val co_induct_of: 'a list -> 'a val strong_co_induct_of: 'a list -> 'a
val morph_fp_bnf_sugar: morphism -> fp_bnf_sugar -> fp_bnf_sugar val morph_fp_co_induct_sugar: morphism -> fp_co_induct_sugar -> fp_co_induct_sugar val morph_fp_ctr_sugar: morphism -> fp_ctr_sugar -> fp_ctr_sugar val morph_fp_sugar: morphism -> fp_sugar -> fp_sugar val transfer_fp_sugar: theory -> fp_sugar -> fp_sugar val fp_sugar_of: Proof.context -> string -> fp_sugar option val fp_sugar_of_global: theory -> string -> fp_sugar option val fp_sugars_of: Proof.context -> fp_sugar list val fp_sugars_of_global: theory -> fp_sugar list val fp_sugars_interpretation: string -> (fp_sugar list -> local_theory -> local_theory) ->
theory -> theory val interpret_fp_sugars: (string -> bool) -> fp_sugar list -> local_theory -> local_theory val register_fp_sugars_raw: fp_sugar list -> local_theory -> local_theory val register_fp_sugars: (string -> bool) -> fp_sugar list -> local_theory -> local_theory
val merge_type_args: BNF_Util.fp_kind -> ''a list * ''a list -> ''a list val type_args_named_constrained_of_spec: (((('a * 'b) * 'c) * 'd) * 'e) * 'f -> 'a val type_binding_of_spec: (((('a * 'b) * 'c) * 'd) * 'e) * 'f -> 'b val mixfix_of_spec: ((('a * 'b) * 'c) * 'd) * 'e -> 'b val mixfixed_ctr_specs_of_spec: (('a * 'b) * 'c) * 'd -> 'b val map_binding_of_spec: ('a * ('b * 'c * 'd)) * 'e -> 'b val rel_binding_of_spec: ('a * ('b * 'c * 'd)) * 'e -> 'c val pred_binding_of_spec: ('a * ('b * 'c * 'd)) * 'e -> 'd val sel_default_eqs_of_spec: 'a * 'b -> 'b
val mk_parametricity_goal: Proof.context -> term list -> term -> term -> term
val flat_corec_preds_predsss_gettersss: 'a list -> 'a listlistlist -> 'a list list list -> 'a list val mk_ctor: typ list -> term -> term val mk_dtor: typ list -> term -> term val mk_bnf_sets: BNF_Def.bnf -> string * term list val liveness_of_fp_bnf: int -> BNF_Def.bnf -> boollist val nesting_bnfs: Proof.context -> typ listlistlist -> typ list -> BNF_Def.bnf list
val massage_simple_notes: string -> (bstring * 'a list * (int -> 'b)) list ->
((binding * 'c list) * ('a list * 'b) list) list val massage_multi_notes: stringlist -> typ list ->
(string * 'a list list * (string -> 'b)) list ->
((binding * 'b) * ('a list * 'c list) list) list
val define_ctrs_dtrs_for_type: string -> typ -> term -> term -> thm -> thm -> int -> int list ->
term -> binding list -> mixfix list -> typ listlist -> local_theory ->
(term listlist * term list * thm * thm list) * local_theory val wrap_ctrs: (string -> bool) -> BNF_Util.fp_kind -> bool -> string -> thm -> int -> int list ->
thm -> thm -> binding list -> binding listlist -> term list -> term list -> thm -> thm list ->
local_theory -> Ctr_Sugar.ctr_sugar * local_theory val derive_map_set_rel_pred_thms: (string -> bool) -> BNF_Util.fp_kind -> int -> typ list ->
typ list -> typ -> typ -> thm list -> thm list -> thm list -> thm list -> thm list ->
thm list -> thm list -> thm list -> thm list -> string -> BNF_Def.bnf -> BNF_Def.bnf list ->
typ -> term -> thm -> thm -> thm -> thm list -> thm -> thm -> thm list -> thm -> thm list ->
thm list -> thm list -> typ listlist -> Ctr_Sugar.ctr_sugar -> local_theory ->
(thm list * thm list * thm listlist * thm list * thm list * thm list * thm list * thm list
* thm list * thm list * thm listlistlistlist * thm listlistlistlist * thm list
* thm list * thm list * thm list * thm list) * local_theory
type lfp_sugar_thms = (thm list * thm * Token.src list) * (thm listlist * Token.src list)
val morph_lfp_sugar_thms: morphism -> lfp_sugar_thms -> lfp_sugar_thms val transfer_lfp_sugar_thms: theory -> lfp_sugar_thms -> lfp_sugar_thms
type gfp_sugar_thms =
((thm list * thm) list * (Token.src list * Token.src list))
* thm listlist
* thm listlist
* (thm listlist * Token.src list)
* (thm listlistlist * Token.src list)
val morph_gfp_sugar_thms: morphism -> gfp_sugar_thms -> gfp_sugar_thms val transfer_gfp_sugar_thms: theory -> gfp_sugar_thms -> gfp_sugar_thms
val mk_co_recs_prelims: Proof.context -> BNF_Util.fp_kind -> typ listlistlist -> typ list ->
typ list -> typ list -> typ list -> int list -> int listlist -> term list ->
term list
* (typ listlist * typ listlistlistlist * term listlist * term listlistlistlist) option
* (string * term list * term listlist
* (((term listlist * term listlist * term listlistlistlist * term listlistlistlist)
