.member\<close>, Type (_, [_, Type (_, [T, _])]))) $ p $ S) $ u) = mkop s T (m, p, S, mk_collect p T (head_of u)) | decomp (Const (s, _) $ u $ ((m as Const (\<^const_name>\<open>Set.member\<close>, Type (_, [_, Type (_, [T, _])]))) $ p $ S)) = mkop s T (m, p, mk_collect p T (head_of u), S) | decomp _ = NONE; val simp = full_simp_tac (put_simpset HOL_basic_ss ctxt addsimps @{thms mem_Collect_eq case_prod_conv}) 1; fun mk_rew t = (case strip_abs_vars t of [] => NONE | xs => (case decomp (strip_abs_body t) of NONE => NONE | SOME (bop, (m, p, S, S')) => SOME (close (Goal.prove ctxt [] []) (Logic.mk_equals (t, fold_rev Term.abs xs (m $ p $ (bop $ S $ S')))) (K (EVERY [resolve_tac ctxt [eq_reflection] 1, REPEAT (resolve_tac ctxt @{thms ext} 1), resolve_tac ctxt @{thms iffI} 1, EVERY [eresolve_tac ctxt @{thms conjE} 1, resolve_tac ctxt @{thms IntI} 1, simp, simp, eresolve_tac ctxt @{thms IntE} 1, resolve_tac ctxt @{thms conjI} 1, simp, simp] ORELSE EVERY [eresolve_tac ctxt @{thms disjE} 1, resolve_tac ctxt @{thms UnI1} 1, simp, resolve_tac ctxt @{thms UnI2} 1, simp, eresolve_tac ctxt @{thms UnE} 1, resolve_tac ctxt @{thms disjI1} 1, simp, resolve_tac ctxt @{thms disjI2} 1, simp]]))) handle ERROR _ => NONE)) in (case strip_comb (Thm.term_of ct) of (h as Const (name, _), ts) => if Symtab.defined tab name then let val rews = map mk_rew ts in if forall is_none rews then NONE else SOME (fold (fn th1 => fn th2 => Thm.combination th2 th1) (map2 (fn SOME r => K r | NONE => Thm.reflexive o Thm.cterm_of ctxt) rews ts) (Thm.reflexive (Thm.cterm_of ctxt h))) end else NONE | _ => NONE) end}; (* only eta contract terms occurring as arguments of functions satisfying p *) fun eta_contract p = let fun eta b (Abs (a, T, body)) = (case eta b body of body' as (f $ Bound 0) => if Term.is_dependent f orelse not b then Abs (a, T, body') else incr_boundvars ~1 f | body' => Abs (a, T, body')) | eta b (t $ u) = eta b t $ eta (p (head_of t)) u | eta b t = t in eta false end; fun eta_contract_thm ctxt p = Conv.fconv_rule (Conv.then_conv (Thm.beta_conversion true, fn ct => Thm.transitive (Thm.eta_conversion ct) (Thm.symmetric (Thm.eta_conversion (Thm.cterm_of ctxt (eta_contract p (Thm.term_of ct)) (***********************************************************) (* rules for converting between predicate and set notation *) (* *) (* rules for converting predicates to sets have the form *) (* P (%x y. (x, y) : s) = (%x y. (x, y) : S s) *) (* *) (* rules for converting sets to predicates have the form *) (* S {(x, y). p x y} = {(x, y). P p x y} *) (* *) (* where s and p are parameters *) (***********************************************************) structure Data = Generic_Data ( type T = {(* rules for converting predicates to sets *) to_set_simps: thm list, (* rules for converting sets to predicates *) to_pred_simps: thm list, (* arities of functions of type t set => ... => u set *) set_arities: (typ * (int list list option list * int list list option)) list Symtab.table, (* arities of functions of type (t => ... => bool) => u => ... => bool *) pred_arities: (typ * (int list list option list * int list list option)) list Symtab.table}; val empty = {to_set_simps = [], to_pred_simps = [], set_arities = Symtab.empty, pred_arities = Symtab.empty}; val extend = I; fun merge ({to_set_simps = to_set_simps1, to_pred_simps = to_pred_simps1, set_arities = set_arities1, pred_arities = pred_arities1}, {to_set_simps = to_set_simps2, to_pred_simps = to_pred_simps2, set_arities = set_arities2, pred_arities = pred_arities2}) : T = {to_set_simps = Thm.merge_thms (to_set_simps1, to_set_simps2), to_pred_simps = Thm.merge_thms (to_pred_simps1, to_pred_simps2), set_arities = Symtab.merge_list (op =) (set_arities1, set_arities2), pred_arities = Symtab.merge_list (op =) (pred_arities1, pred_arities2)}; ); fun name_type_of (Free p) = SOME p | name_type_of (Const p) = SOME p | name_type_of _ = NONE; fun map_type f (Free (s, T)) = Free (s, f T) | map_type f (Var (ixn, T)) = Var (ixn, f T) | map_type f _ = error "map_type"; fun find_most_specific is_inst f eq xs T = find_first (fn U => is_inst (T, f U) andalso forall (fn U' => eq (f U, f U') orelse not (is_inst (T, f U') andalso is_inst (f U', f U))) xs) xs; fun lookup_arity thy arities (s, T) = case Symtab.lookup arities s of NONE => NONE | SOME xs => find_most_specific (Sign.typ_instance thy) fst (op =) xs T; fun lookup_rule thy f rules = find_most_specific (swap #> Pattern.matches thy) (f #> fst) (op aconv) rules; fun infer_arities thy arities (optf, t) fs = case strip_comb t of (Abs (_, _, u), []) => infer_arities thy arities (NONE, u) fs | (Abs _, _) => infer_arities thy arities (NONE, Envir.beta_norm t) fs | (u, ts) => (case Option.map (lookup_arity thy arities) (name_type_of u) of SOME (SOME (_, (arity, _))) => (fold (infer_arities thy arities) (arity ~~ List.take (ts, length arity)) fs handle General.Subscript => error "infer_arities: bad term") | _ => fold (infer_arities thy arities) (map (pair NONE) ts) (case optf of NONE => fs | SOME f => AList.update op = (u, the_default f (Option.map (fn g => inter (op =) g f) (AList.lookup op = fs u))) fs)); (**************************************************************) (* derive the to_pred equation from the to_set equation *) (* *) (* 1. instantiate each set parameter with {(x, y). p x y} *) (* 2. apply %P. {(x, y). P x y} to both sides of the equation *) (* 3. simplify *) (**************************************************************) fun mk_to_pred_inst ctxt fs = map (fn (x, ps) => let val (Ts, T) = strip_type (fastype_of x); val U = HOLogic.dest_setT T; val x' = map_type (K (Ts @ HOLogic.strip_ptupleT ps U ---> HOLogic.boolT)) x; in (dest_Var x, Thm.cterm_of ctxt (fold_rev (Term.abs o pair "x") Ts (HOLogic.Collect_const U $ HOLogic.mk_ptupleabs ps U HOLogic.boolT (list_comb (x', map Bound (length Ts - 1 downto 0)))))) end) fs; fun mk_to_pred_eq ctxt p fs optfs' T thm = let val insts = mk_to_pred_inst ctxt fs; val thm' = Thm.instantiate ([], insts) thm; val thm'' = (case optfs' of NONE => thm' RS sym | SOME fs' => let val U = HOLogic.dest_setT (body_type T); val Ts = HOLogic.strip_ptupleT fs' U; val arg_cong' = Thm.incr_indexes (Thm.maxidx_of thm + 1) arg_cong; val (Var (arg_cong_f, _), _) = arg_cong' |> Thm.concl_of |> dest_comb |> snd |> strip_comb |> snd |> hd |> dest_comb; in thm' RS (infer_instantiate ctxt [(arg_cong_f, Thm.cterm_of ctxt (Abs ("P", Ts ---> HOLogic.boolT, HOLogic.Collect_const U $ HOLogic.mk_ptupleabs fs' U HOLogic.boolT (Bound 0))))] arg_cong' RS sym) end) in Simplifier.simplify (put_simpset HOL_basic_ss ctxt addsimps @{thms mem_Collect_eq case_prod_conv} addsimprocs [\<^simproc>\<open>Collect_mem\<close>]) thm'' |> zero_var_indexes |> eta_contract_thm ctxt (equal p) end; (**** declare rules for converting predicates to sets ****) exception Malformed of string; fun add context thm (tab as {to_set_simps, to_pred_simps, set_arities, pred_arities}) = (case Thm.prop_of thm of Const (\<^const_name>\<open>Trueprop\<close>, _) $ (Const (\<^const_name>\<open>HOL.eq\<close>, Ty (case body_type T of \<^typ>\<open>bool\<close> => let val thy = Context.theory_of