signature TIMING_FUNCTIONS =
sig
type 'a wctxt = {
ctxt: local_theory,
origins: term list,
f: term -> 'a
}
type 'a converter = {
constc : 'a wctxt -> term -> 'a,
funcc : 'a wctxt -> term -> term list -> 'a,
ifc : 'a wctxt -> typ -> term -> term -> term -> 'a,
casec : 'a wctxt -> term -> term list -> 'a,
letc : 'a wctxt -> typ -> term -> (string * typ) list -> term -> 'a
}
val walk : local_theory -> term list -> 'a converter -> term -> 'a
val Iconst : term wctxt -> term -> term
val Ifunc : term wctxt -> term -> term list -> term
val Iif : term wctxt -> typ -> term -> term -> term -> term
val Icase : term wctxt -> term -> term list -> term
val Ilet : term wctxt -> typ -> term -> (string * typ) list -> term -> term
type pfunc = { names : string list, terms : term list, typs : typ list }
val fun_pretty': Proof.context -> pfunc -> Pretty.T
val fun_pretty: Proof.context -> Function.info -> Pretty.T
val print_timing': Proof.context -> pfunc -> pfunc -> unit
val print_timing: Proof.context -> Function.info -> Function.info -> unit
type time_config = {
print: bool,
simp: bool,
partial: bool
}
datatype result = Function of Function.info | PartialFunction of thm
val reg_and_proove_time_func: local_theory -> term list -> term list
-> time_config -> result * local_theory
val reg_time_func: local_theory -> term list -> term list
-> time_config -> result * local_theory
val time_dom_tac: Proof.context -> thm -> thm list -> int -> tactic
end
structure Timing_Functions : TIMING_FUNCTIONS =
struct
(* Configure config variable to adjust the prefix *)
val bprefix = Attrib.setup_config_string @{binding "time_prefix"} (K "T_")
val bprefix_snd = Attrib.setup_config_string @{binding "time_prefix_snd"} (K "T2_")
(* Configure config variable to adjust the suffix *)
val bsuffix = Attrib.setup_config_string @{binding "time_suffix"} (K "")
(* Extracts terms from function info *)
fun terms_of_info (info: Function.info) =
map Thm.prop_of (case #simps info of SOME s => s
| NONE => error "No terms of function found in info")
type pfunc = {
names : string list,
terms : term list,
typs : typ list
}
fun info_pfunc (info: Function.info): pfunc =
let
val {defname, fs, ...} = info;
val T = case hd fs of (Const (_,T)) => T
| (Free (_,T)) => T
| _ => error "Internal error: Invalid info to print"
in
{ names=[Binding.name_of defname], terms=terms_of_info info, typs=[T] }
end
(* Auxiliary functions for printing functions *)
fun fun_pretty' ctxt (pfunc: pfunc) =
let
val {names, terms, typs} = pfunc;
val header_beg = Pretty.str "fun ";
fun prepHeadCont (nm,T) = [Pretty.str (nm ^ " :: "), (Pretty.quote (Syntax.pretty_typ ctxt T))]
val header_content =
List.concat (prepHeadCont (hd names,hd typs) :: map ((fn l => Pretty.str "\nand " :: l) o prepHeadCont) (ListPair.zip (tl names, tl typs)));
val header_end = Pretty.str " where\n ";
val header = [header_beg] @ header_content @ [header_end];
fun separate sep prts =
flat (Library.separate [Pretty.str sep] (map single prts));
val ptrms = (separate "\n| " (map (Syntax.pretty_term ctxt) terms));
in
Pretty.text_fold (header @ ptrms)
end
fun fun_pretty ctxt = fun_pretty' ctxt o info_pfunc
fun print_timing' ctxt (opfunc: pfunc) (tpfunc: pfunc) =
let
val {names, ...} = opfunc;
val poriginal = Pretty.item [Pretty.str "Original function:\n", fun_pretty' ctxt opfunc]
val ptiming = Pretty.item [Pretty.str ("Running time function:\n"), fun_pretty' ctxt tpfunc]
in
Pretty.writeln (Pretty.text_fold [
Pretty.str ("Converting " ^ (hd names) ^ (String.concat (map (fn nm => ", " ^ nm) (tl names))) ^ "\n"),
poriginal, Pretty.str "\n", ptiming])
end
fun print_timing ctxt (oinfo: Function.info) (tinfo: Function.info) =
print_timing' ctxt (info_pfunc oinfo) (info_pfunc tinfo)
fun print_lemma ctxt defs (T_terms: term list) =
let
val names =
defs
|> map snd
|> map (fn s => "_" ^ s)
|> List.foldr (op ^) ""
val begin = "lemma T" ^ names ^ "_simps [simp,code]:\n"
fun convLine T_term =
" \"" ^ Syntax.string_of_term ctxt T_term ^ "\"\n"
val lines = map convLine T_terms
fun convDefs def = " " ^ (fst def)
val proof = " by (simp_all add:" :: (map convDefs defs) @ [")"]
val _ = Pretty.writeln (Pretty.str "Characteristic recursion equations can be derived:")
in
(begin :: lines @ proof)
|> String.concat
(* |> Active.sendback_markup_properties [Markup.padding_fun] *)
|> Pretty.str
|> Pretty.writeln
end
fun contains l e = exists (fn e' => e' = e) l
fun contains' comp l e = exists (comp e) l
(* Split name by . *)
val split_name = String.fields (fn s => s = #".")
