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(* Title: HOL/HOLCF/Tools/Domain/domain_isomorphism.ML
Author: Brian Huffman
Defines new types satisfying the given domain equations.
*)
signature DOMAIN_ISOMORPHISM =
sig
val domain_isomorphism :
(string list * binding * mixfix * typ
* (binding * binding) option) list ->
theory ->
(Domain_Take_Proofs.iso_info list
* Domain_Take_Proofs.take_induct_info) * theory
val define_map_functions :
(binding * Domain_Take_Proofs.iso_info) list ->
theory ->
{
map_consts : term list,
map_apply_thms : thm list,
map_unfold_thms : thm list,
map_cont_thm : thm,
deflation_map_thms : thm list
}
* theory
val domain_isomorphism_cmd :
(string list * binding * mixfix * string * (binding * binding) option) list
-> theory -> theory
end
structure Domain_Isomorphism : DOMAIN_ISOMORPHISM =
struct
val beta_ss =
simpset_of (put_simpset HOL_basic_ss \<^context>
addsimps @{thms simp_thms} addsimprocs [\<^simproc>\<open>beta_cfun_proc\<close>])
fun is_cpo thy T = Sign.of_sort thy (T, \<^sort>\<open>cpo\<close>)
(******************************************************************************)
(************************** building types and terms **************************)
(******************************************************************************)
open HOLCF_Library
infixr 6 ->>
infixr -->>
val udomT = \<^typ>\<open>udom\<close>
val deflT = \<^typ>\<open>udom defl\<close>
val udeflT = \<^typ>\<open>udom u defl\<close>
fun mk_DEFL T =
Const (\<^const_name>\<open>defl\<close>, Term.itselfT T --> deflT) $ Logic.mk_type T
fun dest_DEFL (Const (\<^const_name>\<open>defl\<close>, _) $ t) = Logic.dest_type t
| dest_DEFL t = raise TERM ("dest_DEFL", [t])
fun mk_LIFTDEFL T =
Const (\<^const_name>\<open>liftdefl\<close>, Term.itselfT T --> udeflT) $ Logic.mk_type T
fun dest_LIFTDEFL (Const (\<^const_name>\<open>liftdefl\<close>, _) $ t) = Logic.dest_type t
| dest_LIFTDEFL t = raise TERM ("dest_LIFTDEFL", [t])
fun mk_u_defl t = mk_capply (\<^const>\<open>u_defl\<close>, t)
fun emb_const T = Const (\<^const_name>\<open>emb\<close>, T ->> udomT)
fun prj_const T = Const (\<^const_name>\<open>prj\<close>, udomT ->> T)
fun coerce_const (T, U) = mk_cfcomp (prj_const U, emb_const T)
fun isodefl_const T =
Const (\<^const_name>\<open>isodefl\<close>, (T ->> T) --> deflT --> HOLogic.boolT)
fun isodefl'_const T =
Const (\<^const_name>\<open>isodefl'\, (T ->> T) --> udeflT --> HOLogic.boolT)
fun mk_deflation t =
Const (\<^const_name>\<open>deflation\<close>, Term.fastype_of t --> boolT) $ t
(* splits a cterm into the right and lefthand sides of equality *)
fun dest_eqs t = HOLogic.dest_eq (HOLogic.dest_Trueprop t)
fun mk_eqs (t, u) = HOLogic.mk_Trueprop (HOLogic.mk_eq (t, u))
(******************************************************************************)
(****************************** isomorphism info ******************************)
(******************************************************************************)
fun deflation_abs_rep (info : Domain_Take_Proofs.iso_info) : thm =
let
val abs_iso = #abs_inverse info
val rep_iso = #rep_inverse info
val thm = @{thm deflation_abs_rep} OF [abs_iso, rep_iso]
in
Drule.zero_var_indexes thm
end
(******************************************************************************)
(*************** fixed-point definitions and unfolding theorems ***************)