context; val ctxt = Context.proof_of context; fun factors_of t fs = case strip_abs_body t of Const (\<^const_name>\<open>Set.member\<close>, _) $ u $ S => if is_Free S orelse is_Var S then let val ps = HOLogic.flat_tuple_paths u in (SOME ps, (S, ps) :: fs) end else (NONE, fs) | _ => (NONE, fs); val (h, ts) = strip_comb lhs val (pfs, fs) = fold_map factors_of ts []; val ((h', ts'), fs') = (case rhs of Abs _ => (case strip_abs_body rhs of Const (\<^const_name>\<open>Set.member\<close>, _) $ u $ S => (strip_comb S, SOME (HOLogic.flat_tuple_paths u)) | _ => raise Malformed "member symbol on right-hand side expected") | _ => (strip_comb rhs, NONE)) in case (name_type_of h, name_type_of h') of (SOME (s, T), SOME (s', T')) => if exists (fn (U, _) => Sign.typ_instance thy (T', U) andalso Sign.typ_instance thy (U, T')) (Symtab.lookup_list set_arities s') then (if Context_Position.is_really_visible ctxt then warning ("Ignoring conversion rule for operator " ^ s') else (); tab) else {to_set_simps = Thm.trim_context thm :: to_set_simps, to_pred_simps = Thm.trim_context (mk_to_pred_eq ctxt h fs fs' T' thm) :: to_pred_simps, set_arities = Symtab.insert_list op = (s', (T', (map (AList.lookup op = fs) ts', fs'))) set_arities, pred_arities = Symtab.insert_list op = (s, (T, (pfs, fs'))) pred_arities} | _ => raise Malformed "set / predicate constant expected" end | _ => raise Malformed "equation between predicates expected") | _ => raise Malformed "equation expected") handle Malformed msg => let val ctxt = Context.proof_of context val _ = if Context_Position.is_really_visible ctxt then warning ("Ignoring malformed set / predicate conversion rule: " ^ msg ^ "\n" ^ Thm.string_of_thm ctxt thm) else (); in tab end; val pred_set_conv_att = Thm.declaration_attribute (fn thm => fn ctxt => Data.map (add ctxt thm) ctxt); (**** convert theorem in set notation to predicate notation ****) fun is_pred tab t = case Option.map (Symtab.lookup tab o fst) (name_type_of t) of SOME (SOME _) => true | _ => false; fun to_pred_simproc rules = let val rules' = map mk_meta_eq rules in Simplifier.make_simproc \<^context> "to_pred" {lhss = [anyt], proc = fn _ => fn ctxt => fn ct => lookup_rule (Proof_Context.theory_of ctxt) (Thm.prop_of #> Logic.dest_equals) rules' (Thm.term_of ct)} end; fun to_pred_proc thy rules t = case lookup_rule thy I rules t of NONE => NONE | SOME (lhs, rhs) => SOME (Envir.subst_term (Pattern.match thy (lhs, t) (Vartab.empty, Vartab.empty)) rhs); fun to_pred thms context thm = let val thy = Context.theory_of context; val ctxt = Context.proof_of context; val {to_pred_simps, set_arities, pred_arities, ...} = fold (add context) thms (Data.get context); val fs = filter (is_Var o fst) (infer_arities thy set_arities (NONE, Thm.prop_of thm) []); (* instantiate each set parameter with {(x, y). p x y} *) val insts = mk_to_pred_inst ctxt fs in thm |> Thm.instantiate ([], insts) |> Simplifier.full_simplify (put_simpset HOL_basic_ss ctxt addsimprocs [to_pred_simproc (@{thm mem_Collect_eq} :: @{thm case_prod_conv} :: map (Thm.transfer thy) to_pred_simps)] eta_contract_thm ctxt (is_pred pred_arities) |> Rule_Cases.save thm end; val to_pred_att = Thm.rule_attribute [] o to_pred; (**** convert theorem in predicate notation to set notation ****) fun to_set thms context thm = let val thy = Context.theory_of context; val ctxt = Context.proof_of context; val {to_set_simps, pred_arities, ...