discontinued special treatment of structure Ast: no pervasive content, no inclusion in structure Syntax;
(* Title: HOL/HOLCF/Tools/Domain/domain_constructors.ML
Author: Brian Huffman
Defines constructor functions for a given domain isomorphism
and proves related theorems.
*)
signature DOMAIN_CONSTRUCTORS =
sig
type constr_info =
{
iso_info : Domain_Take_Proofs.iso_info,
con_specs : (term * (bool * typ) list) list,
con_betas : thm list,
nchotomy : thm,
exhaust : thm,
compacts : thm list,
con_rews : thm list,
inverts : thm list,
injects : thm list,
dist_les : thm list,
dist_eqs : thm list,
cases : thm list,
sel_rews : thm list,
dis_rews : thm list,
match_rews : thm list
}
val add_domain_constructors :
binding
-> (binding * (bool * binding option * typ) list * mixfix) list
-> Domain_Take_Proofs.iso_info
-> theory
-> constr_info * theory
end
structure Domain_Constructors : DOMAIN_CONSTRUCTORS =
struct
open HOLCF_Library
infixr 6 ->>
infix -->>
infix 9 `
type constr_info =
{
iso_info : Domain_Take_Proofs.iso_info,
con_specs : (term * (bool * typ) list) list,
con_betas : thm list,
nchotomy : thm,
exhaust : thm,
compacts : thm list,
con_rews : thm list,
inverts : thm list,
injects : thm list,
dist_les : thm list,
dist_eqs : thm list,
cases : thm list,
sel_rews : thm list,
dis_rews : thm list,
match_rews : thm list
}
(************************** miscellaneous functions ***************************)
val simple_ss = HOL_basic_ss addsimps simp_thms
val beta_rules =
@{thms beta_cfun cont_id cont_const cont2cont_APP cont2cont_LAM'} @
@{thms cont2cont_fst cont2cont_snd cont2cont_Pair}
val beta_ss = HOL_basic_ss addsimps (simp_thms @ beta_rules)
fun define_consts
(specs : (binding * term * mixfix) list)
(thy : theory)
: (term list * thm list) * theory =
let
fun mk_decl (b, t, mx) = (b, fastype_of t, mx)
val decls = map mk_decl specs
val thy = Cont_Consts.add_consts decls thy
fun mk_const (b, T, mx) = Const (Sign.full_name thy b, T)
val consts = map mk_const decls
fun mk_def c (b, t, mx) =
(Binding.suffix_name "_def" b, Logic.mk_equals (c, t))
val defs = map2 mk_def consts specs
val (def_thms, thy) =
Global_Theory.add_defs false (map Thm.no_attributes defs) thy
in
((consts, def_thms), thy)
end
fun prove
(thy : theory)
(defs : thm list)
(goal : term)
(tacs : {prems: thm list, context: Proof.context} -> tactic list)
: thm =
let
fun tac {prems, context} =
rewrite_goals_tac defs THEN
EVERY (tacs {prems = map (rewrite_rule defs) prems, context = context})
in
Goal.prove_global thy [] [] goal tac
end
fun get_vars_avoiding
(taken : string list)
(args : (bool * typ) list)
: (term list * term list) =
let
val Ts = map snd args
val ns = Name.variant_list taken (Datatype_Prop.make_tnames Ts)
val vs = map Free (ns ~~ Ts)
val nonlazy = map snd (filter_out (fst o fst) (args ~~ vs))
in
(vs, nonlazy)
end
fun get_vars args = get_vars_avoiding [] args
(************** generating beta reduction rules from definitions **************)
local
fun arglist (Const _ $ Abs (s, T, t)) =
let
val arg = Free (s, T)
val (args, body) = arglist (subst_bound (arg, t))
in (arg :: args, body) end
| arglist t = ([], t)
in
fun beta_of_def thy def_thm =
let
val (con, lam) = Logic.dest_equals (concl_of def_thm)
val (args, rhs) = arglist lam
val lhs = list_ccomb (con, args)
val goal = mk_equals (lhs, rhs)
val cs = ContProc.cont_thms lam
val betas = map (fn c => mk_meta_eq (c RS @{thm beta_cfun})) cs
in
prove thy (def_thm::betas) goal (K [rtac reflexive_thm 1])
end
end
(******************************************************************************)
(************* definitions and theorems for constructor functions *************)
