(* Title: HOL/HOLCF/Tools/fixrec.ML
Author: Amber Telfer and Brian Huffman
Recursive function definition package for HOLCF.
*)
signature FIXREC =
sig
val add_fixrec: (binding * typ option * mixfix) list
-> (bool * (Attrib.binding * term)) list -> local_theory -> local_theory
val add_fixrec_cmd: (binding * string option * mixfix) list
-> (bool * (Attrib.binding * string)) list -> local_theory -> local_theory
val add_matchers: (string * string) list -> theory -> theory
val fixrec_simp_tac: Proof.context -> int -> tactic
end
structure Fixrec : FIXREC =
struct
open HOLCF_Library
infixr 6 ->>
infix -->>
infix 9 `
val def_cont_fix_eq = @{thm def_cont_fix_eq}
val def_cont_fix_ind = @{thm def_cont_fix_ind}
fun fixrec_err s = error ("fixrec definition error:\n" ^ s)
(*************************************************************************)
(***************************** building types ****************************)
(*************************************************************************)
local
fun binder_cfun (Type(@{type_name cfun},[T, U])) = T :: binder_cfun U
| binder_cfun (Type(@{type_name "fun"},[T, U])) = T :: binder_cfun U
| binder_cfun _ = []
fun body_cfun (Type(@{type_name cfun},[_, U])) = body_cfun U
| body_cfun (Type(@{type_name "fun"},[_, U])) = body_cfun U
| body_cfun T = T
in
fun matcherT (T, U) =
body_cfun T ->> (binder_cfun T -->> U) ->> U
end
(*************************************************************************)
(***************************** building terms ****************************)
(*************************************************************************)
val mk_trp = HOLogic.mk_Trueprop
(* splits a cterm into the right and lefthand sides of equality *)
fun dest_eqs t = HOLogic.dest_eq (HOLogic.dest_Trueprop t)
(* similar to Thm.head_of, but for continuous application *)
fun chead_of (Const(@{const_name Rep_cfun},_)$f$_) = chead_of f
| chead_of u = u
infix 1 === val (op ===) = HOLogic.mk_eq
fun mk_mplus (t, u) =
let val mT = Term.fastype_of t
in Const(@{const_name Fixrec.mplus}, mT ->> mT ->> mT) ` t ` u end
fun mk_run t =
let
val mT = Term.fastype_of t
val T = dest_matchT mT
val run = Const(@{const_name Fixrec.run}, mT ->> T)
in
case t of
Const(@{const_name Rep_cfun}, _) $
Const(@{const_name Fixrec.succeed}, _) $ u => u
| _ => run ` t
end
(*************************************************************************)
(************* fixed-point definitions and unfolding theorems ************)
(*************************************************************************)
structure FixrecUnfoldData = Generic_Data
(
type T = thm Symtab.table
val empty = Symtab.empty
val extend = I
fun merge data : T = Symtab.merge (K true) data
)
local
fun name_of (Const (n, _)) = n
| name_of (Free (n, _)) = n
| name_of t = raise TERM ("Fixrec.add_unfold: lhs not a constant", [t])
val lhs_name =
name_of o head_of o fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o Thm.prop_of
in
val add_unfold : attribute =
Thm.declaration_attribute
(fn th => FixrecUnfoldData.map (Symtab.insert (K true) (lhs_name th, th)))
end
fun add_fixdefs
(fixes : ((binding * typ) * mixfix) list)
(spec : (Attrib.binding * term) list)
(lthy : local_theory) =
let
val thy = Proof_Context.theory_of lthy
val names = map (Binding.name_of o fst o fst) fixes
