--- a/src/HOL/Tools/Datatype/datatype_rep_proofs.ML Mon Nov 30 11:42:48 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,747 +0,0 @@
-(* Title: HOL/Tools/datatype_rep_proofs.ML
- Author: Stefan Berghofer, TU Muenchen
-
-Definitional introduction of datatypes with proof of characteristic theorems:
-
- - injectivity of constructors
- - distinctness of constructors
- - induction theorem
-*)
-
-signature DATATYPE_REP_PROOFS =
-sig
- val add_datatype : DatatypeAux.config -> string list -> (string list * binding * mixfix *
- (binding * typ list * mixfix) list) list -> theory -> string list * theory
- val datatype_cmd : string list -> (string list * binding * mixfix *
- (binding * string list * mixfix) list) list -> theory -> theory
-end;
-
-structure DatatypeRepProofs : DATATYPE_REP_PROOFS =
-struct
-
-(** auxiliary **)
-
-open DatatypeAux;
-
-val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of distinct_lemma);
-
-val collect_simp = rewrite_rule [mk_meta_eq mem_Collect_eq];
-
-fun exh_thm_of (dt_info : info Symtab.table) tname =
- #exhaust (the (Symtab.lookup dt_info tname));
-
-val node_name = @{type_name "Datatype.node"};
-val In0_name = @{const_name "Datatype.In0"};
-val In1_name = @{const_name "Datatype.In1"};
-val Scons_name = @{const_name "Datatype.Scons"};
-val Leaf_name = @{const_name "Datatype.Leaf"};
-val Numb_name = @{const_name "Datatype.Numb"};
-val Lim_name = @{const_name "Datatype.Lim"};
-val Suml_name = @{const_name "Sum_Type.Suml"};
-val Sumr_name = @{const_name "Sum_Type.Sumr"};
-
-val In0_inject = @{thm In0_inject};
-val In1_inject = @{thm In1_inject};
-val Scons_inject = @{thm Scons_inject};
-val Leaf_inject = @{thm Leaf_inject};
-val In0_eq = @{thm In0_eq};
-val In1_eq = @{thm In1_eq};
-val In0_not_In1 = @{thm In0_not_In1};
-val In1_not_In0 = @{thm In1_not_In0};
-val Lim_inject = @{thm Lim_inject};
-val Inl_inject = @{thm Inl_inject};
-val Inr_inject = @{thm Inr_inject};
-val Suml_inject = @{thm Suml_inject};
-val Sumr_inject = @{thm Sumr_inject};
-
-
-
-(** proof of characteristic theorems **)
-
-fun representation_proofs (config : config) (dt_info : info Symtab.table)
- new_type_names descr sorts types_syntax constr_syntax case_names_induct thy =
- let
- val descr' = flat descr;
- val big_name = space_implode "_" new_type_names;
- val thy1 = Sign.add_path big_name thy;
- val big_rec_name = big_name ^ "_rep_set";
- val rep_set_names' =
- (if length descr' = 1 then [big_rec_name] else
- (map ((curry (op ^) (big_rec_name ^ "_")) o string_of_int)
- (1 upto (length descr'))));
- val rep_set_names = map (Sign.full_bname thy1) rep_set_names';
-
- 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 Balanced_Tree.make (fn (T, U) => Type ("+", [T, U])) branchTs;
- val arities = remove (op =) 0 (get_arities descr');
- val unneeded_vars =
- subtract (op =) (List.foldr OldTerm.add_typ_tfree_names [] (leafTs' @ branchTs)) (hd tyvars);
- 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, oldTs) = chop (length (hd descr)) recTs;
- val sumT = if null leafTs then HOLogic.unitT
- else Balanced_Tree.make (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 UnivT' = Univ_elT --> HOLogic.boolT;
- val Collect = Const (@{const_name Collect}, UnivT' --> UnivT);
-
- 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 (@{const_name "Sum_Type.Inl"}, T1 --> T) $ (mk_inj' T1 n2 i)
- else
- Const (@{const_name "Sum_Type.Inr"}, T2 --> T) $ (mk_inj' T2 (n - n2) (i - n2))
- end
- in mk_inj' sumT (length leafTs) (1 + find_index (fn T'' => T'' = T') leafTs)