} = fold (add context) thms (Data.get context); val fs = filter (is_Var o fst) (infer_arities thy pred_arities (NONE, Thm.prop_of thm) []); (* instantiate each predicate parameter with %x y. (x, y) : s *) val insts = map (fn (x, ps) => let val Ts = binder_types (fastype_of x); val l = length Ts; val k = length ps; val (Rs, Us) = chop (l - k - 1) Ts; val T = HOLogic.mk_ptupleT ps Us; val x' = map_type (K (Rs ---> HOLogic.mk_setT T)) x in (dest_Var x, Thm.cterm_of ctxt (fold_rev (Term.abs o pair "x") Ts (HOLogic.mk_mem (HOLogic.mk_ptuple ps T (map Bound (k downto 0)), list_comb (x', map Bound (l - 1 downto k + 1)))))) end) fs; in thm |> Thm.instantiate ([], insts) |> Simplifier.full_simplify (put_simpset HOL_basic_ss ctxt addsimps to_set_simps addsimprocs [strong_ind_simproc pred_arities, \<^simproc>\<open>Collect_mem\<close>]) |> Rule_Cases.save thm end; val to_set_att = Thm.rule_attribute [] o to_set; (**** definition of inductive sets ****) fun add_ind_set_def {quiet_mode, verbose, alt_name, coind, no_elim, no_ind, skip_mono} cs intros monos params cnames_syn lthy = let val thy = Proof_Context.theory_of lthy; val {set_arities, pred_arities, to_pred_simps, ...} = Data.get (Context.Proof lthy); fun infer (Abs (_, _, t)) = infer t | infer (Const (\<^const_name>\<open>Set.member\<close>, _) $ t $ u) = infer_arities thy set_arities (SOME (HOLogic.flat_tuple_paths t), u) | infer (t $ u) = infer t #> infer u | infer _ = I; val new_arities = filter_out (fn (x as Free (_, T), _) => member (op =) params x andalso length (binder_types T) > 0 | _ => false) (fold (snd #> infer) intros []); val params' = map (fn x => (case AList.lookup op = new_arities x of SOME fs => let val T = HOLogic.dest_setT (fastype_of x); val Ts = HOLogic.strip_ptupleT fs T; val x' = map_type (K (Ts ---> HOLogic.boolT)) x in (x, (x', (HOLogic.Collect_const T $ HOLogic.mk_ptupleabs fs T HOLogic.boolT x', fold_rev (Term.abs o pair "x") Ts (HOLogic.mk_mem (HOLogic.mk_ptuple fs T (map Bound (length fs downto 0)), x))))) end | NONE => (x, (x, (x, x))))) params; val (params1, (params2, params3)) = params' |> map snd |> split_list ||> split_list; val paramTs = map fastype_of params; (* equations for converting sets to predicates *) val ((cs', cs_info), eqns) = cs |> map (fn c as Free (s, T) => let val fs = the_default [] (AList.lookup op = new_arities c); val (Us, U) = strip_type T |> apsnd HOLogic.dest_setT; val _ = Us = paramTs orelse error (Pretty.string_of (Pretty.chunks [Pretty.str "Argument types", Pretty.block (Pretty.commas (map (Syntax.pretty_typ lthy) Us)), Pretty.str ("of " ^ s ^ " do not agree with types"), Pretty.block (Pretty.commas (map (Syntax.pretty_typ lthy) paramTs)), Pretty.str "of declared parameters"])); val Ts = HOLogic.strip_ptupleT fs U; val c' = Free (s ^ "p", map fastype_of params1 @ Ts ---> HOLogic.boolT) in ((c', (fs, U, Ts)), (list_comb (c, params2), HOLogic.Collect_const U $ HOLogic.mk_ptupleabs fs U HOLogic.boolT (list_comb (c', params1)))) end) |> split_list |>> split_list; val eqns' = eqns @ map (Thm.prop_of #> HOLogic.dest_Trueprop #> HOLogic.dest_eq) (@{thm mem_Collect_eq} :: @{thm case_prod_conv} :: to_pred_simps); (* predicate version of the introduction rules *) val intros' = map (fn (name_atts, t) => (name_atts, t |> map_aterms (fn u => (case AList.lookup op = params' u of SOME (_, (u', _)) => u' | NONE => u)) |> Pattern.rewrite_term thy [] [to_pred_proc thy eqns'] |> eta_contract (member op = cs' orf is_pred pred_arities))) intros; val cnames_syn' = map (fn (b, _) => (Binding.suffix_name "p" b, NoSyn)) cnames_sy val monos' = map (to_pred [] (Context.Proof lthy)) monos; val ({preds, intrs, elims, raw_induct, eqs, ...