val all_names = space_implode "_" names
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)
val cont_thm =
let
val prop = mk_trp (mk_cont functional)
fun err _ = error (
"Continuity proof failed please check that cont2cont rules\n" ^
"or simp rules are configured for all non-HOLCF constants.\n" ^
"The error occurred for the goal statement:\n" ^
Syntax.string_of_term lthy prop)
val rules = Named_Theorems.get lthy @{named_theorems cont2cont}
val fast_tac = SOLVED' (REPEAT_ALL_NEW (match_tac lthy (rev rules)))
val slow_tac = SOLVED' (simp_tac lthy)
val tac = fast_tac 1 ORELSE slow_tac 1 ORELSE err
in
Goal.prove lthy [] [] prop (K tac)
end
fun one_def (Free(n,_)) r =
let val b = Long_Name.base_name n
in ((Binding.name (Thm.def_name b), []), r) end
| one_def _ _ = fixrec_err "fixdefs: lhs not of correct form"
fun defs [] _ = []
| defs (l::[]) r = [one_def l r]
| defs (l::ls) r = one_def l (mk_fst r) :: defs ls (mk_snd r)
val fixdefs = defs lhss fixpoint
val (fixdef_thms : (term * (string * thm)) list, lthy) = lthy
|> fold_map Local_Theory.define (map (apfst fst) fixes ~~ fixdefs)
fun pair_equalI (thm1, thm2) = @{thm Pair_equalI} OF [thm1, thm2]
val tuple_fixdef_thm = foldr1 pair_equalI (map (snd o snd) fixdef_thms)
val P = Var (("P", 0), map Term.fastype_of lhss ---> HOLogic.boolT)
val predicate = lambda_tuple lhss (list_comb (P, lhss))
val tuple_induct_thm = (def_cont_fix_ind OF [tuple_fixdef_thm, cont_thm])
|> Drule.instantiate' [] [SOME (Thm.cterm_of thy predicate)]
|> Local_Defs.unfold lthy @{thms split_paired_all split_conv split_strict}
val tuple_unfold_thm = (def_cont_fix_eq OF [tuple_fixdef_thm, cont_thm])
|> Local_Defs.unfold lthy @{thms split_conv}
fun unfolds [] _ = []
| unfolds (n::[]) thm = [(n, thm)]
| unfolds (n::ns) thm = let
val thmL = thm RS @{thm Pair_eqD1}
val thmR = thm RS @{thm Pair_eqD2}
in (n, thmL) :: unfolds ns thmR end
val unfold_thms = unfolds names tuple_unfold_thm
val induct_note : Attrib.binding * Thm.thm list =
let
val thm_name = Binding.qualify true all_names (Binding.name "induct")
in
((thm_name, []), [tuple_induct_thm])
end
fun unfold_note (name, thm) : Attrib.binding * Thm.thm list =
let
val thm_name = Binding.qualify true name (Binding.name "unfold")
val src = Attrib.internal (K add_unfold)
in
((thm_name, [src]), [thm])
end
val (_, lthy) = lthy
|> fold_map Local_Theory.note (induct_note :: map unfold_note unfold_thms)
in
(lthy, names, fixdef_thms, map snd unfold_thms)
end
(*************************************************************************)
(*********** monadic notation and pattern matching compilation ***********)
(*************************************************************************)
structure FixrecMatchData = Theory_Data
(
type T = string Symtab.table
val empty = Symtab.empty
val extend = I
fun merge data = Symtab.merge (K true) data
)
(* associate match functions with pattern constants *)
fun add_matchers ms = FixrecMatchData.map (fold Symtab.update ms)
fun taken_names (t : term) : bstring list =
let
fun taken (Const(a,_), bs) = insert (op =) (Long_Name.base_name a) bs
| taken (Free(a,_) , bs) = insert (op =) a bs
| taken (f $ u , bs) = taken (f, taken (u, bs))
| taken (Abs(a,_,t), bs) = taken (t, insert (op =) a bs)
| taken (_ , bs) = bs
in
taken (t, [])
end
(* builds a monadic term for matching a pattern *)