(******************************************************************************)
fun mk_projs [] _ = []
| mk_projs (x::[]) t = [(x, t)]
| mk_projs (x::xs) t = (x, mk_fst t) :: mk_projs xs (mk_snd t)
fun add_fixdefs
(spec : (binding * term) list)
(thy : theory) : (thm list * thm list * thm) * theory =
let
val binds = map fst spec
val (lhss, rhss) = ListPair.unzip (map (dest_eqs o snd) spec)
val functional = lambda_tuple lhss (mk_tuple rhss)
val fixpoint = mk_fix (mk_cabs functional)
(* project components of fixpoint *)
val projs = mk_projs lhss fixpoint
(* convert parameters to lambda abstractions *)
fun mk_eqn (lhs, rhs) =
case lhs of
Const (\<^const_name>\<open>Rep_cfun\<close>, _) $ f $ (x as Free _) =>
mk_eqn (f, big_lambda x rhs)
| f $ Const (\<^const_name>\<open>Pure.type\<close>, T) =>
mk_eqn (f, Abs ("t", T, rhs))
| Const _ => Logic.mk_equals (lhs, rhs)
| _ => raise TERM ("lhs not of correct form", [lhs, rhs])
val eqns = map mk_eqn projs
(* register constant definitions *)
val (fixdef_thms, thy) =
(Global_Theory.add_defs false o map Thm.no_attributes)
(map Thm.def_binding binds ~~ eqns) thy
(* prove applied version of definitions *)
fun prove_proj (lhs, rhs) =
let
fun tac ctxt = rewrite_goals_tac ctxt fixdef_thms THEN
(simp_tac (put_simpset beta_ss ctxt)) 1
val goal = Logic.mk_equals (lhs, rhs)
in Goal.prove_global thy [] [] goal (tac o #context) end
val proj_thms = map prove_proj projs
(* mk_tuple lhss == fixpoint *)
fun pair_equalI (thm1, thm2) = @{thm Pair_equalI} OF [thm1, thm2]
val tuple_fixdef_thm = foldr1 pair_equalI proj_thms
val cont_thm =
let
val prop = mk_trp (mk_cont functional)
val rules = Named_Theorems.get (Proof_Context.init_global thy) \<^named_theorems>\<open>cont2cont\<close>
fun tac ctxt = REPEAT_ALL_NEW (match_tac ctxt (rev rules)) 1
in
Goal.prove_global thy [] [] prop (tac o #context)
end
val tuple_unfold_thm =
(@{thm def_cont_fix_eq} OF [tuple_fixdef_thm, cont_thm])
|> Local_Defs.unfold (Proof_Context.init_global thy) @{thms split_conv}
fun mk_unfold_thms [] _ = []
| mk_unfold_thms (n::[]) thm = [(n, thm)]
| mk_unfold_thms (n::ns) thm = let
val thmL = thm RS @{thm Pair_eqD1}
val thmR = thm RS @{thm Pair_eqD2}
in (n, thmL) :: mk_unfold_thms ns thmR end
val unfold_binds = map (Binding.suffix_name "_unfold") binds
(* register unfold theorems *)
val (unfold_thms, thy) =
(Global_Theory.add_thms o map (Thm.no_attributes o apsnd Drule.zero_var_indexes))
(mk_unfold_thms unfold_binds tuple_unfold_thm) thy
in
((proj_thms, unfold_thms, cont_thm), thy)
end
(******************************************************************************)
(****************** deflation combinators and map functions *******************)
(******************************************************************************)
fun defl_of_typ
(thy : theory)
(tab1 : (typ * term) list)
(tab2 : (typ * term) list)
(T : typ) : term =
let
val defl_simps =
Named_Theorems.get (Proof_Context.init_global thy) \<^named_theorems>\<open>domain_defl_simps\<close>
val rules = map (Thm.concl_of #> HOLogic.dest_Trueprop #> HOLogic.dest_eq) (rev defl_simps)
val rules' = map (apfst mk_DEFL) tab1 @ map (apfst mk_LIFTDEFL) tab2
fun proc1 t =
(case dest_DEFL t of
TFree (a, _) => SOME (Free ("d" ^ Library.unprefix "'" a, deflT))
| _ => NONE) handle TERM _ => NONE
