(* Title: HOL/Tools/datatype_rep_proofs.ML ID: $Id: datatype_rep_proofs.ML,v 1.52 2005/09/20 14:17:34 haftmann Exp $ Author: Stefan Berghofer, TU Muenchen Definitional introduction of datatypes Proof of characteristic theorems: - injectivity of constructors - distinctness of constructors - induction theorem *) signature DATATYPE_REP_PROOFS = sig val representation_proofs : bool -> DatatypeAux.datatype_info Symtab.table -> string list -> DatatypeAux.descr list -> (string * sort) list -> (string * mixfix) list -> (string * mixfix) list list -> theory attribute -> theory -> theory * thm list list * thm list list * thm list list * DatatypeAux.simproc_dist list * thm end; structure DatatypeRepProofs : DATATYPE_REP_PROOFS = struct open DatatypeAux; val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of distinct_lemma); (** theory context references **) val f_myinv_f = thm "f_myinv_f"; val myinv_f_f = thm "myinv_f_f"; fun exh_thm_of (dt_info : datatype_info Symtab.table) tname = #exhaustion (the (Symtab.lookup dt_info tname)); (******************************************************************************) fun representation_proofs flat_names (dt_info : datatype_info Symtab.table) new_type_names descr sorts types_syntax constr_syntax case_names_induct thy = let val Datatype_thy = if Context.theory_name thy = "Datatype" then thy else theory "Datatype"; val node_name = "Datatype_Universe.node"; val In0_name = "Datatype_Universe.In0"; val In1_name = "Datatype_Universe.In1"; val Scons_name = "Datatype_Universe.Scons"; val Leaf_name = "Datatype_Universe.Leaf"; val Numb_name = "Datatype_Universe.Numb"; val Lim_name = "Datatype_Universe.Lim"; val Suml_name = "Datatype.Suml"; val Sumr_name = "Datatype.Sumr"; val [In0_inject, In1_inject, Scons_inject, Leaf_inject, In0_eq, In1_eq, In0_not_In1, In1_not_In0, Lim_inject, Suml_inject, Sumr_inject] = map (get_thm Datatype_thy o Name) ["In0_inject", "In1_inject", "Scons_inject", "Leaf_inject", "In0_eq", "In1_eq", "In0_not_In1", "In1_not_In0", "Lim_inject", "Suml_inject", "Sumr_inject"]; val descr' = List.concat descr; val big_name = space_implode "_" new_type_names; val thy1 = add_path flat_names big_name thy; val big_rec_name = big_name ^ "_rep_set"; val rep_set_names = map (Sign.full_name (Theory.sign_of thy1)) (if length descr' = 1 then [big_rec_name] else (map ((curry (op ^) (big_rec_name ^ "_")) o string_of_int) (1 upto (length descr')))); val tyvars = map (fn (_, (_, Ts, _)) => map dest_DtTFree Ts) (hd descr); val leafTs' = get_nonrec_types descr' sorts; val branchTs = get_branching_types descr' sorts; val branchT = if null branchTs then HOLogic.unitT else fold_bal (fn (T, U) => Type ("+", [T, U])) branchTs; val arities = get_arities descr' \ 0; val unneeded_vars = hd tyvars \\ foldr add_typ_tfree_names [] (leafTs' @ branchTs); val leafTs = leafTs' @ (map (fn n => TFree (n, (the o AList.lookup (op =) sorts) n)) unneeded_vars); val recTs = get_rec_types descr' sorts; val newTs = Library.take (length (hd descr), recTs); val oldTs = Library.drop (length (hd descr), recTs); val sumT = if null leafTs then HOLogic.unitT else fold_bal (fn (T, U) => Type ("+", [T, U])) leafTs; val Univ_elT = HOLogic.mk_setT (Type (node_name, [sumT, branchT])); val UnivT = HOLogic.mk_setT Univ_elT; val In0 = Const (In0_name, Univ_elT --> Univ_elT); val In1 = Const (In1_name, Univ_elT --> Univ_elT); val Leaf = Const (Leaf_name, sumT --> Univ_elT); val Lim = Const (Lim_name, (branchT --> Univ_elT) --> Univ_elT); (* make injections needed for embedding types in leaves *) fun mk_inj T' x = let fun mk_inj' T n i = if n = 1 then x else let val n2 = n div 2; val Type (_, [T1, T2]) = T in if i <= n2 then Const ("Sum_Type.Inl", T1 --> T) $ (mk_inj' T1 n2 i) else Const ("Sum_Type.Inr", T2 --> T) $ (mk_inj' T2 (n - n2) (i - n2)) end in mk_inj' sumT (length leafTs) (1 + find_index_eq T' leafTs) end; (* make injections for constructors *) fun mk_univ_inj ts = access_bal (fn t => In0 $ t, fn t => In1 $ t, if ts = [] then Const ("arbitrary", Univ_elT) else foldr1 (HOLogic.mk_binop Scons_name) ts); (* function spaces *) fun mk_fun_inj T' x = let fun mk_inj T n i = if n = 1 then x else let val n2 = n div 2; val Type (_, [T1, T2]) = T; fun mkT U = (U --> Univ_elT) --> T --> Univ_elT in if i <= n2 then Const (Suml_name, mkT T1) $ mk_inj T1 n2 i else Const (Sumr_name, mkT T2) $ mk_inj T2 (n - n2) (i - n2) end in mk_inj branchT (length branchTs) (1 + find_index_eq T' branchTs) end; val mk_lim = foldr (fn (T, t) => Lim $ mk_fun_inj T (Abs ("x", T, t))); (************** generate introduction rules for representing set **********) val _ = message "Constructing representing sets ..."; (* make introduction rule for a single constructor *) fun make_intr s n (i, (_, cargs)) = let fun mk_prem (dt, (j, prems, ts)) = (case strip_dtyp dt of (dts, DtRec k) => let val Ts = map (typ_of_dtyp descr' sorts) dts; val free_t = app_bnds (mk_Free "x" (Ts ---> Univ_elT) j) (length Ts) in (j + 1, list_all (map (pair "x") Ts, HOLogic.mk_Trueprop (HOLogic.mk_mem (free_t, Const (List.nth (rep_set_names, k), UnivT)))) :: prems, mk_lim free_t Ts :: ts) end | _ => let val T = typ_of_dtyp descr' sorts dt in (j + 1, prems, (Leaf $ mk_inj T (mk_Free "x" T j))::ts) end); val (_, prems, ts) = foldr mk_prem (1, [], []) cargs; val concl = HOLogic.mk_Trueprop (HOLogic.mk_mem (mk_univ_inj ts n i, Const (s, UnivT))) in Logic.list_implies (prems, concl) end; val consts = map (fn s => Const (s, UnivT)) rep_set_names; val intr_ts = List.concat (map (fn ((_, (_, _, constrs)), rep_set_name) => map (make_intr rep_set_name (length constrs)) ((1 upto (length constrs)) ~~ constrs)) (descr' ~~ rep_set_names)); val (thy2, {raw_induct = rep_induct, intrs = rep_intrs, ...}) = setmp InductivePackage.quiet_mode (!quiet_mode) (InductivePackage.add_inductive_i false true big_rec_name false true false consts (map (fn x => (("", x), [])) intr_ts) []) thy1; (********************************* typedef ********************************) val thy3 = add_path flat_names big_name (Library.foldl (fn (thy, ((((name, mx), tvs), c), name')) => setmp TypedefPackage.quiet_mode true (TypedefPackage.add_typedef_i false (SOME name') (name, tvs, mx) c NONE (rtac exI 1 THEN QUIET_BREADTH_FIRST (has_fewer_prems 1) (resolve_tac rep_intrs 1))) thy |> #1) (parent_path flat_names thy2, types_syntax ~~ tyvars ~~ (Library.take (length newTs, consts)) ~~ new_type_names)); (*********************** definition of constructors ***********************) val big_rep_name = (space_implode "_" new_type_names) ^ "_Rep_"; val rep_names = map (curry op ^ "Rep_") new_type_names; val rep_names' = map (fn i => big_rep_name ^ (string_of_int i)) (1 upto (length (List.concat (tl descr)))); val all_rep_names = map (Sign.intern_const (Theory.sign_of thy3)) rep_names @ map (Sign.full_name (Theory.sign_of thy3)) rep_names'; (* isomorphism declarations *) val iso_decls = map (fn (T, s) => (s, T --> Univ_elT, NoSyn)) (oldTs ~~ rep_names'); (* constructor definitions *) fun make_constr_def tname T n ((thy, defs, eqns, i), ((cname, cargs), (cname', mx))) = let fun constr_arg (dt, (j, l_args, r_args)) = let val T = typ_of_dtyp descr' sorts dt; val free_t = mk_Free "x" T j in (case (strip_dtyp dt, strip_type T) of ((_, DtRec m), (Us, U)) => (j + 1, free_t :: l_args, mk_lim (Const (List.nth (all_rep_names, m), U --> Univ_elT) $ app_bnds free_t (length Us)) Us :: r_args) | _ => (j + 1, free_t::l_args, (Leaf $ mk_inj T free_t)::r_args)) end; val (_, l_args, r_args) = foldr constr_arg (1, [], []) cargs; val constrT = (map (typ_of_dtyp descr' sorts) cargs) ---> T; val abs_name = Sign.intern_const (Theory.sign_of thy) ("Abs_" ^ tname); val rep_name = Sign.intern_const (Theory.sign_of thy) ("Rep_" ^ tname); val lhs = list_comb (Const (cname, constrT), l_args); val rhs = mk_univ_inj r_args n i; val def = equals T $ lhs $ (Const (abs_name, Univ_elT --> T) $ rhs); val def_name = (Sign.base_name cname) ^ "_def"; val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq (Const (rep_name, T --> Univ_elT) $ lhs, rhs)); val (thy', [def_thm]) = thy |> Theory.add_consts_i [(cname', constrT, mx)] |> (PureThy.add_defs_i false o map Thm.no_attributes) [(def_name, def)]; in (thy', defs @ [def_thm], eqns @ [eqn], i + 1) end; (* constructor definitions for datatype *) fun dt_constr_defs ((thy, defs, eqns, rep_congs, dist_lemmas), ((((_, (_, _, constrs)), tname), T), constr_syntax)) = let val _ $ (_ $ (cong_f $ _) $ _) = concl_of arg_cong; val sg = Theory.sign_of thy; val rep_const = cterm_of sg (Const (Sign.intern_const sg ("Rep_" ^ tname), T --> Univ_elT)); val cong' = standard (cterm_instantiate [(cterm_of sg cong_f, rep_const)] arg_cong); val dist = standard (cterm_instantiate [(cterm_of sg distinct_f, rep_const)] distinct_lemma); val (thy', defs', eqns', _) = Library.foldl ((make_constr_def tname T) (length constrs)) ((add_path flat_names tname thy, defs, [], 1), constrs ~~ constr_syntax) in (parent_path flat_names thy', defs', eqns @ [eqns'], rep_congs @ [cong'], dist_lemmas @ [dist]) end; val (thy4, constr_defs, constr_rep_eqns, rep_congs, dist_lemmas) = Library.foldl dt_constr_defs ((thy3 |> Theory.add_consts_i iso_decls |> parent_path flat_names, [], [], [], []), hd descr ~~ new_type_names ~~ newTs ~~ constr_syntax); (*********** isomorphisms for new types (introduced by typedef) ***********) val _ = message "Proving isomorphism properties ..."; (* get axioms from theory *) val newT_iso_axms = map (fn s => (get_thm thy4 (Name ("Abs_" ^ s ^ "_inverse")), get_thm thy4 (Name ("Rep_" ^ s ^ "_inverse")), get_thm thy4 (Name ("Rep_" ^ s)))) new_type_names; (*------------------------------------------------*) (* prove additional theorems: *) (* inj_on dt_Abs_i rep_set_i and inj dt_Rep_i *) (*------------------------------------------------*) fun prove_newT_iso_inj_thm (((s, (thm1, thm2, _)), T), rep_set_name) = let val sg = Theory.sign_of thy4; val RepT = T --> Univ_elT; val Rep_name = Sign.intern_const sg ("Rep_" ^ s); val AbsT = Univ_elT --> T; val Abs_name = Sign.intern_const sg ("Abs_" ^ s); val inj_Abs_thm = prove_goalw_cterm [] (cterm_of sg (HOLogic.mk_Trueprop (Const ("Fun.inj_on", [AbsT, UnivT] ---> HOLogic.boolT) $ Const (Abs_name, AbsT) $ Const (rep_set_name, UnivT)))) (fn _ => [rtac inj_on_inverseI 1, etac thm1 1]); val setT = HOLogic.mk_setT T val inj_Rep_thm = prove_goalw_cterm [] (cterm_of sg (HOLogic.mk_Trueprop (Const ("Fun.inj_on", [RepT, setT] ---> HOLogic.boolT) $ Const (Rep_name, RepT) $ Const ("UNIV", setT)))) (fn _ => [rtac inj_inverseI 1, rtac thm2 1]) in (inj_Abs_thm, inj_Rep_thm) end; val newT_iso_inj_thms = map prove_newT_iso_inj_thm (new_type_names ~~ newT_iso_axms ~~ newTs ~~ Library.take (length newTs, rep_set_names)); (********* isomorphisms between existing types and "unfolded" types *******) (*---------------------------------------------------------------------*) (* isomorphisms are defined using primrec-combinators: *) (* generate appropriate functions for instantiating primrec-combinator *) (* *) (* e.g. dt_Rep_i = list_rec ... (%h t y. In1 (Scons (Leaf h) y)) *) (* *) (* also generate characteristic equations for isomorphisms *) (* *) (* e.g. dt_Rep_i (cons h t) = In1 (Scons (dt_Rep_j h) (dt_Rep_i t)) *) (*---------------------------------------------------------------------*) fun make_iso_def k ks n ((fs, eqns, i), (cname, cargs)) = let val argTs = map (typ_of_dtyp descr' sorts) cargs; val T = List.nth (recTs, k); val rep_name = List.nth (all_rep_names, k); val rep_const = Const (rep_name, T --> Univ_elT); val constr = Const (cname, argTs ---> T); fun process_arg ks' ((i2, i2', ts, Ts), dt) = let val T' = typ_of_dtyp descr' sorts dt; val (Us, U) = strip_type T' in (case strip_dtyp dt of (_, DtRec j) => if j mem ks' then (i2 + 1, i2' + 1, ts @ [mk_lim (app_bnds (mk_Free "y" (Us ---> Univ_elT) i2') (length Us)) Us], Ts @ [Us ---> Univ_elT]) else (i2 + 1, i2', ts @ [mk_lim (Const (List.nth (all_rep_names, j), U --> Univ_elT) $ app_bnds (mk_Free "x" T' i2) (length Us)) Us], Ts) | _ => (i2 + 1, i2', ts @ [Leaf $ mk_inj T' (mk_Free "x" T' i2)], Ts)) end; val (i2, i2', ts, Ts) = Library.foldl (process_arg ks) ((1, 1, [], []), cargs); val xs = map (uncurry (mk_Free "x")) (argTs ~~ (1 upto (i2 - 1))); val ys = map (uncurry (mk_Free "y")) (Ts ~~ (1 upto (i2' - 1))); val f = list_abs_free (map dest_Free (xs @ ys), mk_univ_inj ts n i); val (_, _, ts', _) = Library.foldl (process_arg []) ((1, 1, [], []), cargs); val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq (rep_const $ list_comb (constr, xs), mk_univ_inj ts' n i)) in (fs @ [f], eqns @ [eqn], i + 1) end; (* define isomorphisms for all mutually recursive datatypes in list ds *) fun make_iso_defs (ds, (thy, char_thms)) = let val ks = map fst ds; val (_, (tname, _, _)) = hd ds; val {rec_rewrites, rec_names, ...} = the (Symtab.lookup dt_info tname); fun process_dt ((fs, eqns, isos), (k, (tname, _, constrs))) = let val (fs', eqns', _) = Library.foldl (make_iso_def k ks (length constrs)) ((fs, eqns, 1), constrs); val iso = (List.nth (recTs, k), List.nth (all_rep_names, k)) in (fs', eqns', isos @ [iso]) end; val (fs, eqns, isos) = Library.foldl process_dt (([], [], []), ds); val fTs = map fastype_of fs; val defs = map (fn (rec_name, (T, iso_name)) => ((Sign.base_name iso_name) ^ "_def", equals (T --> Univ_elT) $ Const (iso_name, T --> Univ_elT) $ list_comb (Const (rec_name, fTs @ [T] ---> Univ_elT), fs))) (rec_names ~~ isos); val (thy', def_thms) = (PureThy.add_defs_i false o map Thm.no_attributes) defs thy; (* prove characteristic equations *) val rewrites = def_thms @ (map mk_meta_eq rec_rewrites); val char_thms' = map (fn eqn => prove_goalw_cterm rewrites (cterm_of (Theory.sign_of thy') eqn) (fn _ => [rtac refl 1])) eqns; in (thy', char_thms' @ char_thms) end; val (thy5, iso_char_thms) = foldr make_iso_defs (add_path flat_names big_name thy4, []) (tl descr); (* prove isomorphism properties *) fun mk_funs_inv thm = let val {sign, prop, ...} = rep_thm thm; val _ $ (_ $ (Const (_, Type (_, [U, _])) $ _ $ S)) $ (_ $ (_ $ (r $ (a $ _)) $ _)) = Type.freeze prop; val used = add_term_tfree_names (a, []); fun mk_thm i = let val Ts = map (TFree o rpair HOLogic.typeS) (variantlist (replicate i "'t", used)); val f = Free ("f", Ts ---> U) in prove_goalw_cterm [] (cterm_of sign (Logic.mk_implies (HOLogic.mk_Trueprop (HOLogic.list_all (map (pair "x") Ts, HOLogic.mk_mem (app_bnds f i, S))), HOLogic.mk_Trueprop (HOLogic.mk_eq (list_abs (map (pair "x") Ts, r $ (a $ app_bnds f i)), f))))) (fn prems => [cut_facts_tac prems 1, REPEAT (rtac ext 1), REPEAT (etac allE 1), rtac thm 1, atac 1]) end in map (fn r => r RS subst) (thm :: map mk_thm arities) end; (* prove inj dt_Rep_i and dt_Rep_i x : dt_rep_set_i *) fun prove_iso_thms (ds, (inj_thms, elem_thms)) = let val (_, (tname, _, _)) = hd ds; val {induction, ...} = the (Symtab.lookup dt_info tname); fun mk_ind_concl (i, _) = let val T = List.nth (recTs, i); val Rep_t = Const (List.nth (all_rep_names, i), T --> Univ_elT); val rep_set_name = List.nth (rep_set_names, i) in (HOLogic.all_const T $ Abs ("y", T, HOLogic.imp $ HOLogic.mk_eq (Rep_t $ mk_Free "x" T i, Rep_t $ Bound 0) $ HOLogic.mk_eq (mk_Free "x" T i, Bound 0)), HOLogic.mk_mem (Rep_t $ mk_Free "x" T i, Const (rep_set_name, UnivT))) end; val (ind_concl1, ind_concl2) = ListPair.unzip (map mk_ind_concl ds); val rewrites = map mk_meta_eq iso_char_thms; val inj_thms' = map (fn r => r RS injD) (map snd newT_iso_inj_thms @ inj_thms); val inj_thm = prove_goalw_cterm [] (cterm_of (Theory.sign_of thy5) (HOLogic.mk_Trueprop (mk_conj ind_concl1))) (fn _ => [(indtac induction THEN_ALL_NEW ObjectLogic.atomize_tac) 1, REPEAT (EVERY [rtac allI 1, rtac impI 1, exh_tac (exh_thm_of dt_info) 1, REPEAT (EVERY [hyp_subst_tac 1, rewrite_goals_tac rewrites, REPEAT (dresolve_tac [In0_inject, In1_inject] 1), (eresolve_tac [In0_not_In1 RS notE, In1_not_In0 RS notE] 1) ORELSE (EVERY [REPEAT (eresolve_tac (Scons_inject :: map make_elim [Leaf_inject, Inl_inject, Inr_inject]) 1), REPEAT (cong_tac 1), rtac refl 1, REPEAT (atac 1 ORELSE (EVERY [REPEAT (rtac ext 1), REPEAT (eresolve_tac (mp :: allE :: map make_elim (Suml_inject :: Sumr_inject :: Lim_inject :: fun_cong :: inj_thms')) 1), atac 1]))])])])]); val inj_thms'' = map (fn r => r RS datatype_injI) (split_conj_thm inj_thm); val elem_thm = prove_goalw_cterm [] (cterm_of (Theory.sign_of thy5) (HOLogic.mk_Trueprop (mk_conj ind_concl2))) (fn _ => [(indtac induction THEN_ALL_NEW ObjectLogic.atomize_tac) 1, rewrite_goals_tac rewrites, REPEAT ((resolve_tac rep_intrs THEN_ALL_NEW ((REPEAT o etac allE) THEN' ares_tac elem_thms)) 1)]); in (inj_thms'' @ inj_thms, elem_thms @ (split_conj_thm elem_thm)) end; val (iso_inj_thms_unfolded, iso_elem_thms) = foldr prove_iso_thms ([], map #3 newT_iso_axms) (tl descr); val iso_inj_thms = map snd