- end;
-
- (* make injections for constructors *)
-
- fun mk_univ_inj ts = Balanced_Tree.access
- {left = fn t => In0 $ t,
- right = fn t => In1 $ t,
- init =
- if ts = [] then Const (@{const_name undefined}, 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 (fn T'' => T'' = T') branchTs)
- end;
-
- fun mk_lim t Ts = fold_rev (fn T => fn t => Lim $ mk_fun_inj T (Abs ("x", T, t))) Ts t;
-
- (************** generate introduction rules for representing set **********)
-
- val _ = message config "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
- (Free (nth rep_set_names' k, UnivT') $ free_t)) :: 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) = fold_rev mk_prem cargs (1, [], []);
- val concl = HOLogic.mk_Trueprop
- (Free (s, UnivT') $ mk_univ_inj ts n i)
- in Logic.list_implies (prems, concl)
- end;
-
- val intr_ts = maps (fn ((_, (_, _, constrs)), rep_set_name) =>
- map (make_intr rep_set_name (length constrs))
- ((1 upto (length constrs)) ~~ constrs)) (descr' ~~ rep_set_names');
-
- val ({raw_induct = rep_induct, intrs = rep_intrs, ...}, thy2) =
- thy1
- |> Sign.map_naming Name_Space.conceal
- |> Inductive.add_inductive_global
- {quiet_mode = #quiet config, verbose = false, alt_name = Binding.name big_rec_name,
- coind = false, no_elim = true, no_ind = false, skip_mono = true, fork_mono = false}
- (map (fn s => ((Binding.name s, UnivT'), NoSyn)) rep_set_names') []
- (map (fn x => (Attrib.empty_binding, x)) intr_ts) []
- ||> Sign.restore_naming thy1
- ||> Theory.checkpoint;
-
- (********************************* typedef ********************************)
-
- val (typedefs, thy3) = thy2 |>
- Sign.parent_path |>
- fold_map (fn ((((name, mx), tvs), c), name') =>
- Typedef.add_typedef false (SOME (Binding.name name')) (name, tvs, mx)
- (Collect $ Const (c, UnivT')) NONE
- (rtac exI 1 THEN rtac CollectI 1 THEN
- QUIET_BREADTH_FIRST (has_fewer_prems 1)
- (resolve_tac rep_intrs 1)))
- (types_syntax ~~ tyvars ~~
- (take (length newTs) rep_set_names) ~~ new_type_names) ||>
- Sign.add_path big_name;
-
- (*********************** 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 (flat (tl descr))));
- val all_rep_names = map (Sign.intern_const thy3) rep_names @
- map (Sign.full_bname thy3) rep_names';
-
- (* isomorphism declarations *)
-
- val iso_decls = map (fn (T, s) => (Binding.name s, T --> Univ_elT, NoSyn))
- (oldTs ~~ rep_names');
-
- (* constructor definitions *)
-
- fun make_constr_def tname T n ((cname, cargs), (cname', mx)) (thy, defs, eqns, i) =
- 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 (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) = fold_rev constr_arg cargs (1, [], []);
- val constrT = (map (typ_of_dtyp descr' sorts) cargs) ---> T;
- val abs_name = Sign.intern_const thy ("Abs_" ^ tname);
- val rep_name = Sign.intern_const thy ("Rep_" ^ tname);
- val lhs = list_comb (Const (cname, constrT), l_args);
- val rhs = mk_univ_inj r_args n i;
- val def = Logic.mk_equals (lhs, Const (abs_name, Univ_elT --> T) $ rhs);
- val def_name = Long_Name.base_name cname ^ "_def";
- val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
- (Const (rep_name, T --> Univ_elT) $ lhs, rhs));
- val ([def_thm], thy') =
- thy
- |> Sign.add_consts_i [(cname', constrT, mx)]
- |> (PureThy.add_defs false o map Thm.no_attributes) [(Binding.name def_name, def)];
-
- in (thy', defs @ [def_thm], eqns @ [eqn], i + 1) end;
-
- (* constructor definitions for datatype *)
-