(* returns true if it's an if term *)
fun is_if (Const (@{const_name "HOL.If"},_)) = true
| is_if _ = false
(* returns true if it's a case term *)
fun is_case (Const (n,_)) = n |> split_name |> List.last |> String.isPrefix "case_"
| is_case _ = false
(* returns true if it's a let term *)
fun is_let (Const (@{const_name "HOL.Let"},_)) = true
| is_let _ = false
(* Convert string name of function to its timing equivalent *)
fun fun_name_to_time' ctxt s second name =
let
val prefix = Config.get ctxt (if second then bprefix_snd else bprefix)
val suffix = (if s then Config.get ctxt bsuffix else "")
fun replace_last_name [n] = [prefix ^ n ^ suffix]
| replace_last_name (n::ns) = n :: (replace_last_name ns)
| replace_last_name _ = error "Internal error: Invalid function name to convert"
val parts = split_name name
in
String.concatWith "." (replace_last_name parts)
end
fun fun_name_to_time ctxt s name = fun_name_to_time' ctxt s false name
(* Count number of arguments of a function *)
fun count_args (Type (n, [_,res])) = (if n = "fun" then 1 + count_args res else 0)
| count_args _ = 0
(* Check if number of arguments matches function *)
fun check_args s (t, args) =
(if length args = count_args (type_of t) then ()
else error ("Partial applications/Lambdas not allowed (" ^ s ^ ")"))
(* Removes Abs *)
fun rem_abs f (Abs (_,_,t)) = rem_abs f t
| rem_abs f t = f t
(* Map right side of equation *)
fun map_r f (pT $ (eq $ l $ r)) = (pT $ (eq $ l $ f r))
| map_r _ _ = error "Internal error: No right side of equation found"
(* Get left side of equation *)
fun get_l (_ $ (_ $ l $ _)) = l
| get_l _ = error "Internal error: No left side of equation found"
(* Get right side of equation *)
fun get_r (_ $ (_ $ _ $ r)) = r
| get_r _ = error "Internal error: No right side of equation found"
(* Return name of Const *)
fun Const_name (Const (nm,_)) = SOME nm
| Const_name _ = NONE
fun is_Used (Type ("Product_Type.prod", _)) = true
| is_Used _ = false
(* Custom compare function for types ignoring variable names *)
fun typ_comp (Type (A,a)) (Type (B,b)) = (A = B) andalso List.foldl (fn ((c,i),s) => typ_comp c i andalso s) true (ListPair.zip (a, b))
| typ_comp (Type _) _ = false
| typ_comp _ (Type _) = false
| typ_comp _ _ = true
fun const_comp (Const (nm,T)) (Const (nm',T')) = nm = nm' andalso typ_comp T T'
| const_comp _ _ = false
type 'a wctxt = {
ctxt: local_theory,
origins: term list,
f: term -> 'a
}
type 'a converter = {
constc : 'a wctxt -> term -> 'a,
funcc : 'a wctxt -> term -> term list -> 'a,
ifc : 'a wctxt -> typ -> term -> term -> term -> 'a,
casec : 'a wctxt -> term -> term list -> 'a,
letc : 'a wctxt -> typ -> term -> (string * typ) list -> term -> 'a
}
(* Walks over term and calls given converter *)
fun list_abs ([], t) = t
| list_abs (a::abs,t) = list_abs (abs,t) |> absfree a
fun walk ctxt (origin: term list) (conv as {ifc, casec, funcc, letc, ...} : 'a converter) (t as _ $ _) =
let
val (f, args) = strip_comb t
val this = (walk ctxt origin conv)
val _ = (case f of Abs _ => error "Lambdas not supported" | _ => ())
val wctxt = {ctxt = ctxt, origins = origin, f = this}
in
(if is_if f then
(case f of (Const (_,T)) =>
(case args of [cond, t, f] => ifc wctxt T cond t f
| _ => error "Partial applications not supported (if)")
| _ => error "Internal error: invalid if term")
else if is_case f then casec wctxt f args
else if is_let f then
(case f of (Const (_,lT)) =>
(case args of [exp, t] =>
let val (abs,t) = Term.strip_abs_eta 1 t in letc wctxt lT exp abs t end
| _ => error "Partial applications not allowed (let)")
| _ => error "Internal error: invalid let term")
else funcc wctxt f args)
end
| walk ctxt origin (conv as {constc, ...}) c =
constc {ctxt = ctxt, origins = origin, f = walk ctxt origin conv} c
fun Ifunc (wctxt: term wctxt) t args = list_comb (#f wctxt t,map (#f wctxt) args)
val Iconst = K I
fun Iif (wctxt: term wctxt) T cond tt tf =
Const (@{const_name "HOL.If"}, T) $ (#f wctxt cond) $ (#f wctxt tt) $ (#f wctxt tf)
fun Icase (wctxt: term wctxt) t cs = list_comb
(#f wctxt t,map (fn c => c |> Term.strip_abs_eta (c |> fastype_of |> strip_type |> fst |> length) ||> #f wctxt |> list_abs) cs)
fun Ilet (wctxt: term wctxt) lT exp abs t =
Const (@{const_name "HOL.Let"}, lT) $ (#f wctxt exp) $ list_abs (abs, #f wctxt t)
(* 1. Fix all terms *)
(* Exchange Var in types and terms to Free *)
fun freeTerms (Var(ixn,T)) = Free (fst ixn, T)
| freeTerms t = t
fun freeTypes (TVar ((t, _), T)) = TFree (t, T)
| freeTypes t = t
fun fix_definition (Const ("Pure.eq", _) $ l $ r) = HOLogic.mk_Trueprop (HOLogic.mk_eq (l,r))
| fix_definition t = t
fun check_definition [t] = [t]
| check_definition _ = error "Only a single definition is allowed"
fun get_terms theory (term: term) =
let
val equations = Spec_Rules.retrieve theory term
|> map #rules
|> map (map Thm.prop_of)
handle Empty => error "Function or terms of function not found"