(* returns (rhs, free variable, used varnames) *)
fun compile_pat match_name pat rhs taken =
let
fun comp_pat p rhs taken =
if is_Free p then (rhs, p, taken)
else comp_con (fastype_of p) p rhs [] taken
(* compiles a monadic term for a constructor pattern *)
and comp_con T p rhs vs taken =
case p of
Const(@{const_name Rep_cfun},_) $ f $ x =>
let val (rhs', v, taken') = comp_pat x rhs taken
in comp_con T f rhs' (v::vs) taken' end
| f $ x =>
let val (rhs', v, taken') = comp_pat x rhs taken
in comp_con T f rhs' (v::vs) taken' end
| Const (c, cT) =>
let
val n = singleton (Name.variant_list taken) "v"
val v = Free(n, T)
val m = Const(match_name c, matcherT (cT, fastype_of rhs))
val k = big_lambdas vs rhs
in
(m`v`k, v, n::taken)
end
| _ => raise TERM ("fixrec: invalid pattern ", [p])
in
comp_pat pat rhs taken
end
(* builds a monadic term for matching a function definition pattern *)
(* returns (constant, (vars, matcher)) *)
fun compile_lhs match_name pat rhs vs taken =
case pat of
Const(@{const_name Rep_cfun}, _) $ f $ x =>
let val (rhs', v, taken') = compile_pat match_name x rhs taken
in compile_lhs match_name f rhs' (v::vs) taken' end
| Free(_,_) => (pat, (vs, rhs))
| Const(_,_) => (pat, (vs, rhs))
| _ => fixrec_err ("invalid function pattern: "
^ ML_Syntax.print_term pat)
fun strip_alls t =
(case try Logic.dest_all t of
SOME (_, u) => strip_alls u
| NONE => t)
fun compile_eq match_name eq =
let
val (lhs,rhs) = dest_eqs (Logic.strip_imp_concl (strip_alls eq))
in
compile_lhs match_name lhs (mk_succeed rhs) [] (taken_names eq)
end
(* this is the pattern-matching compiler function *)
fun compile_eqs match_name eqs =
let
val (consts, matchers) =
ListPair.unzip (map (compile_eq match_name) eqs)
val const =
case distinct (op =) consts of
[n] => n
| [] => fixrec_err "no defining equations for function"
| _ => fixrec_err "all equations in block must define the same function"
val vars =
case distinct (op = o apply2 length) (map fst matchers) of
[vars] => vars
| _ => fixrec_err "all equations in block must have the same arity"
(* rename so all matchers use same free variables *)
fun rename (vs, t) = Term.subst_free (filter_out (op =) (vs ~~ vars)) t
val rhs = big_lambdas vars (mk_run (foldr1 mk_mplus (map rename matchers)))
in
mk_trp (const === rhs)
end
(*************************************************************************)
(********************** Proving associated theorems **********************)
(*************************************************************************)
fun eta_tac i = CONVERSION Thm.eta_conversion i
fun fixrec_simp_tac ctxt =
let
val tab = FixrecUnfoldData.get (Context.Proof ctxt)
val concl = HOLogic.dest_Trueprop o Logic.strip_imp_concl o strip_alls
fun tac (t, i) =
let
val (c, _) =
(dest_Const o head_of o chead_of o fst o HOLogic.dest_eq o concl) t
val unfold_thm = the (Symtab.lookup tab c)
val rule = unfold_thm RS @{thm ssubst_lhs}
in
CHANGED (rtac rule i THEN eta_tac i THEN asm_simp_tac ctxt i)
end
in
SUBGOAL (fn ti => the_default no_tac (try tac ti))
end
(* proves a block of pattern matching equations as theorems, using unfold *)
fun make_simps ctxt (unfold_thm, eqns : (Attrib.binding * term) list) =
let
val rule = unfold_thm RS @{thm ssubst_lhs}
val tac = rtac rule 1 THEN eta_tac 1 THEN asm_simp_tac ctxt 1