- fun dt_constr_defs ((((_, (_, _, constrs)), tname), T), constr_syntax)
- (thy, defs, eqns, rep_congs, dist_lemmas) =
- let
- val _ $ (_ $ (cong_f $ _) $ _) = concl_of arg_cong;
- val rep_const = cterm_of thy
- (Const (Sign.intern_const thy ("Rep_" ^ tname), T --> Univ_elT));
- val cong' =
- Drule.standard (cterm_instantiate [(cterm_of thy cong_f, rep_const)] arg_cong);
- val dist =
- Drule.standard (cterm_instantiate [(cterm_of thy distinct_f, rep_const)] distinct_lemma);
- val (thy', defs', eqns', _) = fold ((make_constr_def tname T) (length constrs))
- (constrs ~~ constr_syntax) (Sign.add_path tname thy, defs, [], 1);
- in
- (Sign.parent_path thy', defs', eqns @ [eqns'],
- rep_congs @ [cong'], dist_lemmas @ [dist])
- end;
-
- val (thy4, constr_defs, constr_rep_eqns, rep_congs, dist_lemmas) =
- fold dt_constr_defs
- (hd descr ~~ new_type_names ~~ newTs ~~ constr_syntax)
- (thy3 |> Sign.add_consts_i iso_decls |> Sign.parent_path, [], [], [], []);
-
-
- (*********** isomorphisms for new types (introduced by typedef) ***********)
-
- val _ = message config "Proving isomorphism properties ...";
-
- val newT_iso_axms = map (fn (_, td) =>
- (collect_simp (#Abs_inverse td), #Rep_inverse td,
- collect_simp (#Rep td))) typedefs;
-
- val newT_iso_inj_thms = map (fn (_, td) =>
- (collect_simp (#Abs_inject td) RS iffD1, #Rep_inject td RS iffD1)) typedefs;
-
- (********* 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 (cname, cargs) (fs, eqns, i) =
- let
- val argTs = map (typ_of_dtyp descr' sorts) cargs;
- val T = nth recTs k;
- val rep_name = nth all_rep_names k;
- val rep_const = Const (rep_name, T --> Univ_elT);
- val constr = Const (cname, argTs ---> T);
-
- fun process_arg ks' dt (i2, i2', ts, Ts) =
- 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 (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) = fold (process_arg ks) cargs (1, 1, [], []);
- 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', _) = fold (process_arg []) cargs (1, 1, [], []);
- 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 (k, (tname, _, constrs)) (fs, eqns, isos) =
- let
- val (fs', eqns', _) =
- fold (make_iso_def k ks (length constrs)) constrs (fs, eqns, 1);
- val iso = (nth recTs k, nth all_rep_names k)
- in (fs', eqns', isos @ [iso]) end;
-
- val (fs, eqns, isos) = fold process_dt ds ([], [], []);
- val fTs = map fastype_of fs;
- val defs = map (fn (rec_name, (T, iso_name)) => (Binding.name (Long_Name.base_name iso_name ^ "_def"),
- Logic.mk_equals (Const (iso_name, T --> Univ_elT),
- list_comb (Const (rec_name, fTs @ [T] ---> Univ_elT), fs)))) (rec_names ~~ isos);
- val (def_thms, thy') =
- apsnd Theory.checkpoint ((PureThy.add_defs 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 => Skip_Proof.prove_global thy' [] [] eqn
- (fn _ => EVERY [rewrite_goals_tac rewrites, rtac refl 1])) eqns;
-
- in (thy', char_thms' @ char_thms) end;
-
- val (thy5, iso_char_thms) = apfst Theory.checkpoint (fold_rev make_iso_defs
- (tl descr) (Sign.add_path big_name thy4, []));
-
- (* prove isomorphism properties *)
-
- fun mk_funs_inv thy thm =
- let
- val prop = Thm.prop_of thm;
- val _ $ (_ $ ((S as Const (_, Type (_, [U, _]))) $ _ )) $
- (_ $ (_ $ (r $ (a $ _)) $ _)) = Type.legacy_freeze prop;
- val used = OldTerm.add_term_tfree_names (a, []);
-
- fun mk_thm i =
- let
- val Ts = map (TFree o rpair HOLogic.typeS)
- (Name.variant_list used (replicate i "'t"));
- val f = Free ("f", Ts ---> U)
- in Skip_Proof.prove_global thy [] [] (Logic.mk_implies