in
equations
|> map (map fix_definition)
|> filter (List.exists
(fn t => typ_comp (t |> get_l |> strip_comb |> fst |> dest_Const |> snd) (term |> strip_comb |> fst |> dest_Const |> snd)))
|> hd
end
fun fixCasecCases _ [t] = [t]
| fixCasecCases wctxt (t::ts) =
let
val num = fastype_of t |> strip_type |> fst |> length
val c' = Term.strip_abs_eta num t |> list_abs
in
c' :: fixCasecCases wctxt ts
end
| fixCasecCases _ _ = error "Internal error: invalid case types/terms"
fun fixCasec wctxt t args =
(check_args "cases" (t,args); list_comb (t,fixCasecCases wctxt args))
fun shortFunc fixedNum (Const (nm,T)) =
Const (nm,T |> strip_type |>> drop fixedNum |> (op --->))
| shortFunc _ _ = error "Internal error: Invalid term"
fun shortApp fixedNum (c, args) =
(shortFunc fixedNum c, drop fixedNum args)
fun shortOriginFunc (term: term list) fixedNum (f as (c as Const (_,_), _)) =
if contains' const_comp term c then shortApp fixedNum f else f
| shortOriginFunc _ _ t = t
fun map_abs f (t as Abs _) = t |> strip_abs ||> f |> list_abs
| map_abs _ t = t
fun fixTerms ctxt (term: term list) (fixedNum: int) (t as pT $ (eq $ l $ r)) =
let
val _ = check_args "args" (strip_comb (get_l t))
val l' = shortApp fixedNum (strip_comb l) |> list_comb
val shortOriginFunc' = shortOriginFunc (term |> map (fst o strip_comb)) fixedNum
val consts = Proof_Context.consts_of ctxt
val net = Consts.revert_abbrevs consts ["internal"] |> hd |> Item_Net.content
(* filter out consts *)
|> filter (is_Const o fst o strip_comb o fst)
(* filter out abbreviations for locales *)
|> filter (fn n => "local"
= (n |> snd |> strip_comb |> fst |> dest_Const_name |> split_name |> hd))
|> filter (fn n => (n |> fst |> strip_comb |> fst |> dest_Const_name |> split_name |> List.last) =
(n |> snd |> strip_comb |> fst |> dest_Const_name |> split_name |> List.last))
|> map (fst #> strip_comb #>> dest_Const_name ##> length)
fun n_abbrev (Const (nm,_)) =
let
val f = filter (fn n => fst n = nm) net
in
if length f >= 1 then f |> hd |> snd else 0
end
| n_abbrev _ = 0
val r' = walk ctxt term {
funcc = (fn wctxt => fn t => fn args =>
let
val n_abb = n_abbrev t
val t = case t of Const (nm,T) => Const (nm, T |> strip_type |>> drop n_abb |> (op --->))
| t => t
val args = drop n_abb args
in
(check_args "func" (t,args);
(#f wctxt t, map (#f wctxt) args) |> shortOriginFunc' |> list_comb)
end),
constc = fn wctxt => map_abs (#f wctxt),
ifc = Iif,
casec = fixCasec,
letc = (fn wctxt => fn expT => fn exp => fn abs => fn t =>
(Const (@{const_name "HOL.Let"},expT) $ (#f wctxt exp) $ list_abs (abs, #f wctxt t)))
} r
in
pT $ (eq $ l' $ r')
end
| fixTerms _ _ _ _ = error "Internal error: invalid term"
(* 2. Check for properties about the function *)
(* 2.1 Check if function is recursive *)
fun or f (a,b) = f a orelse b
fun find_rec ctxt term = (walk ctxt term {
funcc = (fn wctxt => fn t => fn args =>
List.exists (fn term => (Const_name t) = (Const_name term)) term
orelse List.foldr (or (#f wctxt)) false args),
constc = fn wctxt => fn t => case t of
Abs _ => t |> strip_abs |> snd |> (#f wctxt)
| _ => false,
ifc = (fn wctxt => fn _ => fn cond => fn tt => fn tf =>
(#f wctxt) cond orelse (#f wctxt) tt orelse (#f wctxt) tf),
casec = (fn wctxt => fn t => fn cs =>
(#f wctxt) t orelse List.foldr (or (rem_abs (#f wctxt))) false cs),
letc = (fn wctxt => fn _ => fn exp => fn _ => fn t =>
(#f wctxt) exp orelse (#f wctxt) t)
}) o get_r
fun is_rec ctxt (term: term list) = List.foldr (or (find_rec ctxt term)) false
(* 4. Tactic to prove "f_dom n" *)
fun time_dom_tac ctxt induct_rule domintros =
(Induction.induction_tac ctxt true [] [[]] [] (SOME [induct_rule]) []
THEN_ALL_NEW ((K (auto_tac ctxt)) THEN' (fn i => FIRST' (
(if i <= length domintros then [Metis_Tactic.metis_tac [] ATP_Problem_Generate.combsN ctxt [List.nth (domintros, i-1)]] else []) @
[Metis_Tactic.metis_tac [] ATP_Problem_Generate.combsN ctxt domintros]) i)))
(* Register timing function of a given function *)
type time_config = {
print: bool,
simp: bool,
partial: bool
}
datatype result = Function of Function.info | PartialFunction of thm
fun reg_time_func (lthy: local_theory) (term: term list) (terms: term list) (config: time_config) =
let
(* some default values to build terms easier *)
(* Const (@{const_name "Groups.zero"}, HOLogic.natT) *)
val zero = if #partial config then @{term "Some (0::nat)"} else HOLogic.zero
val one = Const (@{const_name "Groups.one"}, HOLogic.natT)
val natOptT = @{typ "nat option"}
val finT = if #partial config then natOptT else HOLogic.natT
val some = @{term "Some::nat \<Rightarrow> nat option"}
(* change type of original function to new type (_ \<Rightarrow> ... \<Rightarrow> _ to _ \<Rightarrow> ... \<Rightarrow> nat)
and replace all function arguments f with (t*T_f) if used *)
fun change_typ' used (Type ("fun", [T1, T2])) =
Type ("fun", [check_for_fun' (used 0) T1, change_typ' (fn i => used (i+1)) T2])