fun proc2 t =
(case dest_LIFTDEFL t of
TFree (a, _) => SOME (Free ("p" ^ Library.unprefix "'" a, udeflT))
| _ => NONE) handle TERM _ => NONE
in
Pattern.rewrite_term thy (rules @ rules') [proc1, proc2] (mk_DEFL T)
end
(******************************************************************************)
(********************* declaring definitions and theorems *********************)
(******************************************************************************)
fun define_const
(bind : binding, rhs : term)
(thy : theory)
: (term * thm) * theory =
let
val typ = Term.fastype_of rhs
val (const, thy) = Sign.declare_const_global ((bind, typ), NoSyn) thy
val eqn = Logic.mk_equals (const, rhs)
val def = Thm.no_attributes (Thm.def_binding bind, eqn)
val (def_thm, thy) = yield_singleton (Global_Theory.add_defs false) def thy
in
((const, def_thm), thy)
end
fun add_qualified_thm name (dbind, thm) =
yield_singleton Global_Theory.add_thms
((Binding.qualify_name true dbind name, thm), [])
(******************************************************************************)
(*************************** defining map functions ***************************)
(******************************************************************************)
fun define_map_functions
(spec : (binding * Domain_Take_Proofs.iso_info) list)
(thy : theory) =
let
(* retrieve components of spec *)
val dbinds = map fst spec
val iso_infos = map snd spec
val dom_eqns = map (fn x => (#absT x, #repT x)) iso_infos
val rep_abs_consts = map (fn x => (#rep_const x, #abs_const x)) iso_infos
fun mapT (T as Type (_, Ts)) =
(map (fn T => T ->> T) (filter (is_cpo thy) Ts)) -->> (T ->> T)
| mapT T = T ->> T
(* declare map functions *)
fun declare_map_const (tbind, (lhsT, _)) thy =
let
val map_type = mapT lhsT
val map_bind = Binding.suffix_name "_map" tbind
in
Sign.declare_const_global ((map_bind, map_type), NoSyn) thy
end
val (map_consts, thy) = thy |>
fold_map declare_map_const (dbinds ~~ dom_eqns)
(* defining equations for map functions *)
local
fun unprime a = Library.unprefix "'" a
fun mapvar T = Free (unprime (fst (dest_TFree T)), T ->> T)
fun map_lhs (map_const, lhsT) =
(lhsT, list_ccomb (map_const, map mapvar (filter (is_cpo thy) (snd (dest_Type lhsT)))))
val tab1 = map map_lhs (map_consts ~~ map fst dom_eqns)
val Ts = (snd o dest_Type o fst o hd) dom_eqns
val tab = (Ts ~~ map mapvar Ts) @ tab1
fun mk_map_spec (((rep_const, abs_const), _), (lhsT, rhsT)) =
let
val lhs = Domain_Take_Proofs.map_of_typ thy tab lhsT
val body = Domain_Take_Proofs.map_of_typ thy tab rhsT
val rhs = mk_cfcomp (abs_const, mk_cfcomp (body, rep_const))
in mk_eqs (lhs, rhs) end
in
val map_specs =
map mk_map_spec (rep_abs_consts ~~ map_consts ~~ dom_eqns)
end
(* register recursive definition of map functions *)
val map_binds = map (Binding.suffix_name "_map") dbinds
val ((map_apply_thms, map_unfold_thms, map_cont_thm), thy) =
add_fixdefs (map_binds ~~ map_specs) thy
(* prove deflation theorems for map functions *)
val deflation_abs_rep_thms = map deflation_abs_rep iso_infos
val deflation_map_thm =
let
fun unprime a = Library.unprefix "'" a
fun mk_f T = Free (unprime (fst (dest_TFree T)), T ->> T)
fun mk_assm T = mk_trp (mk_deflation (mk_f T))
fun mk_goal (map_const, (lhsT, _)) =
let
val (_, Ts) = dest_Type lhsT
val map_term = list_ccomb (map_const, map mk_f (filter (is_cpo thy) Ts))
