(* Title: HOL/Tools/Datatype/datatype_abs_proofs.ML
Author: Stefan Berghofer, TU Muenchen
Datatype package: proofs and definitions independent of concrete
representation of datatypes (i.e. requiring only abstract
properties: injectivity / distinctness of constructors and induction).
*)
signature DATATYPE_ABS_PROOFS =
sig
include DATATYPE_COMMON
val prove_casedist_thms : config -> string list ->
descr list -> (string * sort) list -> thm ->
attribute list -> theory -> thm list * theory
val prove_primrec_thms : config -> string list ->
descr list -> (string * sort) list ->
(string -> thm list) -> thm list list -> thm list list * thm list list ->
thm -> theory -> (string list * thm list) * theory
val prove_case_thms : config -> string list ->
descr list -> (string * sort) list ->
string list -> thm list -> theory -> (thm list list * string list) * theory
val prove_split_thms : config -> string list ->
descr list -> (string * sort) list ->
thm list list -> thm list list -> thm list -> thm list list -> theory ->
(thm * thm) list * theory
val prove_nchotomys : config -> string list -> descr list ->
(string * sort) list -> thm list -> theory -> thm list * theory
val prove_weak_case_congs : string list -> descr list ->
(string * sort) list -> theory -> thm list * theory
val prove_case_congs : string list ->
descr list -> (string * sort) list ->
thm list -> thm list list -> theory -> thm list * theory
end;
structure Datatype_Abs_Proofs: DATATYPE_ABS_PROOFS =
struct
(************************ case distinction theorems ***************************)
fun prove_casedist_thms (config : Datatype_Aux.config)
new_type_names descr sorts induct case_names_exhausts thy =
let
val _ = Datatype_Aux.message config "Proving case distinction theorems ...";
val descr' = flat descr;
val recTs = Datatype_Aux.get_rec_types descr' sorts;
val newTs = take (length (hd descr)) recTs;
val maxidx = Thm.maxidx_of induct;
val induct_Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct)));
fun prove_casedist_thm (i, (T, t)) =
let
val dummyPs = map (fn (Var (_, Type (_, [T', T'']))) =>
Abs ("z", T', Const (@{const_name True}, T''))) induct_Ps;
val P =
Abs ("z", T, HOLogic.imp $ HOLogic.mk_eq (Var (("a", maxidx + 1), T), Bound 0) $
Var (("P", 0), HOLogic.boolT));
val insts = take i dummyPs @ (P :: drop (i + 1) dummyPs);
val cert = cterm_of thy;
val insts' = map cert induct_Ps ~~ map cert insts;
val induct' =
refl RS
(nth (Datatype_Aux.split_conj_thm (cterm_instantiate insts' induct)) i RSN (2, rev_mp));
in
Skip_Proof.prove_global thy []
(Logic.strip_imp_prems t)
(Logic.strip_imp_concl t)
(fn {prems, ...} =>
EVERY
[rtac induct' 1,
REPEAT (rtac TrueI 1),
REPEAT ((rtac impI 1) THEN (eresolve_tac prems 1)),
REPEAT (rtac TrueI 1)])
end;
val casedist_thms =
map_index prove_casedist_thm (newTs ~~ Datatype_Prop.make_casedists descr sorts);
in
thy
|> Datatype_Aux.store_thms_atts "exhaust" new_type_names
(map single case_names_exhausts) casedist_thms
end;
(*************************** primrec combinators ******************************)
fun prove_primrec_thms (config : Datatype_Aux.config) new_type_names descr sorts
injects_of constr_inject (dist_rewrites, other_dist_rewrites) induct thy =
let
val _ = Datatype_Aux.message config "Constructing primrec combinators ...";