(******************************************************************************)
fun add_constructors
(spec : (binding * (bool * typ) list * mixfix) list)
(abs_const : term)
(iso_locale : thm)
(thy : theory)
=
let
(* get theorems about rep and abs *)
val abs_strict = iso_locale RS @{thm iso.abs_strict}
(* get types of type isomorphism *)
val (rhsT, lhsT) = dest_cfunT (fastype_of abs_const)
fun vars_of args =
let
val Ts = map snd args
val ns = Datatype_Prop.make_tnames Ts
in
map Free (ns ~~ Ts)
end
(* define constructor functions *)
val ((con_consts, con_defs), thy) =
let
fun one_arg (lazy, T) var = if lazy then mk_up var else var
fun one_con (_,args,_) = mk_stuple (map2 one_arg args (vars_of args))
fun mk_abs t = abs_const ` t
val rhss = map mk_abs (mk_sinjects (map one_con spec))
fun mk_def (bind, args, mx) rhs =
(bind, big_lambdas (vars_of args) rhs, mx)
in
define_consts (map2 mk_def spec rhss) thy
end
(* prove beta reduction rules for constructors *)
val con_betas = map (beta_of_def thy) con_defs
(* replace bindings with terms in constructor spec *)
val spec' : (term * (bool * typ) list) list =
let fun one_con con (b, args, mx) = (con, args)
in map2 one_con con_consts spec end
(* prove exhaustiveness of constructors *)
local
fun arg2typ n (true, T) = (n+1, mk_upT (TVar (("'a", n), @{sort cpo})))
| arg2typ n (false, T) = (n+1, TVar (("'a", n), @{sort pcpo}))
fun args2typ n [] = (n, oneT)
| args2typ n [arg] = arg2typ n arg
| args2typ n (arg::args) =
let
val (n1, t1) = arg2typ n arg
val (n2, t2) = args2typ n1 args
in (n2, mk_sprodT (t1, t2)) end
fun cons2typ n [] = (n, oneT)
| cons2typ n [con] = args2typ n (snd con)
| cons2typ n (con::cons) =
let
val (n1, t1) = args2typ n (snd con)
val (n2, t2) = cons2typ n1 cons
in (n2, mk_ssumT (t1, t2)) end
val ct = ctyp_of thy (snd (cons2typ 1 spec'))
val thm1 = instantiate' [SOME ct] [] @{thm exh_start}
val thm2 = rewrite_rule (map mk_meta_eq @{thms ex_bottom_iffs}) thm1
val thm3 = rewrite_rule [mk_meta_eq @{thm conj_assoc}] thm2
val y = Free ("y", lhsT)
fun one_con (con, args) =
let
val (vs, nonlazy) = get_vars_avoiding ["y"] args
val eqn = mk_eq (y, list_ccomb (con, vs))
val conj = foldr1 mk_conj (eqn :: map mk_defined nonlazy)
in Library.foldr mk_ex (vs, conj) end
val goal = mk_trp (foldr1 mk_disj (mk_undef y :: map one_con spec'))
(* first rules replace "y = bottom \/ P" with "rep$y = bottom \/ P" *)
val tacs = [
rtac (iso_locale RS @{thm iso.casedist_rule}) 1,
rewrite_goals_tac [mk_meta_eq (iso_locale RS @{thm iso.iso_swap})],
rtac thm3 1]
in
val nchotomy = prove thy con_betas goal (K tacs)
val exhaust =
(nchotomy RS @{thm exh_casedist0})
|> rewrite_rule @{thms exh_casedists}
|> Drule.zero_var_indexes
end
(* prove compactness rules for constructors *)
val compacts =
let
val rules = @{thms compact_sinl compact_sinr compact_spair
compact_up compact_ONE}
val tacs =
[rtac (iso_locale RS @{thm iso.compact_abs}) 1,
REPEAT (resolve_tac rules 1 ORELSE atac 1)]
fun con_compact (con, args) =
let
val vs = vars_of args
val con_app = list_ccomb (con, vs)
val concl = mk_trp (mk_compact con_app)
val assms = map (mk_trp o mk_compact) vs
val goal = Logic.list_implies (assms, concl)
in
prove thy con_betas goal (K tacs)
end
in
map con_compact spec'
end
(* prove strictness rules for constructors *)
local
fun con_strict (con, args) =
let
val rules = abs_strict :: @{thms con_strict_rules}
val (vs, nonlazy) = get_vars args
fun one_strict v' =
let
val bottom = mk_bottom (fastype_of v')
val vs' = map (fn v => if v = v' then bottom else v) vs
val goal = mk_trp (mk_undef (list_ccomb (con, vs')))