in mk_deflation map_term end
val assms = (map mk_assm o filter (is_cpo thy) o snd o dest_Type o fst o hd) dom_eqns
val goals = map mk_goal (map_consts ~~ dom_eqns)
val goal = mk_trp (foldr1 HOLogic.mk_conj goals)
val adm_rules =
@{thms adm_conj adm_subst [OF _ adm_deflation]
cont2cont_fst cont2cont_snd cont_id}
val bottom_rules =
@{thms fst_strict snd_strict deflation_bottom simp_thms}
val tuple_rules =
@{thms split_def fst_conv snd_conv}
val deflation_rules =
@{thms conjI deflation_ID}
@ deflation_abs_rep_thms
@ Domain_Take_Proofs.get_deflation_thms thy
in
Goal.prove_global thy [] assms goal (fn {prems, context = ctxt} =>
EVERY
[rewrite_goals_tac ctxt map_apply_thms,
resolve_tac ctxt [map_cont_thm RS @{thm cont_fix_ind}] 1,
REPEAT (resolve_tac ctxt adm_rules 1),
simp_tac (put_simpset HOL_basic_ss ctxt addsimps bottom_rules) 1,
simp_tac (put_simpset HOL_basic_ss ctxt addsimps tuple_rules) 1,
REPEAT (eresolve_tac ctxt @{thms conjE} 1),
REPEAT (resolve_tac ctxt (deflation_rules @ prems) 1 ORELSE assume_tac ctxt 1)])
end
fun conjuncts [] _ = []
| conjuncts (n::[]) thm = [(n, thm)]
| conjuncts (n::ns) thm = let
val thmL = thm RS @{thm conjunct1}
val thmR = thm RS @{thm conjunct2}
in (n, thmL):: conjuncts ns thmR end
val deflation_map_binds = dbinds |>
map (Binding.prefix_name "deflation_" o Binding.suffix_name "_map")
val (deflation_map_thms, thy) = thy |>
(Global_Theory.add_thms o map (Thm.no_attributes o apsnd Drule.zero_var_indexes))
(conjuncts deflation_map_binds deflation_map_thm)
(* register indirect recursion in theory data *)
local
fun register_map (dname, args) =
Domain_Take_Proofs.add_rec_type (dname, args)
val dnames = map (fst o dest_Type o fst) dom_eqns
fun args (T, _) = case T of Type (_, Ts) => map (is_cpo thy) Ts | _ => []
val argss = map args dom_eqns
in
val thy =
fold register_map (dnames ~~ argss) thy
end
(* register deflation theorems *)
val thy = fold Domain_Take_Proofs.add_deflation_thm deflation_map_thms thy
val result =
{
map_consts = map_consts,
map_apply_thms = map_apply_thms,
map_unfold_thms = map_unfold_thms,
map_cont_thm = map_cont_thm,
deflation_map_thms = deflation_map_thms
}
in
(result, thy)
end
(******************************************************************************)
(******************************* main function ********************************)
(******************************************************************************)
fun read_typ thy str sorts =
let
val ctxt = Proof_Context.init_global thy
|> fold (Variable.declare_typ o TFree) sorts
val T = Syntax.read_typ ctxt str
in (T, Term.add_tfreesT T sorts) end
fun cert_typ sign raw_T sorts =
let
val T = Type.no_tvars (Sign.certify_typ sign raw_T)
handle TYPE (msg, _, _) => error msg
val sorts' = Term.add_tfreesT T sorts
val _ =
case duplicates (op =) (map fst sorts') of
[] => ()
| dups => error ("Inconsistent sort constraints for " ^ commas dups)
in (T, sorts') end
fun gen_domain_isomorphism
(prep_typ: theory -> 'a -> (string * sort) list -> typ * (string * sort) list)
(doms_raw: (string list * binding * mixfix * 'a * (binding * binding) option) list)
(thy: theory)
: (Domain_Take_Proofs.iso_info list
* Domain_Take_Proofs.take_induct_info) * theory =
let
(* this theory is used just for parsing *)
val tmp_thy = thy |>
Sign.add_types_global (map (fn (tvs, tbind, mx, _, _) =>