| change_typ' _ _ = finT
and check_for_fun' true (f as Type ("fun", [_,_])) = HOLogic.mk_prodT (f, change_typ' (K false) f)
| check_for_fun' false (f as Type ("fun", [_,_])) = change_typ' (K true) f
| check_for_fun' _ t = t
val change_typ = change_typ' (K true)
fun time_term ctxt s (Const (nm,T)) =
let
val T_nm = fun_name_to_time ctxt s nm
val T_T = change_typ T
in
(SOME (Syntax.check_term ctxt (Const (T_nm,T_T))))
handle (ERROR _) =>
case Syntax.read_term ctxt (Long_Name.base_name T_nm)
of (Const (T_nm,T_T)) =>
let
fun col_Used i (Type ("fun", [Type ("fun", _), Ts])) (Type ("fun", [T', Ts'])) =
(if is_Used T' then [i] else []) @ col_Used (i+1) Ts Ts'
| col_Used i (Type ("fun", [_, Ts])) (Type ("fun", [_, Ts'])) = col_Used (i+1) Ts Ts'
| col_Used _ _ _ = []
val binderT = change_typ' (contains (col_Used 0 T T_T)) T |> Term.binder_types
val finT = Term.body_type T_T
in
SOME (Const (T_nm, binderT ---> finT))
end
| _ => error ("Timing function of " ^ nm ^ " is not defined")
end
| time_term _ _ _ = error "Internal error: No valid function given"
fun opt_term NONE = zero
| opt_term (SOME t) = t
fun use_origin (Free (nm, T as Type ("fun",_))) = HOLogic.mk_fst (Free (nm,HOLogic.mk_prodT (T, change_typ T)))
| use_origin t = t
(* Conversion of function term *)
fun fun_to_time' ctxt (origin: term list) second (func as Const (nm,T)) =
let
val origin' = map (fst o strip_comb) origin
in
if contains' const_comp origin' func then SOME (Free (func |> Term.term_name |> fun_name_to_time' ctxt true second, change_typ T)) else
if Zero_Funcs.is_zero (Proof_Context.theory_of ctxt) (nm,T) then NONE else
time_term ctxt false func
end
| fun_to_time' _ _ _ (Free (nm,T)) =
SOME (HOLogic.mk_snd (Free (nm,HOLogic.mk_prodT (T,change_typ' (K true) T))))
| fun_to_time' _ _ _ _ = error "Internal error: invalid function to convert"
fun fun_to_time context origin func = fun_to_time' context origin false func
(* Convert arguments of left side of a term *)
fun conv_arg _ (Free (nm,T as Type("fun",_))) =
Free (nm, HOLogic.mk_prodT (T, change_typ' (K false) T))
| conv_arg _ x = x
fun conv_args ctxt = map (conv_arg ctxt)
(* 3. Convert equations *)
(* Some Helper *)
val plusTyp = @{typ "nat => nat => nat"}
fun plus (SOME a) (SOME b) = SOME ((if #partial config then @{term part_add} else Const (@{const_name "Groups.plus"}, plusTyp)) $ a $ b)
| plus (SOME a) NONE = SOME a
| plus NONE (SOME b) = SOME b
| plus NONE NONE = NONE
(* Partial helper *)
val OPTION_BIND = @{term "Option.bind::nat option \<Rightarrow> (nat \<Rightarrow> nat option) \<Rightarrow> nat option"}
fun OPTION_ABS_SUC args = Term.absfree ("_uu", @{typ nat})
(List.foldr (uncurry plus)
(SOME (some $ HOLogic.mk_Suc (Free ("_uu", @{typ nat})))) args |> Option.valOf)
fun build_option_bind term args =
OPTION_BIND $ term $ OPTION_ABS_SUC args
fun WRAP_FUNCTION t =
if (Term.head_of t |> Term.fastype_of |> Term.body_type) = finT
then t
else if #partial config
then some $ t
else @{term "the::nat option \<Rightarrow> nat"} $ t
(* Handle function calls *)
fun build_zero (Type ("fun", [T, R])) = Abs ("uu", T, build_zero R)
| build_zero _ = zero
fun funcc_use_origin (Free (nm, T as Type ("fun",_))) =
HOLogic.mk_fst (Free (nm,HOLogic.mk_prodT (T, change_typ T)))
| funcc_use_origin t = t
fun funcc_conv_arg _ _ (t as (_ $ _)) = map_aterms funcc_use_origin t
| funcc_conv_arg _ u (Free (nm, T as Type ("fun",_))) =
if u then Free (nm, HOLogic.mk_prodT (T, change_typ T))
else HOLogic.mk_snd (Free (nm,HOLogic.mk_prodT (T,change_typ T)))
| funcc_conv_arg wctxt true (f as Const (_,Type ("fun",_))) =
HOLogic.mk_prod (f, funcc_conv_arg wctxt false f)
| funcc_conv_arg wctxt false (f as Const (_,T as Type ("fun",_))) =
Option.getOpt (fun_to_time (#ctxt wctxt) (#origins wctxt) f, build_zero T)
| funcc_conv_arg wctxt false (f as Abs _) =
f
|> Term.strip_abs_eta ((length o fst o strip_type o type_of) f)
||> #f wctxt ||> opt_term
|> list_abs
| funcc_conv_arg wctxt true (f as Abs _) =
let
val f' = f
|> Term.strip_abs_eta ((length o fst o strip_type o type_of) f)
||> map_aterms funcc_use_origin
|> list_abs
in
HOLogic.mk_prod (f', funcc_conv_arg wctxt false f)
end
| funcc_conv_arg _ _ t = t
fun funcc_conv_args _ _ [] = []
| funcc_conv_args wctxt (Type ("fun", [t, ts])) (a::args) =
funcc_conv_arg wctxt (is_Used t) a :: funcc_conv_args wctxt ts args
| funcc_conv_args _ _ _ = error "Internal error: Non matching type"
fun funcc wctxt func args =
let
fun get_T (Free (_,T)) = T
| get_T (Const (_,T)) = T
| get_T (_ $ (Free (_,Type (_, [_, T])))) = T (* Case of snd was constructed *)
| get_T _ = error "Internal error: Forgotten type"
val func = (case fun_to_time (#ctxt wctxt) (#origins wctxt) func
of SOME t => SOME (WRAP_FUNCTION (list_comb (t, funcc_conv_args wctxt (get_T t) args)))
| NONE => NONE)
val args = (map (#f wctxt) args)
in
(if not (#partial config) orelse func = NONE