val tacs = [simp_tac (HOL_basic_ss addsimps rules) 1]
in prove thy con_betas goal (K tacs) end
in map one_strict nonlazy end
fun con_defin (con, args) =
let
fun iff_disj (t, []) = HOLogic.mk_not t
| iff_disj (t, ts) = mk_eq (t, foldr1 HOLogic.mk_disj ts)
val (vs, nonlazy) = get_vars args
val lhs = mk_undef (list_ccomb (con, vs))
val rhss = map mk_undef nonlazy
val goal = mk_trp (iff_disj (lhs, rhss))
val rule1 = iso_locale RS @{thm iso.abs_bottom_iff}
val rules = rule1 :: @{thms con_bottom_iff_rules}
val tacs = [simp_tac (HOL_ss addsimps rules) 1]
in prove thy con_betas goal (K tacs) end
in
val con_stricts = maps con_strict spec'
val con_defins = map con_defin spec'
val con_rews = con_stricts @ con_defins
end
(* prove injectiveness of constructors *)
local
fun pgterm rel (con, args) =
let
fun prime (Free (n, T)) = Free (n^"'", T)
| prime t = t
val (xs, nonlazy) = get_vars args
val ys = map prime xs
val lhs = rel (list_ccomb (con, xs), list_ccomb (con, ys))
val rhs = foldr1 mk_conj (ListPair.map rel (xs, ys))
val concl = mk_trp (mk_eq (lhs, rhs))
val zs = case args of [_] => [] | _ => nonlazy
val assms = map (mk_trp o mk_defined) zs
val goal = Logic.list_implies (assms, concl)
in prove thy con_betas goal end
val cons' = filter (fn (_, args) => not (null args)) spec'
in
val inverts =
let
val abs_below = iso_locale RS @{thm iso.abs_below}
val rules1 = abs_below :: @{thms sinl_below sinr_below spair_below up_below}
val rules2 = @{thms up_defined spair_defined ONE_defined}
val rules = rules1 @ rules2
val tacs = [asm_simp_tac (simple_ss addsimps rules) 1]
in map (fn c => pgterm mk_below c (K tacs)) cons' end
val injects =
let
val abs_eq = iso_locale RS @{thm iso.abs_eq}
val rules1 = abs_eq :: @{thms sinl_eq sinr_eq spair_eq up_eq}
val rules2 = @{thms up_defined spair_defined ONE_defined}
val rules = rules1 @ rules2
val tacs = [asm_simp_tac (simple_ss addsimps rules) 1]
in map (fn c => pgterm mk_eq c (K tacs)) cons' end
end
(* prove distinctness of constructors *)
local
fun map_dist (f : 'a -> 'a -> 'b) (xs : 'a list) : 'b list =
flat (map_index (fn (i, x) => map (f x) (nth_drop i xs)) xs)
fun prime (Free (n, T)) = Free (n^"'", T)
| prime t = t
fun iff_disj (t, []) = mk_not t
| iff_disj (t, ts) = mk_eq (t, foldr1 mk_disj ts)
fun iff_disj2 (t, [], us) = mk_not t
| iff_disj2 (t, ts, []) = mk_not t
| iff_disj2 (t, ts, us) =
mk_eq (t, mk_conj (foldr1 mk_disj ts, foldr1 mk_disj us))
fun dist_le (con1, args1) (con2, args2) =
let
val (vs1, zs1) = get_vars args1
val (vs2, zs2) = get_vars args2 |> pairself (map prime)
val lhs = mk_below (list_ccomb (con1, vs1), list_ccomb (con2, vs2))
val rhss = map mk_undef zs1
val goal = mk_trp (iff_disj (lhs, rhss))
val rule1 = iso_locale RS @{thm iso.abs_below}
val rules = rule1 :: @{thms con_below_iff_rules}
val tacs = [simp_tac (HOL_ss addsimps rules) 1]
in prove thy con_betas goal (K tacs) end
fun dist_eq (con1, args1) (con2, args2) =
let
val (vs1, zs1) = get_vars args1
val (vs2, zs2) = get_vars args2 |> pairself (map prime)
val lhs = mk_eq (list_ccomb (con1, vs1), list_ccomb (con2, vs2))
val rhss1 = map mk_undef zs1
val rhss2 = map mk_undef zs2
val goal = mk_trp (iff_disj2 (lhs, rhss1, rhss2))
val rule1 = iso_locale RS @{thm iso.abs_eq}
val rules = rule1 :: @{thms con_eq_iff_rules}
val tacs = [simp_tac (HOL_ss addsimps rules) 1]
in prove thy con_betas goal (K tacs) end
in
val dist_les = map_dist dist_le spec'
val dist_eqs = map_dist dist_eq spec'
end
val result =
{
con_consts = con_consts,
con_betas = con_betas,
nchotomy = nchotomy,
exhaust = exhaust,
compacts = compacts,
con_rews = con_rews,
inverts = inverts,