- (HOLogic.mk_Trueprop (HOLogic.list_all
- (map (pair "x") Ts, S $ app_bnds f i)),
- HOLogic.mk_Trueprop (HOLogic.mk_eq (list_abs (map (pair "x") Ts,
- r $ (a $ app_bnds f i)), f))))
- (fn _ => EVERY [REPEAT_DETERM_N i (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 *)
-
- val fun_congs = map (fn T => make_elim (Drule.instantiate'
- [SOME (ctyp_of thy5 T)] [] fun_cong)) branchTs;
-
- fun prove_iso_thms ds (inj_thms, elem_thms) =
- let
- val (_, (tname, _, _)) = hd ds;
- val induct = (#induct o the o Symtab.lookup dt_info) tname;
-
- fun mk_ind_concl (i, _) =
- let
- val T = nth recTs i;
- val Rep_t = Const (nth all_rep_names i, T --> Univ_elT);
- val rep_set_name = 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)),
- Const (rep_set_name, UnivT') $ (Rep_t $ mk_Free "x" T i))
- 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 snd newT_iso_inj_thms @
- map (fn r => r RS @{thm injD}) inj_thms;
-
- val inj_thm = Skip_Proof.prove_global thy5 [] []
- (HOLogic.mk_Trueprop (mk_conj ind_concl1)) (fn _ => EVERY
- [(indtac induct [] THEN_ALL_NEW ObjectLogic.atomize_prems_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 :: inj_thms') @ fun_congs) 1),
- atac 1]))])])])]);
-
- val inj_thms'' = map (fn r => r RS @{thm datatype_injI})
- (split_conj_thm inj_thm);
-
- val elem_thm =
- Skip_Proof.prove_global thy5 [] [] (HOLogic.mk_Trueprop (mk_conj ind_concl2))
- (fn _ =>
- EVERY [(indtac induct [] THEN_ALL_NEW ObjectLogic.atomize_prems_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) =
- fold_rev prove_iso_thms (tl descr) ([], map #3 newT_iso_axms);
- val iso_inj_thms = map snd newT_iso_inj_thms @
- map (fn r => r RS @{thm injD}) iso_inj_thms_unfolded;
-
- (* prove dt_rep_set_i x --> x : range dt_Rep_i *)
-
- fun mk_iso_t (((set_name, iso_name), i), T) =
- let val isoT = T --> Univ_elT
- in HOLogic.imp $
- (Const (set_name, UnivT') $ mk_Free "x" Univ_elT i) $
- (if i < length newTs then HOLogic.true_const
- else HOLogic.mk_mem (mk_Free "x" Univ_elT i,
- Const (@{const_name image}, isoT --> HOLogic.mk_setT T --> UnivT) $
- Const (iso_name, isoT) $ Const (@{const_name 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 @{thm range_ex1_eq}))
- iso_inj_thms_unfolded;
-
- val iso_thms = if length descr = 1 then [] else
- drop (length newTs) (split_conj_thm
- (Skip_Proof.prove_global thy5 [] [] iso_t (fn _ => EVERY
- [(indtac rep_induct [] THEN_ALL_NEW ObjectLogic.atomize_prems_tac) 1,
- REPEAT (rtac TrueI 1),
- rewrite_goals_tac (mk_meta_eq choice_eq ::
- symmetric (mk_meta_eq @{thm expand_fun_eq}) :: range_eqs),
- rewrite_goals_tac (map symmetric range_eqs),
- REPEAT (EVERY
- [REPEAT (eresolve_tac ([rangeE, ex1_implies_ex RS exE] @
- maps (mk_funs_inv thy5 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 => fn r => @{thm f_the_inv_into_f} OF [r_inj, r RS mp])
- iso_inj_thms_unfolded iso_thms;
-
- val Abs_inverse_thms = maps (mk_funs_inv thy5) Abs_inverse_thms';
-
- (******************* freeness theorems for constructors *******************)
-
- val _ = message config "Proving freeness of constructors ...";
-
- (* prove theorem Rep_i (Constr_j ...) = Inj_j ... *)
-
- fun prove_constr_rep_thm eqn =
- let
- val inj_thms = map fst newT_iso_inj_thms;
- val rewrites = @{thm o_def} :: constr_defs @ (map (mk_meta_eq o #2) newT_iso_axms)
- in Skip_Proof.prove_global thy5 [] [] eqn (fn _ => EVERY
- [resolve_tac inj_thms 1,
- rewrite_goals_tac rewrites,
- rtac refl 3,
- 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 dist_rewrites' (k, ts) =