fun prove_term t = Goal.prove ctxt [] [] t (K tac)
fun prove_eqn (bind, eqn_t) = (bind, prove_term eqn_t)
in
map prove_eqn eqns
end
(*************************************************************************)
(************************* Main fixrec function **************************)
(*************************************************************************)
local
(* code adapted from HOL/Tools/Datatype/primrec.ML *)
fun gen_fixrec
prep_spec
(raw_fixes : (binding * 'a option * mixfix) list)
(raw_spec' : (bool * (Attrib.binding * 'b)) list)
(lthy : local_theory) =
let
val (skips, raw_spec) = ListPair.unzip raw_spec'
val (fixes : ((binding * typ) * mixfix) list,
spec : (Attrib.binding * term) list) =
fst (prep_spec raw_fixes raw_spec lthy)
val names = map (Binding.name_of o fst o fst) fixes
fun check_head name =
member (op =) names name orelse
fixrec_err ("Illegal equation head. Expected " ^ commas_quote names)
val chead_of_spec =
chead_of o fst o dest_eqs o Logic.strip_imp_concl o strip_alls o snd
fun name_of (Free (n, _)) = tap check_head n
| name_of _ = fixrec_err ("unknown term")
val all_names = map (name_of o chead_of_spec) spec
fun block_of_name n =
map_filter
(fn (m,eq) => if m = n then SOME eq else NONE)
(all_names ~~ (spec ~~ skips))
val blocks = map block_of_name names
val matcher_tab = FixrecMatchData.get (Proof_Context.theory_of lthy)
fun match_name c =
case Symtab.lookup matcher_tab c of SOME m => m
| NONE => fixrec_err ("unknown pattern constructor: " ^ c)
val matches = map (compile_eqs match_name) (map (map (snd o fst)) blocks)
val spec' = map (pair Attrib.empty_binding) matches
val (lthy, _, _, unfold_thms) =
add_fixdefs fixes spec' lthy
val blocks' = map (map fst o filter_out snd) blocks
val simps : (Attrib.binding * thm) list list =
map (make_simps lthy) (unfold_thms ~~ blocks')
fun mk_bind n : Attrib.binding =
(Binding.qualify true n (Binding.name "simps"), @{attributes [simp]})
val simps1 : (Attrib.binding * thm list) list =
map (fn (n,xs) => (mk_bind n, map snd xs)) (names ~~ simps)
val simps2 : (Attrib.binding * thm list) list =
map (apsnd (fn thm => [thm])) (flat simps)
val (_, lthy) = lthy
|> fold_map Local_Theory.note (simps1 @ simps2)
in
lthy
end
in
val add_fixrec = gen_fixrec Specification.check_spec
val add_fixrec_cmd = gen_fixrec Specification.read_spec
end (* local *)
(*************************************************************************)
(******************************** Parsers ********************************)
(*************************************************************************)
val opt_thm_name' : (bool * Attrib.binding) parser =
@{keyword "("} -- @{keyword "unchecked"} -- @{keyword ")"} >> K (true, Attrib.empty_binding)
|| Parse_Spec.opt_thm_name ":" >> pair false
val spec' : (bool * (Attrib.binding * string)) parser =
opt_thm_name' -- Parse.prop >> (fn ((a, b), c) => (a, (b, c)))
val alt_specs' : (bool * (Attrib.binding * string)) list parser =
let val unexpected = Scan.ahead (Parse.name || @{keyword "["} || @{keyword "("})
in Parse.enum1 "|" (spec' --| Scan.option (unexpected -- Parse.!!! @{keyword "|"})) end
val _ =
Outer_Syntax.local_theory @{command_spec "fixrec"} "define recursive functions (HOLCF)"
(Parse.fixes -- (Parse.where_ |-- Parse.!!! alt_specs')
>> (fn (fixes, specs) => add_fixrec_cmd fixes specs))
end