* term listlistlist) * typ list)) option val repair_nullary_single_ctr: typ listlist -> typ listlist val mk_corec_p_pred_types: typ list -> int list -> typ listlist val mk_corec_fun_arg_types: typ listlistlist -> typ list -> typ list -> typ list -> int list ->
int listlist -> term ->
typ listlist
* (typ listlistlistlist * typ listlistlist * typ listlistlistlist * typ list) val define_co_rec_as: BNF_Util.fp_kind -> typ list -> typ -> binding -> term -> local_theory ->
(term * thm) * local_theory val define_rec:
typ listlist * typ listlistlistlist * term listlist * term listlistlistlist ->
(string -> binding) -> typ list -> typ list -> term list -> term -> Proof.context ->
(term * thm) * Proof.context val define_corec: 'a * term list * term list list
* (((term listlist * term listlist * term listlistlistlist * term listlistlistlist)
* term listlistlist) * typ list) -> (string -> binding) -> 'b list -> typ list ->
term list -> term -> local_theory -> (term * thm) * local_theory val mk_induct_raw_prem: (term -> term) -> Proof.context -> typ listlist ->
(string * term list) list -> term -> term -> typ list -> typ list ->
term list * ((term * (term * term)) list * (int * term)) list * term val finish_induct_prem: Proof.context -> int -> term list ->
term list * ((term * (term * term)) list * (int * term)) list * term -> term val mk_coinduct_prem: Proof.context -> typ listlist -> typ listlist -> term list -> term ->
term -> term -> int -> term list -> term listlist -> term list -> term listlist ->
typ listlist -> term val mk_induct_attrs: term listlist -> Token.src list val mk_coinduct_attrs: typ list -> term listlist -> term listlist -> int listlist ->
Token.src list * Token.src list val derive_induct_recs_thms_for_types: (string -> bool) -> BNF_Def.bnf list ->
('a * typ list list list list * term list list * 'b) option -> thm -> thm list ->
BNF_Def.bnf list -> BNF_Def.bnf list -> typ list -> typ list -> typ list ->
typ listlistlist -> thm list -> thm list -> thm list -> term listlist -> thm listlist ->
term list -> thm list -> Proof.context -> lfp_sugar_thms val derive_coinduct_thms_for_types: Proof.context -> bool -> (term -> term) -> BNF_Def.bnf list->
thm -> thm list -> BNF_Def.bnf list -> typ list -> typ list -> typ listlistlist -> int list ->
thm list -> thm list -> (thm -> thm) -> thm listlist -> Ctr_Sugar.ctr_sugar list ->
(thm list * thm) list val derive_coinduct_corecs_thms_for_types: Proof.context -> BNF_Def.bnf list -> string * term list * term listlist
* (((term listlist * term listlist * term listlistlistlist * term listlistlistlist)
* term listlistlist) * typ list) ->
thm -> thm list -> thm list -> thm list -> BNF_Def.bnf list -> typ list -> typ list ->
typ list -> typ listlistlist -> int listlist -> int listlist -> int list -> thm list ->
thm list -> (thm -> thm) -> thm listlist -> Ctr_Sugar.ctr_sugar list -> term list ->
thm list -> gfp_sugar_thms
val co_datatypes: BNF_Util.fp_kind -> (mixfix list -> binding list -> binding list ->
binding list -> binding listlist -> binding list -> (string * sort) list ->
typ list * typ listlist -> BNF_Def.bnf list -> BNF_Comp.absT_info list -> local_theory ->
BNF_FP_Util.fp_result * local_theory) ->
Ctr_Sugar.ctr_options
* ((((((binding option * (typ * sort)) list * binding) * mixfix)
* ((binding, binding * typ) Ctr_Sugar.ctr_spec * mixfix) list) *
(binding * binding * binding))
* term list) list ->
local_theory -> local_theory val co_datatype_cmd: BNF_Util.fp_kind ->
(mixfix list -> binding list -> binding list -> binding list -> binding listlist ->
binding list -> (string * sort) list -> typ list * typ listlist -> BNF_Def.bnf list ->
BNF_Comp.absT_info list -> local_theory -> BNF_FP_Util.fp_result * Proof.context) ->
((Proof.context -> Plugin_Name.filter) * bool)
* ((((((binding option * (string * stringoption)) list * binding) * mixfix)
* ((binding, binding * string) Ctr_Sugar.ctr_spec * mixfix) list)
* (binding * binding * binding)) * stringlist) list ->
Proof.context -> local_theory
val parse_co_datatype_cmd: BNF_Util.fp_kind -> (mixfix list -> binding list -> binding list ->
binding list -> binding listlist -> binding list -> (string * sort) list ->
typ list * typ listlist -> BNF_Def.bnf list -> BNF_Comp.absT_info list -> local_theory ->
BNF_FP_Util.fp_result * local_theory) ->
(local_theory -> local_theory) parser end;