injects = injects,
dist_les = dist_les,
dist_eqs = dist_eqs
}
in
(result, thy)
end
(******************************************************************************)
(**************** definition and theorems for case combinator *****************)
(******************************************************************************)
fun add_case_combinator
(spec : (term * (bool * typ) list) list)
(lhsT : typ)
(dbind : binding)
(con_betas : thm list)
(exhaust : thm)
(iso_locale : thm)
(rep_const : term)
(thy : theory)
: ((typ -> term) * thm list) * theory =
let
(* prove rep/abs rules *)
val rep_strict = iso_locale RS @{thm iso.rep_strict}
val abs_inverse = iso_locale RS @{thm iso.abs_iso}
(* calculate function arguments of case combinator *)
val tns = map fst (Term.add_tfreesT lhsT [])
val resultT = TFree (Name.variant tns "'t", @{sort pcpo})
fun fTs T = map (fn (_, args) => map snd args -->> T) spec
val fns = Datatype_Prop.indexify_names (map (K "f") spec)
val fs = map Free (fns ~~ fTs resultT)
fun caseT T = fTs T -->> (lhsT ->> T)
(* definition of case combinator *)
local
val case_bind = Binding.suffix_name "_case" dbind
fun lambda_arg (lazy, v) t =
(if lazy then mk_fup else I) (big_lambda v t)
fun lambda_args [] t = mk_one_case t
| lambda_args (x::[]) t = lambda_arg x t
| lambda_args (x::xs) t = mk_ssplit (lambda_arg x (lambda_args xs t))
fun one_con f (_, args) =
let
val Ts = map snd args
val ns = Name.variant_list fns (Datatype_Prop.make_tnames Ts)
val vs = map Free (ns ~~ Ts)
in
lambda_args (map fst args ~~ vs) (list_ccomb (f, vs))
end
fun mk_sscases [t] = mk_strictify t
| mk_sscases ts = foldr1 mk_sscase ts
val body = mk_sscases (map2 one_con fs spec)
val rhs = big_lambdas fs (mk_cfcomp (body, rep_const))
val ((case_consts, case_defs), thy) =
define_consts [(case_bind, rhs, NoSyn)] thy
val case_name = Sign.full_name thy case_bind
in
val case_def = hd case_defs
fun case_const T = Const (case_name, caseT T)
val case_app = list_ccomb (case_const resultT, fs)
val thy = thy
end
(* define syntax for case combinator *)
(* TODO: re-implement case syntax using a parse translation *)
local
fun syntax c = Syntax.mark_const (fst (dest_Const c))
fun xconst c = Long_Name.base_name (fst (dest_Const c))
fun c_ast authentic con = Ast.Constant (if authentic then syntax con else xconst con)
fun showint n = string_of_int (n+1)
fun expvar n = Ast.Variable ("e" ^ showint n)
fun argvar n (m, _) = Ast.Variable ("a" ^ showint n ^ "_" ^ showint m)
fun argvars n args = map_index (argvar n) args
fun app s (l, r) = Ast.mk_appl (Ast.Constant s) [l, r]
val cabs = app "_cabs"
val capp = app @{const_syntax Rep_cfun}
val capps = Library.foldl capp
fun con1 authentic n (con,args) =
Library.foldl capp (c_ast authentic con, argvars n args)
fun case1 authentic (n, c) =
app "_case1" (Syntax.strip_positions_ast (con1 authentic n c), expvar n)
fun arg1 (n, (con,args)) = List.foldr cabs (expvar n) (argvars n args)
fun when1 n (m, c) = if n = m then arg1 (n, c) else Ast.Constant @{const_syntax bottom}
val case_constant = Ast.Constant (syntax (case_const dummyT))
fun case_trans authentic =
(if authentic then Syntax.Parse_Print_Rule else Syntax.Parse_Rule)
(app "_case_syntax"
(Ast.Variable "x",
foldr1 (app "_case2") (map_index (case1 authentic) spec)),
capp (capps (case_constant, map_index arg1 spec), Ast.Variable "x"))
fun one_abscon_trans authentic (n, c) =
(if authentic then Syntax.Parse_Print_Rule else Syntax.Parse_Rule)
(cabs (con1 authentic n c, expvar n),
capps (case_constant, map_index (when1 n) spec))
fun abscon_trans authentic =
map_index (one_abscon_trans authentic) spec