val big_name = space_implode "_" new_type_names;
val thy0 = Sign.add_path big_name thy;
val descr' = flat descr;
val recTs = Datatype_Aux.get_rec_types descr' sorts;
val used = fold Term.add_tfree_namesT recTs [];
val newTs = take (length (hd descr)) recTs;
val induct_Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct)));
val big_rec_name' = big_name ^ "_rec_set";
val rec_set_names' =
if length descr' = 1 then [big_rec_name']
else map (prefix (big_rec_name' ^ "_") o string_of_int) (1 upto length descr');
val rec_set_names = map (Sign.full_bname thy0) rec_set_names';
val (rec_result_Ts, reccomb_fn_Ts) = Datatype_Prop.make_primrec_Ts descr sorts used;
val rec_set_Ts =
map (fn (T1, T2) => (reccomb_fn_Ts @ [T1, T2]) ---> HOLogic.boolT) (recTs ~~ rec_result_Ts);
val rec_fns =
map (uncurry (Datatype_Aux.mk_Free "f")) (reccomb_fn_Ts ~~ (1 upto length reccomb_fn_Ts));
val rec_sets' =
map (fn c => list_comb (Free c, rec_fns)) (rec_set_names' ~~ rec_set_Ts);
val rec_sets =
map (fn c => list_comb (Const c, rec_fns)) (rec_set_names ~~ rec_set_Ts);
(* introduction rules for graph of primrec function *)
fun make_rec_intr T rec_set (cname, cargs) (rec_intr_ts, l) =
let
fun mk_prem (dt, U) (j, k, prems, t1s, t2s) =
let val free1 = Datatype_Aux.mk_Free "x" U j in
(case (Datatype_Aux.strip_dtyp dt, strip_type U) of
((_, Datatype_Aux.DtRec m), (Us, _)) =>
let
val free2 = Datatype_Aux.mk_Free "y" (Us ---> nth rec_result_Ts m) k;
val i = length Us;
in
(j + 1, k + 1,
HOLogic.mk_Trueprop (HOLogic.list_all
(map (pair "x") Us, nth rec_sets' m $
Datatype_Aux.app_bnds free1 i $ Datatype_Aux.app_bnds free2 i)) :: prems,
free1 :: t1s, free2 :: t2s)
end
| _ => (j + 1, k, prems, free1 :: t1s, t2s))
end;
val Ts = map (Datatype_Aux.typ_of_dtyp descr' sorts) cargs;
val (_, _, prems, t1s, t2s) = fold_rev mk_prem (cargs ~~ Ts) (1, 1, [], [], []);
in
(rec_intr_ts @
[Logic.list_implies (prems, HOLogic.mk_Trueprop
(rec_set $ list_comb (Const (cname, Ts ---> T), t1s) $
list_comb (nth rec_fns l, t1s @ t2s)))], l + 1)
end;
val (rec_intr_ts, _) =
fold (fn ((d, T), set_name) =>
fold (make_rec_intr T set_name) (#3 (snd d))) (descr' ~~ recTs ~~ rec_sets') ([], 0);
val ({intrs = rec_intrs, elims = rec_elims, ...}, thy1) =
thy0
|> 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 = false, no_ind = true, skip_mono = true, fork_mono = false}
(map (fn (s, T) => ((Binding.name s, T), NoSyn)) (rec_set_names' ~~ rec_set_Ts))
(map dest_Free rec_fns)
(map (fn x => (Attrib.empty_binding, x)) rec_intr_ts) []
||> Sign.restore_naming thy0
||> Theory.checkpoint;
(* prove uniqueness and termination of primrec combinators *)
val _ = Datatype_Aux.message config "Proving termination and uniqueness of primrec functions ...";
fun mk_unique_tac ((((i, (tname, _, constrs)), elim), T), T') (tac, intrs) =
let
val distinct_tac =
if i < length newTs then
full_simp_tac (HOL_ss addsimps (nth dist_rewrites i)) 1
else full_simp_tac (HOL_ss addsimps (flat other_dist_rewrites)) 1;
val inject =
map (fn r => r RS iffD1)
(if i < length newTs then nth constr_inject i else injects_of tname);
fun mk_unique_constr_tac n (cname, cargs) (tac, intr :: intrs, j) =
let
val k = length (filter Datatype_Aux.is_rec_type cargs);
in
(EVERY
[DETERM tac,