open Ctr_Sugar open BNF_FP_Rec_Sugar_Util open BNF_Util open BNF_Comp open BNF_Def open BNF_FP_Util open BNF_FP_Def_Sugar_Tactics
val Eq_prefix = "Eq_";
val case_transferN = "case_transfer"; val ctor_iff_dtorN = "ctor_iff_dtor"; val ctr_transferN = "ctr_transfer"; val disc_transferN = "disc_transfer"; val sel_transferN = "sel_transfer"; val corec_codeN = "corec_code"; val corec_transferN = "corec_transfer"; val map_disc_iffN = "map_disc_iff"; val map_o_corecN = "map_o_corec"; val map_selN = "map_sel"; val pred_injectN = "pred_inject"; val rec_o_mapN = "rec_o_map"; val rec_transferN = "rec_transfer"; val set0N = "set0"; val set_casesN = "set_cases"; val set_introsN = "set_intros"; val set_inductN = "set_induct"; val set_selN = "set_sel";
val transfer_fp_sugar = morph_fp_sugar o Morphism.transfer_morphism;
structure Data = Generic_Data
( type T = fp_sugar Symtab.table; val empty = Symtab.empty; fun merge data : T = Symtab.merge (K true) data;
);
fun fp_sugar_of_generic context = Option.map (transfer_fp_sugar (Context.theory_of context)) o Symtab.lookup (Data.get context);
fun fp_sugars_of_generic context =
Symtab.fold (cons o transfer_fp_sugar (Context.theory_of context) o snd) (Data.get context) [];
val fp_sugar_of = fp_sugar_of_generic o Context.Proof; val fp_sugar_of_global = fp_sugar_of_generic o Context.Theory;
val fp_sugars_of = fp_sugars_of_generic o Context.Proof; val fp_sugars_of_global = fp_sugars_of_generic o Context.Theory;
structure FP_Sugar_Plugin = Plugin(type T = fp_sugar list);
fun fp_sugars_interpretation name f =
FP_Sugar_Plugin.interpretation name (fn fp_sugars => fn lthy =>
f (map (transfer_fp_sugar (Proof_Context.theory_of lthy)) fp_sugars) lthy);
fun quasi_unambiguous_case_names names = let val ps = map (`Long_Name.base_name) names; val dups = Library.duplicates (op =) (map fst ps); fun underscore s = letval ss = Long_Name.explode s in space_implode "_" (drop (length ss - 2) ss) end; in map (fn (base, full) => if member (op =) dups base then underscore full else base) ps
|> Name.variant_list [] end;
fun zipper_map f = let fun zed _ [] = []
| zed xs (y :: ys) = f (xs, y, ys) :: zed (xs @ [y]) ys; in zed [] end;
fun cannot_merge_types fp =
error ("Mutually " ^ co_prefix fp ^ "recursive types must have the same type parameters");
fun merge_type_arg fp T T' = if T = T'then T else cannot_merge_types fp;
fun merge_type_args fp (As, As') = if length As = length As' then map2 (merge_type_arg fp) As As'else cannot_merge_types fp;
fun type_args_named_constrained_of_spec (((((ncAs, _), _), _), _), _) = ncAs; fun type_binding_of_spec (((((_, b), _), _), _), _) = b; fun mixfix_of_spec ((((_, mx), _), _), _) = mx; fun mixfixed_ctr_specs_of_spec (((_, mx_ctr_specs), _), _) = mx_ctr_specs; fun map_binding_of_spec ((_, (b, _, _)), _) = b; fun rel_binding_of_spec ((_, (_, b, _)), _) = b; fun pred_binding_of_spec ((_, (_, _, b)), _) = b; fun sel_default_eqs_of_spec (_, ts) = ts;
fun ctr_sugar_kind_of_fp_kind Least_FP = Datatype
| ctr_sugar_kind_of_fp_kind Greatest_FP = Codatatype;
fun choose_binary_fun fs AB =
find_first (fastype_of #> binder_types #> (fn [A, B] => AB = (A, B))) fs; fun build_binary_fun_app fs t u = Option.map (rapp u o rapp t) (choose_binary_fun fs (fastype_of t, fastype_of u));
fun build_the_rel ctxt Rs Ts A B =
build_rel [] ctxt Ts [] (the o choose_binary_fun Rs) (A, B); fun build_rel_app ctxt Rs Ts t u =
build_the_rel ctxt Rs Ts (fastype_of t) (fastype_of u) $ t $ u;
fun build_set_app ctxt A t = Term.betapply (build_set ctxt A (fastype_of t), t);
fun mk_parametricity_goal ctxt Rs t u = letval prem = build_the_rel ctxt Rs [] (fastype_of t) (fastype_of u) in
HOLogic.mk_Trueprop (prem $ t $ u) end;
val name_of_set = name_of_const "set function" domain_type;
val fundefcong_attrs = @{attributes [fundef_cong]}; val nitpicksimp_attrs = @{attributes [nitpick_simp]}; val simp_attrs = @{attributes [simp]};
val lists_bmoc = fold (fn xs => fn t => Term.list_comb (t, xs));
fun mk_flip (x, Type (_, [T1, Type (_, [T2, T3])])) =
Abs ("x", T1, Abs ("y", T2, Var (x, T2 --> T1 --> T3) $ Bound 0 $ Bound 1));
fun flip_rels ctxt n thm = let val Rs = Term.add_vars (Thm.prop_of thm) []; val Rs' = rev (drop (length Rs - n) Rs); in
infer_instantiate ctxt (map (fn f => (fst f, Thm.cterm_of ctxt (mk_flip f))) Rs') thm end;
fun mk_ctor_or_dtor get_T Ts t = letvalType (_, Ts0) = get_T (fastype_of t) in
Term.subst_atomic_types (Ts0 ~~ Ts) t end;