- let
- fun prove [] = []
- | prove (t :: ts) =
- let
- val dist_thm = Skip_Proof.prove_global thy5 [] [] t (fn _ =>
- EVERY [simp_tac (HOL_ss addsimps dist_rewrites') 1])
- in dist_thm :: Drule.standard (dist_thm RS not_sym) :: prove ts end;
- in prove ts end;
-
- val distinct_thms = map2 (prove_distinct_thms)
- dist_rewrites (DatatypeProp.make_distincts descr sorts);
-
- (* prove injectivity of constructors *)
-
- fun prove_constr_inj_thm rep_thms t =
- let val inj_thms = Scons_inject :: (map make_elim
- (iso_inj_thms @
- [In0_inject, In1_inject, Leaf_inject, Inl_inject, Inr_inject,
- Lim_inject, Suml_inject, Sumr_inject]))
- in Skip_Proof.prove_global thy5 [] [] t (fn _ => EVERY
- [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 ((constr_inject', distinct_thms'), thy6) =
- thy5
- |> Sign.parent_path
- |> store_thmss "inject" new_type_names constr_inject
- ||>> store_thmss "distinct" new_type_names distinct_thms;
-
- (*************************** induction theorem ****************************)
-
- val _ = message config "Proving induction rule for datatypes ...";
-
- val Rep_inverse_thms = (map (fn (_, iso, _) => iso RS subst) newT_iso_axms) @
- (map (fn r => r RS @{thm the_inv_f_f} RS subst) iso_inj_thms_unfolded);
- val Rep_inverse_thms' = map (fn r => r RS @{thm the_inv_f_f}) iso_inj_thms_unfolded;
-
- fun mk_indrule_lemma ((i, _), T) (prems, concls) =
- let
- val Rep_t = Const (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 thy6
- ("Abs_" ^ (nth new_type_names i)), Univ_elT --> T)
- else Const (@{const_name the_inv_into},
- [HOLogic.mk_setT T, T --> Univ_elT, Univ_elT] ---> T) $
- HOLogic.mk_UNIV T $ Const (nth all_rep_names i, T --> Univ_elT)
-
- in (prems @ [HOLogic.imp $
- (Const (nth rep_set_names i, UnivT') $ Rep_t) $
- (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) =
- fold mk_indrule_lemma (descr' ~~ recTs) ([], []);
-
- val cert = cterm_of thy6;
-
- val indrule_lemma = Skip_Proof.prove_global thy6 [] []
- (Logic.mk_implies
- (HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems),
- HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls))) (fn _ => EVERY
- [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_prop = DatatypeProp.make_ind descr sorts;
- val dt_induct = Skip_Proof.prove_global thy6 []
- (Logic.strip_imp_prems dt_induct_prop) (Logic.strip_imp_concl dt_induct_prop)
- (fn {prems, ...} => EVERY
- [rtac indrule_lemma' 1,
- (indtac rep_induct [] THEN_ALL_NEW ObjectLogic.atomize_prems_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 ([dt_induct'], thy7) =
- thy6
- |> Sign.add_path big_name
- |> PureThy.add_thms [((Binding.name "induct", dt_induct), [case_names_induct])]
- ||> Sign.parent_path
- ||> Theory.checkpoint;
-
- in
- ((constr_inject', distinct_thms', dt_induct'), thy7)
- end;
-
-
-
-(** definitional introduction of datatypes **)
-
-fun gen_add_datatype prep_typ config new_type_names dts thy =
- let
- val _ = Theory.requires thy "Datatype" "datatype definitions";
-
- (* this theory is used just for parsing *)
- val tmp_thy = thy |>
- Theory.copy |>
- Sign.add_types (map (fn (tvs, tname, mx, _) =>
- (tname, length tvs, mx)) dts);
-
- val (tyvars, _, _, _)::_ = dts;
- val (new_dts, types_syntax) = ListPair.unzip (map (fn (tvs, tname, mx, _) =>
- let val full_tname = Sign.full_name tmp_thy (Binding.map_name (Syntax.type_name mx) tname)
- in
- (case duplicates (op =) tvs of
- [] =>
- if eq_set (op =) (tyvars, tvs) then ((full_tname, tvs), (tname, mx))