then List.foldr (uncurry plus) func args
else build_option_bind (Option.valOf func) args |> SOME)
end
(* Handle case terms *)
fun casecIsCase (Type (n1, [_,Type (n2, _)])) = (n1 = "fun" andalso n2 = "fun")
| casecIsCase _ = false
fun casecLastTyp (Type (n, [T1,T2])) = Type (n, [T1, change_typ T2])
| casecLastTyp _ = error "Internal error: Invalid case type"
fun casecTyp (Type (n, [T1, T2])) =
Type (n, [change_typ T1, (if casecIsCase T2 then casecTyp else casecLastTyp) T2])
| casecTyp _ = error "Internal error: Invalid case type"
fun casecAbs f (Abs (v,Ta,t)) = (case casecAbs f (subst_bound (Free (v,Ta), t))
of (nconst,t) => (nconst,absfree (v,Ta) t))
| casecAbs f t = (case f t of NONE => (false, opt_term NONE) | SOME t => (true,t))
fun casecArgs _ [t] = (false, [map_aterms use_origin t])
| casecArgs f (t::ar) =
(case casecAbs f t of (nconst, tt) =>
casecArgs f ar ||> (fn ar => tt :: ar) |>> (if nconst then K true else I))
| casecArgs _ _ = error "Internal error: Invalid case term"
fun casec wctxt (Const (t,T)) args =
if not (casecIsCase T) then error "Internal error: Invalid case type" else
let val (nconst, args') = casecArgs (#f wctxt) args in
plus
((#f wctxt) (List.last args))
(if nconst then
SOME (list_comb (Const (t,casecTyp T), args'))
else NONE)
end
| casec _ _ _ = error "Internal error: Invalid case term"
(* Handle if terms -> drop the term if true and false terms are zero *)
fun ifc wctxt _ cond tt ft =
let
val f = #f wctxt
val rcond = map_aterms use_origin cond
val tt = f tt
val ft = f ft
in
plus (f cond) (case (tt,ft) of (NONE, NONE) => NONE | _ =>
if tt = ft then tt else
(SOME ((Const (@{const_name "HOL.If"}, @{typ "bool"} --> finT --> finT --> finT)) $ rcond
$ (opt_term tt) $ (opt_term ft))))
end
fun letc_lambda wctxt T (t as Abs _) =
HOLogic.mk_prod (map_aterms use_origin t,
Term.strip_abs_eta (strip_type T |> fst |> length) t ||> #f wctxt ||> opt_term ||> map_types change_typ |> list_abs)
| letc_lambda _ _ t = map_aterms use_origin t
fun letc wctxt expT exp ([(nm,_)]) t =
plus (#f wctxt exp)
(case #f wctxt t of SOME t' =>
(if Term.used_free nm t'
then
let
val exp' = letc_lambda wctxt expT exp
val t' = list_abs ([(nm,fastype_of exp')], t')
in
Const (@{const_name "HOL.Let"}, [fastype_of exp', fastype_of t'] ---> finT) $ exp' $ t'
end
else t') |> SOME
| NONE => NONE)
| letc _ _ _ _ _ = error "Unknown let state"
fun constc _ (Const ("HOL.undefined", _)) = SOME (Const ("HOL.undefined", finT))
| constc _ _ = NONE
(* The converter for timing functions given to the walker *)
val converter : term option converter = {
constc = constc,
funcc = funcc,
ifc = ifc,
casec = casec,
letc = letc
}
fun top_converter is_rec _ _ =
if #partial config
then (fn t => Option.getOpt (t, zero))
else (opt_term o (fn exp => plus exp (if is_rec then SOME one else NONE)))
(* Use converter to convert right side of a term *)
fun to_time ctxt origin is_rec term =
top_converter is_rec ctxt origin (walk ctxt origin converter term)
(* Converts a term to its running time version *)
fun convert_term ctxt (origin: term list) is_rec (pT $ (Const (eqN, _) $ l $ r)) =
let
val (l_const, l_params) = strip_comb l
in
pT
$ (Const (eqN, finT --> finT --> @{typ "bool"})
$ (list_comb (l_const |> fun_to_time ctxt origin |> Option.valOf, l_params |> conv_args ctxt))
$ (to_time ctxt origin is_rec r))
end
| convert_term _ _ _ _ = error "Internal error: invalid term to convert"
(* 3.5 Support for locales *)
fun replaceFstSndFree ctxt (origin: term list) (rfst: term -> term) (rsnd: term -> term) =
(walk ctxt origin {
funcc = fn wctxt => fn t => fn args =>
case args of
(f as Free _)::args =>
(case t of
Const ("Product_Type.prod.fst", _) =>
list_comb (rfst (t $ f), map (#f wctxt) args)
| Const ("Product_Type.prod.snd", _) =>
list_comb (rsnd (t $ f), map (#f wctxt) args)
| t => list_comb (t, map (#f wctxt) (f :: args)))
| args => list_comb (t, map (#f wctxt) args),
constc = Iconst,
ifc = Iif,
casec = Icase,
letc = Ilet
})
(* 5. Check for higher-order function if original function is used \<rightarrow> find simplifications *)
fun find_used' T_t =
let
val (T_ident, T_args) = strip_comb (get_l T_t)
fun filter_passed [] = []
| filter_passed ((f as Free (_, Type ("Product_Type.prod",[Type ("fun",_), Type ("fun", _)])))::args) =
f :: filter_passed args
| filter_passed (_::args) = filter_passed args
val frees = (walk lthy [] {
funcc = (fn wctxt => fn t => fn args =>
(case t of (Const ("Product_Type.prod.snd", _)) => []
| _ => (if t = T_ident then [] else filter_passed args)
@ List.foldr (fn (l,r) => (#f wctxt) l @ r) [] args)),
constc = (K o K) [],
ifc = (fn wctxt => fn _ => fn cond => fn tt => fn tf => (#f wctxt) cond @ (#f wctxt) tt @ (#f wctxt) tf),
casec = (fn wctxt => fn _ => fn cs => List.foldr (fn (l,r) => (#f wctxt) l @ r) [] cs),
letc = (fn wctxt => fn _ => fn exp => fn _ => fn t => (#f wctxt) exp @ (#f wctxt) t)
}) (get_r T_t)