val mk_ctor = mk_ctor_or_dtor range_type; val mk_dtor = mk_ctor_or_dtor domain_type;
fun mk_bnf_sets bnf = let valType (T_name, Us) = T_of_bnf bnf; val lives = lives_of_bnf bnf; val sets = sets_of_bnf bnf; fun mk_set U =
(case find_index (curry (op =) U) lives of
~1 => Term.dummy
| i => nth sets i); in
(T_name, map mk_set Us) end;
fun mk_xtor_co_recs thy fp fpTs Cs ts0 = let val nn = length fpTs; val (fpTs0, Cs0) = map ((fp = Greatest_FP ? swap) o dest_funT o snd o strip_typeN nn o fastype_of) ts0
|> split_list; val rho = tvar_subst thy (fpTs0 @ Cs0) (fpTs @ Cs); in map (Term.subst_TVars rho) ts0 end;
fun liveness_of_fp_bnf n bnf =
(case T_of_bnf bnf of Type (_, Ts) => map (not o member (op =) (deads_of_bnf bnf)) Ts
| _ => replicate n false);
fun add_nesting_bnf_names Us = let fun add (Type (s, Ts)) ss = letval (needs, ss') = fold_map add Ts ss in ifexists I needs then (true, insert (op =) s ss') else (false, ss') end
| add T ss = (member (op =) Us T, ss); in snd oo add end;
fun nesting_bnfs ctxt ctr_Tsss Us =
map_filter (bnf_of ctxt) (fold (fold (fold (add_nesting_bnf_names Us))) ctr_Tsss []);
fun indexify proj xs f p = f (find_index (curry (op =) (proj p)) xs) p;
fun massage_simple_notes base =
filter_out (null o #2)
#> map (fn (thmN, thms, f_attrs) =>
((Binding.qualify true base (Binding.name thmN), []),
map_index (fn (i, thm) => ([thm], f_attrs i)) thms));
fun massage_multi_notes b_names Ts =
maps (fn (thmN, thmss, attrs) =>
@{map 3} (fn b_name => fn Type (T_name, _) => fn thms =>
((Binding.qualify true b_name (Binding.name thmN), attrs T_name), [(thms, [])]))
b_names Ts thmss)
#> filter_out (null o fst o hd o snd);
fun define_ctrs_dtrs_for_type fp_b_name fpT ctor dtor ctor_dtor dtor_ctor n ks abs ctr_bindings
ctr_mixfixes ctr_Tss lthy = let val ctor_absT = domain_type (fastype_of ctor);
val (((w, xss), u'), _) = lthy
|> yield_singleton (mk_Frees "w") ctor_absT
||>> mk_Freess "x" ctr_Tss
||>> yield_singleton Variable.variant_fixes fp_b_name;
val u = Free (u', fpT);
val ctor_iff_dtor_thm = let val goal =
fold_rev Logic.all [w, u]
(mk_Trueprop_eq (HOLogic.mk_eq (u, ctor $ w), HOLogic.mk_eq (dtor $ u, w))); val vars = Variable.add_free_names lthy goal []; in
Goal.prove_sorry lthy vars [] goal (fn {context = ctxt, ...} =>
mk_ctor_iff_dtor_tac ctxt (map (SOME o Thm.ctyp_of lthy) [ctor_absT, fpT])
(Thm.cterm_of lthy ctor) (Thm.cterm_of lthy dtor) ctor_dtor dtor_ctor)
|> Thm.close_derivation \<^here> end;
val ctr_rhss =
map2 (fn k => fn xs => fold_rev Term.lambda xs (ctor $ mk_absumprod ctor_absT abs n k xs))
ks xss;
fun derive_case_transfer rel_case_thm = let val (S, names_lthy) = yield_singleton (mk_Frees "S") (mk_pred2T C E) names_lthy; val caseA = mk_case As C casex; val caseB = mk_case Bs E casex; val goal = mk_parametricity_goal names_lthy (S :: Rs) caseA caseB; in
Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, prems = _} =>
mk_case_transfer_tac ctxt rel_case_thm case_thms)
|> singleton (Proof_Context.export names_lthy lthy)
|> Thm.close_derivation \<^here> end; in if live = 0 then if plugins transfer_plugin then let val relAsBs = HOLogic.eq_const fpT; val rel_case_thm = derive_rel_case relAsBs [] [];
val case_transfer_thm = derive_case_transfer rel_case_thm;
val notes =
[(case_transferN, [case_transfer_thm], K [])]
|> massage_simple_notes fp_b_name;
val (noted, lthy') = lthy
|> Local_Theory.notes notes;
val subst = Morphism.thm (substitute_noted_thm noted); in
(([], [], [], [], [], [], [], [], [], [], [], [], [], [], [subst case_transfer_thm], [],
[]), lthy') end else
(([], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], []), lthy) else let val mapx = mk_map live As Bs (map_of_bnf fp_bnf); val relAsBs = mk_rel live As Bs (rel_of_bnf fp_bnf); val setAs = map (mk_set As) (sets_of_bnf fp_bnf); val discAs = map (mk_disc_or_sel As) discs; val discBs = map (mk_disc_or_sel Bs) discs; val selAss = map (map (mk_disc_or_sel As)) selss; val selBss = map (map (mk_disc_or_sel Bs)) selss;
val map_ctor_thm = if fp = Least_FP then
fp_map_thm else let val ctorA = mk_ctor As ctor; val ctorB = mk_ctor Bs ctor;
val y_T = domain_type (fastype_of ctorA); val (y as Free (y_s, _), _) = lthy
|> yield_singleton (mk_Frees "y") y_T;
val ctor_cong =
infer_instantiate' lthy [NONE, NONE, SOME (Thm.cterm_of lthy ctorB)] arg_cong; val fp_map_thm' = fp_map_thm
|> infer_instantiate' lthy (replicate live NONE @