}, lthy1) = Inductive.add_ind_def {quiet_mode = quiet_mode, verbose = verbose, alt_name = Binding.empty, coind = coind, no_elim = no_elim, no_ind = no_ind, skip_mono = skip_mono} cs' intros' monos' params1 cnames_syn' lthy; (* define inductive sets using previously defined predicates *) val (defs, lthy2) = lthy1 |> fold_map Local_Theory.define (map (fn (((b, mx), (fs, U, _)), p) => ((b, mx), ((Thm.def_binding b, []), fold_rev lambda params (HOLogic.Collect_const U $ HOLogic.mk_ptupleabs fs U HOLogic.boolT (list_comb (p, params3)))))) (cnames_syn ~~ cs_info ~~ preds)); (* prove theorems for converting predicate to set notation *) val lthy3 = fold (fn (((p, c as Free (s, _)), (fs, U, Ts)), (_, (_, def))) => fn lthy => let val conv_thm = Goal.prove lthy (map (fst o dest_Free) params) [] (HOLogic.mk_Trueprop (HOLogic.mk_eq (list_comb (p, params3), fold_rev (Term.abs o pair "x") Ts (HOLogic.mk_mem (HOLogic.mk_ptuple fs U (map Bound (length fs downto 0)), list_comb (c, params)))))) (K (REPEAT (resolve_tac lthy @{thms ext} 1) THEN simp_tac (put_simpset HOL_basic_ss lthy addsimps [def, @{thm mem_Collect_eq}, @{thm case_prod_conv}]) 1)) in lthy |> Local_Theory.note ((Binding.name (s ^ "p_" ^ s ^ "_eq"), [Attrib.internal (K pred_set_conv_att)]), [conv_thm]) |> snd end) (preds ~~ cs ~~ cs_info ~~ defs) lthy2; (* convert theorems to set notation *) val rec_name = if Binding.is_empty alt_name then Binding.conglomerate (map #1 cnames_syn) else alt_name; val cnames = map (Local_Theory.full_name lthy3 o #1) cnames_syn; (* FIXME *) val spec_name = Binding.conglomerate (map #1 cnames_syn); val (intr_names, intr_atts) = split_list (map fst intros); val raw_induct' = to_set [] (Context.Proof lthy3) raw_induct; val (intrs', elims', eqs', induct, inducts, lthy4) = Inductive.declare_rules rec_name coind no_ind spec_name cnames (map fst defs) (map (to_set [] (Context.Proof lthy3)) intrs) intr_names intr_atts (map (fn th => (to_set [] (Context.Proof lthy3) th, map (fst o fst) (fst (Rule_Cases.get th)), Rule_Cases.get_constraints th)) elims) (map (to_set [] (Context.Proof lthy3)) eqs) raw_induct' lthy3; in ({intrs = intrs', elims = elims', induct = induct, inducts = inducts, raw_induct = raw_induct', preds = map fst defs, eqs = eqs'}, lthy4) end; val add_inductive = Inductive.gen_add_inductive add_ind_set_def; val add_inductive_cmd = Inductive.gen_add_inductive_cmd add_ind_set_def; fun mono_att att = Thm.declaration_attribute (fn thm => fn context => Thm.attribute_declaration att (to_pred [] context thm) context); val mono_add = mono_att Inductive.mono_add; val mono_del = mono_att Inductive.mono_del; (** package setup **) (* attributes *) val _ = Theory.setup (Attrib.setup \<^binding>\<open>pred_set_conv\<close> (Scan.succeed pred_set_conv_att) "declare rules for converting between predicate and set notation" #> Attrib.setup \<^binding>\<open>to_set\<close> (Attrib.thms >> to_set_att) "convert rule to set notation" #> Attrib.setup \<^binding>\<open>to_pred\<close> (Attrib.thms >> to_pred_att) "convert rule to predicate notation" #> Attrib.setup \<^binding>\<open>mono_set\<close> (Attrib.add_del mono_add mono_del) "declare of monotonicity rule for set operators"); (* commands *) val ind_set_decl = Inductive.gen_ind_decl add_ind_set_def; val _ = Outer_Syntax.local_theory \<^command_keyword>\<open>inductive_set\<close> "define induct (ind_set_decl false); val _ = Outer_Syntax.local_theory \<^command_keyword>\<open>coinductive_set\<close> "define coin (ind_set_decl true); end; ¤ Dauer der Verarbeitung: 0.22 Sekunden (vorverarbeitet) ¤ |
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