fun build _ [] = []
| build i (a::args) =
(if contains frees a then [(T_ident,i)] else []) @ build (i+1) args
in
build 0 T_args
end
fun find_simplifyble ctxt term terms =
let
val used =
terms
|> List.map find_used'
|> List.foldr (op @) []
val change =
Option.valOf o fun_to_time ctxt term
fun detect t i (Type ("fun",_)::args) =
(if contains used (change t,i) then [] else [i]) @ detect t (i+1) args
| detect t i (_::args) = detect t (i+1) args
| detect _ _ [] = []
in
map (fn t => t |> type_of |> strip_type |> fst |> detect t 0) term
end
fun define_simp' term simplifyable ctxt =
let
val base_name = case Named_Target.locale_of ctxt of
NONE => ctxt |> Proof_Context.theory_of |> Context.theory_base_name
| SOME nm => nm
val orig_name = term |> dest_Const_name |> split_name |> List.last
val red_name = fun_name_to_time ctxt false orig_name
val name = fun_name_to_time' ctxt true true orig_name
val full_name = base_name ^ "." ^ name
val def_name = red_name ^ "_def"
val def = Binding.name def_name
val canon = Syntax.read_term (Local_Theory.exit ctxt) name |> strip_comb
val canonFrees = canon |> snd
val canonType = canon |> fst |> dest_Const_type |> strip_type |> fst |> take (length canonFrees)
val types = term |> dest_Const_type |> strip_type |> fst
val vars = Variable.variant_fixes (map (K "") types) ctxt |> fst
fun l_typs' i ((T as (Type ("fun",_)))::types) =
(if contains simplifyable i
then change_typ T
else HOLogic.mk_prodT (T,change_typ T))
:: l_typs' (i+1) types
| l_typs' i (T::types) = T :: l_typs' (i+1) types
| l_typs' _ [] = []
val l_typs = l_typs' 0 types
val lhs =
List.foldl (fn ((v,T),t) => t $ Free (v,T)) (Free (red_name,l_typs ---> HOLogic.natT)) (ListPair.zip (vars,l_typs))
fun fixType (TFree _) = HOLogic.natT
| fixType T = T
fun fixUnspecified T = T |> strip_type ||> fixType |> (op --->)
fun r_terms' i (v::vars) ((T as (Type ("fun",_)))::types) =
(if contains simplifyable i
then HOLogic.mk_prod (Const ("HOL.undefined", fixUnspecified T), Free (v,change_typ T))
else Free (v,HOLogic.mk_prodT (T,change_typ T)))
:: r_terms' (i+1) vars types
| r_terms' i (v::vars) (T::types) = Free (v,T) :: r_terms' (i+1) vars types
| r_terms' _ _ _ = []
val r_terms = r_terms' 0 vars types
val full_type = (r_terms |> map (type_of) ---> HOLogic.natT)
val full = list_comb (Const (full_name,canonType ---> full_type), canonFrees)
val rhs = list_comb (full, r_terms)
val eq = (lhs, rhs) |> HOLogic.mk_eq |> HOLogic.mk_Trueprop
val _ = Pretty.writeln (Pretty.block [Pretty.str "Defining simplified version:\n",
Syntax.pretty_term ctxt eq])
val (_, ctxt') = Specification.definition NONE [] [] ((def, []), eq) ctxt
in
((def_name, orig_name), ctxt')
end
fun define_simp simpables ctxt =
let
fun cond ((term,simplifyable),(defs,ctxt)) =
define_simp' term simplifyable ctxt |>> (fn def => def :: defs)
in
List.foldr cond ([], ctxt) simpables
end
fun replace from to =
map (map_aterms (fn t => if t = from then to else t))
fun replaceAll [] = I
| replaceAll ((from,to)::xs) = replaceAll xs o replace from to
fun calculateSimplifications ctxt T_terms term simpables =
let
(* Show where a simplification can take place *)
fun reportReductions (t,(i::is)) =
(Pretty.writeln (Pretty.str
((Term.term_name t |> fun_name_to_time ctxt true)
^ " can be simplified because only the time-function component of parameter "
^ (Int.toString (i + 1)) ^ " is used. "));
reportReductions (t,is))
| reportReductions (_,[]) = ()
val _ = simpables
|> map reportReductions
(* Register definitions for simplified function *)
val (reds, ctxt) = define_simp simpables ctxt
fun genRetype (Const (nm,T),is) =
let
val T_name = fun_name_to_time ctxt true nm |> split_name |> List.last
val from = Free (T_name,change_typ T)
val to = Free (T_name,change_typ' (not o contains is) T)
in
(from,to)
end
| genRetype _ = error "Internal error: invalid term"
val retyping = map genRetype simpables
fun replaceArgs (pT $ (eq $ l $ r)) =
let
val (t,params) = strip_comb l
fun match (Const (f_nm,_),_) =
(fun_name_to_time ctxt true f_nm |> Long_Name.base_name) = (dest_Free t |> fst)
| match _ = false
val simps = List.find match simpables |> Option.valOf |> snd
fun dest_Prod_snd (Free (nm, Type (_, [_, T2]))) =
Free (fun_name_to_time ctxt false nm, T2)
| dest_Prod_snd _ = error "Internal error: Argument is not a pair"
fun rep _ [] = ([],[])
| rep i (x::xs) =
let
val (rs,args) = rep (i+1) xs
in
if contains simps i
then (x::rs,dest_Prod_snd x::args)
else (rs,x::args)
end
val (rs,params) = rep 0 params
fun fFst _ = error "Internal error: Invalid term to simplify"
fun fSnd (t as (Const _ $ f)) =
(if contains rs f
then dest_Prod_snd f
else t)
| fSnd t = t
in
(pT $ (eq
$ (list_comb (t,params))
$ (replaceFstSndFree ctxt term fFst fSnd r
|> (fn t => replaceAll (map (fn t => (t,dest_Prod_snd t)) rs) [t])
|> hd
)
))
end
| replaceArgs _ = error "Internal error: Invalid term"
(* Calculate reduced terms *)