[SOME (Thm.cterm_of lthy (ctorA $ y))])
|> unfold_thms lthy [dtor_ctor]; in
(fp_map_thm' RS ctor_cong RS (ctor_dtor RS sym RS trans))
|> Drule.generalize (Names.empty, Names.make1_set y_s) end;
val map_thms = let fun mk_goal ctrA ctrB xs ys = let val fmap = list_comb (mapx, fs);
fun mk_arg (x as Free (_, T)) (Free (_, U)) = if T = U then x else build_map lthy [] [] (the o AList.lookup (op =) ABfs) (T, U) $ x;
val xs' = map2 mk_arg xs ys; in
mk_Trueprop_eq (fmap $ list_comb (ctrA, xs), list_comb (ctrB, xs')) end;
val goals = @{map 4} mk_goal ctrAs ctrBs xss yss; val goal = Logic.mk_conjunction_balanced goals; val vars = Variable.add_free_names lthy goal []; in
Goal.prove_sorry lthy vars [] goal (fn {context = ctxt, prems = _} =>
mk_map_tac ctxt abs_inverses pre_map_def map_ctor_thm live_nesting_map_id0s ctr_defs'
extra_unfolds_map)
|> Thm.close_derivation \<^here>
|> Conjunction.elim_balanced (length goals) end;
val set0_thms = let fun mk_goal A setA ctrA xs = let val sets = map (build_set_app lthy A)
(filter (exists_subtype_in [A] o fastype_of) xs); in
mk_Trueprop_eq (setA $ list_comb (ctrA, xs),
(if null sets then HOLogic.mk_set A [] else Library.foldl1 mk_union sets)) end;
val goals =
@{map 2} (fn live_A => fn setA => map2 (mk_goal live_A setA) ctrAs xss) live_As setAs
|> flat; in if null goals then
[] else let val goal = Logic.mk_conjunction_balanced goals; val vars = Variable.add_free_names lthy goal []; in
Goal.prove_sorry lthy vars [] goal (fn {context = ctxt, prems = _} =>
mk_set0_tac ctxt abs_inverses pre_set_defs dtor_ctor fp_set_thms
fp_nesting_set_maps live_nesting_set_maps ctr_defs' extra_unfolds_set)
|> Thm.close_derivation \<^here>
|> Conjunction.elim_balanced (length goals) end end; val set_thms = set0_thms
|> map (unfold_thms lthy @{thms insert_is_Un[THEN sym] Un_empty_left Un_insert_left});
val rel_ctor_thm = if fp = Least_FP then
fp_rel_thm else let val ctorA = mk_ctor As ctor; val ctorB = mk_ctor Bs ctor;
val y_T = domain_type (fastype_of ctorA); val z_T = domain_type (fastype_of ctorB); val ((y as Free (y_s, _), z as Free (z_s, _)), _) = lthy
|> yield_singleton (mk_Frees "y") y_T
||>> yield_singleton (mk_Frees "z") z_T; in
fp_rel_thm
|> infer_instantiate' lthy (replicate live NONE @
[SOME (Thm.cterm_of lthy (ctorA $ y)), SOME (Thm.cterm_of lthy (ctorB $ z))])
|> unfold_thms lthy [dtor_ctor]
|> Drule.generalize (Names.empty, Names.make2_set y_s z_s) end;
val rel_inject_thms = let fun mk_goal ctrA ctrB xs ys = let val lhs = list_comb (relAsBs, Rs) $ list_comb (ctrA, xs) $ list_comb (ctrB, ys); val conjuncts = map2 (build_rel_app lthy Rs []) xs ys; in
HOLogic.mk_Trueprop
(if null conjuncts then lhs else HOLogic.mk_eq (lhs, Library.foldr1 HOLogic.mk_conj conjuncts)) end;
val goals = @{map 4} mk_goal ctrAs ctrBs xss yss; val goal = Logic.mk_conjunction_balanced goals; val vars = Variable.add_free_names lthy goal []; in
Goal.prove_sorry lthy vars [] goal (fn {context = ctxt, prems = _} =>
mk_rel_tac ctxt abs_inverses pre_rel_def rel_ctor_thm live_nesting_rel_eqs ctr_defs'
extra_unfolds_rel)
|> Thm.close_derivation \<^here>
|> Conjunction.elim_balanced (length goals) end;
val half_rel_distinct_thmss = let fun mk_goal ((ctrA, xs), (ctrB, ys)) =
HOLogic.mk_Trueprop (HOLogic.mk_not
(list_comb (relAsBs, Rs) $ list_comb (ctrA, xs) $ list_comb (ctrB, ys)));
val rel_infos = (ctrAs ~~ xss, ctrBs ~~ yss);
val goalss = map (map mk_goal) (mk_half_pairss rel_infos); val goals = flat goalss; in
unflat goalss
(if null goals then
[] else let val goal = Logic.mk_conjunction_balanced goals; val vars = Variable.add_free_names lthy goal []; in
Goal.prove_sorry lthy vars [] goal (fn {context = ctxt, prems = _} =>
mk_rel_tac ctxt abs_inverses pre_rel_def rel_ctor_thm live_nesting_rel_eqs
ctr_defs' extra_unfolds_rel)
|> Thm.close_derivation \<^here>
|> Conjunction.elim_balanced (length goals) end) end;
val rel_flip = rel_flip_of_bnf fp_bnf;
fun mk_other_half_rel_distinct_thm thm =
flip_rels lthy live thm RS (rel_flip RS sym RS @{thm arg_cong[of _ _ Not]} RS iffD2);
val other_half_rel_distinct_thmss = map (map mk_other_half_rel_distinct_thm) half_rel_distinct_thmss; val (rel_distinct_thms, _) =
join_halves n half_rel_distinct_thmss other_half_rel_distinct_thmss;
fun mk_rel_intro_thm m thm =
uncurry_thm m (thm RS iffD2) handle THM _ => thm;
val rel_intro_thms = map2 mk_rel_intro_thm ms rel_inject_thms;