REPEAT (etac ex1E 1), rtac ex1I 1,
DEPTH_SOLVE_1 (ares_tac [intr] 1),
REPEAT_DETERM_N k (etac thin_rl 1 THEN rotate_tac 1 1),
etac elim 1,
REPEAT_DETERM_N j distinct_tac,
TRY (dresolve_tac inject 1),
REPEAT (etac conjE 1), hyp_subst_tac 1,
REPEAT (EVERY [etac allE 1, dtac mp 1, atac 1]),
TRY (hyp_subst_tac 1),
rtac refl 1,
REPEAT_DETERM_N (n - j - 1) distinct_tac],
intrs, j + 1)
end;
val (tac', intrs', _) =
fold (mk_unique_constr_tac (length constrs)) constrs (tac, intrs, 0);
in (tac', intrs') end;
val rec_unique_thms =
let
val rec_unique_ts =
map (fn (((set_t, T1), T2), i) =>
Const (@{const_name Ex1}, (T2 --> HOLogic.boolT) --> HOLogic.boolT) $
absfree ("y", T2) (set_t $ Datatype_Aux.mk_Free "x" T1 i $ Free ("y", T2)))
(rec_sets ~~ recTs ~~ rec_result_Ts ~~ (1 upto length recTs));
val cert = cterm_of thy1;
val insts =
map (fn ((i, T), t) => absfree ("x" ^ string_of_int i, T) t)
((1 upto length recTs) ~~ recTs ~~ rec_unique_ts);
val induct' = cterm_instantiate (map cert induct_Ps ~~ map cert insts) induct;
val (tac, _) =
fold mk_unique_tac (descr' ~~ rec_elims ~~ recTs ~~ rec_result_Ts)
(((rtac induct' THEN_ALL_NEW Object_Logic.atomize_prems_tac) 1 THEN
rewrite_goals_tac [mk_meta_eq @{thm choice_eq}], rec_intrs));
in
Datatype_Aux.split_conj_thm (Skip_Proof.prove_global thy1 [] []
(HOLogic.mk_Trueprop (Datatype_Aux.mk_conj rec_unique_ts)) (K tac))
end;
val rec_total_thms = map (fn r => r RS @{thm theI'}) rec_unique_thms;
(* define primrec combinators *)
val big_reccomb_name = space_implode "_" new_type_names ^ "_rec";
val reccomb_names =
map (Sign.full_bname thy1)
(if length descr' = 1 then [big_reccomb_name]
else map (prefix (big_reccomb_name ^ "_") o string_of_int) (1 upto length descr'));
val reccombs =
map (fn ((name, T), T') => list_comb (Const (name, reccomb_fn_Ts @ [T] ---> T'), rec_fns))
(reccomb_names ~~ recTs ~~ rec_result_Ts);
val (reccomb_defs, thy2) =
thy1
|> Sign.add_consts_i (map (fn ((name, T), T') =>
(Binding.name (Long_Name.base_name name), reccomb_fn_Ts @ [T] ---> T', NoSyn))
(reccomb_names ~~ recTs ~~ rec_result_Ts))
|> (Global_Theory.add_defs false o map Thm.no_attributes)
(map
(fn ((((name, comb), set), T), T') =>
(Binding.name (Long_Name.base_name name ^ "_def"),
Logic.mk_equals (comb, absfree ("x", T)
(Const (@{const_name The}, (T' --> HOLogic.boolT) --> T') $ absfree ("y", T')
(set $ Free ("x", T) $ Free ("y", T'))))))
(reccomb_names ~~ reccombs ~~ rec_sets ~~ recTs ~~ rec_result_Ts))
||> Sign.parent_path
||> Theory.checkpoint;
(* prove characteristic equations for primrec combinators *)
val _ = Datatype_Aux.message config "Proving characteristic theorems for primrec combinators ...";
val rec_thms =
map (fn t =>
Skip_Proof.prove_global thy2 [] [] t
(fn _ => EVERY
[rewrite_goals_tac reccomb_defs,
rtac @{thm the1_equality} 1,
resolve_tac rec_unique_thms 1,
resolve_tac rec_intrs 1,
REPEAT (rtac allI 1 ORELSE resolve_tac rec_total_thms 1)]))
(Datatype_Prop.make_primrecs new_type_names descr sorts thy2);
in
thy2
|> Sign.add_path (space_implode "_" new_type_names)
|> Global_Theory.add_thmss [((Binding.name "recs", rec_thms), [Nitpick_Simps.add])]
||> Sign.parent_path
||> Theory.checkpoint
|-> (fn thms => pair (reccomb_names, flat thms))
end;
(***************************** case combinators *******************************)