newT_iso_inj_thms @ iso_inj_thms_unfolded; (* prove x : dt_rep_set_i --> x : range dt_Rep_i *) fun mk_iso_t (((set_name, iso_name), i), T) = let val isoT = T --> Univ_elT in HOLogic.imp $ HOLogic.mk_mem (mk_Free "x" Univ_elT i, Const (set_name, UnivT)) $ (if i < length newTs then Const ("True", HOLogic.boolT) else HOLogic.mk_mem (mk_Free "x" Univ_elT i, Const ("image", [isoT, HOLogic.mk_setT T] ---> UnivT) $ Const (iso_name, isoT) $ Const ("UNIV", HOLogic.mk_setT T))) end; val iso_t = HOLogic.mk_Trueprop (mk_conj (map mk_iso_t (rep_set_names ~~ all_rep_names ~~ (0 upto (length descr' - 1)) ~~ recTs))); (* all the theorems are proved by one single simultaneous induction *) val range_eqs = map (fn r => mk_meta_eq (r RS range_ex1_eq)) iso_inj_thms_unfolded; val iso_thms = if length descr = 1 then [] else Library.drop (length newTs, split_conj_thm (prove_goalw_cterm [] (cterm_of (Theory.sign_of thy5) iso_t) (fn _ => [(indtac rep_induct THEN_ALL_NEW ObjectLogic.atomize_tac) 1, REPEAT (rtac TrueI 1), rewrite_goals_tac (mk_meta_eq choice_eq :: symmetric (mk_meta_eq expand_fun_eq) :: range_eqs), rewrite_goals_tac (map symmetric range_eqs), REPEAT (EVERY [REPEAT (eresolve_tac ([rangeE, ex1_implies_ex RS exE] @ List.concat (map (mk_funs_inv o #1) newT_iso_axms)) 1), TRY (hyp_subst_tac 1), rtac (sym RS range_eqI) 1, resolve_tac iso_char_thms 1])]))); val Abs_inverse_thms' = map #1 newT_iso_axms @ map2 (fn (r_inj, r) => f_myinv_f OF [r_inj, r RS mp]) (iso_inj_thms_unfolded, iso_thms); val Abs_inverse_thms = List.concat (map mk_funs_inv Abs_inverse_thms'); (******************* freeness theorems for constructors *******************) val _ = message "Proving freeness of constructors ..."; (* prove theorem Rep_i (Constr_j ...) = Inj_j ... *) fun prove_constr_rep_thm eqn = let val inj_thms = map (fn (r, _) => r RS inj_onD) newT_iso_inj_thms; val rewrites = o_def :: constr_defs @ (map (mk_meta_eq o #2) newT_iso_axms) in prove_goalw_cterm [] (cterm_of (Theory.sign_of thy5) eqn) (fn _ => [resolve_tac inj_thms 1, rewrite_goals_tac rewrites, rtac refl 1, resolve_tac rep_intrs 2, REPEAT (resolve_tac iso_elem_thms 1)]) end; (*--------------------------------------------------------------*) (* constr_rep_thms and rep_congs are used to prove distinctness *) (* of constructors. *) (*--------------------------------------------------------------*) val constr_rep_thms = map (map prove_constr_rep_thm) constr_rep_eqns; val dist_rewrites = map (fn (rep_thms, dist_lemma) => dist_lemma::(rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0])) (constr_rep_thms ~~ dist_lemmas); fun prove_distinct_thms (_, []) = [] | prove_distinct_thms (dist_rewrites', t::_::ts) = let val dist_thm = prove_goalw_cterm [] (cterm_of (Theory.sign_of thy5) t) (fn _ => [simp_tac (HOL_ss addsimps dist_rewrites') 1]) in dist_thm::(standard (dist_thm RS not_sym)):: (prove_distinct_thms (dist_rewrites', ts)) end; val distinct_thms = map prove_distinct_thms (dist_rewrites ~~ DatatypeProp.make_distincts new_type_names descr sorts thy5); val simproc_dists = map (fn ((((_, (_, _, constrs)), rep_thms), congr), dists) => if length constrs < !DatatypeProp.dtK then FewConstrs dists else ManyConstrs (congr, HOL_basic_ss addsimps rep_thms)) (hd descr ~~ constr_rep_thms ~~ rep_congs ~~ distinct_thms); (* prove injectivity of constructors *) fun prove_constr_inj_thm