val ctr_transfer_thms = let val goals = map2 (mk_parametricity_goal names_lthy Rs) ctrAs ctrBs; val goal = Logic.mk_conjunction_balanced goals; val vars = Variable.add_free_names lthy goal []; in
Goal.prove_sorry lthy vars [] goal
(fn {context = ctxt, prems = _} =>
mk_ctr_transfer_tac ctxt rel_intro_thms live_nesting_rel_eqs)
|> Thm.close_derivation \<^here>
|> Conjunction.elim_balanced (length goals) end;
val (set_cases_thms, set_cases_attrss) = let fun mk_prems assms elem t ctxt =
(case fastype_of t of Type (type_name, xs) =>
(case bnf_of ctxt type_name of
NONE => ([], ctxt)
| SOME bnf =>
apfst flat (fold_map (fn set => fn ctxt => let val T = HOLogic.dest_setT (range_type (fastype_of set)); val new_var = not (T = fastype_of elem); val (x, ctxt') = if new_var then yield_singleton (mk_Frees "x") T ctxt else (elem, ctxt); in
mk_prems (mk_Trueprop_mem (x, set $ t) :: assms) elem x ctxt'
|>> map (new_var ? Logic.all x) end) (map (mk_set xs) (sets_of_bnf bnf)) ctxt))
| T => rpair ctxt
(if T = fastype_of elem then [fold (curry Logic.mk_implies) assms thesis] else [])); in
split_list (map (fn set => let val A = HOLogic.dest_setT (range_type (fastype_of set)); val (elem, names_lthy) = yield_singleton (mk_Frees "e") A names_lthy; val premss = map (fn ctr => let val (args, names_lthy) =
mk_Frees "z" (binder_types (fastype_of ctr)) names_lthy; in
flat (zipper_map (fn (prev_args, arg, next_args) => let val (args_with_elem, args_without_elem) = if fastype_of arg = A then
(prev_args @ [elem] @ next_args, prev_args @ next_args) else
`I (prev_args @ [arg] @ next_args); in
mk_prems [mk_Trueprop_eq (ta, Term.list_comb (ctr, args_with_elem))]
elem arg names_lthy
|> fst
|> map (fold_rev Logic.all args_without_elem) end) args) end) ctrAs; val goal = Logic.mk_implies (mk_Trueprop_mem (elem, set $ ta), thesis); val vars = Variable.add_free_names lthy goal []; val thm =
Goal.prove_sorry lthy vars (flat premss) goal (fn {context = ctxt, prems} =>
mk_set_cases_tac ctxt (Thm.cterm_of ctxt ta) prems exhaust set_thms)
|> Thm.close_derivation \<^here>
|> rotate_prems ~1;
val cases_set_attr =
Attrib.internal \<^here> (K (Induct.cases_pred (name_of_set set)));
val ctr_names = quasi_unambiguous_case_names (flat
(map (uncurry mk_names o map_prod length name_of_ctr) (premss ~~ ctrAs))); in (* TODO: @{attributes [elim?]} *)
(thm, [Attrib.consumes 1, cases_set_attr, Attrib.case_names ctr_names]) end) setAs) end;
val (set_intros_thmssss, set_intros_thms) = let fun mk_goals A setA ctr_args t ctxt =
(case fastype_of t of Type (type_name, innerTs) =>
(case bnf_of ctxt type_name of
NONE => ([], ctxt)
| SOME bnf =>
apfst flat (fold_map (fn set => fn ctxt => let val T = HOLogic.dest_setT (range_type (fastype_of set)); val (y, ctxt') = yield_singleton (mk_Frees "y") T ctxt; val assm = mk_Trueprop_mem (y, set $ t); in
apfst (map (Logic.mk_implies o pair assm)) (mk_goals A setA ctr_args y ctxt') end) (map (mk_set innerTs) (sets_of_bnf bnf)) ctxt))
| T => (if T = A then [mk_Trueprop_mem (t, setA $ ctr_args)] else [], ctxt));
val (goalssss, _) =
fold_map (fn set => letval A = HOLogic.dest_setT (range_type (fastype_of set)) in
@{fold_map 2} (fn ctr => fn xs =>
fold_map (mk_goals A set (Term.list_comb (ctr, xs))) xs)
ctrAs xss end) setAs lthy; val goals = flat (flat (flat goalssss)); in
`(unflattt goalssss)
(if null goals then
[] else let val goal = Logic.mk_conjunction_balanced goals; val vars = Variable.add_free_names lthy goal []; in
Goal.prove_sorry lthy vars [] goal
(fn {context = ctxt, prems = _} => mk_set_intros_tac ctxt set0_thms)
|> Thm.close_derivation \<^here>
|> Conjunction.elim_balanced (length goals) end) end;
val rel_sel_thms = let val n = length discAs; fun mk_conjunct n k discA selAs discB selBs =
(if k = n then [] else [HOLogic.mk_eq (discA $ ta, discB $ tb)]) @
(if null selAs then
[] else
[Library.foldr HOLogic.mk_imp
(if n = 1 then [] else [discA $ ta, discB $ tb],
Library.foldr1 HOLogic.mk_conj (map2 (build_rel_app names_lthy Rs [])
(map (rapp ta) selAs) (map (rapp tb) selBs)))]);
val goals = if n = 0 then
[] else
[mk_Trueprop_eq (build_rel_app names_lthy Rs [] ta tb,
(case flat (@{map 5} (mk_conjunct n) (1 upto n) discAs selAss discBs selBss) of
[] => \<^term>\<open>True\<close>
| conjuncts => Library.foldr1 HOLogic.mk_conj conjuncts))];