fun prove_case_thms (config : Datatype_Aux.config)
new_type_names descr sorts reccomb_names primrec_thms thy =
let
val _ = Datatype_Aux.message config "Proving characteristic theorems for case combinators ...";
val thy1 = Sign.add_path (space_implode "_" new_type_names) thy;
val descr' = flat descr;
val recTs = Datatype_Aux.get_rec_types descr' sorts;
val used = fold Term.add_tfree_namesT recTs [];
val newTs = take (length (hd descr)) recTs;
val T' = TFree (singleton (Name.variant_list used) "'t", HOLogic.typeS);
fun mk_dummyT dt = binder_types (Datatype_Aux.typ_of_dtyp descr' sorts dt) ---> T';
val case_dummy_fns =
map (fn (_, (_, _, constrs)) => map (fn (_, cargs) =>
let
val Ts = map (Datatype_Aux.typ_of_dtyp descr' sorts) cargs;
val Ts' = map mk_dummyT (filter Datatype_Aux.is_rec_type cargs)
in Const (@{const_name undefined}, Ts @ Ts' ---> T') end) constrs) descr';
val case_names = map (fn s => Sign.full_bname thy1 (s ^ "_case")) new_type_names;
(* define case combinators via primrec combinators *)
val (case_defs, thy2) =
fold (fn ((((i, (_, _, constrs)), T), name), recname) => fn (defs, thy) =>
let
val (fns1, fns2) = split_list (map (fn ((_, cargs), j) =>
let
val Ts = map (Datatype_Aux.typ_of_dtyp descr' sorts) cargs;
val Ts' = Ts @ map mk_dummyT (filter Datatype_Aux.is_rec_type cargs);
val frees' = map2 (Datatype_Aux.mk_Free "x") Ts' (1 upto length Ts');
val frees = take (length cargs) frees';
val free = Datatype_Aux.mk_Free "f" (Ts ---> T') j;
in
(free, fold_rev (absfree o dest_Free) frees' (list_comb (free, frees)))
end) (constrs ~~ (1 upto length constrs)));
val caseT = map (snd o dest_Free) fns1 @ [T] ---> T';
val fns = flat (take i case_dummy_fns) @ fns2 @ flat (drop (i + 1) case_dummy_fns);
val reccomb = Const (recname, (map fastype_of fns) @ [T] ---> T');
val decl = ((Binding.name (Long_Name.base_name name), caseT), NoSyn);
val def =
(Binding.name (Long_Name.base_name name ^ "_def"),
Logic.mk_equals (list_comb (Const (name, caseT), fns1),
list_comb (reccomb, (flat (take i case_dummy_fns)) @
fns2 @ (flat (drop (i + 1) case_dummy_fns)))));
val ([def_thm], thy') =
thy
|> Sign.declare_const_global decl |> snd
|> (Global_Theory.add_defs false o map Thm.no_attributes) [def];
in (defs @ [def_thm], thy') end)
(hd descr ~~ newTs ~~ case_names ~~ take (length newTs) reccomb_names) ([], thy1)
||> Theory.checkpoint;
val case_thms =
(map o map) (fn t =>
Skip_Proof.prove_global thy2 [] [] t
(fn _ =>
EVERY [rewrite_goals_tac (case_defs @ map mk_meta_eq primrec_thms), rtac refl 1]))
(Datatype_Prop.make_cases new_type_names descr sorts thy2);
in
thy2
|> Context.theory_map ((fold o fold) Nitpick_Simps.add_thm case_thms)
|> Sign.parent_path
|> Datatype_Aux.store_thmss "cases" new_type_names case_thms
|-> (fn thmss => pair (thmss, case_names))
end;
(******************************* case splitting *******************************)
fun prove_split_thms (config : Datatype_Aux.config)
new_type_names descr sorts constr_inject dist_rewrites casedist_thms case_thms thy =
let
val _ = Datatype_Aux.message config "Proving equations for case splitting ...";
val descr' = flat descr;
val recTs = Datatype_Aux.get_rec_types descr' sorts;
val newTs = take (length (hd descr)) recTs;