rep_thms t = let val inj_thms = Scons_inject :: (map make_elim ((map (fn r => r RS injD) iso_inj_thms) @ [In0_inject, In1_inject, Leaf_inject, Inl_inject, Inr_inject, Lim_inject, Suml_inject, Sumr_inject])) in prove_goalw_cterm [] (cterm_of (Theory.sign_of thy5) t) (fn _ => [rtac iffI 1, REPEAT (etac conjE 2), hyp_subst_tac 2, rtac refl 2, dresolve_tac rep_congs 1, dtac box_equals 1, REPEAT (resolve_tac rep_thms 1), REPEAT (eresolve_tac inj_thms 1), REPEAT (ares_tac [conjI] 1 ORELSE (EVERY [REPEAT (rtac ext 1), REPEAT (eresolve_tac (make_elim fun_cong :: inj_thms) 1), atac 1]))]) end; val constr_inject = map (fn (ts, thms) => map (prove_constr_inj_thm thms) ts) ((DatatypeProp.make_injs descr sorts) ~~ constr_rep_thms); val (thy6, (constr_inject', distinct_thms'))= thy5 |> parent_path flat_names |> store_thmss "inject" new_type_names constr_inject |>>> store_thmss "distinct" new_type_names distinct_thms; (*************************** induction theorem ****************************) val _ = message "Proving induction rule for datatypes ..."; val Rep_inverse_thms = (map (fn (_, iso, _) => iso RS subst) newT_iso_axms) @ (map (fn r => r RS myinv_f_f RS subst) iso_inj_thms_unfolded); val Rep_inverse_thms' = map (fn r => r RS myinv_f_f) iso_inj_thms_unfolded; fun mk_indrule_lemma ((prems, concls), ((i, _), T)) = let val Rep_t = Const (List.nth (all_rep_names, i), T --> Univ_elT) $ mk_Free "x" T i; val Abs_t = if i < length newTs then Const (Sign.intern_const (Theory.sign_of thy6) ("Abs_" ^ (List.nth (new_type_names, i))), Univ_elT --> T) else Const ("Inductive.myinv", [T --> Univ_elT, Univ_elT] ---> T) $ Const (List.nth (all_rep_names, i), T --> Univ_elT) in (prems @ [HOLogic.imp $ HOLogic.mk_mem (Rep_t, Const (List.nth (rep_set_names, i), UnivT)) $ (mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))], concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i]) end; val (indrule_lemma_prems, indrule_lemma_concls) = Library.foldl mk_indrule_lemma (([], []), (descr' ~~ recTs)); val cert = cterm_of (Theory.sign_of thy6); val indrule_lemma = prove_goalw_cterm [] (cert (Logic.mk_implies (HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems), HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls)))) (fn prems => [cut_facts_tac prems 1, REPEAT (etac conjE 1), REPEAT (EVERY [TRY (rtac conjI 1), resolve_tac Rep_inverse_thms 1, etac mp 1, resolve_tac iso_elem_thms 1])]); val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma))); val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else map (Free o apfst fst o dest_Var) Ps; val indrule_lemma' = cterm_instantiate (map cert Ps ~~ map cert frees) indrule_lemma; val dt_induct = prove_goalw_cterm [] (cert (DatatypeProp.make_ind descr sorts)) (fn prems => [rtac indrule_lemma' 1, (indtac rep_induct THEN_ALL_NEW ObjectLogic.atomize_tac) 1, EVERY (map (fn (prem, r) => (EVERY [REPEAT (eresolve_tac Abs_inverse_thms 1), simp_tac (HOL_basic_ss addsimps ((symmetric r)::Rep_inverse_thms')) 1, DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE etac allE 1)])) (prems ~~ (constr_defs @ (map mk_meta_eq iso_char_thms))))]); val (thy7, [dt_induct']) = thy6 |> Theory.add_path big_name |> PureThy.add_thms [(("induct", dt_induct), [case_names_induct])] |>> Theory.parent_path; in (thy7, constr_inject', distinct_thms', dist_rewrites, simproc_dists, dt_induct') end; end;