(tbind, length tvs, mx)) doms_raw)
fun prep_dom thy (vs, t, mx, typ_raw, morphs) sorts =
let val (typ, sorts') = prep_typ thy typ_raw sorts
in ((vs, t, mx, typ, morphs), sorts') end
val (doms : (string list * binding * mixfix * typ * (binding * binding) option) list,
sorts : (string * sort) list) =
fold_map (prep_dom tmp_thy) doms_raw []
(* lookup function for sorts of type variables *)
fun the_sort v = the (AList.lookup (op =) sorts v)
(* declare arities in temporary theory *)
val tmp_thy =
let
fun arity (vs, tbind, _, _, _) =
(Sign.full_name thy tbind, map the_sort vs, \<^sort>\<open>domain\<close>)
in
fold Axclass.arity_axiomatization (map arity doms) tmp_thy
end
(* check bifiniteness of right-hand sides *)
fun check_rhs (_, _, _, rhs, _) =
if Sign.of_sort tmp_thy (rhs, \<^sort>\<open>domain\<close>) then ()
else error ("Type not of sort domain: " ^
quote (Syntax.string_of_typ_global tmp_thy rhs))
val _ = map check_rhs doms
(* domain equations *)
fun mk_dom_eqn (vs, tbind, _, rhs, _) =
let fun arg v = TFree (v, the_sort v)
in (Type (Sign.full_name tmp_thy tbind, map arg vs), rhs) end
val dom_eqns = map mk_dom_eqn doms
(* check for valid type parameters *)
val (tyvars, _, _, _, _) = hd doms
val _ = map (fn (tvs, tname, _, _, _) =>
let val full_tname = Sign.full_name tmp_thy tname
in
(case duplicates (op =) tvs of
[] =>
if eq_set (op =) (tyvars, tvs) then (full_tname, tvs)
else error ("Mutually recursive domains must have same type parameters")
| dups => error ("Duplicate parameter(s) for domain " ^ Binding.print tname ^
" : " ^ commas dups))
end) doms
val dbinds = map (fn (_, dbind, _, _, _) => dbind) doms
val morphs = map (fn (_, _, _, _, morphs) => morphs) doms
(* determine deflation combinator arguments *)
val lhsTs : typ list = map fst dom_eqns
val defl_rec = Free ("t", mk_tupleT (map (K deflT) lhsTs))
val defl_recs = mk_projs lhsTs defl_rec
val defl_recs' = map (apsnd mk_u_defl) defl_recs
fun defl_body (_, _, _, rhsT, _) =
defl_of_typ tmp_thy defl_recs defl_recs' rhsT
val functional = Term.lambda defl_rec (mk_tuple (map defl_body doms))
val tfrees = map fst (Term.add_tfrees functional [])
val frees = map fst (Term.add_frees functional [])
fun get_defl_flags (vs, _, _, _, _) =
let
fun argT v = TFree (v, the_sort v)
fun mk_d v = "d" ^ Library.unprefix "'" v
fun mk_p v = "p" ^ Library.unprefix "'" v
val args = maps (fn v => [(mk_d v, mk_DEFL (argT v)), (mk_p v, mk_LIFTDEFL (argT v))]) vs
val typeTs = map argT (filter (member (op =) tfrees) vs)
val defl_args = map snd (filter (member (op =) frees o fst) args)
in
(typeTs, defl_args)
end
val defl_flagss = map get_defl_flags doms
(* declare deflation combinator constants *)
fun declare_defl_const ((typeTs, defl_args), (_, tbind, _, _, _)) thy =
let
val defl_bind = Binding.suffix_name "_defl" tbind
val defl_type =
map Term.itselfT typeTs ---> map fastype_of defl_args -->> deflT
in
Sign.declare_const_global ((defl_bind, defl_type), NoSyn) thy
end
val (defl_consts, thy) =
fold_map declare_defl_const (defl_flagss ~~ doms) thy
(* defining equations for type combinators *)
fun mk_defl_term (defl_const, (typeTs, defl_args)) =
let
val type_args = map Logic.mk_type typeTs
in
list_ccomb (list_comb (defl_const, type_args), defl_args)
end
val defl_terms = map mk_defl_term (defl_consts ~~ defl_flagss)