val (rel_case_thm, rel_case_attrs) = let val thm = derive_rel_case relAsBs rel_inject_thms rel_distinct_thms; val ctr_names = quasi_unambiguous_case_names (map name_of_ctr ctrAs); in
(thm, [Attrib.case_names ctr_names, Attrib.consumes 1] @ @{attributes [cases pred]}) end;
val case_transfer_thm = derive_case_transfer rel_case_thm;
val sel_transfer_thms = if null selAss then
[] else let val shared_sels = foldl1 (uncurry (inter (op =))) (map (op ~~) (selAss ~~ selBss)); val goals = map (uncurry (mk_parametricity_goal names_lthy Rs)) shared_sels; in if null goals then
[] else let val goal = Logic.mk_conjunction_balanced goals; val vars = Variable.add_free_names lthy goal []; in
Goal.prove_sorry lthy vars [] goal
(fn {context = ctxt, prems = _} =>
mk_sel_transfer_tac ctxt n sel_defs case_transfer_thm)
|> Thm.close_derivation \<^here>
|> Conjunction.elim_balanced (length goals) end end;
val disc_transfer_thms = letval goals = map2 (mk_parametricity_goal names_lthy Rs) discAs discBs in if null goals then
[] else let val goal = Logic.mk_conjunction_balanced goals; val vars = Variable.add_free_names lthy goal []; in
Goal.prove_sorry lthy vars [] goal
(fn {context = ctxt, prems = _} => mk_disc_transfer_tac ctxt
(the_single rel_sel_thms) (the_single exhaust_discs)
(flat (flat distinct_discsss)))
|> Thm.close_derivation \<^here>
|> Conjunction.elim_balanced (length goals) end end;
val map_disc_iff_thms = let val discsB = map (mk_disc_or_sel Bs) discs; val discsA_t = map (fn disc1 => Term.betapply (disc1, ta)) discAs;
fun mk_goal (discA_t, discB) = if head_of discA_t aconv HOLogic.Not orelse is_refl_bool discA_t then
NONE else
SOME (mk_Trueprop_eq (betapply (discB, (Term.list_comb (mapx, fs) $ ta)), discA_t));
val goals = map_filter mk_goal (discsA_t ~~ discsB); in if null goals then
[] else let val goal = Logic.mk_conjunction_balanced goals; val vars = Variable.add_free_names lthy goal []; in
Goal.prove_sorry lthy vars [] goal
(fn {context = ctxt, prems = _} =>
mk_map_disc_iff_tac ctxt (Thm.cterm_of ctxt ta) exhaust (flat disc_thmss)
map_thms)
|> Thm.close_derivation \<^here>
|> Conjunction.elim_balanced (length goals) end end;
val (map_sel_thmss, map_sel_thms) = let fun mk_goal discA selA selB = let val prem = Term.betapply (discA, ta); val lhs = selB $ (Term.list_comb (mapx, fs) $ ta); val lhsT = fastype_of lhs; val map_rhsT =
map_atyps (perhaps (AList.lookup (op =) (map swap live_AsBs))) lhsT; val map_rhs = build_map lthy [] []
(the o (AList.lookup (op =) (live_AsBs ~~ fs))) (map_rhsT, lhsT); val rhs = (case map_rhs of Const (\<^const_name>\<open>id\<close>, _) => selA $ ta
| _ => map_rhs $ (selA $ ta)); val concl = mk_Trueprop_eq (lhs, rhs); in if is_refl_bool prem then concl else Logic.mk_implies (HOLogic.mk_Trueprop prem, concl) end;
val goalss = @{map 3} (map2 o mk_goal) discAs selAss selBss; val goals = flat goalss; in
`(unflat goalss)
(if null goals then
[] else let val goal = Logic.mk_conjunction_balanced goals; val vars = Variable.add_free_names lthy goal []; in
Goal.prove_sorry lthy vars [] goal
(fn {context = ctxt, prems = _} =>
mk_map_sel_tac ctxt (Thm.cterm_of ctxt ta) exhaust (flat disc_thmss)
map_thms (flat sel_thmss) live_nesting_map_id0s)
|> Thm.close_derivation \<^here>
|> Conjunction.elim_balanced (length goals) end) end;
val (set_sel_thmssss, set_sel_thms) = let fun mk_goal setA discA selA ctxt = let val prem = Term.betapply (discA, ta); val sel_rangeT = range_type (fastype_of selA); val A = HOLogic.dest_setT (range_type (fastype_of setA));
fun travese_nested_types t ctxt =
(case fastype_of t of Type (type_name, innerTs) =>
(case bnf_of ctxt type_name of
NONE => ([], ctxt)
| SOME bnf => let fun seq_assm a set ctxt = let val T = HOLogic.dest_setT (range_type (fastype_of set)); val (x, ctxt') = yield_singleton (mk_Frees "x") T ctxt; val assm = mk_Trueprop_mem (x, set $ a); in
travese_nested_types x ctxt'
|>> map (Logic.mk_implies o pair assm) end; in
fold_map (seq_assm t o mk_set innerTs) (sets_of_bnf bnf) ctxt
|>> flat end)
| T => if T = A then ([mk_Trueprop_mem (t, setA $ ta)], ctxt) else ([], ctxt));
val (concls, ctxt') = if sel_rangeT = A then ([mk_Trueprop_mem (selA $ ta, setA $ ta)], ctxt) else travese_nested_types (selA $ ta) ctxt; in if exists_subtype_in [A] sel_rangeT then if is_refl_bool prem then (concls, ctxt') else (map (Logic.mk_implies o pair (HOLogic.mk_Trueprop prem)) concls, ctxt') else
([], ctxt) end;
val (goalssss, _) =