val T_terms_red = T_terms
|> replaceAll retyping
|> map replaceArgs
val _ = print_lemma ctxt reds T_terms_red
val _ =
Pretty.writeln (Pretty.str "If you do not want the simplified T function, use \"time_fun [no_simp]\"")
in
ctxt
end
val _ =
case time_term lthy true (hd term)
handle (ERROR _) => NONE
of SOME _ => error ("Timing function already declared: " ^ (Term.term_name (hd term)))
| NONE => ()
(* Number of terms fixed by locale *)
val fixedNum = term |> hd
|> strip_comb |> snd
|> length
(********************* BEGIN OF CONVERSION *********************)
(* 1. Fix all terms *)
(* Exchange Var in types and terms to Free and check constraints *)
val terms = map
(map_aterms freeTerms
#> map_types (map_atyps freeTypes)
#> fixTerms lthy term fixedNum)
terms
val fixedFrees = (hd term) |> strip_comb |> snd |> take fixedNum
val fixedFreesNames = map (fst o dest_Free) fixedFrees
val term = map (shortFunc fixedNum o fst o strip_comb) term
fun correctTerm term =
let
val get_f = fst o strip_comb o get_l
in
List.find (fn t => (dest_Const_name o get_f) t = dest_Const_name term) terms
|> Option.valOf |> get_f
end
val term = map correctTerm term
(* 2. Find properties about the function *)
(* 2.1 Check if function is recursive *)
val is_rec = is_rec lthy term terms
(* 3. Convert every equation
- Change type of toplevel equation from _ \<Rightarrow> _ \<Rightarrow> bool to nat \<Rightarrow> nat \<Rightarrow> bool
- On left side change name of function to timing function
- Convert right side of equation with conversion schema
*)
fun fFst (t as (Const (_,T) $ Free (nm,_))) =
(if contains fixedFreesNames nm
then Free (nm,strip_type T |>> tl |> (op --->))
else t)
| fFst t = t
fun fSnd (t as (Const (_,T) $ Free (nm,_))) =
(if contains fixedFreesNames nm
then Free (fun_name_to_time lthy false nm,strip_type T |>> tl |> (op --->))
else t)
| fSnd t = t
val T_terms = map (convert_term lthy term is_rec) terms
|> map (map_r (replaceFstSndFree lthy term fFst fSnd))
val simpables = (if #simp config
then find_simplifyble lthy term T_terms
else map (K []) term)
|> (fn s => ListPair.zip (term,s))
(* Determine if something is simpable, if so rename everything *)
val simpable = simpables |> map snd |> exists (not o null)
(* Rename to secondary if simpable *)
fun genRename (t,_) =
let
val old = fun_to_time' lthy term false t |> Option.valOf
val new = fun_to_time' lthy term true t |> Option.valOf
in
(old,new)
end
val can_T_terms = if simpable
then replaceAll (map genRename simpables) T_terms
else T_terms
(* 4. Register function and prove completeness *)
val names = map Term.term_name term
val timing_names = map (fun_name_to_time' lthy true simpable) names
val bindings = map (fn nm => (Binding.name nm, NONE, NoSyn)) timing_names
fun pat_completeness_auto ctxt =
Pat_Completeness.pat_completeness_tac ctxt 1 THEN auto_tac ctxt
val specs = map (fn eq => (((Binding.empty, []), eq), [], [])) can_T_terms
val part_specs = (Binding.empty_atts, hd can_T_terms)
(* Context for printing without showing question marks *)
val print_ctxt = lthy
|> Config.put show_question_marks false
|> Config.put show_sorts false (* Change it for debugging *)
val print_ctxt = List.foldl (fn (term, ctxt) => Variable.add_fixes_implicit term ctxt) print_ctxt (term @ T_terms)
(* Print result if print *)
val _ = if not (#print config) then () else
let
val nms = map (dest_Const_name) term
val typs = map (dest_Const_type) term
in
print_timing' print_ctxt { names=nms, terms=terms, typs=typs }
{ names=timing_names, terms=can_T_terms, typs=map change_typ typs }
end
(* For partial functions sequential=true is needed in order to support them
We need sequential=false to support the automatic proof of termination over dom
*)
fun register seq =
let
val _ = (if seq then warning "Falling back on sequential function..." else ())
val fun_config = Function_Common.FunctionConfig
{sequential=seq, default=NONE, domintros=true, partials=false}
in
if #partial config
then Partial_Function.add_partial_function "option" bindings part_specs lthy |>> PartialFunction o snd
else Function.add_function bindings specs fun_config pat_completeness_auto lthy |>> Function
end
val (info,ctxt) =
register false
handle (ERROR _) =>
register true
| Match =>
register true
val ctxt = if simpable then calculateSimplifications ctxt T_terms term simpables else ctxt
in
(info, ctxt)
end
fun proove_termination (term: term list) terms (T_info: Function.info, lthy: local_theory) =
let
(* Start proving the termination *)
val infos = SOME (map (Function.get_info lthy) term) handle Empty => NONE
val timing_names = map (fun_name_to_time lthy true o Term.term_name) term
(* Proof by lexicographic_order_tac *)
val (time_info, lthy') =
(Function.prove_termination NONE
(Lexicographic_Order.lexicographic_order_tac false lthy) lthy)
handle (ERROR _) =>
let
val _ = warning "Falling back on proof over dom..."