val defl_tab = map fst dom_eqns ~~ defl_terms
val defl_tab' = map fst dom_eqns ~~ map mk_u_defl defl_terms
fun mk_defl_spec (lhsT, rhsT) =
mk_eqs (defl_of_typ tmp_thy defl_tab defl_tab' lhsT,
defl_of_typ tmp_thy defl_tab defl_tab' rhsT)
val defl_specs = map mk_defl_spec dom_eqns
(* register recursive definition of deflation combinators *)
val defl_binds = map (Binding.suffix_name "_defl") dbinds
val ((defl_apply_thms, defl_unfold_thms, defl_cont_thm), thy) =
add_fixdefs (defl_binds ~~ defl_specs) thy
(* define types using deflation combinators *)
fun make_repdef ((vs, tbind, mx, _, _), defl) thy =
let
val spec = (tbind, map (rpair dummyS) vs, mx)
val ((_, _, _, {DEFL, ...}), thy) =
Domaindef.add_domaindef spec defl NONE thy
(* declare domain_defl_simps rules *)
val thy =
Context.theory_map (Named_Theorems.add_thm \<^named_theorems>\<open>domain_defl_simps\<close> DEFL) thy
in
(DEFL, thy)
end
val (DEFL_thms, thy) = fold_map make_repdef (doms ~~ defl_terms) thy
(* prove DEFL equations *)
fun mk_DEFL_eq_thm (lhsT, rhsT) =
let
val goal = mk_eqs (mk_DEFL lhsT, mk_DEFL rhsT)
val DEFL_simps =
Named_Theorems.get (Proof_Context.init_global thy) \<^named_theorems>\<open>domain_defl_simps\<close>
fun tac ctxt =
rewrite_goals_tac ctxt (map mk_meta_eq (rev DEFL_simps))
THEN TRY (resolve_tac ctxt defl_unfold_thms 1)
in
Goal.prove_global thy [] [] goal (tac o #context)
end
val DEFL_eq_thms = map mk_DEFL_eq_thm dom_eqns
(* register DEFL equations *)
val DEFL_eq_binds = map (Binding.prefix_name "DEFL_eq_") dbinds
val (_, thy) = thy |>
(Global_Theory.add_thms o map Thm.no_attributes)
(DEFL_eq_binds ~~ DEFL_eq_thms)
(* define rep/abs functions *)
fun mk_rep_abs ((tbind, _), (lhsT, rhsT)) thy =
let
val rep_bind = Binding.suffix_name "_rep" tbind
val abs_bind = Binding.suffix_name "_abs" tbind
val ((rep_const, rep_def), thy) =
define_const (rep_bind, coerce_const (lhsT, rhsT)) thy
val ((abs_const, abs_def), thy) =
define_const (abs_bind, coerce_const (rhsT, lhsT)) thy
in
(((rep_const, abs_const), (rep_def, abs_def)), thy)
end
val ((rep_abs_consts, rep_abs_defs), thy) = thy
|> fold_map mk_rep_abs (dbinds ~~ morphs ~~ dom_eqns)
|>> ListPair.unzip
(* prove isomorphism and isodefl rules *)
fun mk_iso_thms ((tbind, DEFL_eq), (rep_def, abs_def)) thy =
let
fun make thm =
Drule.zero_var_indexes (thm OF [DEFL_eq, abs_def, rep_def])
val rep_iso_thm = make @{thm domain_rep_iso}
val abs_iso_thm = make @{thm domain_abs_iso}
val isodefl_thm = make @{thm isodefl_abs_rep}
val thy = thy
|> snd o add_qualified_thm "rep_iso" (tbind, rep_iso_thm)
|> snd o add_qualified_thm "abs_iso" (tbind, abs_iso_thm)
|> snd o add_qualified_thm "isodefl_abs_rep" (tbind, isodefl_thm)
in
(((rep_iso_thm, abs_iso_thm), isodefl_thm), thy)
end
val ((iso_thms, isodefl_abs_rep_thms), thy) =
thy
|> fold_map mk_iso_thms (dbinds ~~ DEFL_eq_thms ~~ rep_abs_defs)
|>> ListPair.unzip
(* collect info about rep/abs *)
val iso_infos : Domain_Take_Proofs.iso_info list =
let
fun mk_info (((lhsT, rhsT), (repC, absC)), (rep_iso, abs_iso)) =
{
repT = rhsT,
absT = lhsT,
rep_const = repC,
abs_const = absC,
rep_inverse = rep_iso,
abs_inverse = abs_iso
}
in
map mk_info (dom_eqns ~~ rep_abs_consts ~~ iso_thms)
end
(* definitions and proofs related to map functions *)
val (map_info, thy) =
define_map_functions (dbinds ~~ iso_infos) thy