val trans_rules : Ast.ast Syntax.trrule list =
case_trans false :: case_trans true ::
abscon_trans false @ abscon_trans true
in
val thy = Sign.add_trrules trans_rules thy
end
(* prove beta reduction rule for case combinator *)
val case_beta = beta_of_def thy case_def
(* prove strictness of case combinator *)
val case_strict =
let
val defs = case_beta :: map mk_meta_eq [rep_strict, @{thm cfcomp2}]
val goal = mk_trp (mk_strict case_app)
val rules = @{thms sscase1 ssplit1 strictify1 one_case1}
val tacs = [resolve_tac rules 1]
in prove thy defs goal (K tacs) end
(* prove rewrites for case combinator *)
local
fun one_case (con, args) f =
let
val (vs, nonlazy) = get_vars args
val assms = map (mk_trp o mk_defined) nonlazy
val lhs = case_app ` list_ccomb (con, vs)
val rhs = list_ccomb (f, vs)
val concl = mk_trp (mk_eq (lhs, rhs))
val goal = Logic.list_implies (assms, concl)
val defs = case_beta :: con_betas
val rules1 = @{thms strictify2 sscase2 sscase3 ssplit2 fup2 ID1}
val rules2 = @{thms con_bottom_iff_rules}
val rules3 = @{thms cfcomp2 one_case2}
val rules = abs_inverse :: rules1 @ rules2 @ rules3
val tacs = [asm_simp_tac (beta_ss addsimps rules) 1]
in prove thy defs goal (K tacs) end
in
val case_apps = map2 one_case spec fs
end
in
((case_const, case_strict :: case_apps), thy)
end
(******************************************************************************)
(************** definitions and theorems for selector functions ***************)
(******************************************************************************)
fun add_selectors
(spec : (term * (bool * binding option * typ) list) list)
(rep_const : term)
(abs_inv : thm)
(rep_strict : thm)
(rep_bottom_iff : thm)
(con_betas : thm list)
(thy : theory)
: thm list * theory =
let
(* define selector functions *)
val ((sel_consts, sel_defs), thy) =
let
fun rangeT s = snd (dest_cfunT (fastype_of s))
fun mk_outl s = mk_cfcomp (from_sinl (dest_ssumT (rangeT s)), s)
fun mk_outr s = mk_cfcomp (from_sinr (dest_ssumT (rangeT s)), s)
fun mk_sfst s = mk_cfcomp (sfst_const (dest_sprodT (rangeT s)), s)
fun mk_ssnd s = mk_cfcomp (ssnd_const (dest_sprodT (rangeT s)), s)
fun mk_down s = mk_cfcomp (from_up (dest_upT (rangeT s)), s)
fun sels_of_arg s (lazy, NONE, T) = []
| sels_of_arg s (lazy, SOME b, T) =
[(b, if lazy then mk_down s else s, NoSyn)]
fun sels_of_args s [] = []
| sels_of_args s (v :: []) = sels_of_arg s v
| sels_of_args s (v :: vs) =
sels_of_arg (mk_sfst s) v @ sels_of_args (mk_ssnd s) vs
fun sels_of_cons s [] = []
| sels_of_cons s ((con, args) :: []) = sels_of_args s args
| sels_of_cons s ((con, args) :: cs) =
sels_of_args (mk_outl s) args @ sels_of_cons (mk_outr s) cs
val sel_eqns : (binding * term * mixfix) list =
sels_of_cons rep_const spec
in
define_consts sel_eqns thy
end
(* replace bindings with terms in constructor spec *)
val spec2 : (term * (bool * term option * typ) list) list =
let
fun prep_arg (lazy, NONE, T) sels = ((lazy, NONE, T), sels)
| prep_arg (lazy, SOME _, T) sels =
((lazy, SOME (hd sels), T), tl sels)
fun prep_con (con, args) sels =
apfst (pair con) (fold_map prep_arg args sels)
in
fst (fold_map prep_con spec sel_consts)
end
(* prove selector strictness rules *)
val sel_stricts : thm list =
let
val rules = rep_strict :: @{thms sel_strict_rules}
val tacs = [simp_tac (HOL_basic_ss addsimps rules) 1]
fun sel_strict sel =
let
val goal = mk_trp (mk_strict sel)
in
prove thy sel_defs goal (K tacs)
end
in
map sel_strict sel_consts
end
(* prove selector application rules *)
val sel_apps : thm list =
let
val defs = con_betas @ sel_defs
val rules = abs_inv :: @{thms sel_app_rules}