- else error ("Mutually recursive datatypes must have same type parameters")
- | dups => error ("Duplicate parameter(s) for datatype " ^ quote (Binding.str_of tname) ^
- " : " ^ commas dups))
- end) dts);
- val dt_names = map fst new_dts;
-
- val _ =
- (case duplicates (op =) (map fst new_dts) @ duplicates (op =) new_type_names of
- [] => ()
- | dups => error ("Duplicate datatypes: " ^ commas dups));
-
- fun prep_dt_spec (tvs, tname, mx, constrs) tname' (dts', constr_syntax, sorts, i) =
- let
- fun prep_constr (cname, cargs, mx') (constrs, constr_syntax', sorts') =
- let
- val (cargs', sorts'') = fold_map (prep_typ tmp_thy) cargs sorts';
- val _ =
- (case subtract (op =) tvs (fold (curry OldTerm.add_typ_tfree_names) cargs' []) of
- [] => ()
- | vs => error ("Extra type variables on rhs: " ^ commas vs))
- in (constrs @ [(Sign.full_name_path tmp_thy tname'
- (Binding.map_name (Syntax.const_name mx') cname),
- map (dtyp_of_typ new_dts) cargs')],
- constr_syntax' @ [(cname, mx')], sorts'')
- end handle ERROR msg => cat_error msg
- ("The error above occured in constructor " ^ quote (Binding.str_of cname) ^
- " of datatype " ^ quote (Binding.str_of tname));
-
- val (constrs', constr_syntax', sorts') =
- fold prep_constr constrs ([], [], sorts)
-
- in
- case duplicates (op =) (map fst constrs') of
- [] =>
- (dts' @ [(i, (Sign.full_name tmp_thy (Binding.map_name (Syntax.type_name mx) tname),
- map DtTFree tvs, constrs'))],
- constr_syntax @ [constr_syntax'], sorts', i + 1)
- | dups => error ("Duplicate constructors " ^ commas dups ^
- " in datatype " ^ quote (Binding.str_of tname))
- end;
-
- val (dts', constr_syntax, sorts', i) =
- fold2 prep_dt_spec dts new_type_names ([], [], [], 0);
- val sorts = sorts' @ map (rpair (Sign.defaultS tmp_thy)) (subtract (op =) (map fst sorts') tyvars);
- val dt_info = Datatype_Data.get_all thy;
- val (descr, _) = unfold_datatypes tmp_thy dts' sorts dt_info dts' i;
- val _ = check_nonempty descr handle (exn as Datatype_Empty s) =>
- if #strict config then error ("Nonemptiness check failed for datatype " ^ s)
- else raise exn;
-
- val _ = message config ("Constructing datatype(s) " ^ commas_quote new_type_names);
-
- in
- thy
- |> representation_proofs config dt_info new_type_names descr sorts
- types_syntax constr_syntax (Datatype_Data.mk_case_names_induct (flat descr))
- |-> (fn (inject, distinct, induct) => Datatype_Data.derive_datatype_props
- config dt_names (SOME new_type_names) descr sorts
- induct inject distinct)
- end;
-
-val add_datatype = gen_add_datatype Datatype_Data.cert_typ;
-val datatype_cmd = snd ooo gen_add_datatype Datatype_Data.read_typ default_config;
-
-local
-
-structure P = OuterParse and K = OuterKeyword
-
-fun prep_datatype_decls args =
- let
- val names = map
- (fn ((((NONE, _), t), _), _) => Binding.name_of t | ((((SOME t, _), _), _), _) => t) args;
- val specs = map (fn ((((_, vs), t), mx), cons) =>
- (vs, t, mx, map (fn ((x, y), z) => (x, y, z)) cons)) args;
- in (names, specs) end;
-
-val parse_datatype_decl =
- (Scan.option (P.$$$ "(" |-- P.name --| P.$$$ ")") -- P.type_args -- P.binding -- P.opt_infix --
- (P.$$$ "=" |-- P.enum1 "|" (P.binding -- Scan.repeat P.typ -- P.opt_mixfix)));
-
-val parse_datatype_decls = P.and_list1 parse_datatype_decl >> prep_datatype_decls;
-
-in
-
-val _ =
- OuterSyntax.command "datatype" "define inductive datatypes" K.thy_decl
- (parse_datatype_decls >> (fn (names, specs) => Toplevel.theory (datatype_cmd names specs)));
-
-end;
-
-end;