val { map_consts, map_apply_thms, map_cont_thm, ...} = map_info
(* prove isodefl rules for map functions *)
val isodefl_thm =
let
fun unprime a = Library.unprefix "'" a
fun mk_d T = Free ("d" ^ unprime (fst (dest_TFree T)), deflT)
fun mk_p T = Free ("p" ^ unprime (fst (dest_TFree T)), udeflT)
fun mk_f T = Free ("f" ^ unprime (fst (dest_TFree T)), T ->> T)
fun mk_assm t =
case try dest_LIFTDEFL t of
SOME T => mk_trp (isodefl'_const T $ mk_f T $ mk_p T)
| NONE =>
let val T = dest_DEFL t
in mk_trp (isodefl_const T $ mk_f T $ mk_d T) end
fun mk_goal (map_const, (T, _)) =
let
val (_, Ts) = dest_Type T
val map_term = list_ccomb (map_const, map mk_f (filter (is_cpo thy) Ts))
val defl_term = defl_of_typ thy (Ts ~~ map mk_d Ts) (Ts ~~ map mk_p Ts) T
in isodefl_const T $ map_term $ defl_term end
val assms = (map mk_assm o snd o hd) defl_flagss
val goals = map mk_goal (map_consts ~~ dom_eqns)
val goal = mk_trp (foldr1 HOLogic.mk_conj goals)
val adm_rules =
@{thms adm_conj adm_isodefl cont2cont_fst cont2cont_snd cont_id}
val bottom_rules =
@{thms fst_strict snd_strict isodefl_bottom simp_thms}
val tuple_rules =
@{thms split_def fst_conv snd_conv}
val map_ID_thms = Domain_Take_Proofs.get_map_ID_thms thy
val map_ID_simps = map (fn th => th RS sym) map_ID_thms
val isodefl_rules =
@{thms conjI isodefl_ID_DEFL isodefl_LIFTDEFL}
@ isodefl_abs_rep_thms
@ rev (Named_Theorems.get (Proof_Context.init_global thy) \<^named_theorems>\<open>domain_isodefl\<close>)
in
Goal.prove_global thy [] assms goal (fn {prems, context = ctxt} =>
EVERY
[rewrite_goals_tac ctxt (defl_apply_thms @ map_apply_thms),
resolve_tac ctxt [@{thm cont_parallel_fix_ind} OF [defl_cont_thm, map_cont_thm]] 1,
REPEAT (resolve_tac ctxt adm_rules 1),
simp_tac (put_simpset HOL_basic_ss ctxt addsimps bottom_rules) 1,
simp_tac (put_simpset HOL_basic_ss ctxt addsimps tuple_rules) 1,
simp_tac (put_simpset HOL_basic_ss ctxt addsimps map_ID_simps) 1,
REPEAT (eresolve_tac ctxt @{thms conjE} 1),
REPEAT (resolve_tac ctxt (isodefl_rules @ prems) 1 ORELSE assume_tac ctxt 1)])
end
val isodefl_binds = map (Binding.prefix_name "isodefl_") dbinds
fun conjuncts [] _ = []
| conjuncts (n::[]) thm = [(n, thm)]
| conjuncts (n::ns) thm = let
val thmL = thm RS @{thm conjunct1}
val thmR = thm RS @{thm conjunct2}
in (n, thmL):: conjuncts ns thmR end
val (isodefl_thms, thy) = thy |>
(Global_Theory.add_thms o map (Thm.no_attributes o apsnd Drule.zero_var_indexes))
(conjuncts isodefl_binds isodefl_thm)
val thy =
fold (Context.theory_map o Named_Theorems.add_thm \<^named_theorems>\<open>domain_isodefl\<close>)
isodefl_thms thy
(* prove map_ID theorems *)
fun prove_map_ID_thm
(((map_const, (lhsT, _)), DEFL_thm), isodefl_thm) =
let
val Ts = snd (dest_Type lhsT)
fun is_cpo T = Sign.of_sort thy (T, \<^sort>\<open>cpo\<close>)
val lhs = list_ccomb (map_const, map mk_ID (filter is_cpo Ts))
val goal = mk_eqs (lhs, mk_ID lhsT)
fun tac ctxt = EVERY
[resolve_tac ctxt @{thms isodefl_DEFL_imp_ID} 1,
stac ctxt DEFL_thm 1,
resolve_tac ctxt [isodefl_thm] 1,
REPEAT (resolve_tac ctxt @{thms isodefl_ID_DEFL isodefl_LIFTDEFL} 1)]
in
Goal.prove_global thy [] [] goal (tac o #context)
end
val map_ID_binds = map (Binding.suffix_name "_map_ID") dbinds
val map_ID_thms =
map prove_map_ID_thm
(map_consts ~~ dom_eqns ~~ DEFL_thms ~~ isodefl_thms)
val (_, thy) = thy |>