fold_map (fn set => @{fold_map 2} (fold_map o mk_goal set) discAs selAss)
setAs names_lthy; val goals = flat (flat (flat goalssss)); in
`(unflattt goalssss)
(if null goals then
[] else let val goal = Logic.mk_conjunction_balanced goals; val vars = Variable.add_free_names lthy goal []; in
Goal.prove_sorry lthy vars [] goal
(fn {context = ctxt, prems = _} =>
mk_set_sel_tac ctxt (Thm.cterm_of ctxt ta) exhaust (flat disc_thmss)
(flat sel_thmss) set0_thms)
|> Thm.close_derivation \<^here>
|> Conjunction.elim_balanced (length goals) end) end;
val pred_injects = let val rel_eq_onp_with_tops_of =
Conv.fconv_rule (HOLogic.Trueprop_conv (Conv.arg1_conv
(Conv.top_sweep_rewrs_conv @{thms eq_onp_top_eq_eq[symmetric, THEN eq_reflection]} lthy)));
val eq_onps = map rel_eq_onp_with_tops_of
(map rel_eq_onp_of_bnf fp_bnfs @ fp_nesting_rel_eq_onps @ live_nesting_rel_eq_onps @
fp_nested_rel_eq_onps); val cTs = map (SOME o Thm.ctyp_of lthy) (maps (replicate 2) live_As); val cts = map (SOME o Thm.cterm_of lthy) (map mk_eq_onp Ps);
val get_rhs = Thm.concl_of #> HOLogic.dest_Trueprop #> HOLogic.dest_eq #> snd;
val notes =
(if Config.get lthy bnf_internals then
[(set0N, set0_thms, K [])] else
[]) @
[(case_transferN, [case_transfer_thm], K []),
(ctr_transferN, ctr_transfer_thms, K []),
(disc_transferN, disc_transfer_thms, K []),
(sel_transferN, sel_transfer_thms, K []),
(mapN, map_thms, K (nitpicksimp_attrs @ simp_attrs)),
(map_disc_iffN, map_disc_iff_thms, K simp_attrs),
(map_selN, map_sel_thms, K []),
(pred_injectN, pred_injects, K simp_attrs),
(rel_casesN, [rel_case_thm], K rel_case_attrs),
(rel_distinctN, rel_distinct_thms, K simp_attrs),
(rel_injectN, rel_inject_thms, K simp_attrs),
(rel_introsN, rel_intro_thms, K []),
(rel_selN, rel_sel_thms, K []),
(setN, set_thms, K (case_fp fp nitpicksimp_attrs [] @ simp_attrs)),
(set_casesN, set_cases_thms, nth set_cases_attrss),
(set_introsN, set_intros_thms, K []),
(set_selN, set_sel_thms, K [])]
|> massage_simple_notes fp_b_name;
val (noted, lthy') = lthy
|> uncurry (Spec_Rules.add Binding.empty Spec_Rules.equational)
(`(single o lhs_head_of o hd) map_thms)
|> fp = Least_FP ?
uncurry (Spec_Rules.add Binding.empty Spec_Rules.equational)
(`(single o lhs_head_of o hd) rel_code_thms)
|> uncurry (Spec_Rules.add Binding.empty Spec_Rules.equational)
(`(single o lhs_head_of o hd) set0_thms)
|> plugins code_plugin ? Code.declare_default_eqns (map (rpair true) (rel_code_thms @ map_thms @ set_thms))
|> Local_Theory.notes (anonymous_notes @ notes);
val transfer_gfp_sugar_thms = morph_gfp_sugar_thms o Morphism.transfer_morphism;
fun unzip_recT (Type (\<^type_name>\<open>prod\<close>, [_, TFree x]))
(T as Type (\<^type_name>\<open>prod\<close>, Ts as [_, TFree y])) = if x = y then [T] else Ts
| unzip_recT _ (Type (\<^type_name>\<open>prod\<close>, Ts as [_, TFree _])) = Ts
| unzip_recT _ T = [T];
fun mk_recs_args_types ctxt ctr_Tsss Cs absTs repTs ns mss ctor_rec_fun_Ts = let val Css = map2 replicate ns Cs; val x_Tssss =
@{map 6} (fn absT => fn repT => fn n => fn ms => fn ctr_Tss => fn ctor_rec_fun_T =>
map2 (map2 unzip_recT)
ctr_Tss (dest_absumprodT absT repT n ms (domain_type ctor_rec_fun_T)))
absTs repTs ns mss ctr_Tsss ctor_rec_fun_Ts;
val x_Tsss' = map (map flat_rec_arg_args) x_Tssss; val f_Tss = map2 (map2 (curry (op --->))) x_Tsss' Css;
val ((fss, xssss), _) = ctxt
|> mk_Freess "f" f_Tss
||>> mk_Freessss "x" x_Tssss; in
(f_Tss, x_Tssss, fss, xssss) end;
fun unzip_corecT (Type (\<^type_name>\<open>sum\<close>, _)) T = [T]
| unzip_corecT _ (Type (\<^type_name>\<open>sum\<close>, Ts)) = Ts
| unzip_corecT _ T = [T];
(*avoid "'a itself" arguments in corecursors*) fun repair_nullary_single_ctr [[]] = [[HOLogic.unitT]]
| repair_nullary_single_ctr Tss = Tss;
fun mk_corec_fun_arg_types0 ctr_Tsss Cs absTs repTs ns mss fun_Ts = let val ctr_Tsss' = map repair_nullary_single_ctr ctr_Tsss; val g_absTs = map range_type fun_Ts; val g_Tsss = map repair_nullary_single_ctr (@{map 5} dest_absumprodT absTs repTs ns mss g_absTs); val g_Tssss = @{map 3} (fn C => map2 (map2 (map (curry (op -->) C) oo unzip_corecT)))
Cs ctr_Tsss' g_Tsss; val q_Tssss = map (map (map (fn [_] => [] | [_, T] => [mk_pred1T (domain_type T)]))) g_Tssss; in
(q_Tssss, g_Tsss, g_Tssss, g_absTs) end;
fun mk_corec_p_pred_types Cs ns = map2 (fn n => replicate (Int.max (0, n - 1)) o mk_pred1T) ns Cs;
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