val tacs = [asm_simp_tac (simple_ss addsimps rules) 1]
fun sel_apps_of (i, (con, args: (bool * term option * typ) list)) =
let
val Ts : typ list = map #3 args
val ns : string list = Datatype_Prop.make_tnames Ts
val vs : term list = map Free (ns ~~ Ts)
val con_app : term = list_ccomb (con, vs)
val vs' : (bool * term) list = map #1 args ~~ vs
fun one_same (n, sel, T) =
let
val xs = map snd (filter_out fst (nth_drop n vs'))
val assms = map (mk_trp o mk_defined) xs
val concl = mk_trp (mk_eq (sel ` con_app, nth vs n))
val goal = Logic.list_implies (assms, concl)
in
prove thy defs goal (K tacs)
end
fun one_diff (n, sel, T) =
let
val goal = mk_trp (mk_eq (sel ` con_app, mk_bottom T))
in
prove thy defs goal (K tacs)
end
fun one_con (j, (_, args')) : thm list =
let
fun prep (i, (lazy, NONE, T)) = NONE
| prep (i, (lazy, SOME sel, T)) = SOME (i, sel, T)
val sels : (int * term * typ) list =
map_filter prep (map_index I args')
in
if i = j
then map one_same sels
else map one_diff sels
end
in
flat (map_index one_con spec2)
end
in
flat (map_index sel_apps_of spec2)
end
(* prove selector definedness rules *)
val sel_defins : thm list =
let
val rules = rep_bottom_iff :: @{thms sel_bottom_iff_rules}
val tacs = [simp_tac (HOL_basic_ss addsimps rules) 1]
fun sel_defin sel =
let
val (T, U) = dest_cfunT (fastype_of sel)
val x = Free ("x", T)
val lhs = mk_eq (sel ` x, mk_bottom U)
val rhs = mk_eq (x, mk_bottom T)
val goal = mk_trp (mk_eq (lhs, rhs))
in
prove thy sel_defs goal (K tacs)
end
fun one_arg (false, SOME sel, T) = SOME (sel_defin sel)
| one_arg _ = NONE
in
case spec2 of
[(con, args)] => map_filter one_arg args
| _ => []
end
in
(sel_stricts @ sel_defins @ sel_apps, thy)
end
(******************************************************************************)
(************ definitions and theorems for discriminator functions ************)
(******************************************************************************)
fun add_discriminators
(bindings : binding list)
(spec : (term * (bool * typ) list) list)
(lhsT : typ)
(exhaust : thm)
(case_const : typ -> term)
(case_rews : thm list)
(thy : theory) =
let
fun vars_of args =
let
val Ts = map snd args
val ns = Datatype_Prop.make_tnames Ts
in
map Free (ns ~~ Ts)
end
(* define discriminator functions *)
local
fun dis_fun i (j, (con, args)) =
let
val (vs, nonlazy) = get_vars args
val tr = if i = j then @{term TT} else @{term FF}
in
big_lambdas vs tr
end
fun dis_eqn (i, bind) : binding * term * mixfix =
let
val dis_bind = Binding.prefix_name "is_" bind
val rhs = list_ccomb (case_const trT, map_index (dis_fun i) spec)
in
(dis_bind, rhs, NoSyn)
end
in
val ((dis_consts, dis_defs), thy) =
define_consts (map_index dis_eqn bindings) thy
end
(* prove discriminator strictness rules *)
local
fun dis_strict dis =
let val goal = mk_trp (mk_strict dis)
in prove thy dis_defs goal (K [rtac (hd case_rews) 1]) end
in
val dis_stricts = map dis_strict dis_consts
end
(* prove discriminator/constructor rules *)
local
fun dis_app (i, dis) (j, (con, args)) =
let
val (vs, nonlazy) = get_vars args
val lhs = dis ` list_ccomb (con, vs)
val rhs = if i = j then @{term TT} else @{term FF}
val assms = map (mk_trp o mk_defined) nonlazy
val concl = mk_trp (mk_eq (lhs, rhs))
val goal = Logic.list_implies (assms, concl)
val tacs = [asm_simp_tac (beta_ss addsimps case_rews) 1]
in prove thy dis_defs goal (K tacs) end
fun one_dis (i, dis) =
map_index (dis_app (i, dis)) spec
in
val dis_apps = flat (map_index one_dis dis_consts)
end
(* prove discriminator definedness rules *)
local
fun dis_defin dis =
let
val x = Free ("x", lhsT)
val simps = dis_apps @ @{thms dist_eq_tr}