(Global_Theory.add_thms o map (rpair [Domain_Take_Proofs.map_ID_add]))
(map_ID_binds ~~ map_ID_thms)
(* definitions and proofs related to take functions *)
val (take_info, thy) =
Domain_Take_Proofs.define_take_functions
(dbinds ~~ iso_infos) thy
val { take_consts, chain_take_thms, take_0_thms, take_Suc_thms, ...} =
take_info
(* least-upper-bound lemma for take functions *)
val lub_take_lemma =
let
val lhs = mk_tuple (map mk_lub take_consts)
fun is_cpo T = Sign.of_sort thy (T, \<^sort>\<open>cpo\<close>)
fun mk_map_ID (map_const, (lhsT, _)) =
list_ccomb (map_const, map mk_ID (filter is_cpo (snd (dest_Type lhsT))))
val rhs = mk_tuple (map mk_map_ID (map_consts ~~ dom_eqns))
val goal = mk_trp (mk_eq (lhs, rhs))
val map_ID_thms = Domain_Take_Proofs.get_map_ID_thms thy
val start_rules =
@{thms lub_Pair [symmetric] ch2ch_Pair} @ chain_take_thms
@ @{thms prod.collapse split_def}
@ map_apply_thms @ map_ID_thms
val rules0 =
@{thms iterate_0 Pair_strict} @ take_0_thms
val rules1 =
@{thms iterate_Suc prod_eq_iff fst_conv snd_conv}
@ take_Suc_thms
fun tac ctxt =
EVERY
[simp_tac (put_simpset HOL_basic_ss ctxt addsimps start_rules) 1,
simp_tac (put_simpset HOL_basic_ss ctxt addsimps @{thms fix_def2}) 1,
resolve_tac ctxt @{thms lub_eq} 1,
resolve_tac ctxt @{thms nat.induct} 1,
simp_tac (put_simpset HOL_basic_ss ctxt addsimps rules0) 1,
asm_full_simp_tac (put_simpset beta_ss ctxt addsimps rules1) 1]
in
Goal.prove_global thy [] [] goal (tac o #context)
end
(* prove lub of take equals ID *)
fun prove_lub_take (((dbind, take_const), map_ID_thm), (lhsT, _)) thy =
let
val n = Free ("n", natT)
val goal = mk_eqs (mk_lub (lambda n (take_const $ n)), mk_ID lhsT)
fun tac ctxt =
EVERY
[resolve_tac ctxt @{thms trans} 1,
resolve_tac ctxt [map_ID_thm] 2,
cut_tac lub_take_lemma 1,
REPEAT (eresolve_tac ctxt @{thms Pair_inject} 1), assume_tac ctxt 1]
val lub_take_thm = Goal.prove_global thy [] [] goal (tac o #context)
in
add_qualified_thm "lub_take" (dbind, lub_take_thm) thy
end
val (lub_take_thms, thy) =
fold_map prove_lub_take
(dbinds ~~ take_consts ~~ map_ID_thms ~~ dom_eqns) thy
(* prove additional take theorems *)
val (take_info2, thy) =
Domain_Take_Proofs.add_lub_take_theorems
(dbinds ~~ iso_infos) take_info lub_take_thms thy
in
((iso_infos, take_info2), thy)
end
val domain_isomorphism = gen_domain_isomorphism cert_typ
val domain_isomorphism_cmd = snd oo gen_domain_isomorphism read_typ
(******************************************************************************)
(******************************** outer syntax ********************************)
(******************************************************************************)
local
val parse_domain_iso :
(string list * binding * mixfix * string * (binding * binding) option)
parser =
(Parse.type_args -- Parse.binding -- Parse.opt_mixfix -- (\<^keyword>\<open>=\<close> |-- Parse.typ) --
Scan.option (\<^keyword>\<open>morphisms\<close> |-- Parse.!!! (Parse.binding -- Parse.binding)))
>> (fn ((((vs, t), mx), rhs), morphs) => (vs, t, mx, rhs, morphs))
val parse_domain_isos = Parse.and_list1 parse_domain_iso
in
val _ =
Outer_Syntax.command \<^command_keyword>\<open>domain_isomorphism\<close> "define domain isomorphisms (HOLCF)"
(parse_domain_isos >> (Toplevel.theory o domain_isomorphism_cmd))
end
end
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