fun prove_split_thms ((((((t1, t2), inject), dist_rewrites'), exhaustion), case_thms'), T) =
let
val cert = cterm_of thy;
val _ $ (_ $ lhs $ _) = hd (Logic.strip_assums_hyp (hd (prems_of exhaustion)));
val exhaustion' = cterm_instantiate [(cert lhs, cert (Free ("x", T)))] exhaustion;
val tac =
EVERY [rtac exhaustion' 1,
ALLGOALS (asm_simp_tac (HOL_ss addsimps (dist_rewrites' @ inject @ case_thms')))];
in
(Skip_Proof.prove_global thy [] [] t1 (K tac),
Skip_Proof.prove_global thy [] [] t2 (K tac))
end;
val split_thm_pairs =
map prove_split_thms
((Datatype_Prop.make_splits new_type_names descr sorts thy) ~~ constr_inject ~~
dist_rewrites ~~ casedist_thms ~~ case_thms ~~ newTs);
val (split_thms, split_asm_thms) = ListPair.unzip split_thm_pairs
in
thy
|> Datatype_Aux.store_thms "split" new_type_names split_thms
||>> Datatype_Aux.store_thms "split_asm" new_type_names split_asm_thms
|-> (fn (thms1, thms2) => pair (thms1 ~~ thms2))
end;
fun prove_weak_case_congs new_type_names descr sorts thy =
let
fun prove_weak_case_cong t =
Skip_Proof.prove_global thy [] (Logic.strip_imp_prems t) (Logic.strip_imp_concl t)
(fn {prems, ...} => EVERY [rtac ((hd prems) RS arg_cong) 1]);
val weak_case_congs =
map prove_weak_case_cong (Datatype_Prop.make_weak_case_congs new_type_names descr sorts thy);
in thy |> Datatype_Aux.store_thms "weak_case_cong" new_type_names weak_case_congs end;
(************************* additional theorems for TFL ************************)
fun prove_nchotomys (config : Datatype_Aux.config)
new_type_names descr sorts casedist_thms thy =
let
val _ = Datatype_Aux.message config "Proving additional theorems for TFL ...";
fun prove_nchotomy (t, exhaustion) =
let
(* For goal i, select the correct disjunct to attack, then prove it *)
fun tac i 0 = EVERY [TRY (rtac disjI1 i), hyp_subst_tac i, REPEAT (rtac exI i), rtac refl i]
| tac i n = rtac disjI2 i THEN tac i (n - 1);
in
Skip_Proof.prove_global thy [] [] t
(fn _ =>
EVERY [rtac allI 1,
Datatype_Aux.exh_tac (K exhaustion) 1,
ALLGOALS (fn i => tac i (i - 1))])
end;
val nchotomys =
map prove_nchotomy (Datatype_Prop.make_nchotomys descr sorts ~~ casedist_thms);
in thy |> Datatype_Aux.store_thms "nchotomy" new_type_names nchotomys end;
fun prove_case_congs new_type_names descr sorts nchotomys case_thms thy =
let
fun prove_case_cong ((t, nchotomy), case_rewrites) =
let
val Const ("==>", _) $ tm $ _ = t;
val Const (@{const_name Trueprop}, _) $ (Const (@{const_name HOL.eq}, _) $ _ $ Ma) = tm;
val cert = cterm_of thy;
val nchotomy' = nchotomy RS spec;
val [v] = Term.add_vars (concl_of nchotomy') [];
val nchotomy'' = cterm_instantiate [(cert (Var v), cert Ma)] nchotomy';
in
Skip_Proof.prove_global thy [] (Logic.strip_imp_prems t) (Logic.strip_imp_concl t)
(fn {prems, ...} =>
let val simplify = asm_simp_tac (HOL_ss addsimps (prems @ case_rewrites)) in
EVERY [
simp_tac (HOL_ss addsimps [hd prems]) 1,
cut_facts_tac [nchotomy''] 1,
REPEAT (etac disjE 1 THEN REPEAT (etac exE 1) THEN simplify 1),
REPEAT (etac exE 1) THEN simplify 1 (* Get last disjunct *)]
end)
end;
val case_congs =
map prove_case_cong (Datatype_Prop.make_case_congs
new_type_names descr sorts thy ~~ nchotomys ~~ case_thms);
in thy |> Datatype_Aux.store_thms "case_cong" new_type_names case_congs end;
open Datatype_Aux;
end;