val tacs =
[rtac @{thm iffI} 1,
asm_simp_tac (HOL_basic_ss addsimps dis_stricts) 2,
rtac exhaust 1, atac 1,
DETERM_UNTIL_SOLVED (CHANGED
(asm_full_simp_tac (simple_ss addsimps simps) 1))]
val goal = mk_trp (mk_eq (mk_undef (dis ` x), mk_undef x))
in prove thy [] goal (K tacs) end
in
val dis_defins = map dis_defin dis_consts
end
in
(dis_stricts @ dis_defins @ dis_apps, thy)
end
(******************************************************************************)
(*************** definitions and theorems for match combinators ***************)
(******************************************************************************)
fun add_match_combinators
(bindings : binding list)
(spec : (term * (bool * typ) list) list)
(lhsT : typ)
(exhaust : thm)
(case_const : typ -> term)
(case_rews : thm list)
(thy : theory) =
let
(* get a fresh type variable for the result type *)
val resultT : typ =
let
val ts : string list = map fst (Term.add_tfreesT lhsT [])
val t : string = Name.variant ts "'t"
in TFree (t, @{sort pcpo}) end
(* define match combinators *)
local
val x = Free ("x", lhsT)
fun k args = Free ("k", map snd args -->> mk_matchT resultT)
val fail = mk_fail resultT
fun mat_fun i (j, (con, args)) =
let
val (vs, nonlazy) = get_vars_avoiding ["x","k"] args
in
if i = j then k args else big_lambdas vs fail
end
fun mat_eqn (i, (bind, (con, args))) : binding * term * mixfix =
let
val mat_bind = Binding.prefix_name "match_" bind
val funs = map_index (mat_fun i) spec
val body = list_ccomb (case_const (mk_matchT resultT), funs)
val rhs = big_lambda x (big_lambda (k args) (body ` x))
in
(mat_bind, rhs, NoSyn)
end
in
val ((match_consts, match_defs), thy) =
define_consts (map_index mat_eqn (bindings ~~ spec)) thy
end
(* register match combinators with fixrec package *)
local
val con_names = map (fst o dest_Const o fst) spec
val mat_names = map (fst o dest_Const) match_consts
in
val thy = Fixrec.add_matchers (con_names ~~ mat_names) thy
end
(* prove strictness of match combinators *)
local
fun match_strict mat =
let
val (T, (U, V)) = apsnd dest_cfunT (dest_cfunT (fastype_of mat))
val k = Free ("k", U)
val goal = mk_trp (mk_eq (mat ` mk_bottom T ` k, mk_bottom V))
val tacs = [asm_simp_tac (beta_ss addsimps case_rews) 1]
in prove thy match_defs goal (K tacs) end
in
val match_stricts = map match_strict match_consts
end
(* prove match/constructor rules *)
local
val fail = mk_fail resultT
fun match_app (i, mat) (j, (con, args)) =
let
val (vs, nonlazy) = get_vars_avoiding ["k"] args
val (_, (kT, _)) = apsnd dest_cfunT (dest_cfunT (fastype_of mat))
val k = Free ("k", kT)
val lhs = mat ` list_ccomb (con, vs) ` k
val rhs = if i = j then list_ccomb (k, vs) else fail
val assms = map (mk_trp o mk_defined) nonlazy
val concl = mk_trp (mk_eq (lhs, rhs))
val goal = Logic.list_implies (assms, concl)
val tacs = [asm_simp_tac (beta_ss addsimps case_rews) 1]
in prove thy match_defs goal (K tacs) end
fun one_match (i, mat) =
map_index (match_app (i, mat)) spec
in
val match_apps = flat (map_index one_match match_consts)
end
in
(match_stricts @ match_apps, thy)
end
(******************************************************************************)
(******************************* main function ********************************)
(******************************************************************************)
fun add_domain_constructors
(dbind : binding)
(spec : (binding * (bool * binding option * typ) list * mixfix) list)
(iso_info : Domain_Take_Proofs.iso_info)
(thy : theory) =
let
val dname = Binding.name_of dbind
val _ = writeln ("Proving isomorphism properties of domain "^dname^" ...")