val _ = (if length term > 1 then error "Proof over dom not supported for mutual recursive functions" else ())
fun args (a$(Var ((nm,_),T))) = args a |> (fn ar => (nm,T)::ar)
| args (a$(Const (_,T))) = args a |> (fn ar => ("uu",T)::ar)
| args _ = []
val dom_vars =
terms |> hd |> get_l |> map_types (map_atyps freeTypes)
|> args |> Variable.variant_names lthy
val dom_args =
List.foldl (fn (t,p) => HOLogic.mk_prod ((Free t),p)) (Free (hd dom_vars)) (tl dom_vars)
val {inducts, ...} = case infos of SOME [i] => i | _ => error "Proof over dom failed as no induct rule was found"
val induct = (Option.valOf inducts |> hd)
val domintros = Proof_Context.get_fact lthy (Facts.named (hd timing_names ^ ".domintros"))
val prop = HOLogic.mk_Trueprop (#dom T_info $ dom_args)
(* Prove a helper lemma *)
val dom_lemma = Goal.prove lthy (map fst dom_vars) [] prop
(fn {context, ...} => HEADGOAL (time_dom_tac context induct domintros))
(* Add dom_lemma to simplification set *)
val simp_lthy = Simplifier.add_simp dom_lemma lthy
in
(* Use lemma to prove termination *)
Function.prove_termination NONE
(auto_tac simp_lthy) lthy
end
in
(Function time_info, lthy')
end
fun reg_and_proove_time_func (lthy: local_theory) (term: term list) (terms: term list) (config: time_config) =
case reg_time_func lthy term terms config
of (Function info, lthy') => proove_termination term terms (info, lthy')
| r => r
fun isTypeClass' (Const (nm,_)) =
(case split_name nm |> rev
of (_::nm::_) => String.isSuffix "_class" nm
| _ => false)
| isTypeClass' _ = false
val isTypeClass =
(List.foldr (fn (a,b) => a orelse b) false) o (map isTypeClass')
fun detect_typ (ctxt: local_theory) (term: term) =
let
val class_term = (case term of Const (nm,_) => Syntax.read_term ctxt nm
| _ => error "Could not find term of class")
fun find_free (Type (_,class)) (Type (_,inst)) =
List.foldl (fn ((c,i),s) => (case s of NONE => find_free c i | t => t)) (NONE) (ListPair.zip (class, inst))
| find_free (TFree _) (TFree _) = NONE
| find_free (TFree _) (Type (nm,_)) = SOME nm
| find_free _ _ = error "Unhandled case in detecting type"
in
find_free (type_of class_term) (type_of term)
|> Option.map (hd o rev o split_name)
end
fun set_suffix (fterms: term list) ctxt =
let
val isTypeClass = isTypeClass fterms
val _ = (if length fterms > 1 andalso isTypeClass then error "No mutual recursion inside instantiation allowed" else ())
val suffix = (if isTypeClass then detect_typ ctxt (hd fterms) else NONE)
in
(case suffix of NONE => I | SOME s => Config.put bsuffix ("_" ^ s)) ctxt
end
fun check_opts [] = false
| check_opts ["no_simp"] = true
| check_opts (a::_) = error ("Option " ^ a ^ " is not defined")
(* Converts a function into its timing function using fun *)
fun reg_time_fun_cmd ((opts, funcs), thms) (ctxt: local_theory) =
let
val no_simp = check_opts opts
val fterms = map (Syntax.read_term ctxt) funcs
val ctxt = set_suffix fterms ctxt
val config = { print = true, simp = not no_simp, partial = false }
val (_, ctxt') = reg_and_proove_time_func ctxt fterms
(case thms of NONE => get_terms ctxt (hd fterms)
| SOME thms => thms |> Attrib.eval_thms ctxt |> List.map Thm.prop_of)
config
in ctxt'
end
(* Converts a function into its timing function using function with termination proof provided by user*)
fun reg_time_function_cmd ((opts, funcs), thms) (ctxt: local_theory) =
let
val no_simp = check_opts opts
val fterms = map (Syntax.read_term ctxt) funcs
val ctxt = set_suffix fterms ctxt
val config = { print = true, simp = not no_simp, partial = false }
val ctxt' = reg_time_func ctxt fterms
(case thms of NONE => get_terms ctxt (hd fterms)
| SOME thms => thms |> Attrib.eval_thms ctxt |> List.map Thm.prop_of)
config
|> snd
in ctxt'
end
(* Converts a function definition into its timing function using definition *)
fun reg_time_definition_cmd ((opts, funcs), thms) (ctxt: local_theory) =
let
val no_simp = check_opts opts
val fterms = map (Syntax.read_term ctxt) funcs
val ctxt = set_suffix fterms ctxt
val config = { print = true, simp = not no_simp, partial = false }
val (_, ctxt') = reg_and_proove_time_func ctxt fterms
(case thms of NONE => get_terms ctxt (hd fterms) |> check_definition
| SOME thms => thms |> Attrib.eval_thms ctxt |> List.map Thm.prop_of)
config
in ctxt'
end
(* Converts a a partial function into its timing function using partial_function *)
fun reg_time_partial_function_cmd ((opts, funcs), thms) (ctxt: local_theory) =
let
val no_simp = check_opts opts
val fterms = map (Syntax.read_term ctxt) funcs
val ctxt = set_suffix fterms ctxt
val config = { print = true, simp = not no_simp, partial = true }
val (_, ctxt') = reg_and_proove_time_func ctxt fterms
(case thms of NONE => get_terms ctxt (hd fterms) |> check_definition
| SOME thms => thms |> Attrib.eval_thms ctxt |> List.map Thm.prop_of)
config
in ctxt'
end
val parser = (Parse.opt_attribs >> map (fst o Token.name_of_src))
-- Scan.repeat1 Parse.prop
-- Scan.option (Parse.keyword_improper "equations" -- Parse.thms1 >> snd)
val _ = Toplevel.local_theory
val _ = Outer_Syntax.local_theory @{command_keyword "time_fun"}
"Defines runtime function of a function"
(parser >> reg_time_fun_cmd)
val _ = Outer_Syntax.local_theory @{command_keyword "time_function"}
"Defines runtime function of a function"
(parser >> reg_time_function_cmd)
val _ = Outer_Syntax.local_theory @{command_keyword "time_definition"}
"Defines runtime function of a definition"
(parser >> reg_time_definition_cmd)
val _ = Outer_Syntax.local_theory @{command_keyword "time_partial_function"}
"Defines runtime function of a definition"
(parser >> reg_time_partial_function_cmd)
end