val bindings = map #1 spec
(* retrieve facts about rep/abs *)
val lhsT = #absT iso_info
val {rep_const, abs_const, ...} = iso_info
val abs_iso_thm = #abs_inverse iso_info
val rep_iso_thm = #rep_inverse iso_info
val iso_locale = @{thm iso.intro} OF [abs_iso_thm, rep_iso_thm]
val rep_strict = iso_locale RS @{thm iso.rep_strict}
val abs_strict = iso_locale RS @{thm iso.abs_strict}
val rep_bottom_iff = iso_locale RS @{thm iso.rep_bottom_iff}
val abs_bottom_iff = iso_locale RS @{thm iso.abs_bottom_iff}
val iso_rews = [abs_iso_thm, rep_iso_thm, abs_strict, rep_strict]
(* qualify constants and theorems with domain name *)
val thy = Sign.add_path dname thy
(* define constructor functions *)
val (con_result, thy) =
let
fun prep_arg (lazy, sel, T) = (lazy, T)
fun prep_con (b, args, mx) = (b, map prep_arg args, mx)
val con_spec = map prep_con spec
in
add_constructors con_spec abs_const iso_locale thy
end
val {con_consts, con_betas, nchotomy, exhaust, compacts, con_rews,
inverts, injects, dist_les, dist_eqs} = con_result
(* prepare constructor spec *)
val con_specs : (term * (bool * typ) list) list =
let
fun prep_arg (lazy, sel, T) = (lazy, T)
fun prep_con c (b, args, mx) = (c, map prep_arg args)
in
map2 prep_con con_consts spec
end
(* define case combinator *)
val ((case_const : typ -> term, cases : thm list), thy) =
add_case_combinator con_specs lhsT dbind
con_betas exhaust iso_locale rep_const thy
(* define and prove theorems for selector functions *)
val (sel_thms : thm list, thy : theory) =
let
val sel_spec : (term * (bool * binding option * typ) list) list =
map2 (fn con => fn (b, args, mx) => (con, args)) con_consts spec
in
add_selectors sel_spec rep_const
abs_iso_thm rep_strict rep_bottom_iff con_betas thy
end
(* define and prove theorems for discriminator functions *)
val (dis_thms : thm list, thy : theory) =
add_discriminators bindings con_specs lhsT
exhaust case_const cases thy
(* define and prove theorems for match combinators *)
val (match_thms : thm list, thy : theory) =
add_match_combinators bindings con_specs lhsT
exhaust case_const cases thy
(* restore original signature path *)
val thy = Sign.parent_path thy
(* bind theorem names in global theory *)
val (_, thy) =
let
fun qualified name = Binding.qualified true name dbind
val names = "bottom" :: map (fn (b,_,_) => Binding.name_of b) spec
val dname = fst (dest_Type lhsT)
val simp = Simplifier.simp_add
val case_names = Rule_Cases.case_names names
val cases_type = Induct.cases_type dname
in
Global_Theory.add_thmss [
((qualified "iso_rews" , iso_rews ), [simp]),
((qualified "nchotomy" , [nchotomy] ), []),
((qualified "exhaust" , [exhaust] ), [case_names, cases_type]),
((qualified "case_rews" , cases ), [simp]),
((qualified "compacts" , compacts ), [simp]),
((qualified "con_rews" , con_rews ), [simp]),
((qualified "sel_rews" , sel_thms ), [simp]),
((qualified "dis_rews" , dis_thms ), [simp]),
((qualified "dist_les" , dist_les ), [simp]),
((qualified "dist_eqs" , dist_eqs ), [simp]),
((qualified "inverts" , inverts ), [simp]),
((qualified "injects" , injects ), [simp]),
((qualified "match_rews", match_thms ), [simp])] thy
end
val result =
{
iso_info = iso_info,
con_specs = con_specs,
con_betas = con_betas,
nchotomy = nchotomy,
exhaust = exhaust,
compacts = compacts,
con_rews = con_rews,
inverts = inverts,
injects = injects,
dist_les = dist_les,
dist_eqs = dist_eqs,
cases = cases,
sel_rews = sel_thms,
dis_rews = dis_thms,
match_rews = match_thms
}
in
(result, thy)
end
end